Vorträge in der Woche 20.06.2022 bis 26.06.2022

Montag, 20.06.2022: A new theory for fractional calculus and fractional calculus of varations as well as their numerical methods

Xiaobing Feng (UTK, Knoxville)

In this talk, I shall first briefly review a newly developed theory of weak fractional (differential) calculus and fractional Sobolev spaces. The focus is the introduction of a weak fractional derivative concept which is a natural generalization of integer order weak derivatives and helps to unify multiple existing fractional derivative concepts. Based on the weak fractional derivative concept, new fractional-order Sobolev spaces can be naturally defined and many important properties, such as density theorem, extension theorem, and trace theorem, of those Sobolev spaces can be established. I shall then introduce a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations, the new framework and theory are based on the aforementioned theory of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of fractional differential equations; including new one-side fractional Laplace operators and future value problems. Finally, I shall discuss some new finite element (and DG) methods for approximating the weak fractional derivatives and the solutions of fractional calculus of variations problems and their associated fractional differential equations.

Montag, 20.06.2022: Complete Logarithmic Sobolev inequalities for matrices and matrix-valued functions

Dr. Li Gao (University of Houston, USA)

Logarithmic Sobolev inequalities (LSI) were first introduced by L. Gross in 70s, and later found rich connections to probability, geometry, and information theory. In recent years, logarithmic Sobolev inequalities have also been studied for quantum Markov semigroups, which are noncommutative generalization of Markov semigroups by replacing the underlying probability spaces by matrix algebras. It turns out the LSIs of quantum Markov semigroups on matrix algebras are often related to the LSIs of classical Markov semigroups on matrix valued functions. In this talk, I will present a matrix-valued LSI for the canonical sub-Laplacian on SU(2) and its consequence for quantum Markov semigroups. This talk is based on a joint work with Maria Gordina.

Dienstag, 21.06.2022: Funktorielle Linktypinvarianten und Khovanov-Floer-Theorien

Jonathan Walz (Universität Tübingen)

Im Vortrag soll die Khovanov-Homologie als funktorielle Linktypinvariante und das Konzept der Khovanov-Floer-Theorien erklärt werden. Ein Knoten ist das Bild einer glatten Einbettung einer Kreislinie in den dreidimensionalen Raum und allgemeiner ist ein Link das Bild einer glatten Einbettung mehrerer Kreislinien in den dreidimensionalen Raum. In der Knotentheorie wünscht man solche Knoten und Links bis auf Deformation des umgebenden Raumes zu klassifizieren. Die Unterscheidung verschiedener Links erfolgt vor allem mit Hilfe von Invarianten. Eine berühmte Linktypinvariante ist das Jones-Polynom. Links lassen sich als Objekte einer Kategorie auffassen, wenn man als Morphismen die Cobordismen zwischen Links (modulo Isotopie) betrachtet. Die Khovanov-Homologie, welche eine Verallgemeinerung des Jones-Polynoms ist, ergibt sich dann als funktorielle Linktypinvariante, weshalb man sie auch als Kategorifizierung des Jones-Polynoms bezeichnet. Mit Hilfe von Spektralsequenz-Konstruktionen und Floer-Homologien lassen sich viele bedeutende Eigenschaften der Khovanov-Homologie beweisen, etwa dass diese den Unknoten detektiert. Khovanov-Floer-Theorien bieten eine Möglichkeit, solche Spektralsequenzen wiederum als Funktoren aufzufassen. Abschließend soll die Spektralsequenz der filtrierten Bar-Natan-Theorie vorgestellt werden. Der Satz, dass diese eine Khovanov-Floer-Theorie bildet, vollendet einen neuen, einfacheren Beweis, dass Rasmussens s-Invariante eine Knotentypinvariante ist.

Donnerstag, 23.06.2022: Higher-dimensional Majumdar-Papapetrou black holes

Dr. James Lucietti (University of Edinburgh)

The uniqueness theorems for asymptotically flat, static, black hole solutions to Einstein-Maxwell theory in four and higher dimensions are only valid for non-extreme black holes. In the extreme case the solution must belong to a generalised class of Majumdar-Papapetrou solutions. I will discuss recent work which establishes that the only spacetimes in this class, with a suitably regular event horizon, are the standard multi-black hole solutions. The proof involves a careful analysis of the near-horizon geometry and an extension of the positive mass theorem to Riemannian manifolds with conical singularities.

Donnerstag, 23.06.2022: Length of matrix algebra and some applications in quantum information theory

Yifan Jia (TU München)

The length of a finite generating system for a finite-dimensional associative algebra describes the minimum required length of products (words) of elements of the generating system so that the set of all words until that length include a basis of the algebra. The length of the algebra corresponds to the maximum length among all generating systems. Surprisingly, the length of the most common algebra - the n-by-n matrix algebra over C - has remained mysterious since it was conjectured to be 2n-2 in 1984 by Paz. On the other hand, restricting all the words to be of same length, the new defined WIE-length of the algebra is strongly related to the construction of matrix product states and capacity of quantum channels. In this talk I will present some historical results about length and WIE-length of the matrix algebra Mat(n,C), and specifically discuss their relation and the applications in quantum information theory. Moreover, I will show that, in the generic case, both the length and WIE-length of Mat(n,C) are of order O(log n). Furthermore, we will have an outlook of observing a similar characteristic on a generating system of matrix Lie algebra by counting Lie-brackets with some numerical results provided.

Freitag, 24.06.2022: Die KdV-Hierarchie, Whitham-Deformationen und hydrodynamische Systeme

Jonas Ziefle (Uni Tübingen)