
Search the School of Mathematical SciencesPeople matching "+Differential +equations"Courses matching "+Differential +equations" 
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.
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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 nonlinear 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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Events matching "+Differential +equations" 
Symmetrybreaking and the Origin of Species 15:10 Fri 24 Oct, 2008 :: G03 Napier Building University of Adelaide :: Toby Elmhirst :: ARC Centre of Excellence for Coral Reef Studies, James Cook University
The theory of partial differential equations can say much about generic bifurcations from spatially homogeneous steady states, but relatively little about generic bifurcations from unimodal steady states. In many applications, spatially homogeneous steady states correspond to lowenergy physical states that are destabilized as energy is fed into the system, and in these cases standard PDE theory can yield some impressive and elegant results. However, for many macroscopic biological systems such results are less useful because lowenergy states do not hold the same priviledged position as they do in physical and chemical systems. For example, speciation  the evolutionary process by which new species are formed  can be seen as the destabilization of a unimodal density distribution over phenotype space. Given the diversity of species and environments, generic results are clearly needed, but cannot be gained from PDE theory. Indeed, such questions cannot even be adequately formulated in terms of PDEs. In this talk I will introduce 'Pod Systems' which can provide an answer to the question; 'What happens, generically, when a unimodal steady state loses stability?' In the pod system formalization, the answer involves elements of equivariant bifurcation theory and suggests that new species can arise as the result of broken symmetries. 

Direct "delay" reductions of the Toda equation
13:10 Fri 23 Jan, 2009 :: 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 differentialdifference equations, sometimes referred to as
delaydifferential 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.


Hunting Nonlinear Mathematical Butterflies 15:10 Fri 23 Jan, 2009 :: Napier LG29 :: Prof Nalini Joshi :: University of Sydney
The utility of mathematical models relies on their ability to predict the future from a known set of initial states.
But there are nonlinear systems, like the weather, where future behaviours are unpredictable unless their initial
state is known to infinite precision. This is the butterfly effect. I will show how to analyse functions to overcome
this problem for the classical Painleve equations, differential equations that provide archetypical nonlinear models
of modern physics. 

Strong PredictorCorrector Euler Methods for Stochastic Differential Equations 15:10 Fri 19 Jun, 2009 :: 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
predictorcorrector 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 predictorcorrector Euler
methods. 

Dispersing and settling populations in biology 15:10 Tue 23 Jun, 2009 :: Napier G03 :: Prof Kerry Landman :: University of Melbourne
Partial differential equations are used to model populations (such as cells, animals or molecules) consisting of individuals that undergo two important processes: dispersal and settling. I will describe some general characteristics of these systems, as well as some of our recent projects. 

Conformal geometry of differential equations 13:10 Fri 12 Feb, 2010 :: School Board Room :: Dr Pawel Nurowski :: University of Warsaw


Convolution equations in A^{\infty} for convex domains 13:10 Fri 5 Mar, 2010 :: School Board Room :: Dr Le Hai Khoi :: Nanyang Technological University, Singapore


Vertex algebras and variational calculus I 13:10 Fri 4 Jun, 2010 :: School Board Room :: Dr Pedram Hekmati :: University of Adelaide
A basic operation in calculus of variations is the EulerLagrange 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. 

How to value risk 12:10 Mon 11 Apr, 2011 :: 5.57 Ingkarni Wardli :: Leo Shen :: University of Adelaide
A key question in mathematical finance is: given a future random payoff X, what is its value today? If X represents a loss, one can ask how risky is X. To mitigate risk it must be modelled and quantified. The finance industry has used ValueatRisk and conditional ValueatRisk as measures. However, these measures are not time consistent and ValueatRisk can penalize diversification. A modern theory of risk measures is being developed which is related to solutions of backward stochastic differential equations in continuous time and stochastic difference equations in discrete time.
I first review risk measures used in mathematical finance, including static and dynamic risk measures. I recall results relating to backward stochastic difference equations (BSDEs) associated with a single jump process. Then I evaluate some numerical examples of the solutions of the backward stochastic difference equations and related risk measures. These concepts are new. I hope the examples will indicate how they might be used. 

Probability density estimation by diffusion 15:10 Fri 10 Jun, 2011 :: 7.15 Ingkarni Wardli :: Prof Dirk Kroese :: University of Queensland
Media...One of the beautiful aspects of Mathematics is that seemingly
disparate areas can often have deep connections. This talk is about
the fundamental connection between probability density estimation,
diffusion processes, and partial differential equations. Specifically,
we show how to obtain efficient probability density estimators by
solving partial differential equations related to diffusion processes.
This new perspective leads, in combination with Fast Fourier
techniques, to very fast and accurate algorithms for density
estimation. Moreover, the diffusion formulation unifies most of the
existing adaptive smoothing algorithms and provides a natural solution
to the boundary bias of classical kernel density estimators. This talk
covers topics in Statistics, Probability, Applied Mathematics, and
Numerical Mathematics, with a surprise appearance of the theta
function. This is joint work with Zdravko Botev and Joe Grotowski. 

There are no magnetically charged particlelike solutions of the EinsteinYangMills equations for models with Abelian residual groups 13:10 Fri 19 Aug, 2011 :: B.19 Ingkarni Wardli :: Dr Todd Oliynyk :: Monash University
According to a conjecture from the 90's, globally regular, static, spherically symmetric (i.e. particlelike) solutions with nonzero total magnetic charge are not expected to exist in EinsteinYangMills 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 nonexistence results apply to the most widely studied models with Abelian residual groups. 

Laplace's equation on multiplyconnected domains 12:10 Mon 29 Aug, 2011 :: 5.57 Ingkarni Wardli :: Mr Hayden Tronnolone :: University of Adelaide
Various physical processes take place on multiplyconnected domains
(domains with some number of 'holes'), such as the stirring of a fluid
with paddles or the extrusion of material from a die. These systems may
be described by partial differential equations (PDEs). However, standard
numerical methods for solving PDEs are not wellsuited to such examples:
finite difference methods are difficult to implement on
multiplyconnected domains, especially when the boundaries are irregular
or moving, while finite element methods are computationally expensive.
In this talk I will describe a fast and accurate numerical method for
solving certain PDEs on twodimensional multiplyconnected domains,
considering Laplace's equation as an example. This method takes
advantage of complex variable techniques which allow the solution to be
found with spectral accuracy provided the boundary data is smooth. Other
advantages over traditional numerical methods will also be discussed. 

Stability analysis of nonparallel unsteady flows via separation of variables 15:30 Fri 18 Nov, 2011 :: 7.15 Ingkarni Wardli :: Prof Georgy Burde :: BenGurion University
Media...The problem of variables separation in the linear stability
equations, which govern the disturbance behavior in viscous
incompressible fluid flows, is discussed.
Stability of some unsteady nonparallel threedimensional flows (exact
solutions of the NavierStokes equations)
is studied via separation of variables using a semianalytical, seminumerical approach.
In this approach, a solution with separated variables is defined in a new coordinate system which is sought together with the solution form. As the result, the linear stability problems are reduced to eigenvalue problems for ordinary differential equations which can be solved numerically.
In some specific cases, the eigenvalue
problems can be solved analytically. Those unique examples of exact
(explicit) solution of the nonparallel unsteady flow stability
problems provide a very useful test for methods used in the
hydrodynamic stability theory. Exact solutions of the stability problems for some stagnationtype flows are presented. 

Space of 2D shapes and the WeilPetersson metric: shapes, ideal fluid and Alzheimer's disease 13:10 Fri 25 Nov, 2011 :: B.19 Ingkarni Wardli :: Dr Sergey Kushnarev :: National University of Singapore
The WeilPetersson 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 EulerPoincare 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. 

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, 2012 :: 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. 

The motivic logarithm and its realisations 13:10 Fri 3 Aug, 2012 :: 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 propertieswith 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 nonexplicit. 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 padic 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 nontopological cohomology theories. 

Thinfilm flow in helicallywound channels with small torsion 15:10 Fri 26 Oct, 2012 :: B.21 Ingkarni Wardli :: Dr Yvonne Stokes :: University of Adelaide
The study of flow in open helicallywound channels has application to many natural and industrial flows. We will consider laminar flow down helicallywound channels of rectangular cross section and with small torsion, in which the fluid depth is small. Assuming a steadystate flow that is independent of position along the axis of the channel, the flow solution may be determined in the twodimensional cross section of the channel. A thinfilm approximation yields explicit expressions for the fluid velocity in terms of the freesurface shape. The latter satisfies an interesting nonlinear ordinary differential equation that, for a channel of rectangular cross section, has an analytical solution. The predictions of the thinfilm model are shown to be in good agreement with much more computationally intensive solutions of the smallhelixtorsion NavierStokes equations.
This work has particular relevance to spiral particle separators used in the minerals processing industry. Early work on modelling of particleladen thinfilm flow in spiral channels will also be discussed. 

Thinfilm flow in helicallywound channels with small torsion 15:10 Fri 26 Oct, 2012 :: B.21 Ingkarni Wardli :: Dr Yvonne Stokes :: University of Adelaide
The study of flow in open helicallywound channels has application to many natural and industrial flows. We will consider laminar flow down helicallywound channels of rectangular cross section and with small torsion, in which the fluid depth is small. Assuming a steadystate flow that is independent of position along the axis of the channel, the flow solution may be determined in the twodimensional cross section of the channel. A thinfilm approximation yields explicit expressions for the fluid velocity in terms of the freesurface shape. The latter satisfies an interesting nonlinear ordinary differential equation that, for a channel of rectangular cross section, has an analytical solution. The predictions of the thinfilm model are shown to be in good agreement with much more computationally intensive solutions of the smallhelixtorsion NavierStokes equations.
This work has particular relevance to spiral particle separators used in the minerals processing industry. Early work on modelling of particleladen thinfilm flow in spiral channels will also be discussed. 

A stability theorem for elliptic Harnack inequalities 15:10 Fri 5 Apr, 2013 :: B.18 Ingkarni Wardli :: Prof Richard Bass :: University of Connecticut
Media...Harnack inequalities are an important tool in probability theory,
analysis, and partial differential equations. The classical Harnack
inequality is just the one you learned in your graduate complex analysis
class, but there have been many extensions, to different spaces, such as
manifolds, fractals, infinite graphs, and to various sorts of elliptic operators.
A landmark result was that of Moser in 1961, where he proved the Harnack
inequality for solutions to a class of partial differential equations.
I will talk about the stability of Harnack inequalities. The main result
says that if the Harnack inequality holds for an operator on a space,
then the Harnack inequality will also hold for a large class of other operators
on that same space. This provides a generalization of the result of Moser. 

Mtheory and higher gauge theory 13:10 Fri 12 Apr, 2013 :: Ingkarni Wardli B20 :: Dr Christian Saemann :: HeriotWatt University
I will review my recent work on integrability of Mbrane configurations and
the description of Mbrane models in higher gauge theory. In particular, I
will discuss categorified analogues of instantons and present superconformal equations of motion for the nonabelian tensor multiplet in six dimensions. The latter are derived from considering nonabelian gerbes on certain twistor spaces. 

The Einstein equations with torsion, reduction and duality 12:10 Fri 23 Aug, 2013 :: Ingkarni Wardli B19 :: Dr David Baraglia :: University of Adelaide
We consider the Einstein equations for connections with skew torsion. After some general remarks we look at these equations on principal Gbundles, making contact with string structures and heterotic string theory in the process. When G is a torus the equations are shown to possess a symmetry not shared by the usual Einstein equations  Tduality. This is joint work with Pedram Hekmati. 

A few flavours of optimal control of Markov chains 11:00 Thu 12 Dec, 2013 :: B18 :: Dr Sam Cohen :: Oxford University
Media...In this talk we will outline a general view of optimal control of a continuoustime Markov chain, and how this naturally leads to the theory of Backward Stochastic Differential Equations. We will see how this class of equations gives a natural setting to study these problems, and how we can calculate numerical solutions in many settings. These will include problems with payoffs with memory, with random terminal times, with ergodic and infinitehorizon value functions, and with finite and infinitely many states. Examples will be drawn from finance, networks and electronic engineering. 

Complexifications, Realifications, Real forms and Complex Structures 12:10 Mon 23 Jun, 2014 :: B.19 Ingkarni Wardli :: Kelli FrancisStaite :: University of Adelaide
Media...Italian mathematicians NiccolÃ² Fontana Tartaglia and Gerolamo Cardano introduced complex numbers to solve polynomial equations such as x^2+1=0. Solving a standard real differential equation often uses complex eigenvalues and eigenfunctions. In both cases, the solution space is expanded to include the complex numbers, solved, and then translated back to the real case.
My talk aims to explain the process of complexification and related concepts. It will give vocabulary and some basic results about this important process. And it will contain cute cat pictures.


Modelling the meanfield behaviour of cellular automata 12:10 Mon 4 Aug, 2014 :: B.19 Ingkarni Wardli :: Kale Davies :: University of Adelaide
Media...Cellular automata (CA) are latticebased models in which agents fill the lattice sites and behave according to some specified rule. CA are particularly useful when modelling cell behaviour and as such many people consider CA model in which agents undergo motility and proliferation type events. We are particularly interested in predicting the average behaviour of these models. In this talk I will show how a system of differential equations can be derived for the system and discuss the difficulties that arise in even the seemingly simple case of a CA with motility and proliferation. 

Higher rank discrete Nahm equations for SU(N) monopoles in hyperbolic space 11:10 Wed 8 Apr, 2015 :: Engineering & Maths EM213 :: Joseph Chan :: University of Melbourne
Braam and Austin in 1990, proved that SU(2) magnetic monopoles in hyperbolic space H^3 are the same as solutions of the discrete Nahm equations. I apply equivariant Ktheory to the ADHM construction of instantons/holomorphic bundles to extend the BraamAustin result from SU(2) to SU(N). During its evolution, the matrices of the higher rank discrete Nahm equations jump in dimensions and this behaviour has not been observed in discrete evolution equations before. A secondary result is that the monopole field at the boundary of H^3 determines the monopole. 

IGA Workshop on Symmetries and Spinors: Interactions Between Geometry and Physics 09:30 Mon 13 Apr, 2015 :: Conference Room 7.15 on Level 7 of the Ingkarni Wardli building :: J. FigueroaO'Farrill (University of Edinburgh), M. Zabzine (Uppsala University), et al
Media...The interplay between physics and geometry has lead to stunning advances and enriched the internal structure of each field. This is vividly exemplified in the theory of supergravity, which is a supersymmetric extension of Einstein's relativity theory to the small scales governed by the laws of quantum physics. Sophisticated mathematics is being employed for finding solutions to the generalised Einstein equations and in return, they provide a rich source for new exotic geometries. This workshop brings together worldleading scientists from both, geometry and mathematical physics, as well as young researchers and students, to meet and learn about each others work. 

Harmonic Analysis in Rough Contexts 15:10 Fri 13 May, 2016 :: Engineering South S112 :: Dr Pierre Portal :: Australian National University
Media...In recent years, perspectives on what constitutes the ``natural" framework within which to conduct various forms of mathematical analysis have shifted substantially. The common theme of these shifts can be described as a move towards roughness, i.e. the elimination of smoothness assumptions that had previously been considered fundamental. Examples include partial differential equations on domains with a boundary that is merely Lipschitz continuous, geometric analysis on metric measure spaces that do not have a smooth structure, and stochastic analysis of dynamical systems that have nowhere differentiable trajectories.
In this talk, aimed at a general mathematical audience, I describe some of these shifts towards roughness, placing an emphasis on harmonic analysis, and on my own contributions. This includes the development of heat kernel methods in situations where such a kernel is merely a distribution, and applications to deterministic and stochastic partial differential equations. 

Probabilistic Meshless Methods for Bayesian Inverse Problems 15:10 Fri 5 Aug, 2016 :: Engineering South S112 :: Dr Chris Oates :: University of Technology Sydney
Media...This talk deals with statistical inverse problems that involve partial differential equations (PDEs) with unknown parameters. Our goal is to account, in a rigorous way, for the impact of discretisation error that is introduced at each evaluation of the likelihood due to numerical solution of the PDE. In the context of meshless methods, the proposed, modelbased approach to discretisation error encourages statistical inferences to be more conservative in the presence of significant solver error. In addition, (i) a principled learningtheoretic approach to minimise the impact of solver error is developed, and (ii) the challenge of nonlinear PDEs is considered. The method is applied to parameter inference problems in which nonnegligible solver error must be accounted for in order to draw valid statistical conclusions. 

Predicting turbulence 14:10 Tue 30 Aug, 2016 :: Napier 209 :: Dr Trent Mattner :: School of Mathematical Sciences
Media...Turbulence is characterised by threedimensional unsteady fluid motion over a wide range of spatial and temporal scales. It is important in many problems of technological and scientific interest, such as drag reduction, energy production and climate prediction.
Turbulent flows are governed by the NavierStokes equations, which are a nonlinear system of partial differential equations. Typically, numerical methods are needed to find solutions to these equations. In turbulent flows, however, the resulting computational problem is usually intractable. Filtering or averaging the NavierStokes equations mitigates the computational problem, but introduces new quantities into the equations. Mathematical models of turbulence are needed to estimate these quantities. One promising turbulence model consists of a random collection of fluid vortices, which are themselves approximate solutions of the NavierStokes equations. 

Fault tolerant computation of hyperbolic PDEs with the sparse grid combination technique 15:10 Fri 28 Oct, 2016 :: Ingkarni Wardli 5.57 :: Dr Brendan Harding :: University of Adelaide
Computing solutions to high dimensional problems is challenging because of the curse of dimensionality. The sparse grid combination technique allows one to significantly reduce the cost of computing solutions such that they become manageable on current supercomputers. However, as these supercomputers increase in size the rate of failure also increases. This poses a challenge for our computations. In this talk we look at the problem of computing solutions to hyperbolic partial differential equations with the combination technique in an environment where faults occur. A fault tolerant generalisation of the combination technique will be presented along with results that demonstrate its effectiveness. 

Geometric structures on moduli spaces 12:10 Fri 31 Mar, 2017 :: Napier 209 :: Nicholas Buchdahl :: University of Adelaide
Media...Moduli spaces are used to classify various kinds of objects,
often arising from solutions of certain differential equations on
manifolds; for example, the complex structures on a compact
surface or the antiselfdual YangMills equations on an oriented
smooth 4manifold. Sometimes these moduli spaces carry important
information about the underlying manifold, manifested most
clearly in the results of Donaldson and others on the topology of
smooth 4manifolds. It is also the case that these moduli spaces
themselves carry interesting geometric structures; for example,
the WeilPetersson metric on moduli spaces of compact Riemann
surfaces, exploited to great effect by Maryam Mirzakhani. In this
talk, I shall elaborate on the theme of geometric structures on
moduli spaces, with particular focus on some recentish work done
in conjunction with Georg Schumacher. 

Weil's Riemann hypothesis (RH) and dynamical systems 12:10 Fri 11 Aug, 2017 :: Engineering Sth S111 :: Tuyen Truong :: University of Adelaide
Media...Weil proposed an analogue of the RH in finite fields, aiming at counting asymptotically the number of solutions to a given system of polynomial equations (with coefficients in a finite field) in finite field extensions of the base field. This conjecture influenced the development of Algebraic Geometry since the 1950Ã¢ÂÂs, most important achievements include: Grothendieck et al.Ã¢ÂÂs etale cohomology, and Bombieri and GrothendieckÃ¢ÂÂs standard conjectures on algebraic cycles (inspired by a Kahlerian analogue of a generalisation of WeilÃ¢ÂÂs RH by Serre). WeilÃ¢ÂÂs RH was solved by Deligne in the 70Ã¢ÂÂs, but the finite field analogue of SerreÃ¢ÂÂs result is still open (even in dimension 2). This talk presents my recent work proposing a generalisation of WeilÃ¢ÂÂs RH by relating it to standard conjectures and a relatively new notion in complex dynamical systems called dynamical degrees. In the course of the talk, I will present the proof of a question proposed by Esnault and Srinivas (which is related to a result by Gromov and Yomdin on entropy of complex dynamical systems), which gives support to the finite field analogue of SerreÃ¢ÂÂs result. 

Quantum Airy structures and topological recursion 13:10 Wed 14 Mar, 2018 :: Ingkarni Wardli B17 :: Gaetan Borot :: MPI Bonn
Media...Quantum Airy structures are Lie algebras of quadratic differential operators  their classical limit describes Lagrangian subvarieties in symplectic vector spaces which are tangent to the zero section and cut out by quadratic equations. Their partition function  which is the function annihilated by the collection of differential operators  can be computed by the topological recursion. I will explain how to obtain quantum Airy structures from spectral curves, and explain how we can retrieve from them correlation functions of semisimple cohomological field theories, by exploiting the symmetries. This is based on joint work with Andersen, Chekhov and Orantin. 

Family gauge theory and characteristic classes of bundles of 4manifolds 13:10 Fri 16 Mar, 2018 :: Barr Smith South Polygon Lecture theatre :: Hokuto Konno :: University of Tokyo
Media...I will define a nontrivial characteristic class of bundles of
4manifolds using families of SeibergWitten equations. The basic idea
of the construction is to consider an infinite dimensional
analogue of the Euler class used in the usual theory of characteristic
classes. I will also explain how to prove the nontriviality of this
characteristic class. If time permits, I will mention a relation between
our characteristic class and positive scalar curvature metrics. 

Stability Through a Geometric Lens 15:10 Fri 18 May, 2018 :: Horace Lamb 1022 :: Dr Robby Marangell :: University of Sydney
Focussing on the example of the Fisher/KPP equation, I will show how geometric information can be used to establish (in)stability results in some partial differential equations (PDEs). Viewing standing and travelling waves as fixed points of a flow in an infinite dimensional system, leads to a reduction of the linearised stability problem to a boundary value problem in a linear nonautonomous ordinary differential equation (ODE). Next, by exploiting the linearity of the system, one can use geometric ideas to reveal additional structure underlying the determination of stability. I will show how the Riccati equation can be used to produce a reasonably computable detector of eigenvalues and how such a detector is related to another, wellknown eigenvalue detector, the Evans function. If there is time, I will try to expand on how to generalise these ideas to systems of PDEs. 

How long does it take to get there? 11:10 Fri 19 Oct, 2018 :: Engineering North N132 :: Professor Herbert Huppert :: University of Cambridge
In many situations involving nonlinear partial differential equations, requiring much numerical calculation because there is no analytic solution, it is possible to find a similarity solution to the resulting (still nonlinear) ordinary differential equation; sometimes even analytically, but it is generally independent of the initial conditions. The similarity solution is said to approach the real solution for t >> tau, say. But what is tau? How does it depend on the parameters of the problem and the initial conditions? Answers will be presented for a variety of problems and the audience will be asked to suggest others if they know of them.


An Introduction to Ricci Flow 11:10 Fri 19 Oct, 2018 :: Barr Smith South Polygon Lecture theatre :: Miles Simon :: University of Magdeburg
In these three talks we give an introduction to Ricci flow and present some applications thereof.
After introducing the Ricci flow we present some theorems and arguments from the theory of linear and nonlinear parabolic equations. We explain why this theory guarantees that there is always a solution to the Ricci flow for a short time for any given smooth initial metric on a compact manifold without boundary.
We calculate evolution equations for certain geometric quantities, and present some examples of maximum principle type arguments. In the last lecture we present some geometric results which are derived with the help of the Ricci flow. 
Publications matching "+Differential +equations"Publications 

Dessins d'enfants and differential equations Larusson, Finnur; Sadykov, T, St Petersburg Mathematical Journal 19 (1003–1014) 2008  Model subgrid microscale interactions to accurately discretise stochastic partial differential equations. Roberts, Anthony John,  Computer algebra derives normal forms of stochastic differential equations Roberts, Anthony John,  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 resolves a multitude of microscale interactions to model stochastic partial differential equations Roberts, Anthony John,  Partial differential equations Van Der Hoek, John, Workshop on Mathematical Methods in Finance (2004), Melbourne, Vic, 2004 07/06/04  Edge of the wedge theory in hypoanalytic manifolds Eastwood, Michael; Graham, C, Communications in Partial Differential Equations 28 (2003–2028) 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  Differential equations in spaces of abstract stochastic distributions Filinkov, Alexei; Sorensen, Julian, Stochastics and Stochastic Reports 72 (129–173) 2002  NonSchlesinger deformations of ordinary differential equations with rational coefficients Kitaev, Alexandre, Journal of Physics A: Mathematical and Theoretical (Print Edition) 34 (2259–2272) 2001  Truncationtype 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 
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