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