Colloquium

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.

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.