Uniformly counting rational points of bounded height on certain elliptic curves
Donnerstag, 12.10.23, 10:45-11:45, Raum 404, Ernst-Zermelo-Str. 1
Let E be an elliptic curve defined over a number field k. Canonical height on E in a certain sense measures arithmetic complexity of points of E(k). Given a real number B, it is often useful to have good bounds on the number of points of E(k) with height at most log(B), which we denote by N(B). While classical results give good bounds for a fixed elliptic curve, in general it is hard to get uniform results. This problem can be simplified if we assume the existence of a nontrivial point of prime order \(\bell\) in E(k). We will present a strategy for uniformly bounding N(B) in these families of curves, following methods developed by Bombieri and Zannier and later Naccarato (in the rational case for \(\bell=2\)), as well as new results on how this can be generalized to arbitrary number fields.
Vorstellung der Schwerpunktgebiete des Mathematischen Instituts
Donnerstag, 12.10.23, 12:00-13:00, Hörsaal Weismann-Haus
Vorstellung der Schwerpunktgebiete des Mathematischen Instituts
Donnerstag, 12.10.23, 12:00-13:00, Hörsaal Weismann-Haus
Many examples of abelian varieties satisfying the standard conjecture of Hodge type
Donnerstag, 12.10.23, 13:30-14:30, Raum 404, Ernst-Zermelo-Str. 1
Dimension of period spaces
Donnerstag, 12.10.23, 15:45-16:45, Raum 404, Ernst-Zermelo-Str. 1
Coupled 3D-1D solute transport models: Derivation, model error analysis, and numerical approximation
Dienstag, 17.10.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Starting from full-dimensional models of solute transport, we derive and analyze multi-dimensional (3D-1D) coupled models of time-dependent convection, diffusion, and exchange in and around pulsating vascular and perivascular networks. These models are widely applicable for modelling transport in vascularized tissue, brain perivascular spaces, vascular plants and similar environments. The well-posedness of the full and the multi-dimensional equations is established. In the derivation of the 3D-1D model, a 3D inclusion is reduced to its centerline. Thus, we establish a-priori estimates on the associated modelling errors in evolving Bochner spaces in terms of the inclusion's diameter. We consider both continuous and discontinuous Galerkin approximations to the coupled 3D-1D problems, and we discuss the convergence properties of the numerical schemes. Finally, we present numerical simulations in idealized geometries and in a brain mesh with a large network of vessels on its surface and inside the parenchyma. \n
Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball
Dienstag, 17.10.23, 16:00-17:00, Raum 127, Ernst-Zermelo-Str. 1
Let \(\bSigma\) be a compact hypersurface with a capillary boundary in a unit ball, in this talk, I will discuss the relative isoperimetric problem for such kinds of hypersurfaces. We introduce the relative quermassintegrals for such hypersurfaces from the variational viewpoint. Then by introducing some constrained nonlinear curvature flows, which preserve one geometric quantity invariant and monotone increase another, we obtain the Alexandrov-Fenchel inequality for such hypersurfaces. The talk is based on joint work with Profs. Guofang Wang and Chao Xia.\n
On one of the ends of MMP: Markovian planes
Freitag, 20.10.23, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
About a year ago, I gave a talk at this seminar on degenerations of surfaces with Wahl singularities. I tried by then to explain the explicit birational picture and some connections with exceptional collections of vector bundles. We see in this Mori theory divisorial contractions and flips controlled by Hirzebruch-Jung continued fractions (a summary can be found here https://arxiv.org/abs/1311.4844). As final products of this MMP, we arrive at either nef canonical class, smooth deformations of ruled surfaces, and degenerations of the projective plane (compare with the classical MMP for nonsingular projective surfaces). In this talk, I would like to explain these "Markovian planes". The name comes from the classification of such degenerations , due to Hacking and Prokorov 2010 (after Badescu and Manetti), as partial smoothings of P(a^2,b^2,c^2) where (a,b,c) satisfies the Markov equation x^2+y^2+z^2=3xyz. It turns out that there is a beautiful birational picture behind them, which in particular gives new insights to Markov's uniqueness conjecture. This is a joint work in progress together with Juan Pablo Zúñiga (Ph.D. student at UC Chile). \n\n
Convergence of Star Products on \(T^*G\)
Montag, 23.10.23, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
Star products can be seen as a generalization of a symbol calculus for differential operators. In fact, for cotangent bundles, the global symbol calculus yields a star product of a particular kind. While formal star products have been studied in detail with deep and exciting existence and classification theorems, convergence of the formal star products is still a widely open question. Beside several (classes of) examples, not much is known. In this talk I will focus on a particular class of examples, the cotangent bundles of Lie groups, where a nice convergence scheme has been established. I will try to avoid the technical details as much as possible and focus instead on the principal ideas of the construction. The results are joint work with Micheal Heins and Oliver Roth.
Flexibles Adaptieren im Mathematikunterricht
Dienstag, 24.10.23, 18:30-19:30, Hörsaal II, Albertstr. 23b
Eine möglichst gute Anpassung des Lernangebots an die Voraussetzungen der Schülerinnen und Schüler gilt als eines der zentralen Qualitätskriterien für guten Mathematikunterricht. Doch welche Voraussetzungen sind überhaupt relevant für das Lernen von Mathematik? Was bedeutet dies für die Auswahl bzw. Gestaltung von Aufgaben? Und wie kann man adaptiven Unterricht planen und trotzdem flexibel bleiben? Der Vortrag zeigt auf, welche Erkenntnisse hierzu aus der fachdidaktischen Forschung und der Lehr-Lernforschung vorliegen und wie Lehrkräfte diese nutzen können, um den Möglichkeiten und Herausforderungen beim Umgang mit unterschiedlichen Lernvoraussetzungen zu begegnen
Witten deformation for non-Morse functions and gluing formulas
Montag, 30.10.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
Witten deformation is a versatile tool with numerous applications in\nmathematical physics and geometry. In this talk, we will focus on the analysis\nof Witten deformation for a family of non-Morse functions, leading to a new\nproof of the gluing formula for analytic torsions. Then we could see that the\ngluing formula for analytic torsion can be reformulated as the Bismut-Zhang\ntheorem for non-Morse functions. Furthermore, this approach can be extended to\nanalytic torsion forms, which also provides a new proof of the gluing formula\nfor analytic torsion forms.
Separating \(\bmathsf{DC}(A)\) from \(\bmathsf{AC_\bomega}(A)\)
Dienstag, 31.10.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
The axiom of dependent choice \(\bmathsf{DC}\) and the axiom of countable choice \(\bmathsf{AC_\bomega}\) are two weak forms of the axiom of choice that can be stated for a specific set: \(\bmathsf{DC}(X)\) assets that any total binary relation on \(X\) has an infinite chain; \(\bmathsf{AC_\bomega}(X)\) assets that any countable family of nonempty subsets of \(X\) has a choice function. It is well-known that \(\bmathsf{DC}\) implies \(\bmathsf{AC_\bomega}\). We show that it is consistent with \(\bmathsf{ZF}\) that there is a set \(A\bsubseteq \bmathbb{R}\) such that \(\bmathsf{DC}(A)\) holds but \(\bmathsf{AC_\bomega}(A)\) fails.\n\n
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.
Topological Censorship
Montag, 4.12.23, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
FORMULATING CAPILLARY SURFACES IN THE CONTEXT OF VARIFOLDS
Dienstag, 5.12.23, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
Capillary surface is a fundamental geometric object, describing a particular boundary behavior of surface in a given container. In some recent works of Kagaya-Tonegawa (Hiroshima Math. J. 47(2): 139-153, 2017. https://doi.org/10.32917/hmj/1499392823), De Masi-De Philippis (https://doi.org/10.48550/arXiv.2111.09913), weak capillary surfaces are formulated, using the language of varifolds, and named ”pair of varifolds with fixed contact angle condition”.\n\nIn this talk, we will discuss some recent development in this direction, and prove a strong boundary maximum principle for a specific class of pairs of varifolds with fixed contact angle, which generalizes Li-Zhou’s boundary maximum principle for free boundary varifolds (Comm.\nAnal. Geom. (29): 1509–1521, 2021. https://doi.org/10.4310/CAG.2021.v29.n6.a7). The similar results in the context of rectifiable cones will be discussed as well. If time permits, we will also discuss some applications of the weak formulation, including the establishment of the\nSimon-type monotonicity identities as well as the Li-Yau-type inequalities.
Prescription of Dirac Eigenvalues, Partial Eigenbundles and Surgery
Montag, 11.12.23, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
The prescription of eigenvalues of the Dirac operator on a closed spin manifold requires, besides the usual analytical methods à la Uhlenbeck and Dahl, also surgery methods to transport spectral data along a bordism. In this talk, I will give the necessary basics as well as an overview of the prescription of double eigenvalues on spin manifolds.
Continuum Limit of Nearest Neighbor and Random Long-Range Interactions
Dienstag, 12.12.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The thesis deals with the limit behavior of discrete energies with different types of interactions between points. On the one hand, only nearest neighbor interactions are considered and on the other hand random long-range interactions. For the latter some assumptions on the conductance have to be made. In a last step, we will try to combine these two types of interactions and investigate whether some assumptions can be dropped in this case.\n\n \n\n
Forcings With the Approximation Property
Dienstag, 12.12.23, 14:30-15:30, Raum 232 in der Stochastik
We introduce the approximation property which was implicit in early work of Mitchell and later defined explicitly by Hamkins. In modern set theory, the approximation property has gotten new attention through the ineffable slender list property (ISP), introduced by Weiss in his PhD thesis. In this talk, we give a criterion for the approximation property which is very applicable to variants of Mitchell Forcing, allowing us to obtain several consistency results regarding ISP.
talks about his Master Thesis
Mittwoch, 13.12.23, 16:00-17:00, Raum 226, Hermann-Herder-Str. 10
Arithmetic aspects of quantum cohomology
Freitag, 15.12.23, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Shintanis (wenig bekannte) Vermutung zum 12. Hilbertschen Problem besagt, dass abelsche Erweiterungen von \nZahlkörpern mit Hilfe spezieller Werte von verallgemeinerten Gammafunktionen erzeugt werden können.\n\nDiese Vermutung ist seit gut 50 Jahren ungelöst und weitestgehend unverstanden. Unser Vortrag wird daran leider nichts ändern.\n\nEine Hauptschwierigkeit der Shintanischen Vermutung liegt darin, dass (verallgemeinerte) Gammafunktionen sehr schwer handhabbar sind.\n\nIn unserem (elementar gehaltenen) Vortrag wollen wir erklären, wie der Formalismus der Quantumkohomologie neue Einsichten in das Wesen der Gammafunktionen\nliefern könnte.
String Topology of Compact Symmetric Spaces
Montag, 18.12.23, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
On the homology of the free loop space of a closed manifold M there exists the so-called Chas-Sullivan product. It is a product defined via the concatenation of loops and can, for example, be used to study closed geodesics of Riemannian or Finsler metrics on M. In this talk I will outline how one can use the geometry of symmetric spaces to partially compute the Chas-Sullivan product. In particular, we will see that the powers of certain non-nilpotent homology classes correspond to the iteration of closed geodesics in a symmetric metric. Some triviality results on the Goresky-Hingston cohomology product will also be mentioned. This talk is based on joint work with Maximilian Stegemeyer.
On strong approximation of SDEs with a discontinuous drift coecient
Dienstag, 19.12.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The classical assumption in the literature on numerical approximation of stochastic differential equations (SDEs) is global Lipschitz continuity of the coecients of the equation.\nHowever, many SDEs arising in applications fail to have globally Lipschitz continuous coecients.\nIn the last decade an intensive study of numerical approximation of SDEs with nonglobally Lipschitz continuous coecients has begun. In particular, strong approximation\nof SDEs with a drift coecient that is discontinuous in space has recently gained a lot of\ninterest. Such SDEs arise e.g. in mathematical finance, insurance and stochastic control\nproblems. Classical techniques of error analysis are not applicable to such SDEs and well\nknown convergence results for standard methods do not carry over in general.\n\nIn this talk I will present recent results on strong approximation of such SDEs.\n\nThe talk is based on joint work with Thomas M¨uller-Gronbach (University of Passau).
IP-Mengen, Produktmengen und stabile Formeln
Dienstag, 19.12.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Präsentation der Masterarbeit.
Scalar curvature comparison geometry and the higher mapping degree
Montag, 8.1.24, 16:00-17:00, Ort noch nicht bekannt
Llarull proved in the late '90s that the round \(n\)-sphere is area-extremal in the sense that one can not increase the scalar curvature and the metric simultaneously. Goette and Semmelmann generalized Llarull's work and proved an extremality and rigidity statement for area-non-increasing spin maps \(f\bcolon M\bto N\) of non-zero \(\bhat{A}\)-degree between two closed connected oriented Riemannian manifolds.\n\nIn this talk, I will extend this classical result to maps between not necessarily orientable manifolds and replace the topological condition on the \(\bhat{A}\)-degree with a less restrictive condition involving the so-called higher mapping degree. For that purpose, I will first present an index formula connecting the higher mapping degree and the Euler characteristic of~\(N\) with the index of a certain Dirac operator linear over a \(\bmathrm{C}^\bast\)-algebra. Second, I will use this index formula to show that the topological assumptions, together with our extremal geometric situation, give rise to a family of almost constant sections that can be used to deduce the extremality and rigidity statements.\n
Moduli Spaces of Positive Curvature Metrics
Montag, 15.1.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
Besides the space of positive scalar curvature metrics \(\bmathrm{Riem}^+(M)\), various moduli spaces have gained a lot of attention. \nAmong those, the observer moduli space arguably has the best behaviour from a homotopy-theoretical perspective because the subgroup of \btextit{observer diffeomorphisms} acts freely on the space of Riemannian metrics if the underlying manifold \(M\) is connected.\n\nIn this talk, I will present how to construct non-trivial elements in the second homotopy of the observer moduli space of positive scalar curvature metrics for a large class for four-manifolds. I will further outline how to adapt this construction to produce the first non-trivial elements in higher homotopy groups of the observer moduli space of positive sectional curvature metrics on complex projective spaces.
TBA
Dienstag, 16.1.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
TBA
The p-Laplacien in the setting of multi-valued operators.
Dienstag, 16.1.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\nAbstract: The thesis provides proofs for the existence of solutions for both the stationary problem\n\[ \blambda u + Au \bni f \bin \bOmega \bsubset R^N;~~ u=0 ~\btext{on}~~ \bpartial \bOmega\] \nand the non-stationary problem \n\[dy(t)/dt + Ay(t) \bni f(t) ~ \btext{for}~ t\bin[0,T];~~ y(0)=y_0 \]for A being a maximal monotone/accretive operator on \(L^2(\bOmega)\). It especially considers such operators A that arise as the sub-differential of some energy-functional and also shows some regularity for them in the non-stationary case. As an example the theory is applied to the p-Laplace operator.\n
Elementare Differentialgeometrie zum Anfassen: Vorstellung eines Seminars für Lehramtsstudierende
Dienstag, 16.1.24, 18:30-19:30, Hörsaal II, Albertstr. 23b
Vortrag entfällt leider!
Surgery on fold maps
Montag, 22.1.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
In this talk I will explore the notion of fold maps which are a natural generalization of Morse functions. Morse functions play a central role in the classification of manifolds and getting rid of their critical points is a crucial step in the proof of the h-cobordism theorem. I will describe a similar procedure for eliminating so called fold-singularities. This is similar in spirit but more flexible compared to the above-mentioned removal of critical points as it allows to perform surgery on the underlying manifold. If time permits I will also explain how this can be used to study fiber bundles and their characteristic classes.
A-priori bounds for geometric FE discretizations of a Cosserat rod and simulations for microheterogeneous prestressed rods
Dienstag, 23.1.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In summary, this thesis focuses on developing an a priori theory for geometric finite\nelement discretizations of a Cosserat rod model, which is derived from incompatible\nelasticity. This theory will be supported by corresponding numerical experiments\nto validate the convergence behavior of the proposed method.\nThe main result describes the qualitative behavior of intrinsic H 1 -errors and\n\nL^2 -errors in terms of the mesh diameter 0 < h ≪ 1 of the approximation scheme:\n\nD 1,2 (u, u h ) ≲ h m , d L 2 (u, u h ) ≲ h m+1 ,\n\nfor a sequence of m-order discrete solutions u h and an exact solution u.
(FÄLLT LEIDER WEGEN BAHNSTREIK AUS)
Donnerstag, 25.1.24, 15:00-16:00, Ort noch nicht bekannt
Generalized Seiberg-Witten equations and where to find them
Montag, 29.1.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We explore the framework of Generalized Seiberg-Witten Equations, aimed at deriving fresh invariants for smooth four-manifolds. These generalizations replace the standard spinor bundle with a suitable hyperKähler manifold for the spinor fields. This departure opens up exciting new possibilities for studying the smooth structures of four-dimensional manifolds, while also including a lot of well-known invariants, the most prominent example the Anti-Self-Duality equations and the resulting Donaldson invariants.\n\nWe then present how to compute the solution spaces in on of the most simple cases, where the spinor takes values in a four dimensional hyperKähler manifold, and show how this leads to invariants for four dimensional symplectic and Kähler manifolds, while also giving a geometric interpretation.
Uniform Interpolation for Intuitionistic Logic
Dienstag, 30.1.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
We show that intuitionistic logic has uniform interpolation, a strong property possessed by some logics. For this we introduce the algebraic representation of intuitionistic logic (Heyting algebras) and their dual spaces (Esakia spaces). This duality is similar to that between Boolean algebras and Stone spaces for classical logic. We then show how to prove an open mapping theorem for Esakia spaces, and how uniform interpolation follows from this.
Observations on G2 Moduli Spaces
Montag, 5.2.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
In his two seminal articles, Dominic Joyce not only constructed the first examples of closed manifolds with G2-holonomy metrics, but also proved that the moduli space of all G2-metrics on a closed manifold is itself a finite-dimensional manifold. The statement is, however, only a local one, and the global topological properties of these moduli spaces have remained quite mysterious ever since. Indeed, up to now, we only know that they may be disconnected by the work of Crowley, Goette, and Nordström; the question whether all path components are contractible or not has not been answered yet.\n\nIn this talk, I will give a short introduction to G2 metrics and their moduli spaces and outline a construction of a non-trivial element in the second homotopy of the easier-acceesible observer moduli space of G2 metrics on one of Joyce's examples.\nIf time permits, I will indicate why and how this non-trivial example might also descend to the (full) moduli space.\n\nThis talk is based on ongoing joint work with Sebastian Goette.\n
Convergence of computational homogenization methods based on the fast Fourier transform
Dienstag, 6.2.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Since their inception in the mid 1990s, computational methods based on the fast Fourier transform (FFT) have been established as efficient and powerful tools for computing the effective mechanical properties of composite materials. These methods operate on a regular grid, employ periodic boundary conditions for the displacement fluctuation and utilize the FFT to design matrix-free iterative schemes whose iteration count is (most often) bounded independently of the grid spacing.\nIn the talk at hand, we will take a look both at the convergence of the used iterative schemes and the convergence of the underlying spectral discretization. Remarkably, despite the presence of discontinuous coefficients, the spectral discretization enjoys the same convergence rate as a finite-element discretization on a regular grid. Moreover, the convergence behavior of the effective stresses profit from a superconvergence phenomenon apparently inherent to computational homogenization problems.\n---------------------------------------------------------
Hyperbolicity and rigidity in moduli spaces of polarized manifolds
Dienstag, 6.2.24, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
Neues zur Additivität der Kodaira-Dimension
Freitag, 9.2.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Approximation of the local time of a sticky diffusion and applications
Freitag, 9.2.24, 12:00-13:00, Raum 125, Ernst-Zermelo-Str. 1
We show that the local time of a sticky diffusion can be approximated by certain kind of high-frequency path statistics. This generalizes results of Jacod for smooth diffusions. We prove various form of the result that depend on the type of normalizing sequence we use. We then use the result to: \n1. devise a consistent stickiness parameter estimator, \n2. assess the behavior of number of crossing statistics, \n3. (if time allows) assess the convergence rate of discrete-time hedging strategies in a "sticky Black-Scholes model".\n
On strong approximation of SDEs with a discontinuous drift coefficient
Dienstag, 13.2.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The classical assumption in the literature on numerical approximation of stochastic differential equations (SDEs) is global Lipschitz continuity of the coefficients of the equation.\nHowever, many SDEs arising in applications fail to have globally Lipschitz continuous coefficients.\nIn the last decade an intensive study of numerical approximation of SDEs with nonglobally Lipschitz continuous coefficients has begun. In particular, strong approximation\nof SDEs with a drift coefficient that is discontinuous in space has recently gained a lot of\ninterest. Such SDEs arise e.g. in mathematical finance, insurance and stochastic control\nproblems. Classical techniques of error analysis are not applicable to such SDEs and well\nknown convergence results for standard methods do not carry over in general.\nIn this talk I will present recent results on strong approximation of such SDEs.\nThe talk is based on joint work with Thomas M¨uller-Gronbach (University of Passau)
Counterexamples to Hedetniemi's Conjecture
Montag, 26.2.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
The tensor product \(G\btimes H\) of two finite graphs \(G\) and \(H\) is defined by \(V(G\btimes H) = V(G)\btimes V(H)\) and two vertices \((g_1,h_1)\) and \((g_2,h_2)\) being connected if \(g_1 E_G g_2\) and \(h_1 E_H h_2\). In 1966 Hedetniemi formulated the conjecture that \(\bchi(G\btimes H) = min(\bchi(G),\bchi(H))\). Only in 2019, more then 50 years later, Shitov discovered the existence of counterexamples. We will follow his proof and introduce some interesting notions along the way.
Cohomology of punctual Hilbert schemes of smooth projective surfaces
Montag, 4.3.24, 10:15-11:15, Raum 404, Ernst-Zermelo-Str. 1
The punctual Hilbert scheme X^[n] of a projective scheme X over a field k is a scheme parameterising the closed subschemes of X of length n; loosely speaking, those that have n points when counted with multiplicity. It turns out that if X is smooth, X^[n] is necessarily smooth as well if and only if dim X < 3, making the case where X is a smooth projective surface of particular interest. Remarkable work by L. Göttsche demonstrated that if k is C or the algebraic closure of a finite field, the Betti numbers and the Euler characteristic of X^[n] can be written in terms of explicit generating functions related to modular forms. This talk will review some properties of punctual Hilbert schemes in general, study the special case of projective surfaces and discuss their cohomology.