Colloquium

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.

Classification of tight contact structures on some Seifert fibered manifolds

I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. Tight contact structures have been classified on some 3 manifolds like S^3, R^3, Lens spaces, toric annuli, and almost all Seifert fibered manifolds with 3 exceptional fibers. We look at classification on one example of the Seifert fibered manifold with 4 exceptional fibers. I will explain the Legendrian surgery and convex surface theory which help us calculate the lower bound and upper bound of a number of tight contact structures. If time permits I will show how this method can be generalized for classification on a wider class of Seifert fibered manifolds.

An ergodic approach towards an equidistribution result of Ferrero–Washington

An ergodic approach towards an equidistribution result of Ferrero–Washington

Abstract: An important ingredient in the Ferrero--Washington proof of the vanishing of cyclotomic \mu-invariant for Kubota--Leopoldt p-adic L-functions is an equidistribution result which they established using the Weyl criterion. In joint work with Jungwon Lee, we provide an alternative proof by adopting a dynamical approach. We study an ergodic skew-product map on \Z_p * [0,1], which is then suitably identified as a factor of the 2-sided Bernoulli shift on the alphabet space {0,1,2,…,p-1}.

Generalized Gibbs ensembles of the Calogero fluid

Over recent years there have been widespread activities to understand the hydrodynamic scale of integrable many-body systems. Since such systems have an extensive number of local conservation laws, the standard notion of Gibbs measures has to be extended so to include this very special feature. The Calogero fluid are classical particles in one dimension which interact through the pair potential 1/(sinh) squared. I will explain how the transformation to scattering coordinates relates to the free energy of generalized Gibbs measures.

Alexander polynomial of the mapping torus of a graph map

Alexander polynomial was originally defined as a tool to differentiate knots. Later, its definition was extended to any finitely presented group. In this talk I will define the polynomial. Then I will explain, with the help of an example, how to obtain a determinant formula for the Alexander polynomial of the fundamental group of the mapping torus of a graph map. Joint with Spencer Dowdall and Sam Taylor. 

Total variation and chaos

If we have a chaotic self map f on a metric space X then we may ask how the total variations of the iterates of the chaotic map on the image of a curve in X change. This has been studied for chaotic interval maps and it is known that the total variation of the n-th iterate of the map on the image of any curve on the interval grows to infinity with n. We study this change for the real line and finite graphs which include the circle and finite trees. This is a joint work with Anubrato Bhattacharyya and Subhamoy Mondal.

Dynamical embeddings and approximations

The words 'dynamical embedding' are broadly used to describe a phenomenon where a certain aspect of a dynamical system is embedded in another. In this talk, we explore two instances of this phenomenon in the dynamics of polynomials: firstly, embeddings of families of postcritically finite (pcf) unicritical polynomials of degree n into the same family in degree n+1, and secondly, embeddings of Multibrot sets into the space of bicritical odd polynomials. We will also explore how such embeddings can be used to answer the following question: given an entire function with a dynamical property X, when can f be approximated by polynomials with the same property X?

Abundant existence of minimal hypersurfaces

This is the third talk in a series of three talks on the variational theory of minimal hypersurfaces. In this talk, I will discuss the following theorems. By the works of Marques-Neves and Song, every closed Riemannian manifold M^n, 3 \leq n \leq 7, contains infinitely many closed, minimal hypersurfaces. This was conjectured by Yau. For generic metrics, stronger results have been obtained. Irie, Marques and Neves proved that for a generic metric g on M, the union of all closed, minimal hypersurfaces is dense in (M, g). This theorem was later quantified by Marques, Neves and Song; they proved that for a generic metric there exists an equidistributed sequence of closed, minimal hypersurfaces in (M, g). In higher dimensions, Li proved that every closed Riemannian manifold, equipped with a generic metric, contains infinitely many closed minimal hypersurfaces. The Weyl law for the volume spectrum, proved by Liokumovich, Marques and Neves, played a major role in the proofs of these theorems. Inspired by the abundant existence of closed minimal hypersurfaces, we showed that the number of closed c-CMC hypersurfaces in a closed Riemannian manifold M^n, n \geq 3, tends to infinity as c \rightarrow 0^+.  

Morse theory for the area functional

This is the second talk in a series of three talks on the variational theory of minimal hypersurfaces. In this talk, I will discuss the following theorems. When the ambient dimension 3 \leq n \leq 7, Marques and Neves showed that the index of the min-max minimal hypersurface is bounded from above by the dimension of the parameter space. Zhou proved the multiplicity one property for the min-max minimal hypersurfaces, which was conjectured by Marques and Neves. In the Almgren-Pitts min-max theory, the min-max width is realized by the area of a closed minimal hypersurface, with the possibility that the connected components of the minimal hypersurface can have different multiplicities. The multiplicity one theorem says that for a generic metric, all the min-max minimal hypersurfaces have multiplicity one. Using the Morse index upper bound and multiplicity one theorem, Marques and Neves have proved the following theorem. For a generic metric g, there exists a sequence of closed, embedded, two-sided minimal hypersurfaces {S_p} in (M^n, g) such that the Morse index Ind(S_p) = p and area(S_p) \sim p^{1/n}. In higher dimensions (i.e. when n \geq 8), the Morse index upper bound has been proved by Li.