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Search the School of Mathematical Sciences
People matching "Differential geometry"
Courses matching "Differential geometry"
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Differential Equations 
Most "real life" systems that are described mathematically, be they physical, financial, economic or some other kind, are described by means of differential equations. Our ability to predict the way in which these systems evolve or behave is determined by our ability to find solutions of these equations explicitly or to be able to approximate solutions as accurately as we need. Every differential equation presents its own challenges, but there are various classes of differential equations, and for some of these there are established approaches and methods for solving them. This course presents some of the most important such methods. Topics covered are: first order ordinary differential equations (ODEs), higher order ODEs, numerical techniques for solving ODEs, systems of ODEs, series solutions of ODEs, Laplace transforms, Fourier analysis, solution of linear partial differential equations using the method of separation of variables, and D'Alembert's solution of the wave equation.
More about this course...
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Differential Equations III
Differential equations describe a wide range of practical problems in areas such as biology, engineering, physical sciences, economics and finance. This course aims to provide students with techniques required to solve classes of ordinary and partial differential equations that commonly occur in applications. Topics covered are: methods for the solution of systems of linear and non-linear ordinary differential equations; techniques for the solution of two point boundary value problems for second order linear ordinary differential equations with variable coefficients; classification of partial differential equations and the solution of boundary value problems for these equations using the methods of reduction to ordinary differential equations by use of separation of variables, integral transforms, and characteristics.
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Finite Geometry III
Projective geometry is one of the important modern geometries introduced in the 19th century. Projective geometry is more general than our usual Euclidean geometry, and it has useful applications in Information Security, Statistics, Computer Graphics and Computer Vision. The focus of this course will be primarily on projective planes.
This course will be taught every second year.
Topics covered are: projective planes, homogeneous coordinates, fields, field planes, collineations of projective planes, conics in field planes, projective geometry of general dimension.
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Events matching "Differential geometry"
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Homological algebra and applications - a historical survey 15:10 Fri 19 May 06 :: G08 Mathematics Building University of Adelaide :: Prof. Amnon Neeman
Homological algebra is a curious branch of
mathematics; it is a powerful tool which has been used in many diverse
places, without any clear understanding why it should be so useful.
We will give a list of applications, proceeding chronologically: first
to topology, then to complex analysis, then to algebraic geometry,
then to commutative algebra and finally (if we have time) to
non-commutative algebra. At the end of the talk I hope to be able to
say something about the part of homological algebra on which I have
worked, and its applications. That part is derived categories.
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Finite Geometries: Classical Problems and Recent Developments 15:10 Fri 20 Jul 07 :: G04 Napier Building University of Adelaide :: Prof. Joseph A. Thas :: Ghent University, Belgium
In recent years there has been an increasing interest in finite projective spaces, and important applications to practical topics such as coding theory, cryptography and design of experiments have made the field even more attractive. In my talk some classical problems and recent developments will be discussed. First I will mention Segre's celebrated theorem and ovals and a purely combinatorial characterization of Hermitian curves in the projective plane over a finite field here, from the beginning, the considered pointset is contained in the projective plane over a finite field. Next, a recent elegant result on semiovals in PG(2,q), due to Gács, will be given. A second approach is where the object is described as an incidence structure satisfying certain properties; here the geometry is not a priori embedded in a projective space. This will be illustrated by a characterization of the classical inversive plane in the odd case. Another quite recent beautiful result in Galois geometry is the discovery of an infinite class of hemisystems of the Hermitian variety in PG(3,q^2), leading to new interesting classes of incidence structures, graphs and codes; before this result, just one example for GF(9), due to Segre, was known.
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Mathematical modelling of blood flow in curved arteries 15:10 Fri 12 Sep 08 :: G03 Napier Building University of Adelaide :: Dr Jennifer Siggers :: Imperial College London
Atherosclerosis, characterised by plaques, is the most common arterial
disease. Plaques tend to develop in regions of low mean wall shear
stress, and regions where the wall shear stress changes direction during
the course of the cardiac cycle. To investigate the effect of the
arterial geometry and driving pressure gradient on the wall shear stress
distribution we consider an idealised model of a curved artery with
uniform curvature. We assume that the flow is fully-developed and seek
solutions of the governing equations, finding the effect of the
parameters on the flow and wall shear stress distribution. Most
previous work assumes the curvature ratio is asymptotically small;
however, many arteries have significant curvature (e.g. the aortic arch
has curvature ratio approx 0.25), and in this work we consider in
particular the effect of finite curvature.
We present an extensive analysis of curved-pipe flow driven by a steady
and unsteady pressure gradients. Increasing the curvature causes the
shear stress on the inside of the bend to rise, indicating that the risk
of plaque development would be overestimated by considering only the
weak curvature limit.
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Direct "delay" reductions of the Toda equation
13:10 Fri 23 Jan 09 :: School Board Room :: Prof Nalini Joshi :: University of Sydney
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as
delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painleve equations. The Lax pair associated to this equation is obtained, also by reduction.
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Noncommutative geometry of odd-dimensional quantum spheres 13:10 Fri 27 Feb 09 :: School Board Room :: Dr Partha Chakraborty :: University of Adelaide
We will report on our attempts to understand noncommutative geometry in the lights of the example of quantum spheres. We will see how to produce an equivariant fundamental class and also indicate some of the limitations of isospectral deformations.
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Bibundles 13:10 Fri 6 Mar 09 :: School Board Room :: Prof Michael Murray :: University of Adelaide
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The index theorem for projective families of elliptic operators 13:10 Fri 13 Mar 09 :: School Board Room :: Prof Mathai Varghese :: University of Adelaide
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Geometric analysis on the noncommutative torus 13:10 Fri 20 Mar 09 :: School Board Room :: Prof Jonathan Rosenberg :: University of Maryland
Noncommutative geometry (in the sense of Alain Connes) involves
replacing a conventional space by a "space" in which the algebra of
functions is noncommutative. The simplest truly non-trivial
noncommutative manifold is the noncommutative 2-torus, whose algebra
of functions is also called the irrational rotation algebra. I will
discuss a number of recent results on geometric analysis on the
noncommutative torus, including the study of nonlinear noncommutative
elliptic PDEs (such as the noncommutative harmonic map equation) and
noncommutative complex analysis (with noncommutative elliptic
functions).
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Understanding optimal linear transient growth in complex-geometry flows 15:00 Fri 27 Mar 09 :: Napier LG29 :: Associate Prof Hugh Blackburn :: Monash University
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Classification and compact complex manifolds I 13:10 Fri 17 Apr 09 :: School Board Room :: A/Prof Nicholas Buchdahl :: University of Adelaide
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Classification and compact complex manifolds II 13:10 Fri 24 Apr 09 :: School Board Room :: A/Prof Nicholas Buchdahl :: University of Adelaide
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String structures and characteristic classes for loop group bundles 13:10 Fri 1 May 09 :: School Board Room :: Mr Raymond Vozzo :: University of Adelaide
The Chern-Weil homomorphism gives a geometric method for calculating characteristic classes for principal bundles. In infinite dimensions, however, the standard theory fails due to analytical problems. In this talk I shall give a geometric method for calculating characteristic classes for principal bundle with structure group the loop group of a compact group which side-steps these complications. This theory is inspired in some sense by results on the string class (a certain cohomology class on the base of a loop group bundle) which I shall outline.
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Four classes of complex manifolds 13:10 Fri 8 May 09 :: School Board Room :: A/Prof Finnur Larusson :: University of Adelaide
We introduce the four classes of complex manifolds defined by having few or many holomorphic maps to or from the complex plane. Two of these classes have played an important role in complex geometry for a long time. A third turns out to be too large to be of much interest. The fourth class has only recently emerged from work of Abel Prize winner Mikhail Gromov.
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Lagrangian fibrations on holomorphic symplectic manifolds I: Holomorphic Lagrangian fibrations 13:10 Fri 5 Jun 09 :: School Board Room :: Dr Justin Sawon :: Colorado State University
A compact K{\"a}hler manifold $X$ is a holomorphic symplectic manifold if it admits a non-degenerate holomorphic two-form $\sigma$. According to a theorem of Matsushita, fibrations on $X$ must be of a very restricted type: the fibres must be Lagrangian with respect to $\sigma$ and the generic fibre must be a complex torus. Moreover, it is expected that the base of the fibration must be complex projective space, and this has been proved by Hwang when $X$ is projective. The simplest example of these {\em Lagrangian fibrations\/} are elliptic K3 surfaces. In this talk we will explain the role of elliptic K3s in the classification of K3 surfaces, and the (conjectural) generalization to higher dimensions.
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Chern-Simons classes on loop spaces and diffeomorphism groups 13:10 Fri 12 Jun 09 :: School Board Room :: Prof Steve Rosenberg :: Boston University
The loop space LM of a Riemannian manifold M comes with a family of Riemannian metrics indexed by a Sobolev parameter. We can construct characteristic classes for LM using the Wodzicki residue instead of the usual matrix trace. The Pontrjagin classes of LM vanish, but the secondary or Chern-Simons classes may be nonzero and may distinguish circle actions on M. There are similar results for diffeomorphism groups of manifolds.
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Lagrangian fibrations on holomorphic symplectic manifolds II: Existence of Lagrangian fibrations 13:10 Fri 19 Jun 09 :: School Board Room :: Dr Justin Sawon :: Colorado State University
The Hilbert scheme ${\mathrm Hilb}^nS$ of points on a K3 surface $S$ is a well-known holomorphic symplectic manifold. When does ${\mathrm Hilb}^nS$ admit a Lagrangian fibration? The existence of a Lagrangian fibration places some conditions on the Hodge structure, since the pull back of a hyperplane from the base gives a special divisor on ${\mathrm Hilb}^nS$, and in turn a special divisor on $S$. The converse is more difficult, but using Fourier-Mukai transforms we will show that if $S$ admits a divisor of a certain degree then ${\mathrm Hilb}^nS$ admits a Lagrangian fibration.
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Strong Predictor-Corrector Euler Methods for Stochastic Differential Equations 15:10 Fri 19 Jun 09 :: LG29 :: Prof. Eckhard Platen :: University of Technology, Sydney
This paper introduces a new class of numerical
schemes for the pathwise approximation of solutions of stochastic
differential equations (SDEs). The proposed family of strong
predictor-corrector Euler methods are designed to handle scenario
simulation of solutions of SDEs. It has the potential to overcome
some of the numerical instabilities that are often experienced
when using the explicit Euler method. This is of importance, for
instance, in finance where martingale dynamics arise for solutions
of SDEs with multiplicative diffusion coefficients. Numerical
experiments demonstrate the improved asymptotic stability
properties of the proposed symmetric predictor-corrector Euler
methods.
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Lagrangian fibrations on holomorphic symplectic manifolds III: Holomorphic coisotropic reduction 13:10 Fri 26 Jun 09 :: School Board Room :: Dr Justin Sawon :: Colorado State University
Given a certain kind of submanifold $Y$ of a symplectic manifold $(X,\omega)$ we can form its coisotropic reduction as follows. The null directions of $\omega|_Y$ define the characteristic foliation $F$ on $Y$. The space of leaves $Y/F$ then admits a symplectic form, descended from $\omega|_Y$. Locally, the coisotropic reduction $Y/F$ looks just like a symplectic quotient. This construction also work for holomorphic symplectic manifolds, though one of the main difficulties in practice is ensuring that the leaves of the foliation are compact. We will describe a criterion for compactness, and apply coisotropic reduction to produce a classification result for Lagrangian fibrations by Jacobians.
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Another proof of Gaboriau-Popa 13:10 Fri 3 Jul 09 :: School Board Room :: Prof Greg Hjorth :: University of Melbourne
Gaboriau and Popa showed that a non-abelian free group on finitely many generators has continuum many measure preserving, free, ergodic, actions on standard Borel probability spaces. The original proof used the notion of property (T). I will sketch how this can be replaced by an elementary, and apparently new, dynamical property.
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Generalizations of the Stein-Tomas restriction theorem 13:10 Fri 7 Aug 09 :: School Board Room :: Prof Andrew Hassell :: Australian National University
The Stein-Tomas restriction theorem says that the
Fourier transform of a function in L^p(R^n) restricts to an
L^2 function on the unit sphere, for p in some range [1, 2(n+1)/(n+3)].
I will discuss geometric generalizations of this result, by interpreting
it as a property of the spectral measure of the Laplace operator on
R^n, and then generalizing to the Laplace-Beltrami operator on
certain complete Riemannian manifolds. It turns out that dynamical
properties of the geodesic flow play a crucial role in determining whether
a restriction-type theorem holds for these manifolds.
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Asymmetric Cantor measures and sumsets 13:10 Fri 14 Aug 09 :: School Board Room :: Prof Gavin Brown :: Royal Institution of Australia and University of Adelaide
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Weak Hopf algebras and Frobenius algebras 13:10 Fri 21 Aug 09 :: School Board Room :: Prof Ross Street :: Macquarie University
A basic example of a Hopf algebra is a group algebra: it is the vector space having the group as basis and having multiplication linearly extending that of the group. We can start with a category instead of a group, form the free vector space on the set of its morphisms, and define multiplication to be composition when possible and zero when not. The multiplication has an identity if the category has finitely many objects; this is a basic example of a weak bialgebra. It is a weak Hopf algebra when the category is a groupoid. Group algebras are also Frobenius algebras. We shall generalize weak bialgebras and Frobenius algebras to the context of monoidal categories and describe some of their theory using the geometry of string diagrams.
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Moduli spaces of stable holomorphic vector bundles 13:10 Fri 28 Aug 09 :: School Board Room :: Dr Nicholas Buchdahl :: University of Adelaide
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Defect formulae for integrals of pseudodifferential symbols:
applications to dimensional regularisation and index theory 13:10 Fri 4 Sep 09 :: School Board Room :: Prof Sylvie Paycha :: Universite Blaise Pascal, Clermont-Ferrand, France
The ordinary integral on L^1 functions on R^d unfortunately does not
extend to a translation invariant linear form on the whole algebra of
pseudodifferential symbols on R^d, forcing to work with ordinary linear
extensions which fail to be translation invariant. Defect formulae which express the difference between various linear extensions, show that they differ by local terms involving the noncommutative residue. In particular, we shall show how integrals regularised by a "dimensional regularisation" procedure familiar to physicists differ from Hadamard finite part (or "cut-off" regularised) integrals by a residue. When extended to pseudodifferential operators on closed manifolds, these defect formulae express the zeta regularised traces of a differential
operator in terms of a residue of its logarithm. In particular, we shall express the index of a Dirac type operator on a closed manifold in
terms of a logarithm of a generalized Laplacian, thus giving an a priori local
description of the index and shall discuss further applications.
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Covering spaces and algebra bundles 13:10 Fri 11 Sep 09 :: School Board Room :: Prof Keith Hannabuss :: University of Oxford
Bundles of C*-algebras over a topological space M can be classified by a Dixmier-Douady obstruction in H^3(M,Z). This talk will describe some recent work with Mathai investigating the relationship between algebra bundles on M and on its covering space, where there can be no obstruction, particularly when there is a group acting on M.
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Understanding hypersurfaces through tropical geometry 12:10 Fri 25 Sep 09 :: Napier 102 :: Dr Mohammed Abouzaid :: Massachusetts Institute of Technology
Given a polynomial in two or more variables, one may study the
zero locus from the point of view of different mathematical subjects
(number theory, algebraic geometry, ...). I will explain how tropical
geometry allows to encode all topological aspects by elementary
combinatorial objects called "tropical varieties."
Mohammed Abouzaid received a B.S. in 2002 from the University of Richmond, and a Ph.D. in 2007 from the University of Chicago under the supervision of Paul Seidel. He is interested in symplectic topology and its interactions with algebraic geometry and differential topology, in particular the homological mirror symmetry conjecture. Since 2007 he has been a postdoctoral fellow at MIT, and a Clay Mathematics Institute Research Fellow.
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Stable commutator length 13:40 Fri 25 Sep 09 :: Napier 102 :: Prof Danny Calegari :: California Institute of Technology
Stable commutator length answers the question: "what is the simplest
surface in a given space with prescribed boundary?" where "simplest"
is interpreted in topological terms. This topological definition is
complemented by several equivalent definitions - in group theory, as a
measure of non-commutativity of a group; and in linear programming, as
the solution of a certain linear optimization problem. On the
topological side, scl is concerned with questions such as computing
the genus of a knot, or finding the simplest 4-manifold that bounds a
given 3-manifold. On the linear programming side, scl is measured in
terms of certain functions called quasimorphisms, which arise from
hyperbolic geometry (negative curvature) and symplectic geometry
(causal structures). In these talks we will discuss how scl in free
and surface groups is connected to such diverse phenomena as the
existence of closed surface subgroups in graphs of groups, rigidity
and discreteness of symplectic representations, bounding immersed
curves on a surface by immersed subsurfaces, and the theory of multi-
dimensional continued fractions and Klein polyhedra.
Danny Calegari is the Richard Merkin Professor of Mathematics at the California Institute of Technology, and is one of the recipients of the 2009 Clay Research Award for his work in geometric topology and geometric group theory. He received a B.A. in 1994 from the University of Melbourne, and a Ph.D. in 2000 from the University of California, Berkeley under the joint supervision of Andrew Casson and William Thurston. From 2000 to 2002 he was Benjamin Peirce Assistant Professor at Harvard University, after which he joined the Caltech faculty; he became Richard Merkin Professor in 2007.
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A Fourier-Mukai transform for invariant differential cohomology 13:10 Fri 9 Oct 09 :: School Board Room :: Mr Richard Green :: University of Adelaide
Fourier-Mukai transforms are a geometric analogue of integral transforms playing
an important role in algebraic geometry. Their name derives from the
construction of Mukai involving the Poincare line bundle associated to an
abelian variety. In this talk I will discuss recent work looking at an analogue
of this original Fourier-Mukai transform in the context of differential
geometry, which gives an isomorphism between the invariant differential
cohomology of a real torus and its dual.
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Irreducible subgroups of SO(2,n) 13:10 Fri 16 Oct 09 :: School Board Room :: Dr Thomas Leistner :: University of Adelaide
Berger's classification of irreducibly represented Lie groups that can occur as holonomy groups of semi-Riemannian manifolds is a remarkable result of modern differential geometry. What is remarkable about it is that it is so short and that only so few types of geometry can occur. In Riemannian signature this is even more remarkable, taking into account that any representation of a compact Lie group admits a positive definite invariant scalar product. Hence, for any not too small n there is an abundance of irreducible subgroups of SO(n). We show that in other signatures the situation is quite different with, for example, SO(1,n) having no proper irreducible subgroups. We will show how this and the corresponding result about irreducible subgroups of SO(2,n) follows from the Karpelevich-Mostov theorem. (This is joint work with Antonio J. Di Scala, Politecnico di Torino.)
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Building centralisers in ~A_2 groups 13:10 Fri 23 Oct 09 :: School Board Room :: Prof Guyan Robertson :: University of Newcastle, UK
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Analytic torsion for twisted de Rham complexes 13:10 Fri 30 Oct 09 :: School Board Room :: Prof Mathai Varghese :: University of Adelaide
We define analytic torsion for the twisted de Rham complex, consisting of differential forms on a compact Riemannian manifold X with coefficients in a flat vector bundle E, with a differential given by a flat connection on E plus a closed odd degree differential form on X. The definition in our case is more complicated than in the case discussed by Ray-Singer, as it uses pseudodifferential operators. We show that this analytic torsion is independent of the choice of metrics on X and E, establish some basic functorial properties, and compute it in many examples. We also establish the relationship of an invariant version of analytic torsion for T-dual circle bundles with closed 3-form flux. This is joint work with Siye Wu.
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Upper bounds for the essential dimension of the moduli stack of SL_n-bundles over a curve 11:10 Mon 14 Dec 09 :: School Board Room :: Dr Nicole Lemire :: University of Western Ontario, Canada
In joint work with Ajneet Dhillon, we find upper bounds for the essential dimension of various moduli stacks of SL_n-bundles over a curve. When n is a prime power, our calculation computes the essential dimension of the moduli stack of stable bundles exactly and the essential dimension is not equal to the dimension in this case.
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Critical sets of products of linear forms 13:10 Mon 14 Dec 09 :: School Board Room :: Dr Graham Denham :: University of Western Ontario, Canada
Suppose $f_1,f_2,\ldots,f_n$ are linear polynomials in $\ell$
variables and $\lambda_1,\lambda_2,\ldots,\lambda_n$ are nonzero complex numbers. The product
$$
\Phi_\lambda=\Prod_{i=1}^n f_1^{\lambda_i},
$$
called a master function,
defines a (multivalued) function on $\ell$-dimensional complex space, or more precisely, on the complement of a set of hyperplanes. Then it is easy to ask (but harder to answer) what the set of critical points of a master function looks like, in terms of some properties of the input polynomials and $\lambda_i$'s.
In my talk I will describe the motivation for considering such a question. Then I will indicate how the geometry and combinatorics of hyperplane arrangements can be used to provide at least a partial answer.
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Hartogs-type holomorphic extensions 13:10 Tue 15 Dec 09 :: School Board Room :: Prof Roman Dwilewicz :: Missouri University of Science and Technology
We will review holomorphic extension problems starting with the famous Hartogs extension theorem (1906), via Severi-Kneser-Fichera-Martinelli theorems, up to some recent (partial) results of Al Boggess (Texas A&M Univ.), Zbigniew Slodkowski (Univ. Illinois at Chicago), and the speaker. The holomorphic extension problems for holomorphic or Cauchy-Riemann functions are fundamental problems in complex analysis of several variables. The talk will be very elementary, with many figures, and accessible to graduate and even advanced undergraduate students.
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Group actions in complex geometry, I and II 13:10 Fri 8 Jan 10 :: School Board Room :: Prof Frank Kutzschebauch, IGA Lecturer :: University of Berne
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Group actions in complex geometry, III and IV 10:10 Fri 15 Jan 10 :: School Board Room :: Prof Frank Kutzschebauch, IGA Lecturer :: University of Berne
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Group actions in complex geometry, V and VI 10:10 Fri 22 Jan 10 :: School Board Room :: Prof Frank Kutzschebauch, IGA Lecturer :: University of Berne
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Group actions in complex geometry, VII and VIII 10:10 Fri 29 Jan 10 :: Napier LG 23 :: Prof Frank Kutzschebauch, IGA Lecturer :: University of Berne
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Oka manifolds and Oka maps 13:10 Fri 29 Jan 10 :: Napier LG 23 :: Prof Franc Forstneric :: University of Ljubljana
In this survey lecture I will discuss a
new class of complex manifolds and of holomorphic maps
between them which I introduced in 2009
(F. Forstneric, Oka Manifolds, C. R. Acad. Sci. Paris,
Ser. I, 347 (2009) 1017-1020).
Roughly speaking, a complex manifold Y is said to be
an Oka manifold if Y admits plenty of holomorphic maps
from any Stein manifold (or Stein space) X to Y,
in a certain precise sense. In particular, the inclusion
of the space of holomorphic maps of X to Y into the space of
continuous maps must be a weak homotopy equivalence.
One of the main results is that this class of manifolds
can be characterized by a simple Runge approximation property
for holomorphic maps from complex Euclidean spaces C^n to Y,
with approximation on compact convex subsets of C^n.
This answers in the affirmative a question posed by
M. Gromov in 1989. I will also discuss the Oka properties
of holomorphic maps and their characterization by
approximation properties.
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Proper holomorphic maps from strongly pseudoconvex domains to q-convex manifolds 13:10 Fri 5 Feb 10 :: School Board Room :: Prof Franc Forstneric :: University of Ljubljana
(Joint work with B. Drinovec Drnovsek, Amer. J. Math., in press.)
I will discuss the existence of closed complex subvarieties
of a complex manifold X that are proper holomorphic images
of strongly pseudoconvex Stein domains. The main
sufficient condition is expressed in terms of
the Morse indices and of the number of positive
Levi eigenvalues of an exhaustion function on X.
Examples show that our condition cannot be weakened in general.
I will describe optimal results for subvarieties of this type in
complements of compact complex submanifolds with Griffiths
positive normal bundle; in the projective case these
generalize classical theorems of Remmert, Bishop and
Narasimhan concerning proper holomorphic maps and embeddings
to complex Euclidean spaces.
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Conformal geometry of differential equations 13:10 Fri 12 Feb 10 :: School Board Room :: Dr Pawel Nurowski :: University of Warsaw
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Integrable systems: noncommutative versus commutative 14:10 Thu 4 Mar 10 :: School Board Room :: Dr Cornelia Schiebold :: Mid Sweden University
After a general introduction to integrable systems, we will explain an
approach to their solution theory, which is based on Banach space theory. The
main point is first to shift attention to noncommutative integrable systems and
then to extract information about the original setting via projection techniques.
The resulting solution formulas turn out to be particularly well-suited to the
qualitative study of certain solution classes. We will show how one can obtain
a complete asymptotic description of the so called multiple pole solutions, a
problem that was only treated for special cases before.
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Convolution equations in A^{-\infty} for convex domains 13:10 Fri 5 Mar 10 :: School Board Room :: Dr Le Hai Khoi :: Nanyang Technological University, Singapore
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Holomorphic extension on complex spaces 14:10 Fri 5 Mar 10 :: School Board Room :: Prof Egmont Porten :: Mid Sweden University
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Conformal structures with G_2 ambient metrics 13:10 Fri 19 Mar 10 :: School Board Room :: Dr Thomas Leistner :: University of Adelaide
The n-sphere considered as a conformal manifold can be viewed as the projectivisation of the light cone in n+2 Minkowski space. A construction that generalises this picture to arbitrary conformal classes is the ambient metric introduced by C. Fefferman and R. Graham. In the talk, I will explain the Fefferman-Graham ambient metric construction and how it detects the existence of certain metrics in the conformal class. Then I will present conformal classes of signature (3,2) for which the 7-dimensional ambient metric has the noncompact exceptional Lie group G_2 as its holonomy. This is joint work with P. Nurowski, Warsaw University.
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The fluid mechanics of gels used in tissue engineering 15:10 Fri 9 Apr 10 :: Santos Lecture Theatre :: Dr Edward Green :: University of Western Australia
Tissue engineering could be called 'the science of spare parts'.
Although currently in its infancy, its long-term aim is to grow
functional tissues and organs in vitro to replace those which have
become defective through age, trauma or disease. Recent experiments
have shown that mechanical interactions between cells and the materials
in which they are grown have an important influence on tissue
architecture, but in order to understand these effects, we first need to
understand the mechanics of the gels themselves.
Many biological gels (e.g. collagen) used in tissue engineering have a
fibrous microstructure which affects the way forces are transmitted
through the material, and which in turn affects cell migration and other
behaviours. I will present a simple continuum model of gel mechanics,
based on treating the gel as a transversely isotropic viscous material.
Two canonical problems are considered involving thin two-dimensional
films: extensional flow, and squeezing flow of the fluid between two
rigid plates. Neglecting inertia, gravity and surface tension, in each
regime we can exploit the thin geometry to obtain a leading-order
problem which is sufficiently tractable to allow the use of analytical
methods. I discuss how these results could be exploited practically to
determine the mechanical properties of real gels. If time permits, I
will also talk about work currently in progress which explores the
interaction between gel mechanics and cell behaviour.
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Random walk integrals 13:10 Fri 16 Apr 10 :: School Board Room :: Prof Jonathan Borwein :: University of Newcastle
Following Pearson in 1905, we study the expected distance of a two-dimensional walk in the plane with unit steps in random directions---what Pearson called a "ramble". A series evaluation and recursions are obtained making it possible to explicitly determine this distance for small number of steps. Closed form expressions for all the moments of a 2-step and a 3-step walk are given, and a formula is conjectured for the 4-step walk. Heavy use is made of the analytic continuation of the underlying integral.
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Loop groups and characteristic classes 13:10 Fri 23 Apr 10 :: School Board Room :: Dr Raymond Vozzo :: University of Adelaide
Suppose $G$ is a compact Lie group, $LG$ its (free) loop group and $\Omega G \subseteq LG$ its based loop group. Let $P \to M$ be a principal bundle with structure group one of these loop groups. In general, differential form representatives of characteristic classes for principal bundles can be easily obtained using the Chern-Weil homomorphism, however for infinite-dimensional bundles such as $P$ this runs into analytical problems and classes are more difficult to construct. In this talk I will explain some new results on characteristic classes for loop group bundles which demonstrate how to construct certain classes---which we call string classes---for such bundles. These are obtained by making heavy use of a certain $G$-bundle associated to any loop group bundle (which allows us to avoid the problems of dealing with infinite-dimensional bundles). We shall see that the free loop group case naturally involves equivariant cohomology.
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Moduli spaces of stable holomorphic vector bundles II 13:10 Fri 30 Apr 10 :: School Board Room :: A/Prof Nicholas Buchdahl :: University of Adelaide
In this talk, I shall briefly review the notion of
stability for holomorphic vector bundles on compact
complex manifolds as discussed in the first part of this
talk (28 August 2009). Then I shall attempt to compute
some explicit examples in simple situations, illustrating
the use of basic algebraic-geometric tools.
The level of the talk will be appropriate for graduate
students, particularly those who have been taking part
in the algebraic geometry reading group meetings.
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The caloron transform 13:10 Fri 7 May 10 :: School Board Room :: Prof Michael Murray :: University of Adelaide
The caloron transform is a `fake' dimensional reduction which transforms a G-bundle over certain
manifolds to a loop group of G bundle over a manifold of one lower dimension. This talk will review the
caloron transform and show how it can be best understood using the language of pseudo-isomorphisms
from category theory as well as considering its application to Bogomolny monopoles and string
structures.
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Holonomy groups 15:10 Fri 7 May 10 :: Napier LG24 :: Dr Thomas Leistner :: University of Adelaide
In the first part of the talk I will illustrate some basic concepts of differential geometry that lead to the notion of a holonomy group. Then I will explain Berger's classification of Riemannian holonomy groups and discuss questions that arose from it. Finally, I will focus on holonomy groups of Lorentzian manifolds and indicate briefly why all this is of relevance to present-day theoretical physics.
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Moduli spaces of stable holomorphic vector bundles III 13:10 Fri 14 May 10 :: School Board Room :: A/Prof Nicholas Buchdahl :: University of Adelaide
This talk is a continuation of the talk on 30 April. The same abstract applies:
In this talk, I shall briefly review the notion of
stability for holomorphic vector bundles on compact
complex manifolds as discussed in the first part of this
talk (28 August 2009). Then I shall attempt to compute
some explicit examples in simple situations, illustrating
the use of basic algebraic-geometric tools.
The level of the talk will be appropriate for graduate
students, particularly those who have been taking part
in the algebraic geometry reading group meetings.
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Functorial 2-connected covers 13:10 Fri 21 May 10 :: School Board Room :: David Roberts :: University of Adelaide
The Whitehead tower of a topological space seeks to resolve that space by successively removing homotopy groups from the 'bottom up'. For a path-connected space with no 1-dimensional local pathologies the first stage in the tower can be chosen to be the universal (=1-connected) covering space. This construction also works in the category Diff of manifolds. However, further stages in the two known constructions of the Whitehead tower do not work in Diff, being purely topological - and one of these is non-functorial, depending on a large number of choices. This talk will survey results from my thesis which constructs a new, functorial model for the 2-connected cover which will lift to a generalised (2-)category of smooth objects.
This talk contains joint work with Andrew Stacey of the Norwegian University of Science and Technology.
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On the uniqueness of almost-Kahler structures 13:10 Fri 28 May 10 :: School Board Room :: Dr Paul-Andi Nagy :: University of Auckland
We show uniqueness up to sign of positive, orthogonal almost-Kahler structures on any non-scalar flat Kahler-Einstein surface. This is joint work with A. J. di Scala.
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Vertex algebras and variational calculus I 13:10 Fri 4 Jun 10 :: School Board Room :: Dr Pedram Hekmati :: University of Adelaide
A basic operation in calculus of variations is the Euler-Lagrange variational
derivative, whose kernel determines the extremals of functionals. There exists a
natural resolution of this operator, called the variational complex.
In this talk, I shall explain how to use tools from the theory of vertex
algebras
to explicitly construct the variational complex. This also provides a very
convenient language for classifying and constructing integrable Hamiltonian
evolution equations.
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Vertex algebras and variational calculus II 13:10 Fri 11 Jun 10 :: School Board Room :: Dr Pedram Hekmati :: University of Adelaide
Last time I introduced the variational complex of an algebra of differential
functions and gave a sketchy definition of a vertex algebra. This week I will
make this notion more precise and explain how to apply it to the calculus of
variations.
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On affine BMW algebras 13:10 Fri 25 Jun 10 :: Napier 208 :: Prof Arun Ram :: University of Melbourne
I will describe a family of algebras of tangles (which give rise to link invariants
following the methods of Reshetikhin-Turaev and Jones) and describe some aspects of their
structure and their representation theory. The main goal will be to explain how to use
universal Verma modules for the symplectic group to compute the representation theory
of affine BMW (Birman-Murakami-Wenzl) algebras.
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Introduction to mirror symmetry and the Fukaya category I 13:10 Thu 15 Jul 10 :: Napier G04 :: Dr Mohammed Abouzaid, IGA Lecturer :: Clay Research Fellow, MIT
I shall give an overview of recent progress in homological mirror symmetry, both in clarifying our conceptual understanding of how the sign of the canonical bundle affects the behaviour of the mirror, and in obtaining concrete examples where the mirror conjecture has now been verified. (This is a two-hour talk.)
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Introduction to mirror symmetry and the Fukaya category II 13:10 Fri 16 Jul 10 :: Napier G04 :: Dr Mohammed Abouzaid, IGA Lecturer :: Clay Research Fellow, MIT
I shall give an overview of recent progress in homological mirror symmetry, both in clarifying our conceptual understanding of how the sign of the canonical bundle affects the behaviour of the mirror, and in obtaining concrete examples where the mirror conjecture has now been verified. (This is a two-hour talk.)
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Introduction to mirror symmetry and the Fukaya category III 13:10 Mon 19 Jul 10 :: Napier G04 :: Dr Mohammed Abouzaid, IGA Lecturer :: Clay Research Fellow, MIT
I shall give an overview of recent progress in homological mirror symmetry, both in clarifying our conceptual understanding of how the sign of the canonical bundle affects the behaviour of the mirror, and in obtaining concrete examples where the mirror conjecture has now been verified. (This is a two-hour talk.)
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Introduction to mirror symmetry and the Fukaya category IV 13:10 Tue 20 Jul 10 :: Napier G04 :: Dr Mohammed Abouzaid, IGA Lecturer :: Clay Research Fellow, MIT
I shall give an overview of recent progress in homological mirror symmetry, both in clarifying our conceptual understanding of how the sign of the canonical bundle affects the behaviour of the mirror, and in obtaining concrete examples where the mirror conjecture has now been verified. (This is a two-hour talk.)
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Introduction to mirror symmetry and the Fukaya category V 13:10 Wed 21 Jul 10 :: Napier G04 :: Dr Mohammed Abouzaid, IGA Lecturer :: Clay Research Fellow, MIT
I shall give an overview of recent progress in homological mirror symmetry, both in clarifying our conceptual understanding of how the sign of the canonical bundle affects the behaviour of the mirror, and in obtaining concrete examples where the mirror conjecture has now been verified. (This is a two-hour talk.)
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Higher nonunital Quillen K'-theory 13:10 Fri 23 Jul 10 :: Engineering-Maths G06 :: Dr Snigdhayan Mahanta :: University of Adelaide
Quillen introduced a $K'_0$-theory for possibly nonunital
rings and showed that it
agrees with the usual algebraic $K_0$-theory if the ring is unital. We
shall introduce higher
$K'$-groups for $k$-algebras, where $k$ is a field, and discuss some
elementary properties
of this theory. We shall also show that for stable $C*$-algebras the
higher $K'$-theory agrees
with the topological $K$-theory. If time permits we shall explain how
this provides a formalism
to treat topological $\mathbb{T}$-dualities via Kasparov's bivariant $K$-theory.
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Eynard-Orantin invariants and enumerative geometry 13:10 Fri 6 Aug 10 :: Ingkarni Wardli B20 (Suite 4) :: Dr Paul Norbury :: University of Melbourne
As a tool for studying enumerative problems in geometry Eynard and Orantin associate multilinear differentials to any plane curve. Their work comes from matrix models but does not require matrix models (for understanding or calculations). In some sense they describe deformations of complex structures of a curve and conjectural relationships to deformations of Kahler structures of an associated object. I will give an introduction to their invariants via explicit examples, mainly to do with the moduli space of Riemann surfaces, in which the plane curve has genus zero.
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Counting lattice points in polytopes and geometry 15:10 Fri 6 Aug 10 :: Napier G04 :: Dr Paul Norbury :: University of Melbourne
Counting lattice points in polytopes arises in many areas of pure and applied mathematics. A basic counting problem is this: how many different ways can one give change of 1 dollar into 5,10, 20 and 50 cent coins? This problem counts lattice points in a tetrahedron, and if there also must be exactly 10 coins then it counts lattice points in a triangle. The number of lattice points in polytopes can be used to measure the robustness of a computer network, or in statistics to test independence of characteristics of samples. I will describe the general structure of lattice point counts and the difficulty of calculations. I will then describe a particular lattice point count in which the structure simplifies considerably allowing one to calculate easily. I will spend a brief time at the end describing how this is related to the moduli space of Riemann surfaces.
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Index theory in the noncommutative world 13:10 Fri 20 Aug 10 :: Ingkarni Wardli B20 (Suite 4) :: Prof Alan Carey :: Australian National University
The aim of the talk is to give an overview of the noncommutative geometry approach to index theory.
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A classical construction for simplicial sets revisited 13:10 Fri 27 Aug 10 :: Ingkarni Wardli B20 (Suite 4) :: Dr Danny Stevenson :: University of Glasgow
Simplicial sets became popular in the 1950s as a combinatorial way to
study the homotopy theory of topological spaces. They are more robust
than the older notion of simplicial complexes, which were introduced
for the same purpose. In this talk, which will be as introductory as
possible, we will review some classical functors arising in the theory
of simplicial sets, some well-known, some not-so-well-known. We will
re-examine the proof of an old theorem of Kan in light of these
functors. We will try to keep all jargon to a minimum.
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On some applications of higher Quillen K'-theory 13:10 Fri 3 Sep 10 :: Ingkarni Wardli B20 (Suite 4) :: Dr Snigdhayan Mahanta :: University of Adelaide
In my previous talk I introduced a functor from the category of k-algebras (k field) to abelian groups, called KQ-theory. In this talk I will explain its relationship with
topological (homological) T-dualities and twisted K-theory.
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Contraction subgroups in locally compact groups 13:10 Fri 17 Sep 10 :: Ingkarni Wardli B20 (Suite 4) :: Prof George Willis :: University of Newcastle
For each automorphism, $\alpha$, of the locally compact group $G$ there is a corresponding {\sl contraction subgroup\/}, $\hbox{con}(\alpha)$, which is the set of $x\in G$ such that $\alpha^n(x)$ converges to the identity as $n\to \infty$. Contractions subgroups are important in representation theory, through the Mautner phenomenon, and in the study of convolution semigroups.
If $G$ is a Lie group, then $\hbox{con}(\alpha)$ is automatically closed, can be described in terms of eigenvalues of $\hbox{ad}(\alpha)$, and is nilpotent. Since any connected group may be approximated by Lie groups, contraction subgroups of connected groups are thus well understood. Following a general introduction, the talk will focus on contraction subgroups of totally disconnected groups. A criterion for non-triviality of $\hbox{con}(\alpha)$ will be described (joint work with U.~Baumgartner) and a structure theorem for $\hbox{con}(\alpha)$ when it is closed will be presented (joint with H.~Gl\"oeckner).
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Some algebras associated with quantum gauge theories 13:10 Fri 15 Oct 10 :: Ingkarni Wardli B20 (Suite 4) :: Dr Keith Hannabuss :: Balliol College, Oxford
Classical gauge theories study sections of vector bundles and associated connections and curvature. The corresponding quantum gauge theories are normally written algebraically but can be understood as noncommutative geometries. This talk will describe one approach to the quantum gauge theories which uses braided categories.
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IGA-AMSI Workshop: Dirac operators in geometry, topology, representation theory, and physics 10:00 Mon 18 Oct 10 :: 7.15 Ingkarni Wardli :: Prof Dan Freed :: University of Texas, Austin
Lecture Series by Dan Freed (University of Texas, Austin).
Dirac introduced his eponymous operator to describe electrons in quantum theory.
It was rediscovered by Atiyah and Singer in their study of the index problem on
manifolds. In these lectures we explore new theorems and applications. Several
of these also involve K-theory in its recent twisted and differential
variations.
These lectures will be supplemented by additional talks by invited speakers. For more details, please see the conference webpage:
http://www.iga.adelaide.edu.au/workshops/WorkshopOct2010/
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Higher stacks and homotopy theory II: the motivic context 13:10 Thu 16 Dec 10 :: Ingkarni Wardli B21 :: Mr James Wallbridge :: University of Adelaide and Institut de mathematiques de Toulouse
In part I of this talk (JC seminar May 2008) we presented motivation
and the basic definitions for building homotopy theory into an arbitrary
category by introducing the notion of (higher) stacks. In part II we consider a
specific example on the category of schemes to illustrate how the machinery
works in practice. It will lead us into motivic territory (if we like it or
not).
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Complete quaternionic Kahler manifolds associated to cubic polynomials 13:10 Fri 11 Feb 11 :: Ingkarni Wardli B18 :: Prof Vicente Cortes :: University of Hamburg
We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h = 1} defined by a homogeneous cubic polynomial h such that -\partial^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special
Kahler manifold defines a complete quaternionic Kahler manifold of negative scalar curvature. We classify all complete quaternionic Kahler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.
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Real analytic sets in complex manifolds I: holomorphic closure dimension 13:10 Fri 4 Mar 11 :: Mawson 208 :: Dr Rasul Shafikov :: University of Western Ontario
After a quick introduction to real and complex analytic sets,
I will discuss possible notions of complex dimension of real sets, and then discuss a structure theorem for the holomorphic closure dimension which is defined as the dimension of the smallest complex analytic germ containing the real germ.
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Real analytic sets in complex manifolds II: complex dimension 13:10 Fri 11 Mar 11 :: Mawson 208 :: Dr Rasul Shafikov :: University of Western Ontario
Given a real analytic set R, denote by A the subset of R of points through which there is a nontrivial complex variety contained in R, i.e., A consists of points in R of positive complex dimension. I will discuss the structure of the set A.
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Surface quotients of hyperbolic buildings 13:10 Fri 18 Mar 11 :: Mawson 208 :: Dr Anne Thomas :: University of Sydney
Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons, and the link at each vertex is the complete bipartite graph K_{v,v}. We investigate and mostly determine the set of triples (p,v,g) for which there is a discrete group acting on I(p,v) so that the quotient is a compact orientable surface of genus g. Surprisingly, the existence of such a quotient depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. We use elementary group theory, combinatorics, algebraic topology and number theory. This is joint work with David Futer.
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Lorentzian manifolds with special holonomy 13:10 Fri 25 Mar 11 :: Mawson 208 :: Mr Kordian Laerz :: Humboldt University, Berlin
A parallel lightlike vector field on a Lorentzian manifold X naturally defines a foliation of codimension 1 on X and a 1-dimensional subfoliation. In the first part we introduce Lorentzian metrics on the total space of certain circle bundles in order to construct weakly irreducible Lorentzian manifolds admitting a parallel lightlike vector field such that all leaves of the foliations are compact. Then we study which holonomy representations can be realized in this way. Finally, we consider the structure of arbitrary Lorentzian manifolds for which the leaves of the foliations are compact.
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Operator algebra quantum groups 13:10 Fri 1 Apr 11 :: Mawson 208 :: Dr Snigdhayan Mahanta :: University of Adelaide
Woronowicz initiated the study of quantum groups using C*-algebras. His framework enabled him to deal with compact (linear) quantum groups. In this talk we shall introduce a notion of quantum groups that can handle infinite dimensional examples like SU(\infty). We shall also study some quantum homogeneous spaces associated to this group and compute their K-theory groups. This is joint work with V. Mathai.
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How round is your triangle, square, pentagon, ...? 12:10 Wed 6 Apr 11 :: Napier 210 :: Dr Barry Cox :: University of Adelaide
Media...
Most of us are familiar with the problem of making circular holes in wood or other material. For smaller diameter holes we typically use a drill, and for larger diameter holes a spade-bit, hole-saw or plunge router may be used. However for some applications, like mortise-and-tenon joints, what is needed is a tool that will produce a hole with a cross-section that is something other than a circle. In this talk we look at curves that may be used as the basis for a device that will produce holes with a cross-section of an equilateral triangle, square, or any regular polygon. Along the way we will touch on areas of engineering, algebra, geometry, calculus, Gothic art and architecture.
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Spherical tube hypersurfaces 13:10 Fri 8 Apr 11 :: Mawson 208 :: Prof Alexander Isaev :: Australian National University
We consider smooth real hypersurfaces in a complex vector space. Specifically, we are interested in tube hypersurfaces, i.e., hypersurfaces represented as the direct product of the imaginary part of the space and hypersurfaces lying in its real part. Tube hypersurfaces arise, for instance, as the boundaries of tube domains. The study of tube domains is a classical subject in several complex variables and complex geometry, which goes back to the beginning of the 20th century. Indeed, already Siegel found it convenient to realise certain symmetric domains as tubes.
One can endow a tube hypersurface with a so-called CR-structure, which is the remnant of the complex structure on the ambient vector space. We impose on the CR-structure the condition of sphericity. One way to state this condition is to require a certain curvature (called the CR-curvature of the hypersurface) to vanish identically. Spherical tube hypersurfaces possess remarkable properties and are of interest from both the complex-geometric and affine-geometric points of view. I my talk I will give an overview of the theory of such hypersurfaces. In particular, I will mention an algebraic construction arising from this theory that has applications in abstract commutative algebra and singularity theory. I will speak about these applications in detail in my colloquium talk later today.
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Algebraic hypersurfaces arising from Gorenstein algebras 15:10 Fri 8 Apr 11 :: 7.15 Ingkarni Wardli :: Associate Prof Alexander Isaev :: Australian National University
Media...
To every Gorenstein algebra of finite dimension greater than 1 over a field of characteristic zero, and a projection on its maximal ideal with range equal to the annihilator of the ideal, one can associate a certain algebraic hypersurface lying in the ideal. Such hypersurfaces possess remarkable properties. They can be used, for instance, to help decide whether two given Gorenstein algebras are isomorphic, which for the case of complex numbers leads to interesting consequences in singularity theory. Also, for the case of real numbers such hypersurfaces naturally arise in CR-geometry. In my talk I will discuss these hypersurfaces and some of their applications.
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Centres of cyclotomic Hecke algebras 13:10 Fri 15 Apr 11 :: Mawson 208 :: A/Prof Andrew Francis :: University of Western Sydney
The cyclotomic Hecke algebras, or Ariki-Koike algebras $H(R,q)$, are
deformations of the group algebras of certain complex reflection groups
$G(r,1,n)$, and also are quotients of the ubiquitous affine Hecke algebra.
The centre of the affine Hecke algebra has been understood since
Bernstein in terms of the symmetric group action on the weight lattice.
In this talk I will discuss the proof that over an arbitrary unital
commutative ring $R$, the centre of the affine Hecke algebra maps
\emph{onto} the centre of the cyclotomic Hecke algebra when $q-1$ is
invertible in $R$. This is the analogue of the fact that the centre of
the Hecke algebra of type $A$ is the set of symmetric polynomials in
Jucys-Murphy elements (formerly known as he Dipper-James conjecture). Key
components of the proof include the relationship between the trace
functions on the affine Hecke algebra and on the cyclotomic Hecke algebra,
and the link to the affine braid group. This is joint work with John
Graham and Lenny Jones.
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A strong Oka principle for embeddings of some planar domains into CxC*, I 13:10 Fri 6 May 11 :: Mawson 208 :: Mr Tyson Ritter :: University of Adelaide
The Oka principle refers to a collection of results in
complex analysis which state that there are only topological
obstructions to solving certain holomorphically defined problems
involving Stein manifolds. For example, a basic version of Gromov's
Oka principle states that every continuous map from a Stein manifold
into an elliptic complex manifold is homotopic to a holomorphic map.
In these two talks I will discuss a new result showing that
if we restrict the class of source manifolds to circular domains and
fix the target as CxC* we can obtain a much stronger Oka principle:
every continuous map from a circular domain S into CxC* is homotopic
to a proper holomorphic embedding. This result has close links with
the long-standing and difficult problem of finding proper holomorphic
embeddings of Riemann surfaces into C^2, with additional motivation
from other sources.
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A strong Oka principle for embeddings of some planar domains into CxC*, II 13:10 Fri 13 May 11 :: Mawson 208 :: Mr Tyson Ritter :: University of Adelaide
The Oka principle refers to a collection of results in
complex analysis which state that there are only topological
obstructions to solving certain holomorphically defined problems
involving Stein manifolds. For example, a basic version of Gromov's
Oka principle states that every continuous map from a Stein manifold
into an elliptic complex manifold is homotopic to a holomorphic map.
In these two talks I will discuss a new result showing that
if we restrict the class of source manifolds to circular domains and
fix the target as CxC* we can obtain a much stronger Oka principle:
every continuous map from a circular domain S into CxC* is homotopic
to a proper holomorphic embedding. This result has close links with
the long-standing and difficult problem of finding proper holomorphic
embeddings of Riemann surfaces into C^2, with additional motivation
from other sources.
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Knots, posets and sheaves 13:10 Fri 20 May 11 :: Mawson 208 :: Dr Brent Everitt :: University of York
The Euler characteristic is a nice simple integer invariant that one can attach to a space. Unfortunately, it is not natural: maps between spaces do not induce maps between their Euler characteristics, because it makes no sense to talk of a map between integers. This shortcoming is fixed by homology. Maps between spaces induce maps between their homologies, with the Euler characteristic encoded inside the homology. Recently it has become possible to play the same game with knots and the Jones polynomial: the Khovanov homology of a knot both encodes the Jones polynomial and is a natural invariant of the knot. After saying what all this means, this talk will observe that Khovanov homology is just a special case of sheaf homology on a poset, and we will explore some of the ramifications of this observation. This is joint work with Paul Turner (Geneva/Fribourg).
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Lifting principal bundles and abelian extensions 13:10 Fri 27 May 11 :: Mawson 208 :: Prof Michael Murray :: School of Mathematical Sciences
I will review what it means to lift the structure group of a principal bundle
and the topological obstruction to this in the case of a central extension. I will then discuss
some new results in the case of abelian extensions.
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Natural operations on the Hochschild cochain complex 13:10 Fri 3 Jun 11 :: Mawson 208 :: Dr Michael Batanin :: Macquarie University
The Hochschild cochain complex of an associative algebra provides an important bridge between algebra and geometry.
Algebraically, this is the derived center of the algebra. Geometrically, the Hochschild cohomology of the algebra of smooth functions on a manifold is isomorphic to the graduate space of polyvector fields on this manifold.
There are many important operations acting on the Hochschild complex. It is, however, a tricky question to ask which operations are natural because the Hochschild complex is not a functor. In my talk I will explain how we can overcome this obstacle and compute all possible natural operations on the Hochschild complex. The result leads immediately to a proof of the Deligne conjecture on Hochschild cochains.
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Object oriented data analysis of tree-structured data objects 15:10 Fri 1 Jul 11 :: 7.15 Ingkarni Wardli :: Prof Steve Marron :: The University of North Carolina at Chapel Hill
The field of Object Oriented Data Analysis has made a lot of
progress on the statistical analysis of the variation in populations
of complex objects. A particularly challenging example of this type
is populations of tree-structured objects. Deep challenges arise,
which involve a marriage of ideas from statistics, geometry, and
numerical analysis, because the space of trees is strongly
non-Euclidean in nature. These challenges, together with three
completely different approaches to addressing them, are illustrated
using a real data example, where each data point is the tree of blood
arteries in one person's brain.
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What is... a tensor? 12:10 Mon 25 Jul 11 :: 5.57 Ingkarni Wardli :: Mr Michael Albanese :: School of Mathematical Sciences
Tensors are important objects that are frequently used in a
variety of fields including continuum mechanics, general relativity and
differential geometry. Despite their importance, they are often defined
poorly (if at all) which contributes to a lack of understanding. In this
talk, I will give a concrete definition of a tensor and provide some
familiar examples. For the remainder of the talk, I will discuss some
applications—here I mean applications in the pure maths sense (i.e. more
abstract nonsense, but hopefully still interesting).
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The (dual) local cyclic homology valued Chern-Connes character for some infinite dimensional spaces 13:10 Fri 29 Jul 11 :: B.19 Ingkarni Wardli :: Dr Snigdhayan Mahanta :: School of Mathematical Sciences
I will explain how to construct a bivariant Chern-Connes character on the category of sigma-C*-algebras taking values in Puschnigg's local cyclic homology. Roughly, setting the first (resp. the second) variable to complex numbers one obtains the K-theoretic (resp. dual K-homological) Chern-Connes character in one variable. We shall focus on the dual K-homological Chern-Connes character and investigate it in the example of SU(infty).
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Towards Rogers-Ramanujan identities for the Lie algebra A_n 13:10 Fri 5 Aug 11 :: B.19 Ingkarni Wardli :: Prof Ole Warnaar :: University of Queensland
The Rogers-Ramanujan identities are a pair of q-series identities proved by Leonard Rogers in 1894 which became famous two decades later as conjectures of Srinivasa Ramanujan. Since the 1980s it is known that the Rogers-Ramanujan identities are in fact identities for characters of certain modules for the affine Lie algebra A_1. This poses the obvious question as to whether there exist Rogers-Ramanujan identities for higher rank affine Lie algebras. In this talk I will describe some recent progress on this problem. I will also discuss a seemingly mysterious connection with the representation theory of quivers over finite fields.
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The Selberg integral 15:10 Fri 5 Aug 11 :: 7.15 Ingkarni Wardli :: Prof Ole Warnaar :: University of Queensland
Media...
In this talk I will give a gentle introduction to the mathematics surrounding the Selberg integral. Selberg's integral, which first appeared in two rather unusual papers by Atle Selberg in the 1940s, has become famous as much for its association with (other) mathematical greats such as Enrico Bombieri and Freeman Dyson as for its importance in algebra (Coxeter groups), geometry (hyperplane arrangements) and number theory (the Riemann hypothesis). In this talk I will review the remarkable history of the Selberg integral and discuss some of its early applications. Time permitting I will end the talk by describing some of my own, ongoing work on Selberg integrals related to Lie algebras.
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Horocycle flows at prime times 13:10 Wed 10 Aug 11 :: B.19 Ingkarni Wardli :: Prof Peter Sarnak :: Institute for Advanced Study, Princeton
The distribution of individual orbits of unipotent flows in homogeneous spaces are well
understood thanks to the work work of Marina Ratner. It is conjectured that this property
is preserved on restricting the times from the integers to primes, this being important in the study of prime numbers as well as in such dynamics. We review progress in understanding this conjecture, starting with Dirichlet (a finite system), Vinogradov (rotation of a circle or torus), Green and Tao (translation on a nilmanifold) and Ubis and Sarnak (horocycle flows in the semisimple case).
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K3 surfaces: a crash course 13:10 Fri 12 Aug 11 :: B.19 Ingkarni Wardli :: A/Prof Nicholas Buchdahl :: University of Adelaide
Everything you have ever wanted to know about K3 surfaces! Two talks: 1:10 pm to 3:00 pm.
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There are no magnetically charged particle-like solutions of the Einstein-Yang-Mills equations for models with Abelian residual groups 13:10 Fri 19 Aug 11 :: B.19 Ingkarni Wardli :: Dr Todd Oliynyk :: Monash University
According to a conjecture from the 90's, globally regular, static, spherically symmetric (i.e. particle-like) solutions with nonzero total magnetic charge are not expected to exist in Einstein-Yang-Mills theory. In this talk, I will describe recent work done in collaboration with M. Fisher where we establish the validity of this conjecture under certain restrictions on the residual gauge group. Of particular interest is that our non-existence results apply to the most widely studied models with Abelian residual groups.
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Deformations of Oka manifolds 13:10 Fri 26 Aug 11 :: B.19 Ingkarni Wardli :: A/Prof Finnur Larusson :: University of Adelaide
We discuss the behaviour of the Oka property with respect to deformations of compact complex manifolds. We have recently proved that in a family of compact complex manifolds, the set of Oka fibres corresponds to a G_delta subset of the base. We have also found a necessary and sufficient condition for the limit fibre of a sequence of Oka fibres to be Oka in terms of a new uniform Oka property. The special case when the fibres are tori will be considered, as well as the general case of holomorphic submersions with noncompact fibres.
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Oka properties of some hypersurface complements 13:10 Fri 2 Sep 11 :: B.19 Ingkarni Wardli :: Mr Alexander Hanysz :: University of Adelaide
Oka manifolds can be viewed as the "opposite" of Kobayashi hyperbolic manifolds. Kobayashi conjectured that the complement of a generic algebraic hypersurface of sufficiently high degree is hyperbolic. Therefore it is natural to ask whether the complement is Oka for the case of low degree or non-algebraic hypersurfaces. We provide a complete answer to this question for complements of hyperplane arrangements, and some results for graphs of meromorphic functions.
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IGA-AMSI Workshop: Group-valued moment maps with applications to mathematics and physics 10:00 Mon 5 Sep 11 :: 7.15 Ingkarni Wardli
Media...
Lecture series by Eckhard Meinrenken, University of Toronto.
Titles of individual lectures: 1) Introduction to G-valued moment maps. 2) Dirac geometry and Witten's volume formulas.
3) Dixmier-Douady theory and pre-quantization. 4) Quantization of group-valued moment maps. 5) Application to Verlinde formulas. These lectures will be supplemented by additional talks by invited speakers. For more details, please see the conference webpage.
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Twisted Morava K-theory 13:10 Fri 9 Sep 11 :: 7.15 Ingkarni Wardli :: Dr Craig Westerland :: University of Melbourne
Morava's extraordinary K-theories K(n) are a family of generalized cohomology theories which behave in some ways like K-theory (indeed, K(1) is mod 2 K-theory). Their construction exploits Quillen's description of cobordism in terms of formal group laws and Lubin-Tate's methods in class field theory for constructing abelian extensions of number fields. Constructed from homotopy-theoretic methods, they do not admit a geometric description (like deRham cohomology, K-theory, or cobordism), but are nonetheless subtle, computable invariants of topological spaces. In this talk, I will give an introduction to these theories, and explain how it is possible to define an analogue of twisted K-theory in this setting. Traditionally, K-theory is twisted by a three-dimensional cohomology class; in this case, K(n) admits twists by (n+2)-dimensional classes. This work is joint with Hisham Sati.
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Configuration spaces in topology and geometry 15:10 Fri 9 Sep 11 :: 7.15 Ingkarni Wardli :: Dr Craig Westerland :: University of Melbourne
Media...
Configuration spaces of points in R^n give a family of interesting geometric objects. They and their variants have numerous applications in geometry, topology, representation theory, and number theory. In this talk, we will review several of these manifestations (for instance, as moduli spaces, function spaces, and the like), and use them to address certain conjectures in number theory regarding distributions of number fields.
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Cohomology of higher-rank graphs and twisted C*-algebras 13:10 Fri 16 Sep 11 :: B.19 Ingkarni Wardli :: Dr Aidan Sims :: University of Wollongong
Higher-rank graphs and their $C^*$-algebras were introduced by Kumjian and Pask in 2000. They have provided a rich source of tractable examples of $C^*$-algebras, the most elementary of which are the commutative algebras $C(\mathbb{T}^k)$ of continuous functions on $k$-tori. In this talk we shall describe how to define the homology and cohomology of a higher-rank graph, and how to associate to each higher-rank graph $\Lambda$ and $\mathbb{T}$-valued cocycle on $\Lambda$ a twisted higher-rank graph $C^*$-algebra. As elementary examples, we obtain all noncommutative tori.
This is a preleminary report on ongoing joint work with Alex Kumjian and David Pask.
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T-duality via bundle gerbes I 13:10 Fri 23 Sep 11 :: B.19 Ingkarni Wardli :: Dr Raymond Vozzo :: University of Adelaide
In physics T-duality is a phenomenon which relates certain types of string theories to one another. From a topological point of view, one can view string theory as a duality between line bundles carrying a degree three cohomology class (the H-flux). In this talk we will use bundle gerbes to give a geometric realisation of the H-flux and explain how to construct the T-dual of a line bundle together with its T-dual bundle gerbe.
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T-duality via bundle gerbes II 13:10 Fri 21 Oct 11 :: B.19 Ingkarni Wardli :: Dr Raymond Vozzo :: University of Adelaide
In physics T-duality is a phenomenon which relates certain types of string theories to one another. From a topological point of view, one can view string theory as a duality between line bundles carrying a degree three cohomology class (the H-flux). In this talk we will use bundle gerbes to give a geometric realisation of the H-flux and explain how to construct the T-dual of a line bundle together with its T-dual bundle gerbe.
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Dirac operators on classifying spaces 13:10 Fri 28 Oct 11 :: B.19 Ingkarni Wardli :: Dr Pedram Hekmati :: University of Adelaide
The Dirac operator was introduced by Paul Dirac in 1928 as the formal square
root of the D'Alembert operator. Thirty years later it was rediscovered in
Euclidean signature by Atiyah and Singer in their seminal work on index theory.
In this talk I will describe efforts to construct a Dirac type operator on the
classifying space for odd complex K-theory. Ultimately the aim is to produce a
projective family of Fredholm operators realising elements in twisted K-theory
of a certain moduli stack.
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Staircase to heaven 13:10 Fri 4 Nov 11 :: B.19 Ingkarni Wardli :: Dr Burkard Polster :: Monash University
Media...
How much of an overhang can we produce by stacking identical rectangular blocks at the edge of a table? It has been known for at least 100 years that the overhang can be as large as desired: we arrange the blocks in the form of a staircase. With $n$ blocks of length 2 the overhang can be made to sum to $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n}$. Since the harmonic series diverges, it follows that the overhang can be arranged to be as large as desired, simply by using a suitably large number of blocks.
Recently, a number of interesting twists have been added to this paradoxical staircase. I'll be talking about some of these new developments and in particular about a continuous counterpart of the staircase that I've been pondering together with my colleagues David Treeby and Marty Ross.
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Metric geometry in data analysis 13:10 Fri 11 Nov 11 :: B.19 Ingkarni Wardli :: Dr Facundo Memoli :: University of Adelaide
The problem of object matching under invariances can be
studied using certain tools from metric geometry. The central idea is
to regard
objects as metric spaces (or metric measure spaces). The type of
invariance that one wishes to have in the matching is encoded by the
choice of the metrics with which one endows the objects. The standard
example is matching objects in Euclidean space under rigid isometries:
in this
situation one would endow the objects with the Euclidean metric. More
general scenarios are possible in which the desired invariance cannot
be reflected by the preservation of an ambient space metric. Several
ideas due to M. Gromov are useful for approaching this problem. The
Gromov-Hausdorff distance is a natural candidate for doing this.
However, this metric leads to very hard combinatorial optimization
problems and it is difficult to relate to previously reported
practical approaches to the problem of object matching. I will discuss
different variations of these ideas, and in particular will show a
construction of an L^p version of the Gromov-Hausdorff metric, called
the Gromov-Wassestein distance, which is based on mass transportation
ideas. This new metric directly leads to quadratic optimization
problems on continuous variables with linear constraints.
As a consequence of establishing several lower bounds, it turns out
that several invariants of metric measure spaces turn out to be
quantitatively stable in the GW sense. These invariants provide
practical tools for the discrimination of shapes and connect the GW
ideas to a number of pre-existing approaches.
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Oka theory of blow-ups 13:10 Fri 18 Nov 11 :: B.19 Ingkarni Wardli :: A/Prof Finnur Larusson :: University of Adelaide
This talk is a continuation of my talk last August. I will discuss the recently-obtained answers to the open questions I described then.
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Applications of tropical geometry to groups and manifolds 13:10 Mon 21 Nov 11 :: B.19 Ingkarni Wardli :: Dr Stephan Tillmann :: University of Queensland
Tropical geometry is a young field with multiple origins. These include the work of Bergman on logarithmic limit sets of algebraic varieties; the work of the Brazilian computer scientist Simon on discrete mathematics; the work of Bieri, Neumann and Strebel on geometric invariants of groups; and, of course, the work of Newton on polynomials. Even though there is still need for a unified foundation of the field, there is an abundance of applications of tropical geometry in group theory, combinatorics, computational algebra and algebraic geometry. In this talk I will give an overview of (what I understand to be) tropical geometry with a bias towards applications to group theory and low-dimensional topology.
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Space of 2D shapes and the Weil-Petersson metric: shapes, ideal fluid and Alzheimer's disease 13:10 Fri 25 Nov 11 :: B.19 Ingkarni Wardli :: Dr Sergey Kushnarev :: National University of Singapore
The Weil-Petersson metric is an exciting metric on a space of simple
plane curves. In this talk the speaker will introduce the shape space and
demonstrate the connection with the Euler-Poincare equations on the group
of diffeomorphisms (EPDiff). A numerical method for finding geodesics
between two shapes will be demonstrated and applied to the surface of the hippocampus to study the effects of Alzheimer's disease. As another application the speaker will discuss how to do statistics on the shape space and what should be done to improve it.
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Fluid flows in microstructured optical fibre fabrication 15:10 Fri 25 Nov 11 :: B.17 Ingkarni Wardli :: Mr Hayden Tronnolone :: University of Adelaide
Optical fibres are used extensively in modern telecommunications as they allow the transmission of information at high speeds. Microstructured optical fibres are a relatively new fibre design in which a waveguide for light is created by a series of air channels running along the length of the material. The flexibility of this design allows optical fibres to be created with adaptable (and previously unrealised) optical properties. However, the fluid flows that arise during fabrication can greatly distort the geometry, which can reduce the effectiveness of a fibre or render it useless. I will present an overview of the manufacturing process and highlight the difficulties. I will then focus on surface-tension driven deformation of the macroscopic version of the fibre extruded from a reservoir of molten glass, occurring during fabrication, which will be treated as a two-dimensional Stokes flow problem. I will outline two different complex-variable numerical techniques for solving this problem along with comparisons of the results, both to other models and to experimental data.
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Collision and instability in a rotating fluid-filled torus 15:10 Mon 12 Dec 11 :: Benham Lecture Theatre :: Dr Richard Clarke :: The University of Auckland
The simple experiment discussed in this talk, first conceived by Madden and
Mullin (JFM, 1994) as part of their investigations into the non-uniqueness
of decaying turbulent flow, consists of a fluid-filled torus which is
rotated in an horizontal plane. Turbulence within the contained flow is
triggered through a rapid change in its rotation rate. The flow
instabilities which transition the flow to this turbulent state, however,
are truly fascinating in their own right, and form the subject of this
presentation. Flow features observed in both UK- and Auckland-based
experiments will be highlighted, and explained through both boundary-layer
analysis and full DNS. In concluding we argue that this flow regime, with
its compact geometry and lack of cumbersome flow entry effects, presents an
ideal regime in which to study many prototype flow behaviours, very much in
the same spirit as Taylor-Couette flow.
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Noncritical holomorphic functions of finite growth on algebraic Riemann surfaces 13:10 Fri 3 Feb 12 :: B.20 Ingkarni Wardli :: Prof Franc Forstneric :: University of Ljubljana
Given a compact Riemann surface X and a point p in X,
we construct a holomorphic function without critical points
on the punctured (algebraic) Riemann surface R=X-p
which is of finite order at the point p.
In the case at hand this improves the 1967 theorem of
Gunning and Rossi to the effect that every open
Riemann surface admits a noncritical holomorphic function,
but without any particular growth condition. (Joint work with Takeo Ohsawa.)
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Embedding circle domains into the affine plane C^2 13:10 Fri 10 Feb 12 :: B.20 Ingkarni Wardli :: Prof Franc Forstneric :: University of Ljubljana
We prove that every circle domain in the Riemann sphere admits
a proper holomorphic embedding into the affine plane C^2.
By a circle domain we mean a domain obtained by removing
from the Riemann sphere a finite or countable family
of pairwise disjoint closed round discs.
Our proof also applies to some circle domains with punctures.
The uniformization theorem of He and Schramm (1996)
says that every domain in the Riemann sphere
with at most countably many boundary components is
conformally equivalent to a circle domain, so
our theorem embeds all such domains properly
holomorphically in C^2. (Joint work with Erlend F. Wold.)
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Plurisubharmonic subextensions as envelopes of disc functionals 13:10 Fri 2 Mar 12 :: B.20 Ingkarni Wardli :: A/Prof Finnur Larusson :: University of Adelaide
I will describe new joint work with Evgeny Poletsky. We prove a disc formula for the largest plurisubharmonic subextension of an upper semicontinuous function on a domain $W$ in a Stein manifold to a larger domain $X$ under suitable conditions on $W$ and $X$. We introduce a related equivalence relation on the space of analytic discs in $X$ with boundary in $W$. The quotient is a complex manifold with a local biholomorphism to $X$, except it need not be Hausdorff. We use our disc formula to generalise Kiselman's minimum principle. We show that his infimum function is an example of a plurisubharmonic subextension.
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IGA Workshop: The mathematical implications of gauge-string dualities 09:30 Mon 5 Mar 12 :: 7.15 Ingkarni Wardli :: Prof Rajesh Gopakumar :: Harish-Chandra Research Institute
Media...
Lecture series by Rajesh Gopakumar (Harish-Chandra Research Institute). The lectures will be supplemented by talks by other invited speakers.
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The Lorentzian conformal analogue of Calabi-Yau manifolds 13:10 Fri 16 Mar 12 :: B.20 Ingkarni Wardli :: Prof Helga Baum :: Humboldt University
Calabi-Yau manifolds are Riemannian manifolds with holonomy group SU(m). They are Ricci-flat and Kahler and admit a 2-parameter family of parallel spinors. In the talk we will discuss the Lorentzian conformal analogue of this situation. If on a manifold a class of conformally equivalent metrics [g] is given, then one can consider the holonomy group
of the conformal manifold (M,[g]), which is a subgroup of
O(p+1,q+1) if the metric g has signature (p,q). There is a close relation between algebraic properties of the conformal holonomy group and the existence of Einstein metrics in the conformal class as well as to the existence of conformal Killing spinors. In the talk I will explain classification results for conformal holonomy groups of Lorentzian manifolds. In particular, I will describe Lorentzian manifolds (M,g) with conformal holonomy group SU(1,m), which can be viewed as the conformal analogue of Calabi-Yau manifolds. Such Lorentzian
metrics g, known as Fefferman metrics, appear on S^1-bundles over strictly pseudoconvex CR spin manifolds and admit a 2-parameter family of conformal Killing spinors.
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IGA Workshop: Dualities in field theories and the role of K-theory 09:30 Mon 19 Mar 12 :: 7.15 Ingkarni Wardli :: Prof Jonathan Rosenberg :: University of Maryland
Media...
Lecture series by Jonathan Rosenberg (University of Maryland). There will be additional talks by other invited speakers.
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The de Rham Complex 12:10 Mon 19 Mar 12 :: 5.57 Ingkarni Wardli :: Mr Michael Albanese :: University of Adelaide
Media...
The de Rham complex is of fundamental importance in differential geometry. After first introducing differential forms (in the familiar setting of Euclidean space), I will demonstrate how the de Rham complex elegantly encodes one half (in a sense which will become apparent) of the results from vector calculus. If there is time, I will indicate how results from the remaining half of the theory can be concisely expressed by a single, far more general theorem.
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Financial risk measures - the theory and applications of backward stochastic difference/differential equations with respect to the single jump process 12:10 Mon 26 Mar 12 :: 5.57 Ingkarni Wardli :: Mr Bin Shen :: University of Adelaide
Media...
This is my PhD thesis submitted one month ago. Chapter 1 introduces the backgrounds of the research fields. Then each chapter is a published or an accepted paper.
Chapter 2, to appear in Methodology and Computing in Applied Probability, establishes the theory of Backward Stochastic Difference Equations with respect to the single jump process in discrete time.
Chapter 3, published in Stochastic Analysis and Applications, establishes the theory of Backward Stochastic Differential Equations with respect to the single jump process in continuous time.
Chapter 2 and 3 consist of Part I Theory.
Chapter 4, published in Expert Systems With Applications, gives some examples about how to measure financial risks by the theory established in Chapter 2.
Chapter 5, accepted by Journal of Applied Probability, considers the question of an optimal transaction between two investors to minimize their risks. It's the applications of the theory established in Chapter 3.
Chapter 4 and 5 consist of Part II Applications.
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Bundle gerbes and the Faddeev-Mickelsson-Shatashvili anomaly 13:10 Fri 30 Mar 12 :: B.20 Ingkarni Wardli :: Dr Raymond Vozzo :: University of Adelaide
The Faddeev-Mickelsson-Shatashvili anomaly arises in the quantisation of fermions interacting with external gauge potentials. Mathematically, it can be described as a certain lifting problem for an extension of groups. The theory of bundle gerbes is very useful for studying lifting problems, however it only applies in the case of a central extension whereas in the study of the FMS anomaly the relevant extension is non-central. In this talk I will explain how to describe this anomaly indirectly using bundle gerbes and how to use a generalisation of bundle gerbes to describe the (non-central) lifting problem directly. This is joint work with Pedram Hekmati, Michael Murray and Danny Stevenson.
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New examples of totally disconnected, locally compact groups 13:10 Fri 20 Apr 12 :: B.20 Ingkarni Wardli :: Dr Murray Elder :: University of Newcastle
I will attempt to explain what a totally disconnected,
locally compact group is, and then describe some new work with George
Willis on an attempt to create new examples based on Baumslag-Solitar
groups, which are well known, tried and tested
examples/counterexamples in geometric/combinatorial group theory. I
will describe how to compute invariants of scale and flat rank for
these groups.
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A Problem of Siegel 13:10 Fri 27 Apr 12 :: B.20 Ingkarni Wardli :: Dr Brent Everitt :: University of York
The first explicit examples of orientable hyperbolic 3-manifolds were constructed by Weber,
Siefert, and Lobell in the early 1930's. In the subsequent decades the world
of hyperbolic n-manifolds has grown into an extraordinarily rich one. Its sociology is
best understood through the eyes of invariants, and for hyperbolic manifolds the most
important invariant is volume. Viewed this way the n-dimensional hyperbolic manifolds,
for fixed n, look like a well-ordered subset of the reals (a discrete set even, when n is not 3).
So we are naturally led to the (manifold) Siegel problem: for a given n, determine the minimum
possible volume obtained by an orientable hyperbolic n-manifold. It is a problem with a long
and venerable history. In this talk I will describe a unified solution to the problem in low even
dimensions, one of which at least is new. Joint work with John Ratcliffe and Steve Tschantz (Vanderbilt).
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Acyclic embeddings of open Riemann surfaces into new examples of elliptic manifolds 13:10 Fri 4 May 12 :: Napier LG28 :: Dr Tyson Ritter :: University of Adelaide
In complex geometry a manifold is Stein if there are, in a certain
sense, "many" holomorphic maps from the manifold into C^n. While this
has long been well understood, a fruitful definition of the dual
notion has until recently been elusive. In Oka theory, a manifold is
Oka if it satisfies several equivalent definitions, each stating that
the manifold has "many" holomorphic maps into it from C^n. Related to
this is the geometric condition of ellipticity due to Gromov, who
showed that it implies a complex manifold is Oka.
We present recent contributions to three open questions involving
elliptic and Oka manifolds. We show that affine quotients of C^n are
elliptic, and combine this with an example of Margulis to construct
new elliptic manifolds of interesting homotopy types. It follows that
every open Riemann surface properly acyclically embeds into an
elliptic manifold, extending an existing result for open Riemann
surfaces with abelian fundamental group.
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Index type invariants for twisted signature complexes 13:10 Fri 11 May 12 :: Napier LG28 :: Prof Mathai Varghese :: University of Adelaide
Atiyah-Patodi-Singer proved an index theorem for non-local boundary conditions
in the 1970's that has been widely used in mathematics and mathematical physics.
A key application of their theory gives the index theorem for signature operators on
oriented manifolds with boundary. As a consequence, they defined certain secondary
invariants that were metric independent. I will discuss some recent work with Benameur
where we extend the APS theory to signature operators twisted by an odd degree closed
differential form, and study the corresponding secondary invariants.
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Computational complexity, taut structures and triangulations 13:10 Fri 18 May 12 :: Napier LG28 :: Dr Benjamin Burton :: University of Queensland
There are many interesting and difficult algorithmic problems in
low-dimensional topology. Here we study the problem of finding a taut
structure on a 3-manifold triangulation, whose existence has implications
for both the geometry and combinatorics of the triangulation. We prove
that detecting taut structures is "hard", in the sense that it is NP-complete.
We also prove that detecting taut structures is "not too hard", by showing
it to be fixed-parameter tractable. This is joint work with Jonathan Spreer.
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Unknot recognition and the elusive polynomial time algorithm 15:10 Fri 18 May 12 :: B.21 Ingkarni Wardli :: Dr Benjamin Burton :: The University of Queensland
Media...
What do practical topics such as linear programming and greedy
heuristics have to do with theoretical problems such as unknot
recognition and the Poincare conjecture? In this talk we explore new
approaches to old and difficult computational problems from geometry and
topology: how to tell whether a loop of string is knotted, or whether a
3-dimensional space has no interesting topological features. Although
the best known algorithms for these problems run in exponential time,
there is increasing evidence that a polynomial time solution might be
possible. We outline several promising approaches in which
computational geometry, linear programming and greedy algorithms all
play starring roles.
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On the full holonomy group of special Lorentzian manifolds 13:10 Fri 25 May 12 :: Napier LG28 :: Dr Thomas Leistner :: University of Adelaide
The holonomy group of a semi-Riemannian manifold is defined as the group of parallel transports along loops based at a point. Its connected component, the `restricted holonomy group', is given by restricting in this definition to contractible loops. The restricted holonomy can essentially be described by its Lie algebra and many classification results are obtained in this way. In contrast, the `full' holonomy group is a more global object and classification results are out of reach.
In the talk I will describe recent results with H. Baum and K. Laerz (both HU Berlin) about the full holonomy group of so-called `indecomposable' Lorentzian manifolds.
I will explain a construction method that arises from analysing the effects on holonomy when dividing the manifold by the action of a properly discontinuous group of isometries and present several examples of Lorentzian manifolds with disconnected holonomy groups.
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Geometric modular representation theory 13:10 Fri 1 Jun 12 :: Napier LG28 :: Dr Anthony Henderson :: University of Sydney
Representation theory is one of the oldest areas of algebra, but many basic questions in it are still unanswered. This is especially true in the modular case, where one considers vector spaces over a field F of positive characteristic; typically, complications arise for particular small values of the characteristic. For example, from a vector space V one can construct the symmetric square S^2(V), which is one easy example of a representation of the group GL(V). One would like to say that this representation is irreducible, but that statement is not always true: if F has characteristic 2, there is a nontrivial invariant subspace. Even for GL(V), we do not know the dimensions of all irreducible representations in all characteristics.
In this talk, I will introduce some of the main ideas of geometric modular representation theory, a more recent approach which is making progress on some of these old problems. Essentially, the strategy is to re-formulate everything in terms of homology of various topological spaces, where F appears only as the field of coefficients and the spaces themselves are independent of F; thus, the modular anomalies in representation theory arise because homology with modular coefficients is detecting something about the topology that rational coefficients do not. In practice, the spaces are usually varieties over the complex numbers, and homology is replaced by intersection cohomology to take into account the singularities of these varieties.
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IGA Workshop: Dendroidal sets 14:00 Tue 12 Jun 12 :: Ingkarni Wardli B17 :: Dr Ittay Weiss :: University of the South Pacific
Media...
A series of four 2-hour lectures by Dr. Ittay Weiss.
The theory of dendroidal sets was introduced by Moerdijk and Weiss in 2007 in the study of homotopy operads in algebraic topology. In the five years that have past since then several fundamental and highly non-trivial results were established. For instance, it was established that dendroidal sets provide models for homotopy operads in a way that extends the Joyal-Lurie approach to homotopy categories. It can be shown that dendroidal sets provide new models in the study of n-fold loop spaces. And it is very recently shown that dendroidal sets model all connective spectra in a way that extends the modeling of certain spectra by Picard groupoids.
The aim of the lecture series will be to introduce the concepts mentioned above, present the elementary theory, and understand the scope of the results mentioned as well as discuss the potential for further applications. Sources for the course will include the article "From Operads to Dendroidal Sets" (in the AMS volume on mathematical foundations of quantum field theory (also on the arXiv)) and the lecture notes by Ieke Moerdijk "simplicial methods for operads and algebraic geometry" which resulted from an advanced course given in Barcelona 3 years ago.
No prior knowledge of operads will be assumed nor any knowledge of homotopy theory that is more advanced then what is required for the definition of the fundamental group. The basics of the language of presheaf categories will be recalled quickly and used freely.
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Introduction to quantales via axiomatic analysis 13:10 Fri 15 Jun 12 :: Napier LG28 :: Dr Ittay Weiss :: University of the South Pacific
Quantales were introduced by Mulvey in 1986 in the context of non-commutative topology with the aim of providing a concrete non-commutative framework for the foundations of quantum mechanics. Since then quantales found applications in other areas as well, among others in the work of Flagg. Flagg considers certain special quantales, called value quantales, that are desigend to capture the essential properties of ([0,\infty],\le,+) that are relevant for analysis. The result is a well behaved theory of value quantale enriched metric spaces. I will introduce the notion of quantales as if they were desigend for just this purpose, review most of the known results (since there are not too many), and address a some new results, conjectures, and questions.
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K-theory and unbounded Fredholm operators 13:10 Mon 9 Jul 12 :: Ingkarni Wardli B19 :: Dr Jerry Kaminker :: University of California, Davis
There are several ways of viewing elements of K^1(X). One
of these is via families of unbounded self-adjoint Fredholm operators on X. Each operator will have discrete spectrum, with infinitely many positive and negative eigenvalues of finite multiplicity. One can associate to such a family a geometric object, its graph, and the Chern character and other invariants of the family can be studied from this perspective. By restricting the dimension of the eigenspaces one may sometimes use algebraic topology to completely determine the family up to equivalence. This talk will describe the general framework and some applications to families on low-dimensional manifolds
where the methods work well. Various notions related to spectral flow, the index gerbe and Berry phase play roles which will be discussed. This is joint work with Ron Douglas.
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Complex geometry and operator theory 14:10 Mon 9 Jul 12 :: Ingkarni Wardli B19 :: Prof Ron Douglas :: Texas A&M University
In the study of bounded operators on Hilbert spaces of holomorphic functions, concepts and techniques from complex geometry are important. An anti-holomorphic bundle exists on which one can define the Chern connection. Its curvature turns out to be a complete invariant and various operator notions can't be reframed in terms of geometrical ones which leads to the solution of some problems. We will discuss this approach with an emphasis on natural examples in the one and multivariable case.
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Inquiry-based learning: yesterday and today 15:30 Mon 9 Jul 12 :: Ingkarni Wardli B19 :: Prof Ron Douglas :: Texas A&M University
Media...
The speaker will report on a project to develop and promote approaches to mathematics instruction closely related to the Moore method -- methods which are called inquiry-based learning -- as well as on his personal experience of the Moore method. For background, see the speaker's article in the May 2012 issue of the Notices of the American Mathematical Society. To download the article, click on "Media" above.
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The motivic logarithm and its realisations 13:10 Fri 3 Aug 12 :: Engineering North 218 :: Dr James Borger :: Australian National University
When a complex manifold is defined by polynomial equations, its cohomology groups inherit extra structure. This was discovered by Hodge in the 1920s and 30s. When the defining polynomials have rational coefficients, there is some additional, arithmetic structure on the cohomology. This was discovered by Grothendieck and others in the 1960s. But here the situation is still quite mysterious because each cohomology group has infinitely many different arithmetic structures and while they are not directly comparable, they share many properties---with each other and with the Hodge structure.
All written accounts of this that I'm aware of treat arbitrary varieties. They are beautifully abstract and non-explicit. In this talk, I'll take the opposite approach and try to give a flavour of the subject by working out a perhaps the simplest nontrivial example, the cohomology of C* relative to a subset of two points, in beautifully concrete and explicit detail. Here the common motif is the logarithm. In Hodge theory, it is realised as the complex logarithm; in the crystalline theory, it's as the p-adic logarithm; and in the etale theory, it's as Kummer theory.
I'll assume you have some familiarity with usual, singular cohomology of topological spaces, but I won't assume that you know anything about these non-topological cohomology theories.
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Geometry - algebraic to arithmetic to absolute 15:10 Fri 3 Aug 12 :: B.21 Ingkarni Wardli :: Dr James Borger :: Australian National University
Media...
Classical algebraic geometry is about studying solutions to systems of polynomial equations with complex coefficients. In arithmetic algebraic geometry, one digs deeper and studies the arithmetic properties of the solutions when the coefficients are rational, or even integral. From the usual point of view, it's impossible to go deeper than this for the simple reason that no smaller rings are available - the integers have no proper subrings. In this talk, I will explain how an emerging subject, lambda-algebraic geometry, allows one to do just this and why one might care.
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The importance of being fractal 13:10 Tue 7 Aug 12 :: 7.15 Ingkarni Wardli :: Prof Tony Roberts :: School of Mathematical Sciences
Media...
Euclid's geometry describes the world around us in terms of points, lines and planes. For two thousand years these have formed the limited repertoire of basic geometric objects with which to describe the universe. Fractals immeasurably enhance this world-view by providing a description of much around us that is rough and fragmented---of objects that have structure on many sizes.
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Hodge numbers and cohomology of complex algebraic varieties 13:10 Fri 10 Aug 12 :: Engineering North 218 :: Prof Gus Lehrer :: University of Sydney
Let $X$ be a complex algebraic variety defined over the ring $\mathfrak{O}$ of integers in a number field $K$ and let $\Gamma$ be a group of $\mathfrak{O}$-automorphisms of $X$. I shall discuss how the counting of rational points over reductions mod $p$ of $X$, and an analysis of the Hodge structure of the cohomology of $X$, may be used to determine the cohomology as a $\Gamma$-module. This will include some joint work with Alex Dimca and with Mark Kisin, and some classical unsolved problems.
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The fundamental theorems of invariant theory, classical and quantum 15:10 Fri 10 Aug 12 :: B.21 Ingkarni Wardli :: Prof Gus Lehrer :: The University of Sydney
Media...
Let V = C^n, and let (-,-) be a non-degenerate bilinear form
on V , which is either symmetric or anti-symmetric. Write G for the isometry
group of (V , (-,-)); thus G = O_n (C) or Sp_n (C). The first fundamental
theorem (FFT) provides a set of generators for End_G(V^{\otimes r} ) (r = 1, 2, . . . ),
while the second fundamental theorem (SFT) gives all relations among the
generators. In 1937, Brauer formulated the FFT in terms of his celebrated
'Brauer algebra' B_r (\pm n), but there has hitherto been no similar version of
the SFT. One problem has been the generic non-semisimplicity of B_r (\pm n),
which caused H Weyl to call it, in his work on invariants 'that enigmatic
algebra'. I shall present a solution to this problem, which shows that there is
a single idempotent in B_r (\pm n), which describes all the relations. The proof
is through a new 'Brauer category', in which the fundamental theorems are
easily formulated, and where a calculus of tangles may be used to prove these
results. There are quantum analogues of the fundamental theorems which I
shall also discuss. There are numerous applications in representation theory,
geometry and topology. This is joint work with Ruibin Zhang.
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Differential topology 101 13:10 Fri 17 Aug 12 :: Engineering North 218 :: Dr Nicholas Buchdahl :: University of Adelaide
Much of my recent research been directed at a problem in the
theory of compact complex surfaces---trying to fill in a gap
in the Enriques-Kodaira classification.
Attempting to classify some collection of mathematical
objects is a very common activity for pure mathematicians,
and there are many well-known examples of successful
classification schemes; for example, the classification of
finite simple groups, and the classification of simply
connected topological 4-manifolds.
The aim of this talk will be to illustrate how techniques
from differential geometry can be used to classify compact
surfaces. The level of the talk will be very elementary, and
the material is all very well known, but it is sometimes
instructive to look back over simple cases of a general
problem with the benefit of experience to gain greater
insight into the more general and difficult cases.
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Noncommutative geometry and conformal geometry 13:10 Fri 24 Aug 12 :: Engineering North 218 :: Dr Hang Wang :: Tsinghua University
In this talk, we shall use noncommutative geometry to obtain an index theorem in conformal geometry. This index theorem follows from an explicit and geometric computation of the Connes-Chern character of the spectral triple in conformal geometry, which was introduced recently by Connes and Moscovici. This (twisted) spectral triple encodes the geometry of the group of conformal diffeomorphisms on a spin manifold. The crux of of this construction is the conformal invariance of the Dirac operator. As a result, the Connes-Chern character is intimately related to the CM cocycle of an equivariant Dirac spectral triple. We compute this equivariant CM cocycle by heat kernel techniques. On the way we obtain a new heat kernel proof of the equivariant index theorem for Dirac operators. (Joint work with Raphael Ponge.)
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Holomorphic flexibility properties of compact complex surfaces 13:10 Fri 31 Aug 12 :: Engineering North 218 :: A/Prof Finnur Larusson :: University of Adelaide
I will describe recent joint work with Franc Forstneric (arXiv, July 2012). We introduce a new property, called the stratified Oka property, which fits into a hierarchy of anti-hyperbolicity properties that includes the Oka property. We show that stratified Oka manifolds are strongly dominable by affine spaces. It follows that Kummer surfaces are strongly dominable. We determine which minimal surfaces of class VII are Oka (assuming the global spherical shell conjecture). We deduce that the Oka property and several other anti-hyperbolicity properties are in general not closed in families of compact complex manifolds. I will summarise what is known about how the Oka property fits into the Enriques-Kodaira classification of surfaces.
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Classification of a family of symmetric graphs with complete quotients 13:10 Fri 7 Sep 12 :: Engineering North 218 :: A/Prof Sanming Zhou :: University of Melbourne
A finite graph is called symmetric if its automorphism group is
transitive on the set of arcs (ordered pairs of adjacent vertices) of the
graph. This is to say that all arcs have the same status in the graph. I
will talk about recent results on the classification of a family of
symmetric graphs with complete quotients. The most interesting graphs
arising from this classification are defined in terms of Hermitian unitals
(which are specific block designs), and they admit unitary groups as
groups of automorphisms. I will also talk about applications of our
results in constructing large symmetric graphs of given degree and
diameter.
This talk contains joint work with M. Giulietti, S. Marcugini and F.
Pambianco.
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Geometric quantisation in the noncompact setting 13:10 Fri 14 Sep 12 :: Engineering North 218 :: Dr Peter Hochs :: Leibniz University, Hannover
Traditionally, the geometric quantisation of an action by a compact Lie group on a compact symplectic manifold is defined as the equivariant index of a certain Dirac operator. This index is a well-defined formal difference of finite-dimensional representations, since the Dirac operator is elliptic and the manifold and the group in question are compact. From a mathematical and physical point of view however, it is very desirable to extend geometric quantisation to noncompact groups and manifolds. Defining a suitable index is much harder in the noncompact setting, but several interesting results in this direction have been obtained. I will review the difficulties connected to noncompact geometric quantisation, and some of the solutions that have been proposed so far, mainly in connection to the "quantisation commutes with reduction" principle. (An introduction to this principle will be given in my talk at the Colloquium on the same day.)
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Quantisation commutes with reduction 15:10 Fri 14 Sep 12 :: B.20 Ingkarni Wardli :: Dr Peter Hochs :: Leibniz University Hannover
Media...
The "Quantisation commutes with reduction" principle is an idea from physics, which has powerful applications in mathematics. It basically states that the ways in which symmetry can be used to simplify a physical system in classical and quantum mechanics, are compatible. This provides a strong link between the areas in mathematics used to describe symmetry in classical and quantum mechanics: symplectic geometry and representation theory, respectively. It has been proved in the 1990s that quantisation indeed commutes with reduction, under the important assumption that all spaces and symmetry groups involved are compact. This talk is an introduction to this principle and, if time permits, its mathematical relevance.
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Introduction to pairings in cryptography 13:10 Fri 21 Sep 12 :: Napier 209 :: Dr Naomi Benger :: University of Adelaide
From cryptanalysis to a powerful tool which made identity based cryptography possible, pairings have a range of applications in cryptography. I will present basic background (algebraic geometry) needed to understand pairings, hard problems associated with pairings and protocols which use pairings.
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Supermanifolds and the moduli space of instantons 13:10 Fri 19 Oct 12 :: Engineering North 218 :: Prof Ugo Bruzzo :: International School for Advanced Studies (SISSA), Trieste
I will give an example of an application of supermanifold theory to physics, i.e., how to "superize" the moduli space of instantons on a 4-fold and use it to give a description of the BRST transformations, to compute the "supermeasure" of the moduli space, and the Nekrasov partition function.
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Moduli spaces of instantons in algebraic geometry and physics 15:10 Fri 19 Oct 12 :: B.20 Ingkarni Wardli :: Prof Ugo Bruzzo :: International School for Advanced Studies Trieste
Media...
I will give a quick introduction to the notion of instanton, stressing its role in physics and in mathematics.
I will also show how algebraic geometry provides powerful tools to study the geometry of the moduli spaces of instantons.
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The space of cubic rational maps 13:10 Fri 26 Oct 12 :: Engineering North 218 :: Mr Alexander Hanysz :: University of Adelaide
For each natural number d, the space of rational maps of degree d on the Riemann sphere has the structure of a complex manifold. The topology of these manifolds has been extensively studied. The recent development of Oka theory raises some new and interesting questions about their complex structure. We apply geometric invariant theory to the degree 3 case, studying a double action of the Mobius group on the space of cubic rational maps. We show that the categorical quotient is C, and that the space of cubic rational maps enjoys the holomorphic flexibility properties of strong dominability and C-connectedness.
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Twisted analytic torsion and adiabatic limits 13:10 Wed 5 Dec 12 :: Ingkarni Wardli B17 :: Mr Ryan Mickler :: University of Adelaide
We review Mathai-Wu's recent extension of Ray-Singer analytic torsion to supercomplexes. We explore some new results relating these two torsions, and how we can apply the adiabatic spectral sequence due to Forman and Farber's analytic deformation theory to compute some spectral invariants of the complexes involved, answering some questions that were posed in Mathai-Wu's paper.
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Variation of Hodge structure for generalized complex manifolds 13:10 Fri 7 Dec 12 :: Ingkarni Wardli B20 :: Dr David Baraglia :: University of Adelaide
Generalized complex geometry combines complex and symplectic geometry into a single framework, incorporating also holomorphic Poisson and bi-Hermitian structures. The Dolbeault complex naturally extends to the generalized complex setting giving rise to Hodge structures in twisted cohomology. We consider the variations of Hodge structure and period mappings that arise from families of generalized complex manifolds. As an application we prove a local Torelli theorem for generalized Calabi-Yau manifolds.
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Hyperplane arrangements and tropicalization of linear spaces 10:10 Mon 17 Dec 12 :: Ingkarni Wardli B17 :: Dr Graham Denham :: University of Western Ontario
I will give an introduction to a sequence of ideas in tropical
geometry, the tropicalization of linear spaces. In the beginning, a construction due to De Concini and Procesi (wonderful models, 1995) gave a combinatorially explicit description of various iterated blowups of projective spaces along (proper transforms of) linear subspaces. A decade later, Tevelev's notion of tropical compactifications led to, in particular, a new view of the wonderful models and their intersection theory in terms of the theory of toric varieties (via work of Feichtner-Sturmfels, Feichtner-Yuzvinsky, Ardila-Klivans, and others). Recently, these ideas have played a role in Huh and Katz's proof of a long-standing conjecture in combinatorics.
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Stably Cayley groups over fields of characteristic 0 11:10 Mon 17 Dec 12 :: Ingkarni Wardli B17 :: Dr Nicole Lemire :: University of Western Ontario
A linear algebraic group is called a Cayley group if it is equivariantly
birationally isomorphic to its Lie algebra. It is stably Cayley
if the product of the group and some torus is Cayley. Cayley gave the first
examples of Cayley groups with his Cayley map back in 1846. Over an algebraically closed
field of characteristic 0, Cayley and stably Cayley simple groups were
classified by
Lemire, Popov and Reichstein in 2006.
In recent joint work with Blunk, Borovoi, Kunyavskii and Reichstein, we classify the simple stably Cayley groups over an arbitrary field of
characteristic 0.
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Recent results on holomorphic extension of functions on unbounded domains in C^n 11:10 Fri 21 Dec 12 :: Ingkarni Wardli B19 :: Prof Roman Dwilewicz :: Missouri University of Science and Technology
In the talk there will be given a short review of holomorphic
extension problems starting with the famous Hartogs theorem (1906) up to recent results on global holomorphic extensions for unbounded domains, obtained together with Al Boggess (Arizona State Univ.) and Zbigniew Slodkowski (Univ. Illinois at Chicago). There is an interesting geometry behind the extension problem for unbounded domains, namely (in some cases) it depends on the position of a complex variety in the closure of the domain. The extension problem appeared non-trivial and the work is in progress. However the talk will be illustrated by many figures and pictures and should be accessible also to graduate students.
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Conformally Fedosov manifolds 12:10 Fri 8 Mar 13 :: Ingkarni Wardli B19 :: Prof Michael Eastwood :: Australian National University
Symplectic and projective structures may be compatibly combined. The
resulting structure closely resembles conformal geometry and a manifold endowed
with such a structure is called conformally Fedosov. This talk will present the
basic theory of conformally Fedosov geometry and, in particular, construct a
Cartan connection for them. This is joint work with Jan Slovak.
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Twistor theory and the harmonic hull 15:10 Fri 8 Mar 13 :: B.18 Ingkarni Wardli :: Prof Michael Eastwood :: Australian National University
Media...
Harmonic functions are real-analytic and so automatically extend as functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated conformal geometry. Nothing will be supposed about such matters: I shall base the constructions on an elementary yet mysterious formula of Bateman from 1904. This is joint work with Feng Xu.
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Twistor space for rolling bodies 12:10 Fri 15 Mar 13 :: Ingkarni Wardli B19 :: Prof Pawel Nurowski :: University of Warsaw
We consider a configuration space of two solids rolling on each other
without slipping or twisting, and identify it with an open subset U of
R^5, equipped with a generic distribution D of 2-planes. We will discuss
symmetry properties of the pair (U,D) and will mention that, in the case
of the two solids being balls, when changing the ratio of their radii,
the dimension of the group of local symmetries unexpectedly jumps from 6
to 14. This occurs for only one such ratio, and in such case the local
group of symmetries of the pair (U,D) is maximal. It is maximal not only
among the balls with various radii, but more generally among all (U,D)s
corresponding to configuration spaces of two solids rolling on each
other without slipping or twisting. This maximal group is isomorphic to
the split real form of the exceptional Lie group G2.
In the remaining part of the talk we argue how to identify the space U
from the pair (U,D) defined above with the bundle T of totally null real
2-planes over a 4-manifold equipped with a split signature metric. We
call T the twistor bundle for rolling bodies. We show that the rolling
distribution D, can be naturally identified with an appropriately defined
twistor distribution on T. We use this formulation of the rolling system
to find more surfaces which, when rigidly rolling on each other without
slipping or twisting, have the local group of symmetries isomorphic to
the exceptional group G2.
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Modular forms: a rough guide 12:10 Mon 18 Mar 13 :: B.19 Ingkarni Wardli :: Damien Warman :: University of Adelaide
Media...
I recently found the need to learn a little about what I had naively believed to be an abstruse branch of number theory, but which turns out to be a ubiquitous and intriguing theory.
I'll introduce some of the geometry underlying the elementary theory of modular functions and modular forms. We'll look at some pictures and play with sage, time permitting.
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On the chromatic number of a random hypergraph 13:10 Fri 22 Mar 13 :: Ingkarni Wardli B21 :: Dr Catherine Greenhill :: University of New South Wales
A hypergraph is a set of vertices and a set of hyperedges, where each
hyperedge is a subset of vertices. A hypergraph is r-uniform if every
hyperedge contains r vertices. A colouring of a hypergraph is an
assignment of colours to vertices such that no hyperedge is monochromatic.
When the colours are drawn from the set {1,..,k}, this defines a
k-colouring.
We consider the problem of k-colouring a random r-uniform hypergraph
with n vertices and cn edges, where k, r and c are constants and n tends
to infinity. In this setting, Achlioptas and Naor showed that for the
case of r = 2, the chromatic number of a random graph must have one of two
easily computable values as n tends to infinity.
I will describe some joint work with Martin Dyer (Leeds) and Alan Frieze
(Carnegie Mellon), in which we generalised this result to random uniform
hypergraphs. The argument uses the second moment method, and applies a
general theorem for performing Laplace summation over a lattice. So the
proof contains something for everyone, with elements from combinatorics,
analysis and algebra.
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Gauge groupoid cocycles and Cheeger-Simons differential characters 13:10 Fri 5 Apr 13 :: Ingkarni Wardli B20 :: Prof Jouko Mickelsson :: Royal Institute of Technology, Stockholm
Groups of gauge transformations in quantum field theory are typically
extended by a 2-cocycle with values in a certain abelian group due to chiral symmetry breaking. For these extensions there exist a global explicit construction since the 1980's. I shall study the higher group cocycles following a recent paper by F. Wagemann and C. Wockel, but extending to the transformation groupoid
setting (motivated by QFT) and discussing potential obstructions in the
construction due to a nonvanishing of low dimensional homology groups
of the gauge group. The resolution of the obstruction is obtained
by an application of the Cheeger-Simons differential characters.
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M-theory and higher gauge theory 13:10 Fri 12 Apr 13 :: Ingkarni Wardli B20 :: Dr Christian Saemann :: Heriot-Watt University
I will review my recent work on integrability of M-brane configurations and
the description of M-brane models in higher gauge theory. In particular, I
will discuss categorified analogues of instantons and present superconformal equations of motion for the non-abelian tensor multiplet in six dimensions. The latter are derived from considering non-abelian gerbes on certain twistor spaces.
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A glimpse at the Langlands program 15:10 Fri 12 Apr 13 :: B.18 Ingkarni Wardli :: Dr Masoud Kamgarpour :: University of Queensland
Media...
Abstract: In the late 1960s, Robert Langlands made a series of surprising conjectures relating fundamental concepts from number theory, representation theory, and algebraic geometry. Langlands' conjectures soon developed into a high-profile international research program known as the Langlands program. Many fundamental problems, including the Shimura-Taniyama-Weil conjecture (partially settled by Andrew Wiles in his proof of the Fermat's Last Theorem), are particular cases of the Langlands program. In this talk, I will discuss some of the motivation and results in this program.
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Conformal Killing spinors in Riemannian and Lorentzian geometry 12:10 Fri 19 Apr 13 :: Ingkarni Wardli B19 :: Prof Helga Baum :: Humboldt University
Conformal Killing spinors are the solutions of the conformally covariant twistor equation on spinors. Special cases are parallel and Killing spinors, the latter appear as eigenspinors of the Dirac operator on compact Riemannian manifolds of positive scalar curvature for the smallest possible positive eigenvalue. In the talk I will discuss geometric properties of manifolds admitting (conformal) Killing spinors. In particular, I will explain a local classification of the special geometric structures admitting conformal Killing spinors without zeros in the Riemannian as well as in the Lorentzian setting.
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An Oka principle for equivariant isomorphisms 12:10 Fri 3 May 13 :: Ingkarni Wardli B19 :: A/Prof Finnur Larusson :: University of Adelaide
I will discuss new joint work with Frank Kutzschebauch (Bern) and Gerald Schwarz (Brandeis). Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$, which are locally $G$-biholomorphic over a common categorical quotient $Q$. When is there a global $G$-biholomorphism $X\to Y$?
In a situation that we describe, with some justification, as generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch.
We prove that $X$ and $Y$ are $G$-biholomorphic if $X$ is $K$-contractible, where $K$ is a maximal compact subgroup of $G$, or if there is a $G$-diffeomorphism $X\to Y$ over $Q$, which is holomorphic when restricted to each fibre of the quotient map $X\to Q$. When $G$ is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of $G$-biholomorphisms from $X$ to $Y$ over $Q$. This sheaf can be badly singular, even in simply defined examples.
Our work is in part motivated by the linearisation problem for actions on $\C^n$. It follows from one of our main results that a holomorphic $G$-action on $\C^n$, which is locally $G$-biholomorphic over a common quotient to a generic linear action, is linearisable.
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Diffeological spaces and differentiable stacks 12:10 Fri 10 May 13 :: Ingkarni Wardli B19 :: Dr David Roberts :: University of Adelaide
The category of finite-dimensional smooth manifolds gives rise to interesting structures outside of itself, two examples being mapping spaces and classifying spaces. Diffeological spaces are a notion of generalised smooth space which form a cartesian closed category, so all fibre products and all mapping spaces of smooth manifolds exist as diffeological spaces. Differentiable stacks are a further generalisation that can also deal with moduli spaces (including classifying spaces) for objects with automorphisms. This talk will give an introduction to this circle of ideas.
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Crystallographic groups I: the classical theory 12:10 Fri 17 May 13 :: Ingkarni Wardli B19 :: Dr Wolfgang Globke :: University of Adelaide
A discrete isometry group acting properly discontinuously on the n-dimensional
Euclidean space with compact quotient is called a crystallographic group.
This name reflects the fact that in dimension n=3 their compact fundamental
domains resemble a space-filling crystal pattern.
For higher dimensions, Hilbert posed his famous 18th problem:
"Is there in n-dimensional Euclidean space only a finite number of essentially
different kinds of groups of motions with a [compact] fundamental region?"
This problem was solved by Bieberbach when he proved that in every
dimension n there exists only a finite number of isomorphic crystallographic groups
and also gave a description of these groups.
From the perspective of differential geometry these results are of major importance,
as crystallographic groups are precisely the fundamental groups of
compact flat Riemannian orbifolds.
The quotient is even a manifold if the fundamental group is required to be torsion-free,
in which case it is called a Bieberbach group.
Moreover, for a flat manifold the fundamental group completely determines the
holonomy group.
In this talk I will discuss the properties of crystallographic groups, study examples in
dimension n=2 and n=3, and present the three Bieberbach theorems on the
structure of crystallographic groups.
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Crystallographic groups II: generalisations 12:10 Fri 24 May 13 :: Ingkarni Wardli B19 :: Dr Wolfgang Globke :: University of Adelaide
The theory of crystallographic groups acting cocompactly on Euclidean space
can be extended and generalised in many different ways.
For example, instead of studying discrete groups of Euclidean isometries, one
can consider groups of isometries for indefinite inner products.
These are the fundamental groups of compact flat pseudo-Riemannian manifolds.
Still more generally, one might study group of affine transformation on n-space
that are not required to preserve any bilinear form.
Also, the condition of cocompactness can be dropped.
In this talk, I will present some of the results obtained for these generalisations,
and also discuss some of my own work on flat homogeneous pseudo-Riemannian
spaces.
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A strong Oka principle for proper immersions of finitely connected planar domains into CxC* 12:10 Fri 31 May 13 :: Ingkarni Wardli B19 :: Dr Tyson Ritter :: University of Adelaide
Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. In previous work we showed that, given a continuous map from X to the elliptic manifold CxC*, where X is a finitely connected planar domain without isolated boundary points, a stronger Oka property holds whereby the map is homotopic to a proper holomorphic embedding. If the planar domain is additionally permitted to have isolated boundary points the problem becomes more difficult, and it is not yet clear whether a strong Oka property for embeddings into CxC* continues to hold. We will discuss recent results showing that every continuous map from a finitely connected planar domain into CxC* is homotopic to a proper immersion that, in most cases, identifies at most finitely many pairs of distinct points. This is joint work with Finnur Larusson.
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A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces 12:10 Fri 7 Jun 13 :: Ingkarni Wardli B19 :: Prof Thierry Coulhon :: Australian National University
On doubling metric measure spaces endowed with a Dirichlet form and satisfying the Davies-Gaffney estimate, we show some characterisations of pointwise upper bounds
of the heat kernel in terms of one-parameter weighted inequalities which correspond respectively to the Nash inequality and to a Gagliardo-Nirenberg type inequality when the volume growth is polynomial. This yields a new and simpler proof of the well-known equivalence between classical heat kernel upper bounds and the relative Faber-Krahn inequalities. We are also able to treat more general pointwise estimates where the heat kernel rate of decay is not necessarily governed by the volume growth. This is a joint work with Salahaddine Boutayeb and Adam Sikora.
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Heat kernel estimates on non-compact Riemannian manifolds: why and how? 15:10 Fri 7 Jun 13 :: B.18 Ingkarni Wardli :: Prof Thierry Coulhon :: Australian National University
Media...
We will describe what is known and remains to be known about the connection between the large scale geometry of non-compact Riemannian manifolds
(and more general metric measure spaces) and large time estimates of their heat kernel. We will show how some of these estimates can be characterised in terms of Sobolev inequalities and give applications to the boundedness of Riesz transforms.
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TBA 12:10 Fri 21 Jun 13 :: Ingkarni Wardli B19 :: Dr Jarod Alper :: Australian National University
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IGA/AMSI Workshop: Representation theory and operator algebras 10:00 Mon 1 Jul 13 :: 7.15 Ingkarni Wardli :: Prof Nigel Higson :: Pennsylvania State University
Media...
This interdisciplinary workshop will be about aspects of representation theory (in the sense of Harish-Chandra), aspects of noncommutative geometry (in the sense of Alain Connes) and aspects of operator K-theory (in the sense of Gennadi Kasparov). It features the renowned speaker, Professor Nigel Higson (Penn State University) http://www.iga.adelaide.edu.au/workshops/WorkshopJuly2013/ All are welcome.
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K-homology and the quantization commutes with reduction problem 12:10 Fri 5 Jul 13 :: 7.15 Ingkarni Wardli :: Prof Nigel Higson :: Pennsylvania State University
The quantization commutes with reduction problem for Hamiltonian actions of compact Lie groups was solved by Meinrenken in the mid-1990s using geometric techniques, and solved again shortly afterwards by Tian and Zhang using analytic methods. In this talk I shall outline some of the close links that exist between the problem, the two solutions, and the geometric and analytic versions of K-homology theory that are studied in noncommutative geometry. I shall try to make the case for K-homology as a useful conceptual framework for the solutions and (at least some of) their various generalizations.
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Quantization, Representations and the Orbit Philosophy 15:10 Fri 5 Jul 13 :: B.18 Ingkarni Wardli :: Prof Nigel Higson :: Pennsylvania State University
Media...
This talk will be about the mathematics of quantization and about representation theory, where the concept of quantization seems to be especially relevant. It was discovered by Kirillov in the 1960's that the representation theory of nilpotent Lie groups (such as the group that encodes Heisenberg's commutation relations) can be beautifully and efficiently described using a vocabulary drawn from geometry and quantum mechanics. The description was soon adapted to other classes of Lie groups, and the expectation that it ought to apply almost universally has come to be called the "orbit philosophy." But despite early successes, the orbit philosophy is in a decidedly unfinished state. I'll try to explain some of the issues and some possible new directions.
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News matching "Differential geometry"
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Stoneham Prize The inaugural Stoneham Prize, awarded for the best poster by a graduate student in the first two years of their candidature, was awarded on the 4th of April. The winner was Ric Green, for his poster "What is Geometry?". Two Viewers' Choice prizes were also awarded to Ray Vozzo for his poster "The 7 Bridges of Koenigsberg - The 1st Theorem in Topology" and David Butler for his poster "The Queen of Hearts Plays Noughts and Crosses". Posted Sun 13 Apr 08.
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Workshop on Complex Geometry The Institute for Geometry and its Applications will host a Workshop on Complex Geometry at the University of Adelaide from Monday 16 February to Friday 20 February 2009. Click here for full details. Posted Wed 17 Sep 08.
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Mini Winter School on Geometry and Physics The Institute for Geometry and its Applications will host a Winter School on Geometry and Physics on 20-22 July 2009. There will be three days of expository lectures aimed at 3rd year and honours students interested in postgraduate studies in pure mathematics or mathematical physics. Posted Wed 24 Jun 09.More information...
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Go8-Germany Research Cooperation Scheme Congratulations to Thomas Leistner whose application under the Go8-Germany Research Co-operation Scheme is one of 24 across Australia to be funded in 2011-2012. Thomas will work with Professor Helga Baum of Humbolt University in Berlin on spinor field equations in global Lorentzian geometry. Posted Thu 4 Nov 10.
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IGA-AMSI Workshop: Group-valued moment maps with applications to mathematics and physics (5–9 September 2011) Lecture series by Eckhard Meinrenken, University of Toronto. Titles of
individual lectures: 1) Introduction to G-valued moment maps. 2) Dirac
geometry and Witten's volume formulas. 3) Dixmier-Douady theory and
pre-quantization. 4) Quantization of group-valued moment maps. 5)
Application to Verlinde formulas. These lectures will be supplemented by
additional talks by invited speakers. For more details, please see the
conference webpage
Posted Wed 27 Jul 11.More information...
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ARC Grant Success Congratulations to the following staff who were successful in securing funding from the Australian Research Council Discovery Projects Scheme. Associate Professor Finnur Larusson awarded $270,000 for his project Flexibility and symmetry in complex geometry; Dr Thomas Leistner, awarded $303,464 for his project Holonomy groups in Lorentzian geometry, Professor Michael Murray Murray and Dr Daniel Stevenson (Glasgow), awarded $270,000 for their project Bundle gerbes: generalisations and applications; Professor Mathai Varghese, awarded $105,000 for his project Advances in index theory and Prof Anthony Roberts and Professor Ioannis Kevrekidis (Princeton) awarded $330,000 for their project Accurate modelling of large multiscale dynamical systems for engineering and scientific
simulation and analysis Posted Tue 8 Nov 11.
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Dualities in field theories and the role of K-theory Between Monday 19 and Friday 23 March 2012, the Institute for Geometry and its Applications will host a lecture series by Professor Jonathan Rosenberg from the University of Maryland. There
will be additional talks by other invited speakers. Posted Tue 6 Dec 11.More information...
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The mathematical implications of gauge-string dualities Between Monday 5 and Friday 9 March 2012, the Institute for Geometry and its Applications will host a lecture series by Rajesh Gopakumar from the Harish-Chandra Research Institute. These lectures will be supplemented by talks by other invited speakers. Posted Tue 6 Dec 11.More information...
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Publications matching "Differential geometry"
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A characterisation of the lines external to an oval cone in PG(3, q), q even
Barwick, Susan; Butler, David, Journal of Geometry 93 (21–27) 2009 |
Portfolio risk minimization and differential games
Elliott, Robert; Siu, T, Nonlinear Analysis-Theory Methods & Applications In Press (–) 2009 |
A markovian regime-switching stochastic differential game for portfolio risk minimization
Elliott, Robert; Siu, T, 2008 American Control Conference, Washington 11/06/08 |
Metric connections in projective differential geometry
Eastwood, Michael; Matveev, V, Symmetries and Overdetermined Systems of Partial Differential Equations, USA 17/07/08 |
Notes on projective differential geometry
Eastwood, Michael, Symmetries and Overdetermined Systems of Partial Differential Equations, USA 17/07/08 |
Dessins d'enfants and differential equations
Larusson, Finnur; Sadykov, T, St Petersburg Mathematical Journal 19 (1003–1014) 2008 |
Equivariant and fractional index of projective elliptic operators
Varghese, Mathai; Melrose, R; Singer, I, Journal of Differential Geometry 78 (465–473) 2008 |
Invariant differential pairings
Kroeske, Jens, Universitas Comeniana. Acta Mathematica 77 (215–244) 2008 |
The basic bundle gerbe on unitary groups
Murray, Michael; Stevenson, Daniel, Journal of Geometry and Physics 58 (1571–1590) 2008 |
Model subgrid microscale interactions to accurately discretise stochastic partial differential equations.
Roberts, Anthony John, |
Monogenic functions in conformal geometry
Eastwood, Michael; Ryan, J, Symmetry, Integrability and Geometry: Methods and Applications 84 (1–14) 2007 |
On the geometry of regular hyperbolic fibrations
Brown, Matthew; Ebert, G; Luyckz, D, European Journal of Combinatorics 28 (1626–1636) 2007 |
Projective ovoids and generalized quadrangles
Brown, Matthew, Advances in Geometry 7 (65–81) 2007 |
Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras
Eastwood, Michael; Somberg, P; Soucek, V, Journal of Geometry and Physics 57 (2539–2546) 2007 |
Symmetries and invariant differential pairings
Eastwood, Michael, Symmetry, Integrability and Geometry: Methods and Applications 113 (1–10) 2007 |
T-Duality in type II string theory via noncommutative geometry and beyond
Varghese, Mathai, Progress of Theoretical Physics Supplement 171 (237–257) 2007 |
Computer algebra derives normal forms of stochastic differential equations
Roberts, Anthony John, |
Towards the fractional quantum Hall effect: a noncummutative geometry perspective
Marcolli, M; Varghese, Mathai, chapter in Noncommutative geometry and number theory (Vieweg, Springer Science+Business Media) 235–262, 2006 |
Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds
Leistner, Thomas, Differential Geometry and its Applications 24 (458–478) 2006 |
Formal adjoints and canonical form for linear operations
Eastwood, Michael; Gover, A, Conformal Geometry and Dynamics 10 (285–287) 2006 |
Fractional analytic index
Varghese, Mathai; Melrose, R; Singer, I, Journal of Differential Geometry 74 (265–292) 2006 |
Quantum Hall effect and noncommutative geometry
Carey, Alan; Hannabuss, K; Varghese, Mathai, Journal of Geometry and Symmetry in Physics 6 (16–36) 2006 |
Screen bundles of Lorentzian manifolds and some generalisations of pp-waves
Leistner, Thomas, Journal of Geometry and Physics 56 (2117–2134) 2006 |
Some Penrose transforms in complex differential geometry
Anco, S; Bland, J; Eastwood, Michael, Science in China Series A-Mathematics Physics Astronomy 49 (1599–1610) 2006 |
Resolving the multitude of microscale interactions accurately models stochastic partial differential equations
Roberts, Anthony John, London Mathematical Society. Journal of Computation and Mathematics 9 (193–221) 2006 |
Computer algebra derives discretisations of the stochastically forced Burgers' partial differential equation
Roberts, Anthony John, |
Dynamics of CP1 lumps on a cylinder
Romao, Nuno, Journal of Geometry and Physics 54 (42–76) 2005 |
The index of projective families of elliptic operators
Varghese, Mathai; Melrose, R; Singer, I, Geometry & Topology Online 9 (341–373) 2005 |
On the analysis of a case-control study with differential measurement error
Glonek, Garique, 20th International Workshop on Statistical Modelling, Sydney, Australia 10/07/05 |
Computer algebra resolves a multitude of microscale interactions to model stochastic partial differential equations
Roberts, Anthony John, |
A geometrical construction of the oval(s) associated with an a-flock
Brown, Matthew; Thas, J, Advances in Geometry 4 (9–17) 2004 |
Geometrical contributions to secret sharing theory
Jackson, Wen-Ai; Martin, K; O'Keefe, Christine, Journal of Geometry 79 (102–133) 2004 |
Gerbes, Clifford Modules and the index theorem
Murray, Michael; Singer, Michael, Annals of Global Analysis and Geometry 26 (355–367) 2004 |
Holonomy on D-branes
Carey, Alan; Johnson, Stuart; Murray, Michael, Journal of Geometry and Physics 52 (186–216) 2004 |
Partial differential equations
Van Der Hoek, John, Workshop on Mathematical Methods in Finance (2004), Melbourne, Vic, 2004 07/06/04 |
Towards a Classification of Homogeneous Tube Domains in C(4)
Eastwood, Michael; Ezhov, Vladimir; Isaev, A, Journal of Differential Geometry 68 (553–569) 2004 |
Compact Khler surfaces with trivial canonical bundle
Buchdahl, Nicholas, Annals of Global Analysis and Geometry 23 (189–204) 2003 |
Edge of the wedge theory in hypo-analytic manifolds
Eastwood, Michael; Graham, C, Communications in Partial Differential Equations 28 (2003–2028) 2003 |
Hyperbolic monopoles and holomorphic spheres
Murray, Michael; Norbury, Paul; Singer, Michael, Annals of Global Analysis and Geometry 23 (101–128) 2003 |
Stochastic Differential Equations in Hilbert Spaces
Filinkov, Alexei; Maizurna, Isna; Sorenson, J; Van Der Hoek, John, chapter in Applicable Mathematics in the Golden Age (Morgan & Claypool) 32–169, 2003 |
A step towards holistic discretisation of stochastic partial differential equations
Roberts, Anthony John, The ANZIAM Journal 45 (C1–C15) 2003 |
Evidence for a Differential Cellular Distribution of Inward Rectifier K Channels in the Rat Isolated Mesenteric Artery
Crane, Glenis Jayne; Walker, S; Dora, K; Garland, C, Journal of Vascular Research 40 (159–168) 2003 |
The geometry and physics of the Seiberg-Witten equations
Wu, Siye, chapter in Geometric analysis and applications to quantum field theory (Birkhauser) 157–203, 2002 |
Differential equations in spaces of abstract stochastic distributions
Filinkov, Alexei; Sorensen, Julian, Stochastics and Stochastic Reports 72 (129–173) 2002 |
The Andr/Bruck and Bose representation of conics in Baer subplanes of PG(2, q2)
Quinn, Catherine, Journal of Geometry 74 (123–138) 2002 |
Some special geometry in dimension six
Eastwood, Michael; Cap, A, Czech Winter School on Geometry and Physics (22nd: 2002:, Srn'i, Czechoslovakia), |
Phase transitions in shape memory alloys with hyperbolic heat conduction and differential-algebraic models
Melnik, R; Roberts, Anthony John; Thomas, K, Computational Mechanics 29 (16–26) 2002 |
A proof of Atiyah's conjecture on configurations of four points in Euclidean three-space
Eastwood, Michael; Norbury, Paul, Geometry & Topology 5 (885–893) 2001 |
Equivariant Seiberg-Witten Floer homology
Marcolli, M; Wang, Bai-Ling, Communications in Analysis and Geometry 9 (451–639) 2001 |
Generalising a characterisation of Hermitian curves
Barwick, Susan; Quinn, Catherine, Journal of Geometry 70 (1–7) 2001 |
Non-Schlesinger deformations of ordinary differential equations with rational coefficients
Kitaev, Alexandre, Journal of Physics A: Mathematical and Theoretical (Print Edition) 34 (2259–2272) 2001 |
Truncation-type methods and Bcklund transformations for ordinary differential equations: The third and fifth Painlev equations
Gordoa, P; Joshi, Nalini; Pickering, A, Glasgow Mathematical Journal 43A (23–32) 2001 |
Conformally invariant differential operators on spin bundles
Eastwood, Michael, chapter in Further advances in twistor theory. Vol. III, Curved twistor spaces (Chapman & Hall/CRC) 72–74, 2001 |
A note on higher cohomology groups of Khler quotients
Wu, Siye, Annals of Global Analysis and Geometry 18 (569–576) 2000 |
Local Constraints on Einstein-Weyl geometries: The 3-dimensional case
Eastwood, Michael; Tod, K, Annals of Global Analysis and Geometry 18 (1–27) 2000 |
The determination of ovoids of PG(3, q) containing a pointed conic
Brown, Matthew, Journal of Geometry 67 (61–72) 2000 |
Unitals which meet Baer subplanes in 1 modulo q points
Barwick, Susan; O'Keefe, Christine; Storme, L, Journal of Geometry 68 (16–22) 2000 |
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