**Title:** Local laws for polynomials of Wigner matrices.

**Abstract:** We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have constant variance. Under some numerically checkable conditions, we establish the optimal local law, i.e., we show that the empirical spectral distribution on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. First, we give a brief introduction to the linearization technique that allows to transform the polynomial model into a linear one, which has simpler correlation structure but higher dimension. After that we show that the local law holds up to the optimal scale for the generalized resolvent of the linearized model, which also yields the local law for polynomials. Finally, we show how the above results can be applied to prove the optimal bulk local law for two concrete families of polynomials: general quadratic forms in Wigner matrices and symmetrized products of independent matrices with i.i.d. entries.

This is a joint work with Laszlo Erdos and Torben Kruger.

**Title:** Rigidity of eigenvalues for beta ensemble in multi-cut regime

**Abstract:** I will first talk about the background and some well known results of beta ensemble. Then I will introduce the rigidity of eigenvalues for beta ensemble in multi-cut regime, i.e., the fact that each eigenvalue in the bulk is very close to its "classical location". The probability that the distance between the eigenvalue and its classical location is larger than N^{-1+r} is exponentially small where r is an arbitrarily small positive number. The model is an generalization of the beta ensemble in one-cut regime for which the rigidity of eigenvalues was proved by Bourgade, Erdos and Yau. Finally I will explain the difference between the proof in one-cut case and the proof in multi-cut case.

**Title:** On shocks in the TASEP

**Abstract:** The TASEP particle system runs into traffic jams when the initial particle density increases in the direction of flow. It servers as a microscopic model of shocks in Burgers' equation. I will describe work with Jeremy Quastel on a specialization of TASEP, where we identify limiting joint fluctuations of particles at the shock by using determinantal formulae for its correlation functions. The limit is described in terms of random matrix laws.

**Title:** Optimal Clustering Algorithms in Block Markov Chains

**Abstract:** In this talk, we consider cluster detection in Block Markov Chains. These Markov chains are characterized by a block structure in their transition matrix. More precisely, the $n$ possible states are divided into a finite number of $K$ groups or clusters, such that states in the same cluster exhibit the same transition rates to other states. One observes a trajectory of the Markov chain, and the objective is to recover, from this observation only, the (initially unknown) clusters. In this paper we devise a clustering procedure that accurately, efficiently, and provably detects the clusters. We first derive a fundamental information-theoretical lower bound on the detection error rate satisfied under any clustering algorithm. This bound identifies the parameters of the Block Markov Chain and trajectory lengths, for which it is possible to accurately detect the clusters. We next develop two clustering algorithms that can together accurately recover the cluster structure from the shortest possible trajectories, whenever the parameters allow detection. These algorithms thus reach the fundamental detectability limit, and are optimal in that sense.

Joint work with Jaron Sanders and Seyoung Yun

**Title:** Universal Local Statistics of Lyapunov exponents

**Abstract:** We consider the product of M complex Ginibre random matrices of sizes
N × N, being a simple toy model for chaotic dynamical systems.
While the behaviour of the Lyapunov exponents of the product matrix was
known in the limit M to inifinity at finite N, taking deterministic
values,
recent progress has been made at finite M and N. The corresponding
exponents follow a determinantal point process and thus all correlation
functions of the squared singular values of the product matrix are
known. This has allowed us to take a double scaling limit for N,M to infinity,
where the Lyapunov exponents exhibits a rich variety of correlations,
interpolating between deterministic behaviour and the sine- or
Airy-kernel of a single random matrix, depending on the location in the
spectrum. Surprisingly, in the bulk we find the same limiting
interpolating kernel as for Dyson's Brownian motion for certain initial
conditions as constructed by Johansson.
This is joint work with Zdzislaw Burda and Mario Kieburg
[arXiv:1809.05905 [math-ph]].

**Title: **Level crossings in deterministic and random matrix pencils.

**Abstract:** Collisions of energy levels (alias level crossings

) in pencils of linear operators is a physically important phenomenon studied since the down of quantum mechanics. In this talk we discuss two intriguing concrete examples coming from the so-called quasi-exactly solvable potentials and mainly concentrate on some recent results about the distribution of level crossings in families A+t B, where A and B and independently taken from some standard random matrix ensembles.

**Title: **Gaussian Approximation of the Distribution of Strongly Repelling Particles on the Unit Circle

**Abstract:** Abstract: In this talk, we will consider an interacting particle system on the unit circle with stronger repelling than that of Circular beta ensemble. I will prove the Gaussian approximation of the distribution of the particles in this model. This is joint work with Alexander Soshnikov.

**Title:** SLEs and related conformally invariant processes

**Abstract:** I will give an informal introduction to the Schramm-Loewner evolution and (depending on time) connections to related topics such as Gaussian fields, Brownian loops and conformal field theory.

**OBS: Special Time: 13:00, F11**

**Title:** Random walks on groups and free convolutions.

**Abstract:** We start with a result of Harry Kesten stating that a symmetric random walk on the free group of order d is transient for d>1. I will sketch a proof that leads us to the concept of free independence of non-commutative random variables as introduced by Dan Voiculescu. A main tool in the proof is the free additive convolution of probability measures, an operation associated with the addition of freely independent random variables. Kesten's result follows from a special case of the free convolution which is accessible via explicit calculations. I will then discuss properties of the free convolution of generic probability measures and present some recent regularity results. I will conclude by establishing a link with random matrix theory and explain the role of free probability in some on-going research projects.