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.
Liquid Tensor Experiment
Donnerstag, 25.11.21, 17:00-18:00, Hörsaal Pharmazie, Hermann-Herder-Str. 7
In December 2020, Peter Scholze posed a challenge to formally verify\nthe main theorem on liquid \(\bmathbb{R}\)-vector spaces,\nwhich is part of his joint work with Dustin Clausen on condensed\nmathematics.\nI took up this challenge with a team of mathematicians\nto verify the theorem in the Lean proof assistant.\nHalf a year later, we reached a major milestone,\nand our expectation is that in a couple of months\nwe will have completed the full challenge.\n\nIn this talk I will give a brief motivation for condensed/liquid\nmathematics,\na demonstration of the Lean proof assistant,\nand discuss our experiences formalizing state-of-the-art research in\nmathematics.
Optimal stochastic control of a path-dependent risk indicator for insurance companies
Dienstag, 18.1.22, 08:30-09:30, Zoom Meeting
The drawdown of a stochastic process (modelling the surplus of a company) is the absolute distance to its historical high water mark. It can therefore be interpreted as a "relative loss" and\nis a risk and performance measure widely used in financial applications: whilst large and long-\nlasting drawdowns might manifest existing financial and reputational risks, small and infrequent\ndrawdowns can be considered a sign of economic strength and stability. For this reason,\nminimising drawdowns is desirable for companies - especially in insurance, where customer trust\nis the basis for success. In this talk, we consider a stochastic control problem inspired by the\nminimisation of the drawdown size and "recovery time" for insurance companies. By exploiting\nconnections to Laplace transforms of passage times, Hamilton-Jacobi-Bellman equations and\nreflected stochastic differential equations, we find value functions and optimal strategies. We\ndiscuss our results and implications of the model in explicit examples.
On Absence of Arbitrage and Propagation of Chaos
Dienstag, 18.1.22, 10:30-11:30, Zoom Meeting
In the talk I discuss two recent research projects. The first part is related to mathematical finance. More precisely, I consider a single asset model whose (discounted) price process is assumed to be a non-negative semimartingale diffusion. The important new feature of this model is that the diffusion is not assumed to have an SDE representation, which allows possible local time effects such as sticky points. For this financial model I discuss explicit deterministic sufficient and necessary conditions for the existence and absence of arbitrage in the sense of NFLVR. The proof of the result is based on the concept of separating times, which I also shortly explain. In the second part of my talk I discuss a propagation of chaos result for a system of (weakly) interacting stochastic PDEs. More precisely, under quite mild continuity and linear growth conditions, I present a law of large numbers and the corresponding McKean-Vlasov limit. The first part of my talk is based on joint work with Mikhail Urusov (U Duisburg-Essen).
Dependence Structures in Finance: Applications in Credit Risk Modeling and Pairs Trading
Dienstag, 18.1.22, 14:00-15:00, Zoom Meeting
In the first part of the talk, we develop a generalized interacting intensity-based contagious credit risk model with hidden Markov state process. The main contribution is that the model, as well as the closed-form default distributions derived are applicable to a wide class of default intensities with various forms of dependence structures. A number of practical problems can then be solved efficiently with these explicit formulas for the distribution of default times. In the second part of the talk, we discuss optimal pairs trading strategies under symmetric and non-symmetric trading constraints. Under the assumption that the price spread of a pair of correlated securities is mean-reverting and follows a Ornstein-Uhlenbeck process, closed-form trading strategies under each of the constraints are obtained in a mean-variance framework. Numerical results indicate that our pairs trading strategies have fairly good performance.
Donnerstag, 3.2.22, 17:00-18:00, Hörsaal II, Albertstr. 23b
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.
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.