Analysis

The Analysis seminar meets on Wednesdays at 12:00 in room 70 of the Ross Building.
2020 Dec 09

Analysis seminar (SPECIAL TIME): Yuri Lvovsky (HUJI) — Bounded multiplicity of eigenvalues of the vibrating clamped circular plate

11:00am to 12:00pm


It was shown by C. L. Siegel (1929) that the eigenvalues of the vibrating membrane problem has no non-trivial multiplicities. In this talk we consider the eigenvalues of the vibrating clamped plate problem. This is a fourth order problem. We show that its eigenvalues
have multiplicity at most six. The proof is based on a new recursion formula for
a Bessel-like function and on Siegel-Shidlovskii Theory.
If time permits we also consider the problem of determining the density of the
nodal sets of a clamped plate.
2021 Jan 20

Analysis Seminar: Dirk Hundertmark (Karlsruhe) "Quantum Systems at the Brink: Helium"

12:00pm to 1:00pm

Location: 

Zoom

Title: Quantum Systems at the Brink: Helium 


Abstract: Perturbation theoryworks well for the the discrete spectrum below the essential spectrum. Whathappens if a parameter of a quantum system is tuned in such a way that abound state energy (e.g. the ground state energy) hits the bottom of theessential spectrum? Does the eigenvalue survive, i.e., the correspondingeigenfunction stays $L^2$, or does it dissolve into the continuumenergies? 

It turns out the 

2020 Dec 09

Analysis seminar (SPECIAL TIME!): Nadav Dym (Duke) — Phase retrieval stability via notions of graph connectivity

4:00pm to 5:00pm

Phase retrieval is the inverse problem of reconstructing a signal from linear measurements, when the phase of the measurements is lost and only the magnitude is known. This problem occurs in many applications including crystallography, optics, and acoustics.

2020 Nov 18

Analysis Seminar: Hynek Kovarik (Brescia) "Absence of positive eigenvalues of magnetic Schroedinger operators"

12:00pm to 1:00pm


ABSTRACT:
We study sufficient conditions for the absence of positive eigenvalues of magnetic Schroedinger operators in R^n. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller-Simon shows that our result is sharp as far as the decay of the magnetic field is concerned.
The talk is based on a joint work with Silvana Avramska-Lukarska and Dirk Hundertmark.
2020 Nov 11

Analysis Seminar: Adam Dor-On (Copenhagen) "Operator algebras for subshifts and random walks"

12:00pm to 1:00pm

Location: 

zoom

Abstract:

There is a rich history of studying dynamical systems through the lens of operator algebras, and particularly through C*-algebras. For instance, in the work of Giordano, Matui, Putnam and Skau, C*-algebras were used as a key tool for classifying Cantor minimal $\mathbb{Z}^d$ systems up to various notions of orbit equivalence. Another successful study was conducted by Cuntz and Krieger, where subshifts of finite type (SFTs) are interpreted through C*-algebras of directed graphs, and invariants studied in symbolic dynamics naturally arise from these C*-algebras.

2020 Oct 28

Analysis seminar: Hans Knüpfer (Heidelberg) — Γ-limit for zigzag domain walls in thin ferromagnetic films

12:00pm to 1:00pm

Charged domain walls are a type of transition layers in thin ferromagnetic films which appear due to global topological constraints. The underlying micromagnetic energy is determined by a competition between a diffuse interface energy and the long-range magnetostatic interaction. The underlying model is non-convex and vectorial. In the macroscopic limit we show that the energy Γ-converges to a limit model where jump discontinuities of the magnetization are penalized anisotropically. In particular, we identify a supercritical regime which allows for tangential variation of the domain walls.
2021 Jan 13

Analysis seminar: Tracey Balehowsky (Helsinki) — An inverse problem for the relativistic Boltzmann equation

12:00pm to 1:00pm

In this talk, we consider the following problem: Given the source-to-solution map for a relativistic Boltzmann equation on a neighbourhood $V$ of an observer in a Lorentzian spacetime $(M,g)$ and knowledge of $g|_V$, can we determine (up to diffeomorphism) the spacetime metric $g$ on the domain of causal influence for the set $V$?

2020 Nov 04

Analysis Seminar: Xavier Lamy (Toulouse) — On relaxed harmonic maps with anisotropy

12:00pm to 1:00pm

Location: 

Zoom
Consider maps $u:R^n\to R^k$ with values constrained in a fixed submanifold, and minimizing (locally) the energy $E(u)=\int W(
abla u)$. Here $W$ is a positive definite quadratic form on matrices. Compared to the isotropic case $W(
abla u)=|
abla u|^2$ this may look like a harmless generalization, but the regularity theory for general $W$'s is widely open. I will explain why, and describe results with Andres Contreras on a relaxed problem, where the manifold-valued constraint is replaced by an integral penalization.
Zoom link:
2020 Nov 25

Analysis Seminar: Massimiliano Morini (Parma) — The surface diffusion flow with elasticity in two and three dimensions

12:00pm to 1:00pm

Location: 

Zoom
We show short-time existence and uniqueness for the surface
diffusion flow with a nonlocal forcing of elastic type. We also
establish long-time existence and asymptotic behavior for a suitable
class of strictly stable initial data. To the best of our knowledge
these are the first rigorous results for a surface diffusion evolution
equation with elastic stress and without curvature regularization.
2020 Oct 21

Analysis Seminar: Klas Modin (Chalmers University and the University of Gothenburg) — The structure of groups of diffeomorphisms

12:00pm to 1:00pm

Location: 

Ross 70/Zoom
In 1966 V. Arnold made an astonishing discovery: the incompressible Euler equations describe Riemannian geodesics on the infinite-dimensional “Lie group" of volume-preserving diffeomorphisms. This discovery led to geometric hydrodynamics – a field that today encompasses many equations of mathematical physics, information theory, shape analysis, etc. In this talk I shall address the infinite-dimensional manifold and group structures assigned to spaces of diffeomorphisms. Having such structures in place often enable out-of-the-box local existence and uniqueness results.
2020 Jun 24

Analysis Seminar: Angkana Rüland (Heidelberg) "Microstructure in Shape-Memory Alloys: Rigidity, Flexibility and Some Numerical Simulations"

12:00pm to 1:00pm

Location: 

zoom

Title: Microstructure in Shape-Memory Alloys: Rigidity,Flexibility and Some Numerical Simulations

 

Abstract: In this talk I will discuss a striking dichotomy whichoccurs in the mathematical analysis of microstructures in shape-memory alloys:On the one hand, some models for these materials display a very rigid structurewith only very specific microstructures, if one assumes that surface energiesare penalised. On the other hand, without this penalisation, for the samemodels a plethora of very ``wild'' solutions exists.

Pages