Colloquium

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.

On geometry of hyperbolic trees of spaces and Cannon-Thurston maps

We shall begin with an outline of a new proof of the Bestvina-Feighn combination theorem for trees of hyperbolic metric spaces. The proof naturally gives rise to a sort of description of uniform quasi geodesics in this space.  In the second half of the talk we will use some of these ideas to show that the Cannon-Thurston map exists from any subtree of space to the whole tree of space. Time permitting we shall discuss some natural applications of this result and related results. 

This is based on a joint work with Misha Kapovich.

Critical exponents for three-dimensional percolation models with long-range dependence

The talk will report on recent progress regarding the near-critical behavior of certain statistical mechanics models in dimension three. Our results deal with the phase transition associated with two percolation problems involving the Gaussian free field (GFF) in 3D. In one case, they determine a unique “fixed point” corresponding to the transition, which is proved to obey Fisher’s scaling law. This is one of several relations classically conjectured by physicists to hold on the grounds of a corresponding scaling ansatz. 

Dominating PSL(n,C)-representations of punctured-surface groups

For a closed and oriented surface S, Deroin-Tholozan had proved that for any representation of the fundamental group of S into PSL(2,C), there is a dominating Fuchsian representation. Here, domination is a notion that can be defined in terms of the marked length spectrum of the representation. They also conjectured a generalization in the context of Higgs bundles. I shall motivate and describe these, and talk of the following result for the case when S has punctures:  for a generic representation of the punctured-surface group into PSL(n,C), there is a dominating Hitchin representation in the same relative representation variety. The proof uses Fock-Goncharov coordinates for the moduli space of framed representations. 

Some remarks on properties of Loewner chains in terms of its driving function

Loewner's theory was developed by K. Loewner in an attempt to solve Bieberbach's conjecture. This theory resurfaced with the work of O. Schramm which led to the invention of Schramm-Loewner-Evolutions (SLEs). Loewner's theory gives a one-to-one correspondence between a certain family of compact sets in the upper half plane (a.k.a. Loewner chains) and real valued continuous functions. In this talk we will address how to study various properties of Loewner chains in terms of its driver. This talk will be based on various joint works with Y. Wang, F. Viklund, H. Tran, Y. Yuan, V. Margarint.