Idisplays

Kolloquium über Geschichte und Didaktik der Mathematik

After the two historical descriptions by Einstein and Langevin of Brownian motion, the now well known generators acting on p-forms, are on one side the Witten Laplacian (Einstein) and on the other side Bismut's hypoelliptic Laplacian (Langevin). The accurate computation of exponentially small eigenvalues has many applications, in particular for the design of effective molecular dynamics algorithms. In the case of the Witten Laplacian, I will present the result obtained a few years ago with D. Le Peutrec and C. Viterbo, which makes the connection between the various exponential scales of small eigenvalues and the bar code of persistent homology. This provides a natural topological extension of the well known Arrhenius law in the scalar case, for general potential functions not assumed to be Morse. I will also present the more recent result obtained with X. Sang and F. White, which provides the same determination of the different spectral exponential scales in terms of the persistent homology bar code, in the double asymptotic regime of large friction and small temperature for Bismut's hypoelliptic Laplacian.

Abstract: Gromov proved the following ''Limit theorem'': Let g be a C^2 Riemannian metric on a smooth manifold M (without boundary). If a sequence of C^2 Riemannian metrics on M converges to g in the C^0 sense, and each scalar curvature is bounded from below by k. Then the scalar curvature of the limiting metric g is also bounded from below by k. In this talk, I'd like to talk about some total-scalar-curvature-version theorems of this limit theorem. I also consider the limit theorem for an weighted total scalar curvature and as a corollary, I give a definition of scalar curvature lower bound in a weak sense. To prove these, we use the Ricci flow. If I have time, I also would like to talk about limit theorems for the upper bound of the total scalar curvature. Compared to the above results, we use a different geometric flow: the Yamabe flow.

Wie in jedem Semester gibt es eine Kurzvorstellung der Projektseminare und Seminare des Folgesemesters, um sich aus dem Angebot das Interessanteste und Passendste heraussuchen zu können. Bis zum Vorstellungstermin 3.7.2024 sollten auch alle Kurzvorstellungen in Form von Foliensätzen etc. in Learnwebkurs hinterlegt sein, dann startet auch die Bewerbungszeit um Projektseminarplätze, über die Buchungstool im Learnweb-Kurs.
S.a. Kurs PSemInfo im Learnweb

We propose a partial differential equation framework for modeling distribution shift of a strategic population interacting with a learning algorithm. We consider two particular settings; one, where the objective of the algorithm and population are aligned, and two, where the algorithm and population have opposite goals. We present convergence analysis for both settings, including three timescale settings for the opposing-goal objective dynamics. We illustrate how our framework can accurately model real-world data and show via synthetic examples how it captures sophisticated distribution changes which cannot be modeled with simpler methods.

We study stochastic differential equations with simple additive noise under the impact of a shear force. We fully characterize the behaviour of these systems, depending on the shear function, in terms of global and local properties of the stochastic flow.
In more detail, we show criteria for weak and strong completeness and the existence of random set and point attractors. We also demonstrate the possibility of negative and positive Lyapunov exponents, in combination with all other behaviours, indicating synchronization or chaos as characterizations of the random attractors if they exist.

Abstract: In joint work with Rachael Boyd and Corey Bregman we study the classifying space B Diff(M) of the diffeomorphism group of a connected, compact, orientable 3-manifold M. By a theorem of Milnor every such M has a unique prime decomposition as a connected sum of prime 3-manifolds. The purpose of this talk is to explain how one can compute the moduli space B Diff(M) in terms of the moduli spaces of prime factors. We show that certain space of systems of reducing spheres is contractible. (This can be thought of as saying that the modular infinity-operad of 3-manifolds is freely generated by irreducible manifolds.) We use this to prove that if M has non-empty boundary, then B Diff(M rel boundary) has the homotopy type of a finite CW complex, as was conjectured by Kontsevich.