Vorträge in der Woche 17.07.2023 bis 23.07.2023

Montag, 17.07.2023: How it all started

Prof. Dr. Kai Zehmisch (Ruhr-Universität Bochum)

16:00-16:45 Uhr Vor-Kolloquium im N14 für Studierende und Promovierende, moderiert von Karim Mosani, Vortragender Prof. Dr. Kai Zehmisch The moduli space The approach to Gromov's „Nonsqueezing Theorem“ via holomorphic discs taken from the book „A Course on Holomorphic Discs“ (https://link.springer.com/book/9783031360633) requires knowledge from the theory of non-linear elliptic partial differential equations. I’ll highlight the essentials in order to unterstand the Cauchy-Riemann operator of the related boundary value problem. Kolloquium 17:15 Uhr „How it all started“ In 1985 Gromov proved that the unit ball does not embed symplectically into a symplectic cylinder of radius smaller than 1. Because volume preserving embeddings can do so, the theorem is call the „Nonsqueezing Theorem“. In my talk I will present a student friendly approach to the Nonsqueezing Theorem via holomorphic discs base on the forthcoming book „A Course on Holomorphic Discs“ (https://link.springer.com/book/9783031360633) jointly written with Geiges.

Dienstag, 18.07.2023: Solid Groups II

Anton Deitmar

Donnerstag, 20.07.2023: Concepts of quasi-local mass and quasi-local size

Gerhard Huisken (Universität Tübingen)

We explore different notions of quasi-local mass and their relation to a conjecture of Kip Thorne.

Donnerstag, 20.07.2023: Canonical Typicality For Other Ensembles Than Micro-Canonical

Cornelia Vogel (Tübingen)

https://www.math.uni-tuebingen.de/de/forschung/maphy/lehre/ss-2023/oberseminar-mathematical-physics/cornelia-vogel

Donnerstag, 20.07.2023: Scalar curvature rigidity of convex polytopes

Simon Brendle (Columbia University)

We prove a scalar curvature rigidity theorem for convex polytopes. The proof uses the Fredholm theory for Dirac operators on manifolds with boundary. A variant of a theorem of Fefferman and Phong plays a central role in our analysis.

Donnerstag, 20.07.2023: Computing the spectrum of Hamiltonians with finite local complexity

Paul Hege (Tübingen)

The theory of computability asks which problems can be solved by a computer algorithm. For example, the square root of a number can be approximated to any desired accuracy, but the famous Mandelbrot set, for example, is not computable in this sense. After some definitions of what computability means for real computations, we will describe some new results on the computability of the spectrum for Hamiltonians (self-adjoint operators) on an infinite-dimensional Hilbert space. It is known that if the operator is given as an infinite matrix, the spectrum cannot be computed from the entries. We argue that this impossibility result has more to do with the problem statement and less with the intrinsic difficulty of computing the spectrum. In fact we can show that the spectrum is computable for Hamiltonians of finite local complexity. That is, for any error e, we can find an algorithm that computes an approximation that has Hausdorff distance less than e to the exact spectrum of any infinite operator. Our method can be applied in practice, for example to compute the spectrum of quasiperiodic Hamiltonians, about which not many rigorous results were known before. For example, we can prove the existence of spectral gaps in operators such as the Hofstadter Hamiltonian defined on a quasicrystal.

Freitag, 21.07.2023: Convergence and non-convergence of some self-interacting random walks to Brownian motion perturbed at extrema

Prof. Elena Kosygina (Baruch College and the CUNY Graduate Center)

Generalized Ray-Knight theorems for edge local times proved to be a very useful tool for studying the limiting behavior of several classes of self-interacting random walks (SIRWs) on integers. Examples include some reinforced random walks, excited random walks, rotor walks with defects. I shall describe two classes of SIRWs introduced and studied by Balint Toth (1996), asymptotically free and polynomially self-repelling SIRWs, and discuss new results which resolve an open question posed in Toth’s paper. We show that in the asymptotically free case the rescaled SIRWs converge to a perturbed Brownian motion (conjectured by Toth) while in the polynomially self-repelling case the convergence to the conjectured process fails in spite of the fact that generalized Ray-Knight theorems clearly identify the unique candidate in the class of all perturbed Brownian motions. This negative result was somewhat unexpected. The question whether there is convergence in the polynomially self-repelling case and, if yes, then how to describe the limiting process remains open. This is joint work with Thomas Mountford (EPFL, Switzerland) and Jonathon Peterson (Purdue University, USA).

Freitag, 21.07.2023: An Introduction to Nonnegativity and Polynomial Optimization

Timo de Wolff (TU Braunschweig)

In science and engineering, we regularly face polynomial optimization problems, that is: minimize a real, multivariate polynomial under polynomial constraints. Solving these problems is essentially equivalent to certifying of nonnegativity of real polynomials -- a key problem in real algebraic geometry since the 19th century. Since this is a notoriously hard to solve problem, one is interested in certificates that imply nonnegativity and are easier to check than nonnegativity itself. In particular, a polynomial is nonnegative if it is a sums of squares (SOS) of other polynomials. Being an SOS can be detected effectively via semidefinite programming (SDP) in practice. In 2014, Iliman and I introduced a new certificate of nonnegativity based on sums of nonnegative circuit polynomials (SONC), which I have developed further since then both in theory and practice joint with different coauthors. In particular, these certificates are independent of sums of squares and can be computed via relative entropy programs. In this talk, I will give an introduction to polynomial optimization, nonnegativity, SOS, and SONC.