Colloquium

Diophantine approximation, averaging operators, and the spectrum of unitary representations

We discuss the problem of Diophantine approximation on algebraic groups and relate this problem to the behavior of certain averaging operators acting on homogeneous spaces. It turns out that the problem of approximation is closely linked to understanding the spectral decomposition of automorphic representations. 

Currently, establishing the optimal approximation presents a significant challenge due to the presence of representations with slow decay rates. We discuss a method that overcomes this difficulty based on multiplicity estimates. The talk is based on joint work with M. Francyk, A. Ghosh, and A. Nevo.

On geometric properties of conformally invariant curves

How to construct a canonical random conformally invariant path in two dimensions? Motivated by Loewner's classical theory of dynamics of slit domains, Schramm introduced random Loewner evolutions to model canonical random curves via evolutions of conformal maps. While the initial usage of such Schramm-Loewner evolutions (SLEs) was to describe critical interfaces in statistical physics models and their relation to conformal field theory (CFT), SLE type curves quickly turned out to be ubiquitous in various problems in probability theory and mathematical physics, and to have intricate connections to complex geometry and beyond. 

This talk highlights some geometric aspects emerging from the study of SLE curves and CFT. As examples, we shall mention versions of the Loewner energy (the anticipated action functional of these canonical curve models, or more rigorously, the rate function in large deviations principles for the random curves), classification problems of covering maps with prescribed critical points, and if time permits, the emergence of the Virasoro algebra (the symmetry algebra of CFTs) from complex deformations of boundaries of bordered Riemann surfaces (i.e., loops).

The eta invariant of a circle bundle on a Fano manifold

The eta invariant of Atiyah-Patodi-Singer is formally the signature of a Dirac operator and is in general difficult to compute. We present a computation of the eta invariant; namely for the spin-c Dirac operator on the unit circle bundle of a positive line bundle over a Fano manifold. The computation is in terms of Zhang’s value of its adiabatic limit and extends an earlier computation of the author from small to arbitrary values of the adiabatic parameter.

Quantitative statistical properties of dynamical systems

I will introduce the transfer operator techniques to compute quantitative statistical properties of dynamical systems, giving some literature references where the quantitative analysis can be done to arbitrary precision. I will then talk about my result (in collaboration with Olivar Butterley and Giovanni Canestrari) on non-Markov interval maps which show obstruction in the analysis to arbitrary precision.

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.