Colloquium

Min-max construction of minimal hypersurfaces

In the 1960s, Almgren developed a min-max theory to construct closed minimal submanifolds in an arbitrary closed Riemannian manifold. The regularity theory in the co-dimension 1 case was further developed by Pitts and Schoen-Simon. In particular, by the combined works of Almgren, Pitts and Schoen-Simon, in every closed Riemannian manifold M^n, n \geq 3, there exists at least one closed, minimal hypersurface. Recently, the Almgren-Pitts min-max theory has been further developed to show that minimal hypersurfaces exist in abundance.

In addition to the Almgren-Pitts min-max theory, there is an alternative PDE based approach for the min-max construction of minimal hypersurfaces. This approach was introduced by Guaraco and further developed by Gaspar and Guaraco. It is based on the study of the limiting behaviour of solutions to the Allen-Cahn equation. In my talk, I will briefly describe the Almgren-Pitts min-max theory and the Allen-Cahn min-max theory and discuss the question to what extent these two theories agree.

Gromov-Tischler theorem for symplectic stratified spaces

Singular symplectic spaces appear naturally as examples of reduced Hamiltonian phase spaces in physics as well as singular projective algebraic varieties in mathematics. We give a unified and geometric definition for these objects, and prove a singular variant of the Gromov-Tischler theorem: such a space with an integral symplectic form can always be embedded symplectically inside the complex projective space. On the way we discuss the topology of stratified spaces, symplectic reduction and h-principles. 

This is joint work with Mahan Mj.

The approximation by conjugation method in smooth ergodic theory

Limit sets of paths in Outer space

In analogy to the mapping class group acting on the Teichmuller space, we have the group of outer automorphisms of the free group acting on Culler-Vogtmann's `Outer space'. The limit sets of geodesics in Teichmuller space exhibit very interesting and varied phenomena with respect to the Teichmuller metric, Thurston metric and Weil Petersson metric. In this talk, we will look for similar results for `folding/unfolding' paths in Outer space.

Weil-Petersson geometry of Teichmuller space

The Weil-Petersson metric is a negatively curved, incomplete Riemannian metric on the Teichmuller space with connections to hyperbolic geometry. In this talk we present some results about the behavior of geodesics of the metric and its relation to subsurface coefficients in analogy with continued fraction expansions.

Dynamics on spaces of discrete sets

The Dimer Model in 3 dimensions

The dimer model, also referred to as domino tilings or perfect matching, are tilings of the Z^d lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very well-studied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture is a little daunting because most of these tools are limited to two dimensions. In this talk I will describe what techniques can be extended to higher dimensions and give a brief account of a large deviations principle for dimer tilings in three dimensions that we prove analogous to the results by Cohn, Kenyon and Propp (2000). 

This is joint work with Scott Sheffield and Catherine Wolfram.

A higher dimensional analog of Margulis' construction of expanders

The first explicit example of a family of expander graphs was quotients of the Cayley graph of a group G, having Property (T), by subgroups of finite index. This construction is due to Margulis, in a special case, and Alon-Milman in general.  We will discuss a higher dimensional analog of this result that can be obtained by replacing 'expander graphs' by 'higher spectral expanders', 'group having Property (T)'  by 'strongly n-Kazhdan group' and and 'Cayley graph' by 'n-skeleton of the universal cover of a K(G,1) simplicial complex'.  New examples of 2-dimensional spectral expanders are obtained using this construction. 

Universality in Random Growth Processes

Universality in disordered systems has always played a central role in the direction of research in Probability and Mathematical Physics, a classical example being the Gaussian universality class (the central limit theorem). In this talk, I will describe a different universality class for random growth models, called the KPZ universality class. Since Kardar, Parisi and Zhang introduced the KPZ equation in their seminal paper in 1986, the equation has made appearances everywhere from bacterial growth, fire front, coffee stain to the top edge of a randomized game of Tetris; and this field has become a subject of intense research interest in Mathematics and Physics for the last 15 to 20 years. The random growth processes that are expected to have the same scaling and asymptotic fluctuations as the KPZ equation and converge to the universal limiting object called the KPZ fixed point, are said to lie in the KPZ universality class, though this KPZ universality conjecture has been rigorously proved for only a handful of models till now. Here, I will talk about some recent results on universal geometric properties of the KPZ fixed point and the underlying landscape and show that the KPZ equation and exclusion processes converge to the KPZ fixed point under the 1:2:3 scaling, establishing the KPZ universality conjecture for these models, which were long-standing open problems in this field.

The talk is based on joint works with Jeremy Quastel, Balint Virag and Duncan Dauvergne.

Stochastic geometry beyond independence and its applications

The classical paradigm of randomness  is the model of independent and identically distributed (i.i.d.) random variables, and venturing beyond i.i.d. is often considered  a challenge to be overcome. In this talk, we will explore a different perspective, wherein stochastic systems with constraints in fact aid in understanding fundamental problems. Our constrained systems are well-motivated from statistical physics, including models like the random critical points and determinantal probability measures. These will be used to shed important light on natural questions of relevance in understanding data, including problems of likelihood maximization and dimensionality reduction. En route, we will explore connections to spiked random matrix models and novel asymptotics for the fluctuations of spectrally constrained random  systems. Based on the joint works below.

[1] Gaussian determinantal processes: A new model for directionality in data, with P. Rigollet, Proceedings of the National Academy of Sciences, vol. 117, no. 24 (2020), pp. 13207--13213.

[2] Fluctuation and Entropy in Spectrally Constrained random fields, with K. Adhikari, J.L. Lebowitz, Communications in Math. Physics, 386, 749–780 (2021).

[3] Maximum Likelihood under constraints: Degeneracies and Random Critical Points, with S. Chaudhuri, U. Gangopadhyay, submitted.