Abstract: We will discuss the question: for a random walk in a random environment, when should one expect a central limit theorem, i.e. that after appropriate scaling, the random walk converges to Brownian motion? The answer will turn out to involve the spectral theory of unbounded operators. All notions will be defined in the talk. Joint work with Balint Toth.

I will present an overview of some recent progress on the study of the nodal sets of Steklov eigenfunctions. In particular, I will discuss sharp estimates on the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary obtained in my joint work with D. Sher and J. Toth.

Abstract: Consider a random matrix with i.i.d. normal entries. Since its distribution is invariant under rotations, any normalized eigenvector is uniformly distributed over the unit sphere. For a general distribution of the entries, this is no longer true. Yet, if the size of the matrix is large, the eigenvectors are distributed approximately uniformly. This property, called delocalization, can be quantified in various senses. In these lectures, we will discuss recent results on delocalization for general random matrices.

Title: No-gaps delocalization for general random matrices. Abstract:

20 years ago Weiss constructed a collection of non-trivial translation invariant probability measures on the space of entire functions using tools from dynamical systems. In this talk, we will present another elementary construction of such a measure, and give upper and lower bounds for the possible growth of entire functions in the support of such measures.

On the Asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential. T.Bothner, P.Deift, A.Its and I.Krasovsky Abstract: We study the partition function Z of a Coulomb gas of particles with an external potential 2v applied to the particles in an interval of length L. When v is infinite, Z describes the gap probability for GUE eigenvalues in the bulk scaling limit, and has been well-studied for many years. Here we study the the behavior of Z in the full (v,L) plane.

Abstract: Consider a random matrix with i.i.d. normal entries. Since its distribution is invariant under rotations, any normalized eigenvector is uniformly distributed over the unit sphere. For a general distribution of the entries, this is no longer true. Yet, if the size of the matrix is large, the eigenvectors are distributed approximately uniformly. This property, called delocalization, can be quantified in various senses. In these lectures, we will discuss recent results on delocalization for general random matrices.

A Gaussian stationary function (GSF) is a random f: R --> R whose distribution is shift-invariant and all its finite marginals have centered multi-normal distribution. It is a simple and popular model for noise, for which the mean number of zeroes was computed already in the 1940's by Kac and Rice. However, it is far more complicated to estimate the probability of a significant deficiency or abundance in the number of zeroes in a long interval (compared to the expectation). We do so for a specific family of GSFs with additional smoothness and absolutely

Abstract. We will give a sketch of the proof of the fact formulated in the title.

Title: a random Schroedinger operator stemming from reinforced process Abstract: We will explain the relationship between a toy model to Anderson localization, called the H^{2|2} model (according to Zirnbauer) and edge reinforced random walk. The latter is a random walk in which, at each step, the walker prefers traversing previously visited edges, with a bias proportional to the number of times the edge was traversed. Recent study on this random walk showed that it is equivalent to Zirnbauer's model and we will show some consequences once this equivalence is established.

Title: Asymptotics for Chebyshev Polynomials of Infinite Gap Sets on the Real Line Abstract: The Chebyshev Polynomials of a compact subset, e, of the complex plane are the monic polynomials minimizing the sup over e. We prove Szego--Widom asymptotics for the Chebyshev Polynomials of a compact subset of R which is regular for potential theory and obeys the Parreau--Widom and DCT conditions. We give indications why these sufficient conditions may also be necessary.

Let us consider the heat equation: $u_t+Lu=0$ in a domain $\Omega$. Here, $L$ will be a self-adjoint Schrodinger-type operator of the form abla^*

Abstract: Given a self-adjoint bounded operator, its spectrum is a compact subset of the real numbers. The space of compact subsets of the real numbers is naturally equipped with the Hausdorff metric. Let $T$ be a topological (metric) space and $(A_t)$ be a family of self-adjoint, bounded operators. In the talk, we study the (Hölder-)continuity of the map assigning to each $t\in T$ the spectrum of the operator $A_t$.

I will discuss the asymptotic behaviour (both on and off the diagonal) of the spectral function of a Schroedinger operator with smooth bounded potential when energy becomes large. I formulate the conjecture that the local density of states (i.e. the spectral function on the diagonal) admits the complete asymptotic expansion and discuss the known results, mostly for almost-periodic potentials.

Abstract: Consider a random matrix with i.i.d. normal entries. Since its distribution is invariant under rotations, any normalized eigenvector is uniformly distributed over the unit sphere. For a general distribution of the entries, this is no longer true. Yet, if the size of the matrix is large, the eigenvectors are distributed approximately uniformly. This property, called delocalization, can be quantified in various senses. In these lectures, we will discuss recent results on delocalization for general random matrices.