Seminars

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.
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 in the context of this general framework and present various open problems.
2016 Dec 29

Analysis and PDEs Seminar: Svetlana Jitomirskaya (UC Irvine)

1:00pm to 2:00pm

Location: 

Ross 70
Title: Sharp arithmetic spectral transitions and universal hierarchical structure of quasiperiodic eigenfunctions. Abstract: A very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite signifi cant numerical studies and
2017 May 24

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 27

Jerusalem Analysis Seminar: "Optimal Gaussian Partitions", Elchanan Mossel (MIT)

12:00pm to 1:00pm

Location: 

Ross 70

 How should we partition the Gaussian space into k parts in a way that minimizes Gaussian surface area, maximize correlation or simulate a specific distribution. The problem of Gaussian partitions was studied since the 70s first as a generalization of the isoperimetric problem in the context of the heat equation. It found a renewed interest in the context of the double bubble theorem proven in geometric measure theory and due to connection to problems in theoretical computer science and social choice theory. 

2017 Apr 27

PDE and Analysis Seminar: Grzegorz Swiderski (Wroclaw)

1:00pm to 2:00pm

Location: 

Ross 70
Title: Asymptotics of Christoffel functions in an unbounded setting Abstract: Consider a measure $\mu$ supported on the real line with all moments finite. Let $(p_n : n \geq 0)$ be the corresponding sequence of orthonormal polynomials. This sequence satisfies the three-term recurrence relation \[ a_{n-1} p_{n-1}(x) + b_n p_n(x) a_n p_{n+1}(x) = x p_n(x) \quad (n > 0) \] for some sequences $a$ and $b$. One defines the $n$th Christoffel function by \[ \lambda_n(x) = \left[ \sum_{k=0}^n p_k(x)^2 \right]^{-1}. \] In the talk, under some regularity hypotheses on $a$ and $b$, we show
2017 Dec 06

Jerusalem Analysis and PDEs seminar: "Asymptotics of the ground state energy for relativistic heavy atoms and molecules" Victor Ivrii (Toronto)

12:00pm to 1:00pm

Location: 

Ross 70.
We discuss sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, without magnetic field or with the self-generated magnetic field, and, in particular, relativistic Scott correction term and also Dirac, Schwinger and relativistic correction terms. In particular, we conclude that the Thomas-Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.
2016 Nov 10

Analysis and PDEs - Maurice Duits (KTH) Title: Global fluctuations for non-colliding processes

1:00pm to 2:00pm

Location: 

Ross 70
In this talk we will discuss the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. By viewing the paths as level lines these systems give rise to random (stepped) surfaces. When the number of paths is large a limit shape appears. The fluctuations for the random surfaces are believed to be universally described by the Gaussian Free Field.

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