Colloquium

Orientable maps and polynomial invariants of free-by-cyclic groups

Given a graph map from a graph to itself, we can associate two numbers to it: geometric stretch factor and homological stretch factor. I will define a notion of orientability for graph maps and use it to characterise when the two numbers are equal. The notion of orientability can be upgraded for certain automorphisms of free groups as well. A (fully irreducible) automorphism of a free group determines a free-by-cyclic group to which we can associate two polynomial invariants: the McMullen polynomial and the Alexander polynomial. These polynomials determine the stretch factor and homological stretch factor of f. We will see how orientability helps us to relate these two polynomials. 

This is joint work with Spencer Dowdall and Samuel Taylor.

Spectral properties of random perturbations of non-self-adjoint operators

Understanding spectral properties of non-self-adjoint operators are of significant importance as they arise in many problems such as scattering systems, open or damped quantum systems, and the analysis of the stability of solutions to nonlinear PDEs. Absence of suitable methods (e.g. variational methods) renders the study of the spectrum of such operators to be difficult. On the other hand,  its high sensitivity to small perturbations leads to serious numerical errors. Motivated by problems in different fields such as numerical analysis, semiclassical analysis, fluid dynamics, and mathematical physics, during the last fifteen years there have been several works in understanding the spectral properties of random perturbations of non-self-adjoint operators. In this talk, we will focus on random perturbations of large dimensional non-self-adjoint Toeplitz matrices, and discuss (i) Weyl type law for the empirical measure of its eigenvalues, (ii) limiting eigenvalue density inside the zone of spectral instability (i.e. limit law for outlier eigenvalues), and (iii) localization/delocalization of its eigenvectors, and the universality and non-universality of these features. I will also present some fun pictures and simulations. Based on joint works with Elliot Paquette, Martin Vogel, and Ofer Zeitouni. 

Inhomogeneous quadratic forms

An inhomogeneous quadratic form is a quadratic form along with a shift. These forms arise in a variety of situations in number theory, dynamics, and in quantum chaos. I will explain these connections and then discuss some recent progress on understanding the values taken by such forms at integer points, using a variety of ergodic, geometric and analytic tools.

Deformation space analogies between Kleinian reflection groups and rational maps

We will describe an explicit correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps acting on the Riemann sphere. This correspondence has several dynamical and parameter space consequences. To illustrate some of these, we will discuss many striking similarities between the deformation spaces of these two classes of conformal dynamical systems, including an analogue of Thurston’s compactness theorem for anti-holomorphic rational maps and relations between the global topology of the corresponding deformation spaces.

Totally geodesic submanifolds of real and complex hyperbolic manifolds

After some history and motivation, I will discuss recent works with Bader, Miller and Stover in which we prove finiteness of maximal totally geodesic submanifolds in real and complex hyperbolic spaces. 

Commensurators and arithmeticity of hyperbolic manifolds

The commensurator of a Riemannian manifold M encodes symmetries between all the finite covers of M, and lifts to a subgroup of isometries of the universal cover of M. In case M is an (irreducible) finite volume locally symmetric space, the commensurator is thus a subgroup of a simple Lie group G. Margulis proved that if the commensurator is dense in G, then M is arithmetic. Shalom asked if the same is true for infinite volume M? I will report on recent progress on this question when M regularly covers a finite volume hyperbolic manifold. This is joint work with D. Fisher and M. Mj.

Multifractal analysis of Birkhoff averages for non-uniformly expanding Markov interval maps

Anosov representations, Hodge theory, and Lyapunov exponents

Discrete subgroups of semisimple Lie groups arise in a variety of contexts, sometimes "in nature" as monodromy groups of families of algebraic manifolds, and other times in relation to geometric structures and associated dynamical systems. I will discuss a class of such discrete subgroups that arise from certain variations of Hodge structure and lead to Anosov representations, thus relating algebraic and dynamical situations. Among many consequences of these relations, I will explain Torelli theorems for certain families of Calabi-Yau manifolds, uniformization results for domains of discontinuity of the associated discrete groups, and also a proof of a conjecture of Eskin, Kontsevich, Moller, and Zorich on Lyapunov exponents. The necessary context and background will be explained.

Two problems on homogenization in geometry

In this talk, I show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map. This is joint work with Vlad Marković.

Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation III