Colloquium

4:00-5:00 pm India time

Title: SU(2) Representations of Three-Manifold groups

Abstract: 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.


4:00-5:00 pm India time

Title: Generating the liftable mapping class groups of regular cyclic covers

Abstract: 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. 

  

6:30-7:30 pm India time

Title: Atypical stars on a directed landscape geodesic

Abstract: 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.

Video 


6:30-7:30 pm India time

Title: On the Cheeger constant of hyperbolic surfaces 

Abstract: 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.


6:30-7:30 pm India time

Title: Endperiodic maps via pseudo-Anosov flows

Abstract: 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.


4:00-5:00 pm India time

Title: Phase transition for percolation with axes-aligned defects

Abstract: 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.


4:00-5:00 pm India time

Title: Multiscale decompositions and random walks on convex bodies

Abstract: 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).


2:00-3:00 pm India time, AG - 66 (Please note the unusual time)

Title: Private Optimization and Statistical Physics: Low-Rank Matrix Approximation

Abstract: 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.


4:00-5:00 pm India time

Title: Classification of tight contact structures on some Seifert fibered manifolds.

Abstract: 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.


4:30-5:30 pm India time

Title: 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}.

Slides 


11:30-12:30 pm India time

Title: Generalized Gibbs ensembles of the Calogero fluid

Abstract: 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.


4:30-5:30 pm India time

Title: Alexander polynomial of the mapping torus of a graph map

Abstract: 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. 


4:30-5:30 pm India time

Title: Dynamical embeddings and approximations

Abstract: 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?


4:30-5:30 pm India time

Title: Abundant existence of minimal hypersurfaces.

Abstract: 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^+.  


4:30-5:30 pm India time

Title: Morse theory for the area functional

Abstract: 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.

Video 


4:30-5:30 pm India time

Title: Min-max construction of minimal hypersurfaces

Abstract: 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.

Video 


4:30-5:30 pm India time

Title: Gromov-Tischler theorem for symplectic stratified spaces

Abstract: 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.

Video 


4:30-5:30 pm India time

Title: Limit sets of paths in Outer space

Abstract: 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.

Video 


4:30-5:30 pm India time

Title: Weil-Petersson geometry of Teichmuller space

Abstract: 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.

Video 


4:30-5:30 pm India time

Title: The Dimer Model in 3 dimensions

Abstract: 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.

Slides 


4:00-5:00 pm India time

Title: A higher dimensional analog of Margulis' construction of expanders

Abstract: 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. 

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Date and Time: To be announced

Title: Localization schemes: A framework for the analysis of sampling algorithms

Abstract: Two recent and seemingly-unrelated techniques for proving mixing bounds for Markov chains are: (i) the framework of "Spectral Independence", introduced by Anari, Liu and Oveis Gharan, and its numerous extensions, which have given rise to several breakthroughs in the analysis of mixing times of discrete Markov chains and (ii) the Stochastic Localization technique which has proven useful in establishing mixing and expansion bounds for both log-concave measures and for measures on the discrete hypercube. In this talk, I'll present a framework which aims to both unify and extend those techniques, thus providing an approach that gives bounds for sampling algorithms in both discrete and continuous settings. In its center is the concept of a ``localization scheme'' which, to every probability measure on some space $\Omega$ (which will usually be either the discrete hypercube or R^n), assigns a martingale of probability measures which ``localize'' in space as time evolves. As it turns out, every such scheme can be associated with a Markov chain, and many chains of interest (such as Glauber dynamics) appear naturally in this framework. This viewpoint provides tools for deriving mixing bounds for the dynamics through the analysis of the corresponding localization process. Generalizations of the concept of Spectral Independence naturally arise from our definitions, and in particular we will show how to recover the main theorems in the spectral independence framework via simple martingale arguments (completely bypassing the need to use the theory of high-dimensional expanders). We demonstrate how to apply our machinery towards simple proofs to many mixing bounds in the recent literature. We will briefly discuss some applications, among which are obtaining the first $O(n \log n)$ bound for mixing time of the hardcore-model (of arbitrary degree) in the tree-uniqueness regime, under Glauber dynamics and to proving a KL-divergence decay bound for log-concave sampling via the Restricted Gaussian Oracle, which achieves optimal mixing under any $\exp(n)$-warm start. Based on a joint work with Yuansi Chen.


3:00-4:00 pm India time

Title: Universality in Random Growth Processes

Abstract: 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.

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Slides 


4:00-5:00 pm India time

Title: Stochastic geometry beyond independence and its applications

Abstract: 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.

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4:00-5:00 pm India time

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

Abstract: 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.

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4:00-5:00 pm India time

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

Abstract: 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. 

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4:00-5:00 pm India time

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

Abstract: 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. 

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4:00-5:00 pm India time

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

Abstract: 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.

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8:00-9:00 pm India time

Title: Orientable maps and polynomial invariants of free-by-cyclic groups

Abstract: 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.

Slides 


4:00-5:00 pm India time

Title: Spectral properties of random perturbations of non-self-adjoint operators

Abstract: 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. 

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4:00-5:00 pm India time

Title: Inhomogeneous quadratic forms

Abstract: 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.

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4:00-5:00 pm India time

Title: Deformation space analogies between Kleinian reflection groups and rational maps

Abstract: 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.

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8:00-9:00 pm India time

Title: Totally geodesic submanifolds of real and complex hyperbolic manifolds

Abstract: After some history and motivation, I will discuss recent works with Bader, Miller and Stover in which we prove finiteness of maximal totally geodesic submanifolds in real and complex hyperbolic spaces. 

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8:00-9:00 pm India time

Title:  Commensurators and arithmeticity of hyperbolic manifolds

Abstract: The commensurator of a Riemannian manifold M encodes symmetries between all the finite covers of M, and lifts to a subgroup of isometries of the universal cover of M. In case M is an (irreducible) finite volume locally symmetric space, the commensurator is thus a subgroup of a simple Lie group G. Margulis proved that if the commensurator is dense in G, then M is arithmetic. Shalom asked if the same is true for infinite volume M? I will report on recent progress on this question when M regularly covers a finite volume hyperbolic manifold. This is joint work with D. Fisher and M. Mj.

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7:00-8:00 pm India time

Title: Anosov representations, Hodge theory, and Lyapunov exponents

Abstract: Discrete subgroups of semisimple Lie groups arise in a variety of contexts, sometimes "in nature" as monodromy groups of families of algebraic manifolds, and other times in relation to geometric structures and associated dynamical systems. I will discuss a class of such discrete subgroups that arise from certain variations of Hodge structure and lead to Anosov representations, thus relating algebraic and dynamical situations. Among many consequences of these relations, I will explain Torelli theorems for certain families of Calabi-Yau manifolds, uniformization results for domains of discontinuity of the associated discrete groups, and also a proof of a conjecture of Eskin, Kontsevich, Moller, and Zorich on Lyapunov exponents. The necessary context and background will be explained.

Slides 


4:00-5:00 pm India time

Title: Two problems on homogenization in geometry

Abstract: In this talk, I show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map. This is joint work with Vlad Marković.

Slides 


4:00-5:00 pm India time

Title : Lower tail bounds beyond Chernoff for very small deviations from the mean.

Abstract: Concentration bounds for sums of independent random variables are ubiquitous in probabilistic setups. The most famous ones of these, the Chernoff-Hoeffding bounds provide guarantees of the following type: given a binomial random variable with expectation k, what is the probability that the variable deviates by a factor of more than (1+y) from k? The (nearly tight) answer is approx exp(-ky^2). However, what if we are interested in small additive deviations from the expectation k? In particular, what is the probability that the variable takes a value strictly less than k? Now, for large k, the Chernoff-Hoeffding bounds will give only the vacuous bound of 1. However, Jogdeo and Samuels showed in 1968 that the right answer is actually 1/2. In this talk, I will describe this result and a helpful result of Hoeffding which shows that the worst case is the Poisson case. I will finally talk about our result from this year (with Nikhil Bansal, Majid Farhadi and Prasad Tetali) which extends the bound of Jogdeo and Samuels for the case of small deviations from k/t for t close to and at least 1.


4:00-5:00 pm India time

Title: Large deviations for random hives and the spectrum of the sum of two random matrices.

Abstract: 

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6:30-7:30 pm India time

Title: Asymptotic Length Saturation for Zariski Dense Surfaces.

Abstract: The lengths of closed geodesics on a hyperbolic manifold are determined by the traces of its fundamental group. When the latter is a Zariski dense subgroup of an arithmetic group, the trace set is contained in the ring of integers of a number field, and may have some local obstructions. We say that the surface's length set "saturates" (resp. "asymptotically saturates") if every (resp. almost every) sufficiently large admissible trace appears. In joint work with Xin Zhang, we prove the first instance of asymptotic length saturation for punctured covers of the modular surface, in the full range of critical exponent exceeding one-half (below which saturation is impossible).

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7:30-8:30 pm India time

Title: Translation surfaces and their connections

Abstract: A translation surface is a Riemann surface equipped with a metric that is locally Euclidean away from finitely many points, which are cone points with angle an integer multiple of 2 pi. This talk will present connections that translations surfaces have with billiards in polygons, smooth flows on surfaces and geometric topology.

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4:00-5:00 pm India time

Title: Spectrum of the laplacian on self-similar groups and holomorphic dynamics

Abstract: In this talk based on a joint work with Rostislav Grigorchuk and Mikhail Lyubich, I will explain how the calculation of the spectrum of the laplacian on the Grigorchuk group and the Hanoi group is related to the iteration of particular rational maps in two complex variables and their ergodic properties.

Video 

Slides 


 4:00-5:00 pm India time

Title: Measurement phase transitions and the statistical mechanics of tree tensor networks

Abstract: A many-body quantum system that is continually monitored by an external observer can be in two distinct dynamical phases, distinguished by whether or not repeated local measurements (throughout the bulk of the system) prevent the build-up of long-range quantum entanglement. I will describe the key features of such “measurement phase transitions” and briefly sketch theoretical approaches to their critical properties that make connections with topics in classical statistical mechanics, such as percolation and disordered magnetism. Finally I will discuss random tensor networks with a tree geometry. These arise in a simple limit of the measurement problem, and they show an entanglement transition that can be solved exactly by a mapping to a problem of traveling waves.


4:00-5:00 pm India time

Title: Metric Ergodicity and its applications in Rigidity Theory.

Abstract: An action is called Metrically Ergodic (ME) if the acted space admits no equivariant maps to metric spaces. This is a far reaching strengthening of usual Ergodicity. For probability measure preserving actions, ME is equivalent to Weak Mixing, but in general it is different and could be thought of as a generalization of Mautner phenomenon. It is a remarkable fact that every locally compact second countable group admits an action which is both Amenable and Metrically Ergodic. For example, the celebrated fixed point theorem of Ryll-Nardzewski could be deduced easily from this fact. In another direction, it implies Super-Rigidity for Lattices in Products. It is also a main player in the proof of the Simplicity of the Lyapunov Spectra in various situations. In my talk I will provide a gentle introduction to ME and discuss its applications.

The talk is based on a joint work with Alex Furman.

Slides 


4:00-5:00 pm India time

Title: Discrete stationary random subgroups and application to discrete subgroups of Lie groups

Abstract: The notion of invariant random subgroups (IRS) has proven extremely useful during the last decade, particularly to the study of asymptotic invariants of lattices. However, the scope of problems that one can investigate when restricting to invariant measures (on the space of subgroups)is limited. It was recently realised that the notion of stationary random subgroups (SRS), which is much more general, is still extremely powerful and opens up new paths to attacking problems that previously seemed to be out of our reach. 

  In this talk, using the notion of SRS, I will explain a proof of the following conjecture of Margulis: Let G be a higher rank simple Lie group and Λ ⊂ G a discrete subgroup. Then the orbifold Λ\G/K has finite volume if and only if it has bounded injectivity radius. This is a far-reaching generalisation of the celebrated Normal Subgroup Theorem of Margulis, and while it is new even for subgroups of lattices, it is completely general.    

This is a joint work with Mikolaj Fraczyk.

Video 


4:00-5:00 pm India time

Title: Invariant random subgroups and applications to lattices

Abstract: In mathematics in general, it is fruitful to allow randomness. Indeed, it is often easier to deal with random rather than deterministic objects. It seems miraculous, however, when we are able to say more about deterministic objects by considering them as random ones.

The notion of invariant random subgroups (IRS) has proven extremely useful  to the study of asymptotic invariants of lattices. In this talk I will demonstrate several results (old and new) about lattices in semisimple Lie groups which are proved by considering them as IRS.

Video 


4:00-5:00 pm India time

Title: Uniformization of quasiconformal trees

Abstract: Quasisymmetric maps are generalizations of conformal maps and may be viewed as a global versions of quasiconformal maps. Originally, they were introduced in the context of geometric function theory, but appear now in geometric group theory and analysis on metric spaces among others. The quasisymmetric uniformization problem asks when a given metric space is quasisymmetric to some model space. Here we consider this uniformization problem for certain trees. In particular, we consider the "continuum self-similar tree'' (CSST) and give necessary and sufficient conditions for another tree to be quasisymmetrically equivalent to the CSST. One motivation to study the CSST is that it is almost surely homeomorphic to the continuum random tree introduced by Aldous. This is joint work with Mario Bonk.


4:00-5:00 pm India time

Title: Volume vs. complexity of hyperbolic groups

Abstract: In this talk we will discuss the relation between the volume of a quotient X/G of a (Gromov) hyperbolic graph X by a group G acting freely and cocompactly on X, and the "complexity" of the group G. We will then show how to use this relation to study finite-index subgroups of cubulated hyperbolic groups.

Slides 


4:00-5:00 pm India time

Title: Brolin’s theorem for finitely generated polynomial semigroups

Abstract: In this talk, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh--Sibony measure) in terms of potential theory. The existence of this measure follows from a very general result of Dinh--Sibony applied to a holomorphic correspondence that one can naturally associate with a semigroup of the above type. We are interested in a precise description of this invariant measure. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Our result generalizes the classical result by Brolin. Along the way, we establish the continuity of the logarithmic potential for the Dinh--Sibony measure, which might be of independent interest. If time permits, we shall also present some bounds on the capacity and diameter of the Julia sets of such semigroups, which uses the F-functional of Mhaskar and Saff.

Slides 


6:30-7:30 pm India time

Title: Perturbing the action of a hyperbolic group on its boundary

Abstract:  A hyperbolic group G comes with an action by homeomorphisms on its Gromov boundary.  In general this boundary is some compact metrizable space which can have complicated local topology, but sometimes it is a manifold (for example if G is the fundamental group of a closed negatively curved manifold).  We show that the action of a torsion free hyperbolic group on its boundary is topologically stable, assuming that boundary is a manifold.  

This is joint work with Kathryn Mann.

Slides 


6:00-7:00 pm India time

Title: Random trees from conformal welding

Abstract: Conformal welding is a way of gluing Riemann surfaces along their boundary via a specified equivalence relation. Even in the case that the resulting boundary interface is a simple curve, the existence and uniqueness of the resulting conformal structure is in general difficult to determine; this is the conformal welding problem.

We give criteria for the solution of the welding problem in the case that the boundary interface is a dendrite. In particular we prove that a natural conformal welding problem associated with the continuum random tree (CRT) has a solution, giving rise to a `canonical’ embedding of the CRT in the plane.

Joint work with Steffen Rohde.

Slides


4:00-5:00 pm India time

Title: Hot spots problem for convex planar domains

Abstract: In this talk I will first give a brief introduction to the hot spots problem for planar domains. I will then recall the old results on this problem, and, in the last part of the talk I will discuss some recent developments.


4:00-5:00 pm India time

Title: `Quantization' and Topological Aspects of the Space of Renormalization Group flows in 2-dim Quantum Field Theory

Abstract: In this talk we will discuss a `quantization' of the renormalization group equations by adding a gaussian noise term and converting them into stochastic differential equations. We will discuss the case of two dim. unitary QFTs where a Zamolodchikov c-function exists and the `drift term' is a gradient of the c-function. Quantization leads to supersymmetric quantum mechanics which can be studied in the `semi-classical' approximation. In particular one can attempt to characterise the topology of the space of paths in the path integral  using Morse theory using the Zamolodchikov c-function as a Morse function. Assuming the validity of Morse inequalities in the infinite dimensional case we calculate, as an illustration, the Betti numbers of the space of flows of the c < 1 unitary minimal models of 2-dm conformal field theory. This talk is based on work with S. R. Das and G. Mandal: "Stochastic differential equations on 2-dim. theory space and Morse theory", Mod. Phys. Letts A, Vol 4 No.8 (1989).

Slides 


4:00-5:00 pm India time

Title:  Convex Hulls of Two Dimensional Stochastic Processes

Abstract: Convex hull of a set of points in two dimensions roughly describes the shape of the set. In this talk, I will discuss the statistical properties of the convex hull of several stochastic processes in two dimensions. By adapting Cauchy's formula to random curves, we develop a formalism to compute explicitly the mean perimeter and the mean area of the convex hull of arbitrary two dimensional stochastic processes of a fixed duration. Our result makes an interesting and general connection between random geometry and extreme value statistics. I will discuss two examples in detail (i) a set of n independent planar Brownian paths (ii) planar branching Brownian motion with death. The first problem has application in estimating the home range of an animal population of size n, while the second is useful to estimate the spatial extent of the outbreak of animal epidemics. Finally I will also discuss two other recent examples of planar stochastic processes: (a) active run-and-tumble process and (b) resetting Brownian motion.


10:30-11:30 am India time

Title: Yang-Mills on the lattice: New results and open problems

Abstract: Quantum Yang-Mills theories have mathematically well-defined formulations on lattices, known as lattice gauge theories. I will give a brief introduction to lattice gauge theories and a survey of existing results, followed by an overview of a number of longstanding open problems and recent progress on some of these questions.


4:00-5:00 pm India time

Title: CAT(0) cube complexes for the working geometer


2:30-3:30 pm India time

Title: Four-point functions in the Fortuin-Kasteley cluster model

Abstract: The determination of four-point correlation functions of two-dimensional lattice models is of fundamental importance in statistical physics. In the limit of an infinite lattice, this question can be formulated in terms of conformal field theory (CFT). For the so-called minimal models the problem was solved more than 30 years ago, by using that the existence of singular states implies that the correlation functions must satisfy certain differential equations. This settles the issue for models defined in terms of local degrees of freedom, such as the Ising and 3-state Potts models. However, for geometrical observables in the Fortuin-Kasteleyn cluster formulation of the Q-state Potts model, for generic values of Q, there is in general no locality and no singular states, and so the question remains open. As a warm-up to solving this problem, we discuss which states propagate in the s-channel of such correlation functions, when the four points are brought together two by two. To this end we combine CFT methods with algebraic and numerical approaches to the lattice model. We then outline work in progress that aims at solving the problem entirely, through an interchiral conformal bootstrap setup that makes contact with time-like Liouville field theory and a number of profound algebraic results.

 

4:00-5:00 pm India time

Title: Logarithmic correlations in percolation and other geometrical critical phenomena

Abstract: The purpose of renormalisation group and quantum field theory approaches to critical phenomena is to diagonalise the dilatation operator. Its eigenvalues are the critical exponents that determine the power law decay of correlation functions. However, in many realistic situations the dilatation operator is, in fact, not diagonalisable. Examples include geometrical critical phenomena, such as percolation, in which the correlation functions describe fluctuating random interfaces. These situations are described instead by logarithmic (conformal) field theories, in which the power-law behavior of correlation functions is modified by logarithms. Such theories can be obtained as limits of ordinary quantum field theories, and the logarithms originate from a resonance phenomenon between two or more operators whose critical exponents collide in the limit. We illustrate this phenomenon on the geometrical Q-state Potts model (Fortuin-Kasteleyn random cluster model), where logarithmic correlation functions arise in any dimension. The amplitudes of the logarithmic terms are universal and can be computed exactly in two dimensions, in fine agreement with numerical checks. In passing we provide a combinatorial classification of bulk operators in the Potts model in any dimension.


4:00-5:00 pm India time

Title: Bose Fermi duality in matter Chern Simons theories

Abstract: In four and higher spacetime dimensions Bosons and Fermions are irreducibly different. This difference blurs out in three spacetime dimensions. This blurring allows a nontrivial phenomenon. There is now substantial evidence that Bosons coupled to Chern Simons theories are dual to Fermions coupled to (level rank dual) Chern Simons theories. I will review what is understood - and what is not - about these dualities.


4:00-5:00 pm India time

Title: Extreme Value Statistics: An overview and perspectives

Abstract: Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of `uncorrelated' variables is well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In thistalk, I will give an overview and perspectives on this interdisciplinary and rapidly evolving area of research.


6:00-7:00 pm India time

Title: Uniform exponential growth for CAT(0) cube complexes

Abstract: Kar and Sageev showed that if a group acts freely on a CAT(0) square complex, then it either has uniform exponential growth or it is virtually abelian. The behavior, in this sense, of a group that acts by isometries on a higher dimensional CAT(0) cube complex is not known. In this talk, I will present some generalizations of their theorem. On the one hand we allow the action to be proper instead of free and on the other hand we assume our space has isolated flats. I will define exponential growth and also present the general strategy to obtain a result like that of Kar--Sageev. This is joint work with Kasia Jankiewicz and Thomas Ng.


6:00-7:00 pm India time

Title: On geometrically finite degenerations

Abstract: The Sullivan dictionary provides a conceptual framework for understanding the connections between dynamics of rational maps and Kleinian groups. In this talk, I will discuss some recent development of geometrically finite degenerations of rational maps motivated by the dictionary. 

In particular, I will explain how to use this theory to understand the bumping structure of hyperbolic components and how to construct self-bumps, a phenomenon first discovered by McMullen on the Bers boundary and later generalized and studied by Bromberg, Anderson, Canary and McCullough.

I will also talk about how the analogues of a double limit theorem and Thurston’s compactness theorem can be deduced for this setting.

Slides 


4:00-5:00 pm India time

Title: Jackiw Teitelboim Gravity and Random Matrix Theory

Abstract: Two dimensional gravity is a fascinating subject of interest in the study of quantum gravity, statistical mechanics and random matrix theory. In this talk we will introduce a model of two dimensional gravity called Jackiw-Teitelboim gravity which has received considerable attention recently, motivated partly by the study of spin systems in condensed matter physics. We will discuss the classical solutions of the theory and quantise it using the path integral formalism. It will turn out that the theory is equivalent to a quantum mechanical system with a random Hamiltonian. The connection will involve recursion relations which were obtained by Mirzakhani in her study  of the moduli space of bordered Riemann surfaces. 

      The talk is intended  to be non-technical and accessible to a broad audience. It will be based on the following three papers:

1) D. Stanford and E. Witten, ``Fermionic Localisation of the Schwarzian Theory", arXiv: 1703.04612 

2) P. Saad, S. Shenker and D. Stanford, ``JT gravity as a Matrix Integral", arXiv: 1903.1115 

3) U. Moitra, S. Sake and S. P. Trivedi, ``JT gravity in the second order formalism", arXiv: 210.00596    

Video 


Title: A universal Cannon-Thurston map and the surviving complex.

Abstract: The fundamental group of a surface (closed or with punctures) acts on the curve complex of the surface with one additional puncture via the Birman Exact Sequence.  I will describe a construction of a continuous, equivariant map from a subset of the circle at infinity of the universal cover of the surface onto the Gromov boundary of the curve complex (along the way, explaining what these objects and actions are).  This map is universal with respect to all Cannon-Thurston maps coming from type-preserving Kleinian representations without accidental parabolics.  In the case of closed surfaces, this map was constructed in joint work with Mj and Schleimer, and in this talk I will talk about an extension to the case of punctured surfaces obtained in joint work with Gultepe and Pho-On.  The proof for punctured surfaces involves constructing a continuous equivariant map to the Gromov boundary of a "larger" complex called the surviving complex.  I will describe this complex, its Gromov boundary, and the construction of the map.

Slides 


6:00-7:00 pm India time

Title: Gromov compactness revisited.

Abstract: Gromov's compactness theorem for pseudoholomorphic curves is a fundamental result in almost-complex geometry which finds many applications in symplectic topology. The usual proofs of this theorem show sequential compactness of the relevant moduli space. I will sketch the proof of a quantitative version of this theorem, blackboxing some of the analytical estimates and focusing on the more combinatorial aspects of the proof. At the end, I will also outline an application (in progress) of this quantitative compactness theorem.

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6:00-7:00 pm India time

Title: Dynamics on the moduli space of point configurations on the Riemann sphere

Abstract: A rational function f(z) is called post-critically finite (PCF) if every critical point is either pre-periodic or periodic. PCF rational functions have been studied for their special dynamics, and their special distribution within the moduli space of all rational maps. By works of W. Thurston and S. Koch, (“almost") every PCF map arises as an isolated fixed point of an algebraic dynamical system on the moduli space M_{0,n} of point-configurations on P^1; these dynamical systems are called Hurwitz correspondences. I will give an overview of the study of PCF rational maps, introduce Hurwitz correspondences and present results on their dynamics.


3:00-5:00 pm India time

Title: Rotational invariance in planar percolation

Abstract: In this series of two talks, we will discuss the rotational invariance of critical Bernoulli percolation on the square lattice. We will start by reviewing the state of the art on criticality for this very classical model, and will then discuss the recent progress in the understanding of the large scale properties of the model at criticality.

Video I 

Video II 


3:00-4:00 pm India time

Title: Emergent symmetries in statistical physics systems

Abstract: A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millenium, the mathematical understanding of this fact has progressed tremendously in two dimensions with the introduction of the Schramm-Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models. In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. We will also explain some recent and partial progress in the direction of proving conformal invariance for a large class of such models.

Video 


4:00-5:30 pm India time

Title: The word and Riemannian metrics on lattices of semisimple groups II.

Video 


4:00-5:30 pm India time

Title: The word and Riemannian metrics on lattices of semisimple groups I.


Title: Patterson-Sullivan measures for Anosov subgroups II.


 6:00 pm India time

Title: Convex co-compact representations of non-Gromov hyperbolic groups

Abstract: Convex co-compact representations are a generalization of convex co-compact Kleinian groups. These are discrete faithful representations into the projective linear group whose image acts convex co-compactly on a Hilbert geometry (i.e. a properly convex domain in real projective space). In this talk, I will discuss such representations of relatively hyperbolic groups and closed 3-manifold groups. We will study them by developing analogies between Hilbert geometry and CAT(0) geometry. Using this approach, I will prove a geometric characterization of relative hyperbolicity and also classify convex co-compact representations of closed 3-manifold groups.

      Video


4:00 pm India time

Title: Patterson-Sullivan measures for Anosov subgroups

Abstract: Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.

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4:00 pm India time

Title: Manifold learning and an inverse problem for a wave equation

Abstract: We consider invariant manifold learning and its applications in wave imaging. The invariant manifold learning problem, also known as the geometric Whitney problem, means the construction of a manifold $M$ and its Riemannian metric $g$ using a discrete metric space $(X,d_X)$ that approximates the manifold in the Gromov-Hausdorff sense. This problem is closely related to  manifold interpolation where a smooth $n$-dimensional surface $S\subset \mathbb R^m$, $m>n$ needs to be constructed to approximate a point cloud in $\mathbb R^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. As an example, we consider an inverse problem for a wave equation $(\partial_t^2-\Delta_g)u(x,t)=F(x,t)$ on a Riemannian manifold $(M,g)$. We assume that we are given an open subset $V$ of $M$ and the source-to-solution map that maps a source supported in $V\times \mathbb R_+$ to the restriction of the solution $u$ in the set $V\times \mathbb R_+$. This map corresponds to the measurements made on the set $V$. The results on the first problem are done in collaboration with C. Fefferman, S. Ivanov, Y. Kurylev, and H. Narayanan, and the results on the second problem with R. Bosi and Y. Kurylev.

Video 


4:00 pm India time

Title: Positive metric entropy in nearly integrable Hamiltonian systems

Abstract: The celebrated Kolmogorov-Arnold-Moser (KAM) theorem asserts that a small perturbation of an integrable Hamiltonian system preserves the quasi-periodicity of trajectories on a set of large measure. The question remains: how chaotic the system's behavior can be on the remaining "small" set? I will speak on a recent result of Dima Burago, Dong Chen and myself saying that every integrable system can be perturbed so that the resulting Hamiltonian system has positive measure-theoretic entropy.

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4:00 pm India time

Title: Counting closed geodesics on hyperbolic surfaces II

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4:00 pm India time

Title: Counting closed geodesics on hyperbolic surfaces I

Abstract : In these talks, we will discuss the exponential growth rate of the number of closed geodesics of length  at most $T$ on a compact hyperbolic surface, when $T$ goes to infinity. In a first large audience talk, I will introduce the context, some history of this kind of statement, and, if time allows, a few important ingredients of the proof. In a second talk, I will explain how to extend it on infinite volume surfaces, as proven in a recent work with S. Tapie. If time allows, I will briefly mention recent works allowing a description of the error term in this kind of asymptotics.

 Video 

 Slides 


6:00 pm India time

Title: Variance estimates for geometric counting problems III

Video 

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6:00 pm India time

Title: Variance estimates for geometric counting problems II

Video 

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4:00 pm India time

Title: Variance estimates for geometric counting problems I

Abstract: We'll discuss three examples (lattices, translation surfaces, hyperbolic surfaces) of computing the variance of natural counting problems for random geometric structures. All of them can be viewed as generalizations of the space of flat structures on the two-dimensional torus. We'll carefully discuss this base example, and the three different generalizations. These talks should be accessible to those with some familiarity with real and complex analysis at a first-year graduate level. In the first talk, we'll discuss lattices, and some joint work with G. Margulis; in the second, translation surfaces, and joint work with Y. Cheung and H. Masur, and in the third, hyperbolic surfaces, and joint work with F. Arana-Herrera. 

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Slides