Das allgemeine Kolloquium des mathematischen Instituts findet während der Vorlesungszeit donnerstags um 17:15 Uhr im Hilbertraum (Raum 05-432) statt. Ab 16:45 Uhr gibt es Kaffee und Kuchen.
19.04. (12 Uhr) Vasily Golyshev (ICTP Trieste):
Lifting differential equations, in-dept
Abstract: I will introduce generalized Heun's equations following Boalch-Katz-Simpson,
and show how two equations in BKS correspondence share the same kernell.
This is joint work in progress with Ilia Gaiur.
2.5. Antrittsvorlesung (16:00 h, Audimax)
Prof. Dr. Hendrik Ranocha (Mainz):
Numerische Mathematik
16.5. Prof. Dr. Matthias Lesch (Univ. Bonn):
The product formula for Fredholm determinants and related questions
Abstract: Fredholm determinants are the natural generalization of Linear Algebra determinants to certain operators acting on a Hilbert or Banach space being naturally related to the trace resp. trace ideals (e.g. B.Simon's Book on trace ideals). Its product formula (for operators of the form I+A with A in a higher trace ideal) has only quite recently be fully understood and it exhibits quite some interesting combinatorial features. I will report on a proof which was developed in a recent Masters Thesis with my student Nikolaos Koutsonikos-Kouloumpis (https://arxiv.org/abs/2202.12923). If time permits I will hint at another a priori quite different generalization of determinant in the context of elliptic operators; a posteriori there are interesting relations between the two notions. That latter is joint work with Luiz Hartmann (J. Funct. Anal. 283 (2022))
13.6. Prof. Dr. Frank den Hollander (Univ. Leiden):
The Friendship Paradox for Social Networks
Abstract: In 1991, the American sociologist Scott Feld discovered the paradoxical phenomenon that `your friends are more popular than you are'. This statement means the following. Consider a group of individuals who form a social network. For each individual in the group, compute the difference between the average number of friends of friends and the number of friends (all friendships are mutual), and average these numbers over all the individuals in the group. It turns out that the latter average is always non-negative, and is strictly positive as soon as not all individuals have exactly the same number of friends. This bias, which at first glance seems counterintuitive, goes under the name of friendship paradox, even though it is a hard fact. In this lecture we model the social network as a sparse random graph. We explain where the bias comes from, how it can be quantified, and illustrate our findings with two examples.
Based on joint work with R.S. Hazra and A. Parvaneh.
20.6. Prof. Dr. Jochen Schütz (Hasselt University):
A class of parallel-in-time multi-derivative time integrators
Abstract: In this talk, we present a class of parallel-in-time multi-derivative time integrators for ordinary (and partial) differential equations. The distinguishing feature of multi-derivative schemes is that they do not only use the information on the time derivative u_t of the unknown solution u (in an ODE, this information could be u_t = f(u)), but also information on higher-order time derivatives u_tt (in the example, u_tt = f'(u) f(u)), u_ttt and so on. We show how this can in practice be used to generate high-order, parallel-in-time schemes that are suitable to integrate highly stiff equations in time. A focus is set on stability of these methods, both classical linear stability as well as asymptotic stability in the stiff limit are treated.
11.7. Prof. Dr. Peter Ullrich (Univ. Koblenz):
Von Euler über Dirichlet zur Riemannschen Zeta-Funktion
Abstract: Die Zeta-Reihe, die die nach Bernhard Riemann (1826–1866) benannte Zeta-Funktion definiert, wurde bereits von Leonhard Euler (1707–1783) untersucht: Zum einen bestimmte er deren Werte für gerade natürliche Argumente, zum anderen gab er die nach ihm benannte Produktdarstellung für sie an.
Beide Ergebnisse publizierte er auch in Lehrbüchern.
Auf die Herleitung der Produktdarstellung in der Introductio in analysin infinitorum nahm Gustav Lejeune Dirichlet (1805–1859) im Jahr 1837 bei seinem Beweis des Primzahlsatzes für arithmetische Progressionen expliziten Bezug. Weiterhin gab er von seinem Wissen über die mathematischen Schriften Eulers an Riemann weiter, der 1859 in seiner Arbeit "Über die Anzahl der Primzahlen unter einer gegebenen Größe" ebenfalls auf die Produktdarstellung verwies.
Im Vortrag wird analysiert, wie die genannten Ergebnisse zur Entstehung der analytischen Zahlentheorie beigetragen haben.
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