Colloquium

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.

Phase transition for percolation with axes-aligned defects

In this talk we will review a model that was first introduced by Jonasson, Mossel and Peres. Starting with the usual square lattice on Z^2, entire rows (respectively columns) of edges extending along the horizontal (respectively vertical) direction are removed independently at random. On the remaining thinned lattice, Bernoulli bond percolation is performed, giving rise to a percolation model with infinite range dependencies under the annealed law. In 2005, Hoffman solved the main conjecture around this model: proving that this percolation process indeed undergoes a nontrivial phase transition. In this talk, besides reviewing this surprisingly challenging problem, we will present a novel proof, which replaces the dynamic renormalization presented previously by a static version. This makes the proof easier to follow and to extend to other models. We finally present some remarks on the sharpness of Hoffman’s result as well as a list of interesting open problems that we believe can provide a renewed interest in this family of questions.

This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis.

Multiscale decompositions and random walks on convex bodies

Running a random walk in a convex body K ⊆ Rⁿ is a standard approach to sample approximately uniformly from the body. The requirement is that from a suitable initial distribution, the distribution of the walk comes close to the uniform distribution π on K after a number of steps polynomial in the dimension n and the aspect ratio R/r (i.e., when the body is contained in a ball of radius R and contains a ball of radius r). 

Proofs of rapid mixing of such walks often require that the initial distribution from which the random walk starts should be somewhat diffuse: formally, the probability density η₀ of the initial distribution with respect to π should be at most polynomial in the dimension n: this is called a "warm start". Achieving a warm start often requires non-trivial pre-processing before starting the random walk. 

This motivates proving rapid mixing from a "cold start", where the initial density η₀ with respect to π can be exponential in the dimension n. Unlike warm starts, a cold start is usually trivial to achieve. However, a random walk need not mix rapidly from a cold start: an example being the well-known "ball walk". On the other hand, Lovász and Vempala proved that the "hit-and-run" random walk mixes rapidly from a cold start. For the related *coordinate* hit-and-run (CHR) walk, which has been found to be promising in computational experiments, rapid mixing from a warm start was proved only recently but the question of rapid mixing from a cold start remained open. 

We construct a family of random walks inspired by the classical Whitney decomposition of subsets of Rⁿ into countably many axis-aligned dyadic cubes. We show that even with a cold start, the mixing times of these walks are bounded by a polynomial in n and the aspect ratio. Our main technical ingredient is an isoperimetric inequality for K for a metric that magnifies distances between points close to the boundary of K. As a corollary, we show that the coordinate hi-and-run walk also mixes rapidly both from a cold start and even from any initial point not too close to the boundary of K.

Joint work with Hariharan Narayanan (TIFR) and Amit Rajaraman (IIT Bombay).

Private Optimization and Statistical Physics: Low-Rank Matrix Approximation

In this talk, I will discuss the following connections between private optimization and statistical physics in the context of the low-rank matrix approximation problem: 

1) An efficient algorithm to privately compute a low-rank approximation and how it leads to an efficient way to sample from Harish-Chandra-Itzykson-Zuber densities studied in physics and mathematics, and 

2) An improved analysis of the "utility" of the  "Gaussian Mechanism" for private low-rank approximation using Dyson Brownian motion.