Vorträge in der Woche 11.12.2023 bis 17.12.2023

Montag, 11.12.2023: Informationsveranstaltung zu Auslandsstudien (Erasmus, CIVIS, Trento, ...)

Daniele Agostini + FBZ

Die Veranstaltung richtet sich an Studierende in den Studiengängen des Fachbereich Mathematik, die sich über die Möglichkeiten für ein Auslandsstudium informieren möchten.

Donnerstag, 14.12.2023: Infima of the Energy Functional in Homotopy Classes of Mappings of CP^N

Dr. Joseph Hoisington (Max-Planck-Institut für Mathematik - Bonn)

We will determine the infimum of the energy in all homotopy classes of mappings from complex projective spaces to Riemannian manifolds, by showing that it is proportional to the infimal area in the homotopy class of mappings of the 2-sphere representing the induced action on the second homotopy group. We will also discuss some background and related results about stable harmonic mappings of complex projective spaces.

Donnerstag, 14.12.2023: Die Plurigenera torischer Hyperflächen

Julius Giesler (Universität Tübingen)

Die Plurigenera sind wichtige birationale Invarianten glatter algebraischer Varietäten beziehungsweise Varietäten mit milden Singularitäten. Wir präsentieren eine Formel, die die Plurigenera (minimaler) torischer Hyperflächen in Abhängigkeit von Gitterpunkten und einem Polytop, dem sogenannten Fine Interior, ausdrückt. Wir illustrieren diese Formel in niedrigen Dimensionen anhand von glatten algebraischen Kurven, minimalen algebraischen Flächen sowie anhand von Calabi-Yau torischen Hyperflächen

Donnerstag, 14.12.2023: Differential Chain Approach for the Unitary Dirac Evolution between Cauchy Surfaces

Ozan Semin (Tübingen)

In this talk, we aim to show a rigorous proof of the unitarity of the Dirac evolution between two arbitrary Cauchy surfaces. Unlike traditional constraints, we refrain from imposing differentiability requirements on these surfaces. Our proof employs two important tools. Firstly, we introduce 'Differential Chains', a concept pioneered by Jenny Harrison, enabling the extension of geometric theorems to less regular surfaces. Secondly, we utilize Bohmian trajectories as a complementary tool. As a corollary, we demonstrate the validity of the equivariance principle in relativistic Bohmian Mechanics. For simplicity, we focus our analysis on the one-particle case, noting that the proof for the non-interacting N-particle case is almost identical. Specifically, assuming the surface wave functions themselves are differentiable and evolve according to the Dirac equation, we demonstrate the preservation of unitarity in wave function evolution between two arbitrary Cauchy surfaces, even those lacking differentiability, while confirming the consistency of the equivariance principle in Bohmian Mechanics.

Donnerstag, 14.12.2023: Q-curvature, 4th order mass

Dr. Nicolas Marque (Elie Cartan Institute of Lorraine - Nancy)

The ADM mas is a scalar metric quantity computed at the infinity of an Asymptotically Euclidean manifold. It has a physical meaning (mass of an isolated stellar object), a geometric weight (positivity and rigidity) and an analytic importance (in the Green function of the Yamabe operator), and it comes from the analysis of conserved quantities from a relativistic approach of gravitation. If one changes the gravitational theory for a fourth order Lagrangian, the same analysis of conserved quantities yieds a fourth order mass. One can then show it has similar properties to the ADM mass, and a deeper analysis reveals its link to a fundamental geometric quantity: the Q-curvature.

Donnerstag, 14.12.2023: On the stability of many-particle systems

Kajetan Söhnen (LMU/Tübingen)

Although the Vlasov-Poisson equation has established itself as a helpful tool yielding strong results, it still lacks a rigorous justification. So far the mathematical results either worked with a cut-off on the force or used a slightly weaker interaction all together. In this talk we will present a stability property that the typical many particle system satisfies. When adding a new particle to a N body system, it will remain stable. How the system reacts to small variations regarding the position of the added particle can be estimated using mean-field theory. This has interesting consequences for preceding results and probabilistic estimates.

Freitag, 15.12.2023: Konvergenz von Gradientenmethoden unter der Polyak-Lojasiewicz-Bedinggung, Teil II

Johannes Laub (Uni Tübingen)