Mathematisches Kolloquium
Das Mathematische Kolloquium ist eine gemeinsame wissenschaftliche Veranstaltung des gesamten Mathematischen Instituts. Es steht allen Interessierten offen und richtet sich neben den Mitgliedern und Mitarbeitern des Instituts auch an die Studierenden. Das Kolloquium findet dreimal im Semester am Donnerstag um 15:00 s.t. im Hörsaal II, Albertstr. 23b statt. Danach (gegen 16:15) gibt es Kaffee und Kekse, zu dem der vortragende Gast und alle Besucher eingeladen sind.
Wintersemester 2024
Vorträge
Zeit und Ort
Donnerstag, 9.1.25, 15:00–16:00, Hörsaal II, Albertstr. 23b
Zusammenfassung
The Sobolev inequality is a paradigmatic example of a functional inequality with many applications in the Calculus of Variations, Geometric Analysis and PDEs. In some of these applications the optimal value of the constant is of importance, as is a characterization of the set of optimizers. The stability question is whether functions whose Sobolev quotient is almost minimal are close to minimizers of the inequality and, if so, in which sense. We give a gentle introduction to this question and review some recent results on the Sobolev inequality and other functional inequalities of a similar nature.
Zeit und Ort
Donnerstag, 7.11.24, 15:00–16:00, Hörsaal II, Albertstr. 23b
Zusammenfassung
As an example for a random walk in random environment, we study biased random walk for dynamical percolation on the d-dimensional lattice. We establish a law\nof large numbers and an invariance principle for this random walk using regeneration times.\nMoreover, we verify that the Einstein relation holds, and we investigate the speed of the walk\nas a function of the bias. While for d = 1 the speed is increasing, we show that in general this\nfails in dimension d ≥ 2. As our main result, we establish two regimes of parameters, separated\nby a critical curve, such that the speed is either eventually strictly increasing or eventually\nstrictly decreasing. This is in sharp contrast to the biased random walk on a static supercritical\npercolation cluster, where the speed is known to be eventually zero.\n\nBased on joint work with Sebastian Andres, Dominik Schmid and Perla Sousi.