Uncertainty in Credit Risk
Montag, 7.3.22, 11:15-12:15, Raum 232
Sonderkolloquium Reine Mathematik
Mittwoch, 9.3.22, 08:00-09:00, Online per BBB, Link wird noch bekannt gegeben.
TBA
The two faces of scalar curvature
Mittwoch, 9.3.22, 08:00-09:00, https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl
Ort: https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl\n\nBy Gromov’s h-principle there are no global obstructions against Riemannian metrics with prescribed curvature bounds on non-compact connected manifolds. Under additional assumptions, such as metric completeness or specific boundary conditions, this flexibility is challenged by rigidity phenomena which lead to classification patterns in terms of algebraic topological and metric invariants. \n\nThe geometry of Riemannian manifolds of positive scalar curvature lies at the border between the flexible and rigid worlds. I will illustrate this dual nature by some exemplary ideas and results.
The gradient-flow structure of mean curvature flow: from algorithms to basic analysis
Mittwoch, 9.3.22, 11:00-12:00, https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl
Ort: https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl\n\nMean curvature flow is one of the most fundamental geometric evolution equations and arises in various problems from geometry, physics, data science, and many other fields. In this talk, I will show how the gradient-flow structure sheds new light on numerical methods for this equation. Then I will show how these results give rise to more basic questions regarding the very definition of solutions to (multiphase) mean curvature flow and corresponding existence and uniqueness theories. Starting again from the gradient-flow structure, we will define a generalization of calibrations to this dynamic setting, which allows to reduce uniqueness questions to the construction of such a gradient flow calibration. In particular, our results imply that solutions to mean curvature flow are characterized by a single inequality.
Special values of L-functions
Mittwoch, 9.3.22, 14:30-15:30, https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl
Ort: https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl\n\nL-functions are one of the central objects of study in number theory. There are many beautiful theorems, and many deep open conjectures, linking the values of these functions to all kinds of arithmetic problems; the Birch--Swinnerton-Dyer conjecture, which is one of the Clay millennium problems, is just one of these. I will explain how these mysterious functions arise, and describe some of the progress that has recently been made towards understanding their values.
Sonderkolloquium Reine Mathematik
Freitag, 11.3.22, 08:00-09:00, Online per BBB, Link wird noch bekannt gegeben.
TBA
Discrete subgroups of semisimple Lie groups: Anosov groups, higher rank Teichmüller theories and beyond
Freitag, 11.3.22, 08:00-09:00, https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl
Ort: https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl\n\nIn recent years, the study of discrete subgroups of semisimple Lie groups has undergone exciting developments inspired from different areas of mathematics: low dimensional topology, complex and symplectic geometry, dynamical systems, real algebra.. I will discuss the framework in which to encompass these achievements as well as some of my contributions to the area.
Enumerative Geometry of Hyperkähler varieties
Freitag, 11.3.22, 11:00-12:00, https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl
Ort: https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl\n\nK3 surfaces are complex algebraic surfaces whose geometry has been studied for over 200 years by Cayley, Kummer, Klein and many others. In higher dimension K3 surfaces are generalized by the notion of hyperkähler varieties. In this talk I will give an overview about my work on the enumerative geometry of hyperkähler varieties. This yields insights into their Chow theory, and also provides interesting relations to modular forms, in particular, Jacobi forms.
Closed Lorentzian manifolds with large conformal group
Freitag, 11.3.22, 14:30-15:30, https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl
Ort: https://bbb.uni-freiburg.de/b/dan-qk9-oqr-msl\n\nThe celebrated Ferrand-Obata Theorem says that a compact Riemannian manifold with noncompact conformal group is conformally diffeomorphic to the round sphere. The Lorentzian Lichnerowicz Conjecture seeks to establish an analogue of this theorem in Lorentzian signature. I will present my result with C. Frances proving this conjecture for analytic, 3-dimensional Lorentzian manifolds. This establishes the local geometry of such spaces admitting an essential conformal group. I will discuss some current work-in-progress on their topology.