Seminar Talks

Abstract: The multiplication of two differential operators in an open set of R^n is explicitly known in terms of their standard symbols: This is a motivating point for the theory of deformation quantization. On a differentiable manifold equipped with a connection ∇ in the tangent bundle the same formula --where partial derivatives are replaced with symmetrized covariant derivatives-- will be wrong in general, and one has to correct it by terms containing torsion and curvature and their covariant derivatives. In the talk, based on our work with my PhD student Hamilton Menezes de Araujo 2021, we shall give an `explicit formula' of the corrected formula in the more algebraic framework of Lie-Rinehart algebras L (G.S.Rinehart, 1963, which are now being used in the study of singular foliations) over a commutative unital K-algebra A (where the ground ring K should contain the rational numbers as a subring). Similar results have been obtained independently by M.Al-Kaabi, K.Ebrahimi-Fard, D.Manchon, and H.Munthe-Kaas (2022) and X.Bekaert, N.Kowalzig, and P.Saracco (2022). L generalizes the Lie algebra of vector fields (more generally a Lie algebroid) and A the algebra of smooth functions in differential geometry. The enveloping algebra of L over A introduced by Rinehart plays the rôle of the differential operator algebra. The arising combinatorial problems can conveniently be treated in terms of the fibrewise shuffle comultiplication in the free algebra generated by L over A and the associated convolution products. The torsion and curvature terms arise in a morphism of Lie-Rinehart algebras Z from the free Lie algebra generated by L over A equipped with a Lie-Rinehart bracket isomorphic to the one on M.Kapranov's `free path Lie algebroid' (2007) to L which are related to the (infinitesimal) holonomy of the connection. Z is obtained by a simple explicit linear recursion.

Herbert Koch (Universität Bonn)

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Oberseminar Analysis-Probability at MPI Leipzig, E1 05 (Leibniz-Saal), 3:15 PM

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Chloé Rouyer (U Potsdam)

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Research Seminar Mathematical Statistics at WIAS Berlin, Erhard-Schmidt lecture room, 10 AM

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Azadeh Parvaneh (Bielefeld University)

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Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

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Tom ter Elst (Department of Mathematics, University of Auckland, New Zealand)

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Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

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Lukas Wessels (Université Côte d’Azur, Nice)

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Berlin Probability Colloquium at IMoS (TU Berlin), 4:15 PM

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Federico Cannerozzi (Bielefeld)

Singular Mean-Field Control via singular Mean-Field Games

Research Seminar Stochastic Analysis & Finance at HU Berlin, Rudower Chaussee 25, Room 1.115, 4:15 PM

Abstract: Mean-field control (MFC) and mean-field games (MFGs) provide two different descriptions of large interacting populations: in MFC, a central planner optimizes the collective performance of the system, while in MFGs each representative player optimizes individually against a given population distribution, which is then determined through a consistency condition. In general, the corresponding optimal controls and equilibria need not coincide.

A much tighter connection is known for potential mean-field games: an MFG equilibrium can be obtained by solving an associated auxiliary mean-field control problem. This naturally suggests the reverse question: can a mean-field control problem be solved by means of an auxiliary potential mean-field game?

We discuss this approach in two settings of mean-field control with singular controls, where the associated potential MFG can provide a tractable route to the original MFC problem. In a stationary Ornstein-Uhlenbeck model with two-sided singular controls, we obtain a two-way correspondence between the MFC problem and an associated potential MFG. In a finite-horizon framework with general dependence on the state distribution, we show that equilibria of the associated potential MFG yield optimal controls for the MFC problem, and illustrate the approach through a mean-field monotone follower problem.

Ziju Huang (HU Berlin)

TBA

Research Seminar Stochastic Analysis & Finance at HU Berlin, Rudower Chaussee 25, Room 1.115, 5:15 PM

Abstract: TBA

Kolyan Ray (Imperial College London)

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Research Seminar Mathematical Statistics at WIAS Berlin, Erhard-Schmidt lecture room, 10 AM

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Nicolas Verzelen (INRAE)

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Research Seminar Mathematical Statistics at WIAS Berlin, Erhard-Schmidt lecture room, 10 AM

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Lars Schröder (University of Twente)

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Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

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René Aïd (Université Paris-Dauphine)

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Berlin Probability Colloquium at IMoS (TU Berlin), 5:15 PM

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Philipp Schmocker (ETH Zürich)

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Research Seminar Stochastic Analysis & Finance at HU Berlin, Rudower Chaussee 25, Room 1.115, 4:15 PM

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Christoph Czichowsky (LSE)

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Research Seminar Stochastic Analysis & Finance at HU Berlin, Rudower Chaussee 25, Room 1.115, 5:15 PM

Abstract: TBA

Mark Podolskij (University of Luxembourg)

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Research Seminar Mathematical Statistics at WIAS Berlin, Erhard-Schmidt lecture room, 10 AM

Abstract: TBA

Martin Kružík (Czech Academy of Sciences Prague, CZ)

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Langenbach-Seminar at WIAS Berlin, Room 405-406, 2:15 PM

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Federica Masiero (Università degli Studi di Milano-Bicocca)

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Research Seminar Stochastic Analysis & Finance at HU Berlin, Rudower Chaussee 25, Room 1.115, 5:15 PM

Abstract: TBA

Ulrik Thinggaard Hansen (University of Innsbruck)

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Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

Abstract: TBA

Gianna Götzmann (U Augsburg)

Perturbed minimizing movements of time-dependent functionals on metric spaces

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: Minimizing movement schemes are a fundamental tool for approximating evolution equations that have a gradient-flow structure with respect to an energy. Motivated by homogenization problems involving, for instance, geo metries that evolve in time, we consider Γ-converging perturbations of time dependent driving energies. We study the convergence of the corresponding minimizing movement scheme and provide sufficient conditions under which limits of the scheme are (time-dependent) curves of maximal slope for the limit energy. We further exhibit examples showing that, in general, such a stability property may fail, with the limit motion depending on the rela tion between the time-discretization parameter τ and the perturbation pa rameter ε. This is joint work with Jan-Frederik Pietschmann and Antonio Tribuzio.