Adapted Wasserstein distance for SDEs with irregular coefficients
Dienstag, 7.11.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We solve an optimal transport problem under probabilistic constraints, where the marginals are laws of solutions of stochastic differential equations with irregular, that is non-globally Lipschitz continuous coefficients. Numerical methods are employed as a theoretical tool\nto bound the adapted Wasserstein distance. This opens the door for\ncomputing the adapted Wasserstein distance in a simple way.\n\n\nJoint work with B. Robinson (University of Vienna).
Die Tiefenstrukturen als Qualitätsmerkmal von Mathematikunterricht am Beispiel der Produktregel
Dienstag, 7.11.23, 18:30-19:30, Hörsaal II, Albertstr. 23b
Es ist wissenschaftlich anerkannt, dass die fachübergreifenden Tiefenstrukturen ein wesentliches Qualitätsmerkmal von lernwirksamem Unterricht sind. Anhand des vom IBBW (Institut für Bildungsanalysen in Baden-Württemberg) entwickelten Unterrichtsfeedbackbogens Tiefenstrukturen (UFB) werden diese Tiefenstrukturen beobachtbar. Wie sich die Tiefenstrukturen anhand der Items, die im UFB beschrieben sind, mathematik-spezifisch in einer Unterrichtsstunde ausdifferenzieren, wird in diesem Vortrag am Beispiel „Einführung der Produktregel, Klasse 11, Gymnasium“ gezeigt. Der Vortrag führt in die Tiefenstrukturen und den UFB ein, führt eine fachdidaktische Analyse zur Produktregel durch und zeigt konkrete Unterrichtselemente auf, in denen diese Aspekte lernwirksam verknüpft sind.
(FÄLLT WEGEN KRANKHEIT AUS)
Donnerstag, 9.11.23, 15:00-16:00, Ort noch nicht bekannt
Three-dimensional Riemannian manifolds are called asymptotically Euclidean if, outside a compact set, they are diffeomorphic to the exterior region of a ball in Euclidean space, and if the Riemannian metric converges to the Euclidean metric as the Euclidean radial coordinate \(r\) tends to infinity. In 1996, Huisken and Yau proved existence of a foliation by constant mean curvature (CMC) surfaces in the asymptotic end of an asymptotically Euclidean Riemannian three-manifold. Their work has inspired the study of various other foliations in asymptotic ends, most notably the foliations by constrained Willmore surfaces (Lamm—Metzger—Schulze) and by constant expansion/null mean curvature surfaces in the context of asymptotically Euclidean initial data sets in General Relativity (Metzger, Nerz).\n \nAfter a rather extensive introduction of the central concepts and ideas, I will present a new foliation by constant spacetime mean curvature surfaces (STCMC), also in the context of asymptotically Euclidean initial data sets in General Relativity (joint work with Anna Sakovich). This STCMC-foliation is well-suited to consistently define the center of mass of an isolated system in General Relativity and thereby answers some previously open questions of relevance in General Relativity. At the end, I will touch upon subtle convergence issues for the center of mass (joint work with Christopher Nerz and with Melanie Graf and Jan Metzger).
!! FÄLLT WEGEN KRANKHEIT AUS !!
Freitag, 10.11.23, 12:00-13:00, Raum 125, Ernst-Zermelo-Str. 1
Abstract: Motivated by model uncertainty and stochastic control problems, we develop a systematic theory for convex monotone semigroups on spaces of continuous functions. The present approach is self-contained and does, in particular, not rely on the theory of viscosity solutions. Instead, we provide a comparison principle for semigroups related to Hamilton-Jacobi-Bellman equations which uniquely determines the semigroup by its infinitesimal generator evaluated at smooth functions. While the statement itself resembles the classical analogue for linear semigroups, the proof requires the introduction of several novel analytical concepts such as the Lipschitz set and the \(\bGamma\)-generator. Furthermore, starting with a generating family \((I(t))_{t\bgeq 0}\) of operators, we show that the limit\n\(S(t)f:=\blim_n \bto \binfty I(\bfrac{t}{n})^nf\) \ndefines a semigroup which is uniquely determined by the time derivative \(I’(0)f\) for smooth functions \(f\). We identify explicit conditions for the generating family that are transferred to the semigroup and can easily be verified in applications. The abstract results are illustrated by emphasizing the structural link between approximation schemes for convex monotone semigroups and law of large numbers and central limit theorem type results for convex expectations. Furthermore, the limit can be represented as a stochastic control problem.\n
Geomod Conference in Model Theory, day 1
Montag, 13.11.23, 09:30-10:30, Hörsaal II, Albertstr. 23b
Monday, 13.11.2023\n\nSchedule:\n9:30-10:30 Nick Ramsey (Notre Dame), Higher amalgamation in PAC fields\n\n\n10:40-11:15 Coffee Break \n\n11:20-12:20 Nadja Hempel Valentin (Düsseldorf), Pushing Properties for NIP Groups and Fields up the n-dependent\nhierarchy\n\n12:20-13:20 Omar Leon Sánchez (Manchester), Some remarks on differentially large fields and CODFs\n\n13:20-15:30 Lunch Break\n\n15:30-16:30 Silvain Rideau-Kikuchi (ENS Ulm), An imaginary Ax-Kochen-Ershov principle\n\n16:30-17:20 Coffee Break \n\n17:20-17:45 Simone Ramello (Münster), The Kaplansky theory of non-inversive valued difference fields\n\n17:50-18:15 Paul Wang (ENS Ulm), On groups and fields interpretable in various NTP2 fields \n\n18:30-20:00 Reception and Poster Session\n
String topology of the space of paths with endpoints in a submanifold
Montag, 13.11.23, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
String topology is the study of algebraic structures on the homology of the free loop space of a closed manifold.\nThe most famous operation is the Chas-Sullivan product which is a graded commutative and unital product on the homology of the free loop space.\nIn this talk we study the space of paths in a manifold whose endpoints lie in a chosen submanifold.\nIt turns out that the homology of this space also admits a product which is defined similarly to the one of Chas and Sullivan.\nMoreover, the homology of this path space is a module over the Chas-Sullivan ring. \nWe will see that in some situations both structures together form an algebra - i.e. the product on homology of the path space with endpoints in a submanifold is an algebra over the Chas-Sullivan ring - but that this property does not hold in general.
Geomod Conference in Model Theory, day 2
Dienstag, 14.11.23, 09:30-10:30, Hörsaal II, Albertstr. 23b
Tuesday 14.11.2023 \n\nSchedule\n\n9:30-10:30 Tom Scanlon (Berkeley), TBA\n\n10:40-11:15 Coffee Break\n\n11:20-12:20 Adele Padgett (McMaster) Some equations involving the Gamma function\n\n12:20-13:20 Itay Kaplan (Jerusalem), A result on the chromatic number of stable graphs. \n\n13:20-15:30 Lunch Break\n\n15:30-15:55 Shezad Mohamed (Manchester), Very slim differential fields\n\n16:00-16:25 Neer Bhardwaj (Rehovot), Approximate Pila-Wilkie type counting for complex analytic sets\n\n16:30-17:20 Coffee Break\n\n17:20-17:45 Sebastian Eterovic (Leeds), Solutions to equations involving the modular j function and its\nderivatives\n\n17:50-18:15 Haydar Göral (Izmir), Lehmer’s conjecture via model theory\n\n
1D approximation in Wasserstein spaces
Dienstag, 14.11.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Given a Borel probability measure, we seek to approximate it with a measure uniformly\ndistributed over a 1-dimensional set. With this end, we minimize the Wasserstein distance of this fixed measure to all probability measures uniformly distributed to connected 1 dimensional sets and a regularization term given by their length. To show existence of solution to this problem, one cannot easily resort to the direct method in the calculus of variations due to concentration of mass effects. Therefore, we propose a relaxed problem in the space of probability measures which always admits a solution. In the sequel, we show that whenever the initial measureis absolutely continuous w.r.t. the 1-Hausdorff measure (in particular for absolutely continuous measures w.r.t. Lebesgue measure in R^d) then the solution will be a rectiable measure. This allows us to perform a blow-up argument that, in dimension 2, shows that the solution has a uniform density, being therefore a solution to the original problem. Finally, we prove a phase-field approximation for this problem in the form of a Gamma-convergence result of a functional reminiscent of the Ambrosio-Tortorelli approximation for the Mumford-Shah problem, with the additional property of enforcing connectivity of the 1-dimensional sets that emerges from the approximation. This last feature is achieved with the connectivity functional introduced by Dondl and Wojtowytsch.
Geomod Conference in Model Theory, day 3
Mittwoch, 15.11.23, 09:30-10:30, Hörsaal II, Albertstr. 23b
Wednesday 15.11.2023\n\nSchedule\n\n9:30-10:30 Benjamin Castle (Be’er Sehva), Zilber's Restricted Trichotomy for o-minimal Structures in Higher\nDimensions\n\n10:40-11:15 Coffee Break\n\n11:20-12:20 Zoé Chatzidakis (CNRS-IMJ Paris), Some remarks about difference-differential fields\n\n12:20-13:20 Daniel Palacín (Complutense Madrid), Algebraic structures without the CBP\n\n
Geomod Conference in Model Theory, day 4
Donnerstag, 16.11.23, 09:30-10:30, Hörsaal II, Albertstr. 23b
Thursday 16.11.2023\n\nSchedule\n\n9:30-10:30 Anand Pillay (Notre Dame), Invariant measures on automorphism groups of prime models\n\n10:40-11:15 Coffee Break\n\n11:20-12:20 Isabel Müller (Cairo), On the Model Theory of Generic Nilpotent Groups and Lie Algebras\n\n12:20-13:20 Christian d’Elbée (Leeds), Free amalgamation of Lazard Lie Algebras\n\n13:20-15:30 Lunch Break\n\n15:30-16:30 Rémi Jaoui (CNRS-Lyon 1), On the Galois group of the equation of one-forms of a differential field\nextension\n\n16:30-17:20 Coffee Break\n\n17:20-17:45 Giuseppina Terzo (Naples), Generic derivations on Algebraically Bounded Structures\n\n17:50-18:15 Aris Papadopoulos (Leeds), Zarankiewicz's Problem in Presburger Arithmetic
Geomod Conference in Model Theory, day 5
Freitag, 17.11.23, 09:30-10:30, Hörsaal II, Albertstr. 23b
Friday 17.11.2023\n\nSchedule\n\n9:30-10:30 Raf Cluckers (Lille), Motivic integration and Mellin transforms.\n\n\n10:40-11:15 Coffee Break\n\n11:20-12:20 Rosario Mennuni (Pisa), O-minimality, domination, and preorders\n\n12:20-13:20 Konstantinos Kartas (IMJ, Paris), On C_i fields of mixed characteristic\n\n13:20-15:00 Lunch Break\n\n15:00-16:00 Martin Ziegler (Freiburg), Pairs of algebraically closed fields and the Hilbert scheme.
Positive intermediate Ricci curvature, surgery and Gromov's Betti number bound
Montag, 20.11.23, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Positive intermediate Ricci curvature is a family of interpolating curvature conditions between positive sectional and positive Ricci curvature. While many results that hold for positive sectional or positive Ricci curvature have been extended to these intermediate conditions, only relatively few examples are known so far. In this talk I will present several extensions of construction techniques from positive Ricci curvature to these curvature conditions, such as surgery, gluing and bundle techniques. As an application we obtain a large class of new examples of manifolds with a metric of positive intermediate Ricci curvature, including all homotopy spheres that bound a parallelisable manifold, and show that Gromov's Betti number bound for manifolds of non-negative sectional curvature does not hold from positive Ricci curvature up to roughly halfway towards positive sectional curvature. This is joint work with David Wraith.\n
Automorphismentürme von Gruppen
Dienstag, 21.11.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Eine Gruppe mit trivialem Zentrum lässt sich in ihre Automorphismengruppe einbetten, die selbst auch triviales Zentrum hat. Durch Iteration bekommt man so den Automorphismenturm einer Gruppe. Wieland hat gezeigt, dass er für endliche Gruppen nach endlich vielen Schritten stationär wird. Simon Thomas hat dies auf unendliche Gruppen verallgemeinert. Der Vortrag präsentiert einige Ergebnisse aus diesem Umfeld.
Stochastik verstehen – wie kann das funktionieren?
Dienstag, 21.11.23, 18:30-19:30, Hörsaal II, Albertstr. 23b
Die Stochastik umfasst bekanntlich die beschreibende Statistik, die Wahrscheinlichkeitsrechnung sowie die beurteilende Statistik. Für die Entwicklung eines stochastischen Verständnisses sollten in einem verständnisorientierten Mathematikunterricht diese drei Gebiete nicht isoliert betrachtet werden. Die zentralen Begriffe sind nicht einfach zu verstehen. So ist die Wahrscheinlichkeit ein schwieriges und kontraintuitives Konzept. In dem Vortrag werden einige Aufgaben vorgestellt, die (hoffentlich) geeignet sind, dass Schülerinnen und Schüler geeignete Grundvorstellungen für die mathematischen Begriffe ausbilden. Hierbei spielen Simulationen eine wichtige Rolle. Dies wird durch den Einsatz von GeoGebra exemplarisch vorgestellt. Falls Sie einen Rechner mit einer GeoGebra-App zufällig mitbringen, können Sie auch experimentieren. Es werden alltagstaugliche und bewährte Problemstellungen aus beiden Sekundarstufen vorgestellt und hoffentlich lebhaft diskutiert. Im Zentrum steht dabei das 1/√n -Gesetz. Für eine vertiefende Beschäftigung werden Materialien zur Verfügung gestellt.
Resolvent estimates for one-dimensional Schroedinger operators with complex potentials
Dienstag, 28.11.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We study one-dimensional Schroedinger operators with unbounded complex potentials of various growths (from iterated logs to super-exponentials). We derive asymptotic formulas for the norm of the resolvent as the spectral parameter diverges along the imaginary and real axes. In each case, our analysis yields an explicit leading order term as well as an optimal estimate of the remainder. We also discuss several extensions of the main results, their interrelation with the complementary estimates based on non-semiclassical pseudomode construction in [KS-19] and several examples.\n\nThe talk is based on the joint work [AS-23] with A. Arnal.\n\nReferences:\n\n[AS-23] A. Arnal and P. Siegl: Resolvent estimates for one-dimensional Schroedinger operators with complex potentials, 2023, J. Funct. Anal. 284, 109856\n\n[KS-19] D. Krejcirik and P. Siegl: Pseudomodes for Schroedinger operators with complex potentials, 2019, J. Funct. Anal. 276, 2856-2900
Poisson structures from corners of field theories
Mittwoch, 29.11.23, 16:15-17:15, Hörsaal II, Albertstr. 23b
The BV formalism and its shifted versions in field theory have a nice compatibility with boundary structures. Namely, one such structure in the bulk induces a shifted (possibly degenerated) version on its boundary, which can be interpreted as a Poisson structure (up to homotopy). I will present the results for some field theories, in particular, 4D BF theory and 4D gravity.\n\n
A cohomological view of quantum field theory
Donnerstag, 30.11.23, 15:00-16:00, Hörsaal II, Albertstr. 23b
In physics, fields (e.g., the electromagnetic field) are quantities that depend on time. In mathematics, they correspond to functions, vector fields, sections of sheaves, depending on the context. Classical and quantum field theory describe the evolution of fields and study how to compute their properties. One useful approach takes as its starting point the action, a functional on the space of fields.\nIn the classical theory, the PDEs the fields have to satisfy (e.g., the Maxwell equations) are the critical points of this functional; in\nthe quantum theory, one also has to study fluctuations around them, and one heuristic approach is the functional integral, where one formally integrate over the space of fields.\n \nAn interesting situation occurs when there are symmetries: vector fields on the space of fields under which the action functional is invariant. In classical physics, one is then interested in the space of critical points modulo symmetries. In the functional integral approach to quantum theory, one is morally interested in integrating over the quotient of the space of fields by the symmetries, but this is too complicated (and too singular). Instead one considers a section, called a gauge fixing, i.e., a submanifold that intersects the symmetries transversally, and integrates over it. Invariance under the choice of this section is a fundamental question.\n \nThe BV formalism is a cohomological procedure that solves the two goals: it gives a resolution of the critical locus modulo symmetries and allows showing the formal independence of the functional integral from deformations of the gauge fixing. In addition to these properties, in this talk I will recall other important aspects. One is the study of field theories on manifolds with boundaries (or, more generally, higher-codimensional stratifications) where the BV formalism is nicely coupled with the BFV formalism—responsible for the cohomological resolution of the reduced phase space (roughly speaking, the space of initial conditions).\nAnother aspect is the BV pushforward (i.e., a partial integration) which plays a role in defining effective theories, in casting renormalization à la Wilson for gauge theories, and in constructing nontrivial observables.