Colloquium

The eta invariant of a circle bundle on a Fano manifold

Orbit equivalence of pseudo-Anosov flows on 3-manifolds

This is a two-part minicourse on recent amazing work of mostly Barthelmé, Mann, and Frankel. We will talk about the result that given two pseudo-Anosov flows, at least one of which is transitive, and so that they have the same spectrum, and at least one of them does not have a tree of scalloped regions, then the flows are (isotopically) orbitally equivalent. The spectrum is the set of conjugacy classes in the fundamental group, which are represented by periodic orbits of the corresponding flow. There is a refined statement that takes into account the existence of trees of scalloped regions, where one still gets an orbitally equivalence result. The proof is done by analyzing the actions on the corresponding orbit spaces, each of which is a bifoliated plane.

A great part of the first lecture will be on the setup and properties. This part is accessible to a general mathematical audience.
 

Graph homomorphisms, some cohomology and the word problem

When can a given subset S be tiled by rectangular tiles in Z^2? This is an old question and for answering it Conway and Lagarias introduced certain groups which are now known as the Conway-Lagarias group (1990). Together with some analytic conditions, Thurston noticed that these questions can be answered completely in certain situations. This later formed the basis for understanding what uniformly random tiling of regions looked like (for instance in the results from Cohn, Kenyon and Propp from 2001). In 1995, Klaus Schmidt realised that this was related to a certain cohomology of dynamical systems and initiated its formal study. We will talk about such questions in the context of graph homomorphisms on the Z^d lattice and see how it relates to the word problem of finitely presented groups. The talk should be accessible to a general audience.

Proper affine actions of hyperbolic groups

Classification of crystallographic groups by Bieberbach gave rise to the Auslander conjecture which states that any affine crystallographic group is virtually polycyclic. The conjecture is known to be true in lower dimensions but is still open in the general case. However, if one eases the assumption of cocompactness, then Margulis in a celebrated work showed that the new conjecture fails to hold. He showed that non-abelian free groups can act properly as affine transformations. In this talk, I will give an overview of the history and present some recent developments.

Polyhedral-like approximations of strongly C-convex domains

Polyhderal approximations of convex bodies have been studied extensively in both affine and stochastic geometry. Of particular interest are the asymptotics of the approximation error as a function of the complexity of the approximating polyhedra. This analysis yields invariant combinatorial and geometric data associated to the underlying convex body. In this talk, we will discuss the motivation to study polyhedral-like approximations of domains satisfying notions of convexity that are suited for complex analysis. In particular, we will focus on the notion of $\C$-convexity, which is a natural analogue of convexity in complex projective spaces. We will introduce a suitable notion of polyhedra in this context, and present some (optimal and random) approximation results in the spirit of several results in real convex geometry.

SU(2) Representations of Three-Manifold groups

By the resolution of the Poincare conjecture in 3D, we know that the only closed three-manifold with trivial fundamental group is the three-sphere. In light of it, one can ask the following question: Suppose M is a closed three-manifold with the property that the only representation \pi_1(M) --> SU(2) is the trivial one. Does this imply that \pi_1(M) is trivial? The class of manifolds M for which this question is interesting (and open) are integer homology spheres. We prove a result in this direction: the half-Dehn surgery on any fibered knot K in S^3 admits an irreducible representation. The proof uses instanton floer homology. I will give a brief introduction to instanton floer homology and sketch the proof. This is based on work in progress, some jointly with Zhenkun Li and Fan Ye.

Generating the liftable mapping class groups of regular cyclic covers

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g \geq 1$. We show that the liftable mapping class group $\mathrm{LMod}_k(S_g)$ of the $k$-sheeted regular cyclic cover of $S_g$ is self-normalizing in $\mathrm{Mod}(S_g)$ and that $\mathrm{LMod}_k(S_g)$ is maximal in $\mathrm{Mod}(S_g)$ when $k$ is prime. Moreover, we establish the existence of a normal series of $\mathrm{LMod}_k(S_g)$ that generalizes a well-known normal series of congruence subgroups in $\mathrm{SL}(2,\mathbb{Z})$. Furthermore, we give an explicit finite generating set for $\mathrm{LMod}_k{S_g)$ for $g \geq 3$ and $k \geq 2$, and when $(g,k) = (2,2)$. As an application, we provide a finite generating set for the liftable mapping class group of the infinite-sheeted regular cyclic covering of $S_g$ for $g \geq 3$ by the infinite ladder surface. 

Atypical stars on a directed landscape geodesic

In random geometry, a recurring theme is that all geodesics emanating from a typical point merge into each other close to their starting point, and we call such points as 1-stars. However, the measure zero set of atypical stars, the points where such coalescence fails, is typically uncountable and the corresponding Hausdorff dimensions of these sets have been heavily investigated for a variety of models including the directed landscape, Liouville quantum gravity and the Brownian map. In this talk, we will consider the directed landscape -- the scaling limit of last passage percolation as constructed in the work Dauvergne-Ortmann-Virág  and look into the Hausdorff dimension of the set of atypical stars lying on a geodesic. The main result we will discuss is that the above dimension is almost surely equal to 1/3. This is in contrast to Ganguly-Zhang where it was shown that the set of atypical stars on the line {x=0} has dimension 2/3. This reduction of the dimension from 2/3 to 1/3 yields a quantitative manifestation of the smoothing of the environment around a geodesic with regard to exceptional behaviour.

On the Cheeger constant of hyperbolic surfaces

It is a well-known result due to Bollobas that the maximal Cheeger constant of large d-regular graphs cannot be close to the Cheeger constant of the d-regular tree. We shall prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by 2/π ≈ 0, 63.... which is strictly less than the Cheeger constant of the hyperbolic plane. The proof uses a random construction based on a Poisson-Voronoi tessellation of the surface with a vanishing intensity and makes an interesting object appear: the pointless Poisson--Voronoi  tessellation of the hyperbolic plane.

Joint work with Thomas Budzinski and Bram Petri.

Endperiodic maps via pseudo-Anosov flows

We show that every atoroidal, endperiodic map of an infinite-type surface is isotopic to a homeomorphism that is naturally the first return map of a pseudo-Anosov suspension flow on a fibered manifold. Morally, these maps are all obtained by “spinning” fibers around a surface in the boundary of the fibered cone. The structure associated to these spun pseudo-Anosov maps allows for several applications. These include defining and characterizing stretch factors of endperiodic maps, relating Cantwell—Conlon foliation cones to Thurston’s fibered cones, and defining a convex entropy function on these cones that extends log(stretch factor). 

This is joint work with Michael Landry and Yair Minsky.