Title: Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics
Date: 24 November 2023, 10:10am
Abstract: Using results from the theory of harmonic maps, Kotschick proved that a closed hyperbolic four-manifold cannot admit a complex structure. We give a new proof which instead relies on properties of Einstein metrics in dimension four. The benefit of this new approach is that it generalizes to prove that another class of aspherical four-manifolds (graph manifolds with positive Euler characteristic) also fail to admit complex structures. This is joint work with Luca Di Cerbo.
Title: Translation surfaces and isoperiodic foliation
Date: 10 November 2023, 10:10am
Abstract: Translation surfaces are surfaces obtained by gluing sides of polygons with translations. The space of these surfaces, their moduli space, lies at the crossroads of dynamical systems and hyperbolic geometry on surfaces. It appears naturally when exploring both billiards trajectories and geodesics in the moduli space of curves. I will give an introduction to these objects and explain these connections. The moduli space of translation surfaces is naturally stratified by prescribing conical singularities, and carries a foliation called the isoperiodic foliation. I will give a characterisation of the leaves that intersect a given stratum.
Title: Bundle gerbes in all dimensions
Date: 20 October 2023, 10:10am
Abstract: Bundle gerbes were introduced by Michael Murray as a geometric interpretation of integral degree 3 cohomology generalising the interpretation of degree 2 cohomology in terms of vector bundles. It generalises the gauge field interpretation of electromagnetism to some of the fields in string theory. In his thesis Danny Stevenson extended the theory to degree 4 cohomology with the concept of bundle 2-gerbes. In this talk I will show how to define bundle gerbes in all dimensions in a way that recovers the key results of the theory. This work was developed in conjunction with Michael and Danny.
Title: Change in framing of links in 3-manifolds via ambient isotopy
Date: 1 September 2023, 10:10am
Abstract: In this talk we will present work on the change of framing of knots and links via ambient isotopy in 3-manifolds by using D. McCullough's results on a generalized definition of Dehn twist homeomorphisms. In particular, we will discuss work by P. Cahn, V. Chernov, and R. Sadykov for framed knots and then the generalization of their work (joint work with R. P. Bakshi, J, H. Przytycki, G. Montoya-Vega, and D. Weeks) to links by showing that the only way of changing the framing of a link by ambient isotopy in an oriented 3-manifold is when the manifold admits a properly embedded non-separating S^2. We will then use spin structures to show that the ambient isotopy is a composition of even powers of Dehn homeomorphisms along the disjoint union of non-separating 2-spheres.
Title: Equivariant index on toric contact manifolds
Date: 25 August 2023, 10:10am
Abstract: I will discuss the equivariant index of the horizontal Dolbeault operator on compact toric contact manifolds of Reeb type. This operator is transversally elliptic to the Reeb foliation and it features notably in calculations of partition functions of cohomologically twisted gauge theories. I will describe how to evaluate the index in general and give an explicit expression for it in terms of the moment cone. This is joint work with Pedram Hekmati.
Title: Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds
Date: 2 June 2023, 10:10am
Abstract: In this talk we examine the Ricci flow of initial metric spaces that are Reifenberg and locally bi-Lipschitz to Euclidean space. We show that any two solutions starting from such an initial metric space, whose Ricci curvatures are uniformly bounded from below and whose curvatures are bounded by $c\cdot t^{-1}$, are exponentially in time close to one another in the appropriate gauge. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott. This is joint work with Alix Deruelle and Felix Schulze.
Title: Separation of variables for spaces of constant curvature
Date: 26 May 2023, 10:10am
Abstract: I will discuss orthogonal
separation of variables for spaces of constant curvature,
with the emphasis on pseudo-Riemannian metrics. I
will give a local description of all possible
separating coordinates, and write the
transformation to flat coordinates. The
problem was actively studied since at least hundred years:
different aspects of separations were considered by Stäckel
in the 19th century and by Eisenhart in the beginning of the
20th and in a series of papers and books of E.
Kalnins, J. Kress and W. Miller, I will
give a historical overview.
The new results are joint with A. Bolsinov and A. Konyaev
and appeared within the Nijenhuis Geometry project and
are a by-product of our classification of compatible
inhomogeneous geometric Poisson brackets of degree 3+1;
separation of variables is closely related to compatible
pencils of Poisson brackets of degree 1. They are also
closely related to geodesically equivalent metrics; all
these will be mentioned if the time allows.
Title: Curvature Aspects of Hyperbolicity in Complex Geometry
Date: 19 May 2023, 10:10am
Abstract: A compact complex manifold X is said to be Kobayashi hyperbolic if every holomorphic map from the complex plane to X is constant. An extension of a conjecture of Kobayashi predicts that all compact Kobayashi hyperbolic manifolds are projective and admit a Kähler--Einstein metric with negative Ricci curvature. We will discuss the curvature aspects of Kobayashi hyperbolic manifolds and their study via Gauduchon connections. We will also present a "positive analog" of the Kobayashi conjecture.
Title: Complex Manifolds of Hyperbolic and Non-Hyperbolic-Type
Date: 18 May 2023, 11:10am
Abstract: A general overview of the role played by curvature in the study of complex manifolds of hyperbolic-type and non-hyperbolic-type.
Title: Generalised spectral dimensions in non-perturbative quantum gravity
Date: 12 May 2023, 10:10am
Title: Loop spaces, Lie bialgebras and variations on Teichmüller space
Date: 5 May 2023, 10:10am
Abstract: A fundamental idea in
“Grothendieck-Teichmüller theory” is that one can study the
absolute Galois group of the rationals by studying the
actions of the Galois group on the “Teichmüller tower”, i.e.
the collection of (geometric) fundamental groups of all the
moduli spaces $\mathcal{M}_{g,n}$ and the natural maps
between them. This transforms a difficult arithmetic problem
into a geometric problem.
The goal of this talk is to explain an extension of this
idea where we replace one mysterious group (the absolute
Galois group) with a family of mysterious groups (the higher
genus Kashiwara-Vergne groups). The associated “KV tower” is
a tower of fundamental groups of the free loop spaces on
various punctured Reimann surfaces. We aim to describe how
this variation on Teichmüller space turns a problem in Lie
theory (the Kashiwara-Vernge problem) into a problem in
topology.
Title: Cohomogeneity One Ancient Ricci Flows
Date: 31 March 2023, 10:10am
Title: Arbitrarily high order concentration-compactness for curvature flow
Date: 17 March 2023, 10:10am
Abstract: We extend Struwe and
Kuwert-Sch\"atzle's concentration-compactness method for the
analysis of geometric evolution equations to flows of
arbitrarily high order, with the geometric polyharmonic heat
flow (GPHF) of surfaces, a generalisation of surface
diffusion flow, as exemplar. For the (GPHF) we apply the
technique to deduce localised energy and interior estimates,
a concentration-compactness alternative, pointwise curvature
estimates, a gap theorem, and study the blowup at a singular
time. This gives general information on the behaviour of the
flow for any initial data. Applying this for initial data
satisfying $||A^o||_2^2 < \varepsilon$ where
$\varepsilon$ is a universal constant, we perform global
analysis to obtain exponentially fast full convergence of
the flow in the smooth topology to a standard round
sphere.The talk will focus mostly on the concentration
compactness duality and how it translates from its initial
appearance in elliptic setting in the literature to the
parabolic setting of flows and finally to our geometric
polyharmonic heat flow.
This is joint work with James McCoy, Scott Parkins, and Glen
Wheeler.
Title: Scattering for wave equations with sources close to the light cone
Date: 3 March 2023, 10:10am