Biased random walk on dynamical percolation
Donnerstag, 7.11.24, 15:00-16:00, Hörsaal II, Albertstr. 23b
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.
Weyl formulae for some singular metrics with application to acoustic modes in gas giants
Montag, 11.11.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
We prove eigenvalue asymptotics of the Laplace-Beltrami operator for certain singular Riemannian metrics. This is motivated by the study of propagation of soundwaves in gas planets. This is joint work with Yves Colin de Verdière, Maarten de Hoop and Emmanuel Trélat.
Rigorous justification of kinetic equations: Recent progress and finite size corrections
Dienstag, 12.11.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The justification of kinetic equations for long times is a longstanding mathematical challenge; in fact, Hilbert's 6th problem refers specifically to the Boltzmann equation. In my talk I will discuss the case of hard spheres and the very recent progress by Deng & Hani. Finally, I will present results on finite size corrections.
On (non-)elimination of imaginaries
Dienstag, 12.11.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Imaginaries are a way of introducing canonical representatives of equivalence classes. Given a theory, an important question is whether equivalence classes are already coded in the models of the theory so that one can avoid working with imaginaries (i.e. we say that the theory then eliminates imaginaries).\n\nIn this talk, we want to present a criterion that yields the failure of elimination of imaginaries due to equivalence classes that arise as cosets of subgroups. We will illustrate the main ideas by considering the example of the theory of beautiful pairs of algebraically closed fields. Pillay and Vassiliev proved that this theory does not have elimination of imaginaries. In this talk, we will present a different proof that generalizes to more theories of fields.
Central limit theorems under non-stationarity via relative weak convergence
Freitag, 15.11.24, 13:00-14:00, Raum 232, Ernst-Zermelo-Str. 1
Traditional central limit theorems (CLTs) have limited applicability to non-stationary sequences, restricting their use in real-world, dynamic data contexts. We introduce relative weak convergence, a generalisation of classical weak convergence that provides a natural framework for CLTs in non-stationary settings. Within this framework we can prove multivariate, uniform, and sequential relative CLTs for dynamic sequences under general assumptions. These results bridge a critical gap in asymptotic statistics, enabling researchers to rigorously study the asymptotic behaviour of dynamic sequences that do not satisfy traditional assumptions of stationarity.
Symplectic topology and rectangular peg problem
Montag, 18.11.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
The rectangular peg problem, an extension of the square peg problem which has a long history, is easy to outline but challenging to prove through elementary methods. I will report the recent progress on the existence and multiplicity results, utilizing advanced concepts from symplectic topology, e.g. J-holomorphic curves and Floer theory.
Numbers on barcodes and torsion theory
Montag, 25.11.24, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
Morse function on a manifold M is called strong if all its critical points have different critical values. Given a strong Morse function f and a field F we construct a bunch of elements of F, which we call Bruhat numbers (they're defined up to sign). More concretely, Bruhat number is written on each bar in the barcode of f. It turns out that if homology of M over F is that of a sphere, then the product of all the numbers is independent of f. We then construct the barcode and Bruhat numbers with twisted (a.k.a. local) coefficients and prove that mentioned product equals the Reidemeister torsion of M. In particular, it's again independent of f. This way we link Morse theory to the Reidemeister torsion via barcodes. Based on a joint work with Petya Pushkar. \n
Das Rigidity Phänomen einer Artificial Venus Flytrap
Dienstag, 26.11.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Wir werden ein Modell zur Beschreibung einer Artificial Venus Flytrap formulieren, bei dem das Material als rigid angenommen wird. Während dieses rigide Modell numerisch das Phänomen der Curvature Inversion erfasst, werden wir sehen, dass die Annahme der Rigidity dazu führt, dass die planare Lösung die einzige exakte Lösung ist. Darauf aufbauend werden wir nicht-planare Lösungen betrachten, sobald wir die Rigidity-Annahme fallenlassen. Schließlich werden wir besprechen, wie sich eine Beschreibung einer Limit-Theorie angehen lässt.
Omega-kategorische Ringe
Dienstag, 26.11.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1