Page 378 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 378
SPECTRAL THEORY AND INTEGRABLE SYSTEMS (MS-57)
An almost periodic model for general reflectionless spectral data
Roman Bessonov, bessonov@pdmi.ras.ru
St. Petersburg State University, Russian Federation
A basic feature of second order almost periodic differential and finite-difference operators is
the reflectionless property of their Weyl functions. Conversely, each regular enough pair of
reflectionless Nevanlinna functions generate an almost periodic operator on the real line whose
half-line Weyl functions coincide with the given pair. Until recently, for operators with un-
bounded spectra, this scheme worked under quite restrictive assumptions on the "quality" of the
spectrum. We extend it to cover all homogeneous spectra, and, more generally, to all (possibly,
unbounded) spectra satisfying Widom and DCT conditions. Joint work with M. Lukic and P.
Yuditskii.
Beyond the Strong Szegö Limit Theorem
Maurice Duits, duits@kth.se
Royal Institute of Technology, Sweden
In random matrix theory it is well-established that the Strong Szegö Limit Theorem for Toeplitz
determinants implies a CLT for linear statistics for eigenvalues of a CUE matrix. The purpose
of this talk will be to discuss an extension of the Strong Szegö Limit Theorem to determinants
of truncated exponentials of banded matrices such as Jacobi and CMV matrices. This extension
shows that the second term in the Strong Szegö Limit Theorem is universal, providing a general
CLT for more general classes of determinantal point processes including orthogonal polynomial
ensembles on the real line and unit circle. A time-dependent analogue can be used tot establish
Gaussian Free Field fluctuations in certain non-colliding process and random tilings of planar
domains. The talk aims to present an overview of various results based on this idea.
Control of eigenfunctions on negatively curved surfaces
Semyon Dyatlov, dyatlov@math.mit.edu
MIT, United States
Coauthors: Long Jin, Stéphane Nonnenmacher
Given an L2-normalized eigenfunction with eigenvalue λ2 on a compact Riemannian manifold
(M, g) and a nonempty open set subset Ω of M , what lower bound can we prove on the L2-mass
of the eigenfunction on Ω? The unique continuation principle gives a bound for any Ω which
is exponentially small as λ goes to infinity. On the other hand, microlocal analysis gives a
λ-independent lower bound if Ω is large enough, i.e. it satisfies the geometric control condition.
This talk presents a λ-independent lower bound for any set Ω in the case when M is a
negatively curved surface, or more generally a surface with Anosov geodesic flow. The proof
uses microlocal analysis, the chaotic behavior of the geodesic flow, and a new ingredient from
harmonic analysis called the Fractal Uncertainty Principle. Applications include control for
Schrodinger equation and exponential decay of damped waves. Joint work with Jean Bourgain,
Long Jin, and Stéphane Nonnenmacher.
376
An almost periodic model for general reflectionless spectral data
Roman Bessonov, bessonov@pdmi.ras.ru
St. Petersburg State University, Russian Federation
A basic feature of second order almost periodic differential and finite-difference operators is
the reflectionless property of their Weyl functions. Conversely, each regular enough pair of
reflectionless Nevanlinna functions generate an almost periodic operator on the real line whose
half-line Weyl functions coincide with the given pair. Until recently, for operators with un-
bounded spectra, this scheme worked under quite restrictive assumptions on the "quality" of the
spectrum. We extend it to cover all homogeneous spectra, and, more generally, to all (possibly,
unbounded) spectra satisfying Widom and DCT conditions. Joint work with M. Lukic and P.
Yuditskii.
Beyond the Strong Szegö Limit Theorem
Maurice Duits, duits@kth.se
Royal Institute of Technology, Sweden
In random matrix theory it is well-established that the Strong Szegö Limit Theorem for Toeplitz
determinants implies a CLT for linear statistics for eigenvalues of a CUE matrix. The purpose
of this talk will be to discuss an extension of the Strong Szegö Limit Theorem to determinants
of truncated exponentials of banded matrices such as Jacobi and CMV matrices. This extension
shows that the second term in the Strong Szegö Limit Theorem is universal, providing a general
CLT for more general classes of determinantal point processes including orthogonal polynomial
ensembles on the real line and unit circle. A time-dependent analogue can be used tot establish
Gaussian Free Field fluctuations in certain non-colliding process and random tilings of planar
domains. The talk aims to present an overview of various results based on this idea.
Control of eigenfunctions on negatively curved surfaces
Semyon Dyatlov, dyatlov@math.mit.edu
MIT, United States
Coauthors: Long Jin, Stéphane Nonnenmacher
Given an L2-normalized eigenfunction with eigenvalue λ2 on a compact Riemannian manifold
(M, g) and a nonempty open set subset Ω of M , what lower bound can we prove on the L2-mass
of the eigenfunction on Ω? The unique continuation principle gives a bound for any Ω which
is exponentially small as λ goes to infinity. On the other hand, microlocal analysis gives a
λ-independent lower bound if Ω is large enough, i.e. it satisfies the geometric control condition.
This talk presents a λ-independent lower bound for any set Ω in the case when M is a
negatively curved surface, or more generally a surface with Anosov geodesic flow. The proof
uses microlocal analysis, the chaotic behavior of the geodesic flow, and a new ingredient from
harmonic analysis called the Fractal Uncertainty Principle. Applications include control for
Schrodinger equation and exponential decay of damped waves. Joint work with Jean Bourgain,
Long Jin, and Stéphane Nonnenmacher.
376
