Schedule

• September 17

  Gaëtan Borot (Max Planck Institute for Mathematics)

  Towards modular invariant measures via a geometric recursion

  Abstract: n the end, I will sketch a mechanism to construct modular invariant Radon measures on the Teichmuller space of surfaces of arbitrary topologies, by induction on the Euler characteristic from initial data attached to pairs of pants and tori with one boundary. Before, I will focus on a simpler setup which yields, by a similar recursion, continuous functions on the moduli space of bordered Riemann surfaces (this is the quotient of the Teichmuller space by the modular group) and yet retains some of the key ideas of the recursive construction. For instance, the constant function 1 can be produced in this way due to an identity of Mirazkhani. We prove a generalization of Mirzakhani's identity to get a recursion on statistics of hyperbolic lengths of simple closed curves. In general, integrating over the moduli space the functions produced by this ''geometric recursion'' give functions of the boundary lengths that satisfy the topological recursion of Eynard and Orantin, and this result has a converse. I will allude to potential uses of these ideas in 2d quantum gravity.

  The talk is based on joint ongoing works with Andersen and Orantin.

• September 24

  Benjamin Landon (MIT)

  Local ergodicity of Dyson Brownian motion

  Abstract: Dyson Brownian motion is a stochastic random matrix dynamics used in proofs of eigenvalue universality. We will briefly review the notion of universality in random matrix theory as well as the role played by Dyson Brownian motion in its study. We then focus on recent approaches to proving local ergodicity of Dyson Brownian motion, including coupling and homogenization techniques. This talk is based on joint work with P. Sosoe and H.-T. Yau.

• October 1

  Jean-Bernard Zuber (Sorbonne University)

  Revisiting Horn's problem

  Abstract: Horn's problem deals with the following question: what can be said about the spectrum of eigenvalues of the sum C=A+B of two Hermitian matrices of given spectrum? The support of the spectrum of C is well understood, after a long series of works from Weyl (1912) to Klyachko (1998) and Knutson and Tao (1999).

  In this talk, after a short review of the problem, I show how to compute the probability distribution function of the eigenvalues of C, when A and B are independently and uniformly distributed on their orbit under the action of the unitary group. Comparison with the similar problem for real symmetric matrices and the action of the orthogonal group reveals unexpected differences...

• October 8

  Columbus day

• October 12

  CHARLES RIVER LECTURE SERIES

• October 15

  Victor Kleptsyn (Institute of Mathematical Research of Rennes)

  Furstenberg theorem: now with a parameter!

  Abstract: My talk will be devoted to our joint work with Anton Gorodetski. Consider a random product of i.i.d. matrices, randomly chosen from SL(2,R): T_n = A_n... A₂ A₁, where the random matrices A_i are i.i.d.. A classical Furstenberg theorem then implies, that under some very mild nondegeneracy conditions (no finite common invariant set of lines, no common invariant metric) for the law of A_i's the norm of such a product almost surely grows exponentially.

  Now, what happens if each of these matrices A_i(s) depends on an additional parameter s, and hence so does their product T_n(s)? For each individual s, the Furstenberg theorem is still applicable. However, what can be said almost surely for the random products T_n(s), depending on a parameter?

  We will impose a few reasonable additional assumptions, of which the most important is that the dependence of angle is monotonous w.r.t. the parameter: increasing the parameter rotates all the directions clockwise.

  It turns out that, under these assumptions:

  1. Almost surely for all the parameter values, except for a zero Hausdorff dimension (random) set, the Lyapunov exponent exists andequals to the Furstenberg one.
  2. Almost surely for all the parameter values the upper Lyapunov exponent equals to the Furstenberg one.
  3. At the same time, in the no-uniform-hyperbolicity parameter region there exists a dense subset of parameters, for each of which the lower Lyapunov exponent takes any fixed value between 0 and the Furstenberg exponent.

  Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof.

• October 22

  Xin Sun (Columbia)

  Natural measures on random planar fractals

  Abstract: Random fractals arise naturally as the scaling limit of large discrete models at criticality. These fractals usually exhibit strong self similarity and spacial independence. In this talk, we will explain how these additional properties should give the existence of a natural occupation measure on the fractal, defined to be the limit of the properly rescaled Lebesgue measure restricted to small neighborhoods of the fractal. Moreover, the occupation measure is also the scaling limit of the normalized counting measure over the corresponding discrete set. In two dimension, when putting an independent Liouville quantum gravity background over such a planar fractal, the quantum version of the occupation measure still exists, where the scaling dimension is related to the Euclidean one via the famous KPZ relation due to Knizhnik-Polyakov-Zamolodchikov and Duplantier-Sheffield. The quantum occupation measure is supposed to be the scaling limit of the normalized counting measure of the corresponding discrete set on certain random planar maps. The picture described above is expected to be true in great generality yet it is only established for a few models to various extents. In this talk, we report a fairly complete picture for planar percolation on the regular and random triangular lattice. Based on joint works with N. Holden, G. Lawler and X. Li.

• October 29

  Ankur Moitra (MIT)

  A New Approach to Approximate Counting and Sampling

  Abstract: Over the past sixty years, many remarkable connections have been made between statistical physics, probability, analysis and theoretical computer science through the study of approximate counting. While tight phase transitions are known for many problems with pairwise constraints, much less is known about problems with higher-order constraints.

  Here we introduce a new approach for approximately counting and sampling in bounded degree systems. Our main result is an algorithm to approximately count the number of solutions to a CNF formula where the degree is exponential in the number of variables per clause. Our algorithm extends straightforwardly to approximate sampling, which shows that under Lovasz Local Lemma-like conditions, it is possible to generate a satisfying assignment approximately uniformly at random. In our setting, the solution space is not even connected and we introduce alternatives to the usual Markov chain paradigm.

• November 5

  Yilin Wang (ETH Zurich)

  Loewner energy via renormalization of Brownian loop measure

  Abstract: We have introduced Loewner energy to describe the large deviation behavior of chordal Schramm-Loewner evolution with vanishing parameter. The intriguing parametrization-independence of its generalization to Jordan curves was explained by an intrinsic description using zeta-regularized determinants of Laplacians and the fact that it is a Kahler potential for the Weil-Petersson metric in the universal Teichmueller space. These identities suggest that we may also interpret the Loewner energy as (minus) the total mass of the Brownian loops attached to the curve. I will show that the cut-off by equipotentials provides an appropriate renormalization to make sense of this statement.

• November 12

  Veteran's Day

• November 19

  Soumik Pal (University of Washington)

  The Aldous diffusion on continuum trees

  Abstract: Consider a binary tree with n labeled leaves. Randomly select a leaf for removal and then reinsert it back on an edge selected at random from the remaining structure. This produces a Markov chain on the space of n-leaved binary trees whose invariant distribution is the uniform distribution. David Aldous, who introduced and analyzed this Markov chain, conjectured the existence of a continuum limit of this process if we remove labels from leaves, scale edge-length and time appropriately with n, and let n go to infinity. The conjectured diffusion will have an invariant distribution given by the so-called Brownian Continuum Random Tree. In a series of papers, co-authored with N. Forman, D. Rizzolo, and M. Winkel, we construct this continuum limit. This talk will be an overview of our construction and describe the path behavior of this limiting object.

• November 26

  Aukosh Jagannath (Harvard University)

  Algorithmic thresholds for tensor principal component analysis

  Abstract: Consider the problem of recovering a rank 1 tensor of order k that has been subject to Gaussian noise. The log-likelihood for this problem is highly non-convex. It is information theoretically possible to recover the tensor with a finite number of samples via maximum likelihood estimation, however, it is expected that one needs a polynomially diverging number of samples to efficiently recover it. What is the cause of this large statistical-to-algorithmic gap? To study this question, we investigate the thresholds for efficient recovery for a simple family of algorithms, Langevin dynamics and gradient descent. We view this problem as a member of a broader class of problems which correspond to recovering a signal from a non-linear observation that has been perturbed by isotropic Gaussian noise. We propose a mechanism for success/failure of recovery of such algorithms in terms of the strength of the signal on the high entropy region of the initialization. Joint work with G. Ben Arous (NYU) and R. Gheissari (NYU).

• December 3

  Jian Ding (University of Pennsylvania)

  Random walk among Bernoulli obstacles

  Abstract: Consider a discrete time simple random walk on Z^d, d ≥ 2 with random Bernoulli obstacles, where the random walk will be killed when it hits an obstacle. We show that the following holds for a typical environment (for which the origin is in an infinite cluster free of obstacles): conditioned on survival up to time n, the random walk will be localized in a single island. In addition, the limiting shape of the island is a ball and the asymptotic volume is also determined. This is based on joint works with Changji Xu. Time permitting, I will also describe a recent result in the annealed case, which is a joint work with Ryoki Fukushima, Rongfeng Sun and Changji Xu.