2017 May 18

# Mark Rudelson: Delocalization of the eigenvectors of random matrices.

1:00pm to 2:00pm

## Location:

Ross 70
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.
2016 Mar 31

# PDE & Analysis: Mark Ruselson (UMichigan) - No-gaps delocalization for general random matrices.

1:00pm to 2:30pm

## Location:

Ross 70
Title: No-gaps delocalization for general random matrices.
Abstract:
2017 Dec 20

# Jerusalem Analysis Seminar: "Translation invariant probability measures on the space of entire functions." Adi Glucksam

12:00pm to 1:00pm

## Location:

Ross 70
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.
2016 Dec 22

# Analysis and PDE's Seminar -- Percy Deift (Courant)

1:00pm to 2:00pm

## Location:

Ross 70
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.
2017 May 17

# Mark Rudelson: Delocalization of the eigenvectors of random matrices.

2:00pm to 3:00pm

## Location:

רוס 63
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.
2017 Dec 13

# Jerusalem Analysis Seminar: "Exponential concentration of zeroes of Gaussian stationary functions" Naomi Feldheim (Weizmann Institute)

12:00pm to 1:00pm

## Location:

Ross 70
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).
2016 Nov 17

# Analysis and PDEs- A. Logunov "The zero set of a nonconstant harmonic function in R^3 has infinite area"

1:00pm to 2:00pm

## Location:

Ross 70
Abstract. We will give a sketch of the proof of the fact formulated in the title.
2017 Mar 23

# Xiaolin Zeng (TAU)

1:00pm to 2:00pm

## Location:

Ross 70
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.
2017 Jun 29

# Barry Simon (Caltech)

1:00pm to 2:00pm

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.
2016 Nov 03

# Analysis and PDEs - Baptiste Devyver (Technion) - "Heat kernel estimates of Schrodinger-type operators"

1:00pm to 2:00pm

## Location:

Ross 70
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^*
2017 Nov 01

# Jerusalem Analysis Seminar "When do the spectra of self-adjoint operators converge?" Siegfried Beckus (Technion)

12:00pm to 1:00pm

## Location:

Ross 63

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 $A_t$.
2017 Mar 09

# Analysis and PDEs: Leonid Parnovski (London) - Local Density of states and the spectral function for almost periodic operators

1:00pm to 2:00pm

## Location:

Ross 70
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.
2017 May 25

# Mark Rudelson: Delocalization of the eigenvectors of random matrices.

1:00pm to 2:00pm

## Location:

Ross 70
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.
2016 Jun 16

# Jerusalem Analysis and PDEs - Gilbert Weinstein (Ariel)

1:00pm to 2:00pm

## Location:

Ross 70
Title: Harmonic maps with prescribed singularities and applications to general relativity
Abstract: We will present a general theory of existence and uniqueness for harmonic maps with prescribed singularities into Riemannian manifolds with non-positive curvature. The singularities are prescribed along submanifolds of co-dimension 2. This result generalizes one from 1996, and is motivated by a number of recent applications in general relativity including:
* a lower bound on the ADM mass in terms of charge and angular momentum for multiple black holes;
2017 Nov 08

# Jerusalem Analysis Seminar: "To infinity and back (a bit)", Jonathan Breuer (HUJI)

12:00pm to 1:00pm

## Location:

Ross 63

Let H be a self-adjoint operator defined on an infinite dimensional Hilbert space. Given some
spectral information about H, such as the continuity of its spectral measure, what can be said about
the asymptotic spectral properties of its finite dimensional approximations? This is a natural (and
general) question, and can be used to frame many specific problems such as the asymptotics of zeros of
orthogonal polynomials, or eigenvalues of random matrices. We shall discuss some old and new results