see program
Thursday, 11.4.24, 09:50-10:50, Basel, Hörsaal 120 in Kollegienhaus (Petersplatz 1)
Minimal geodescis
Monday, 22.4.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
A geodesic \(c:\bmathbb{R}\bto M\) is called minimal if a lift to the universal covering globally minimizes distance. On the \(2\)-dimensional torus with an arbitrary Riemannian metric there are uncountably many minimal geodesic. In dimension at least \(3\), there may be very few minimal geodesics. Let us assume that \(M\) is closed. In 1990 Victor Bangert has shown that the number of geometrically distinct minimal geodesics is bounded below by the first Betti number \(b_1\).\n\nIn joint work with Clara Löh, we improve Bangert's lower bound and we show that this number is at least \(b_1^2+2b_1\).\n\nThe talk will have many ties to previous research done in Freiburg many years ago: to the research of Victor Bangert, to the Diploma thesis I have written in Freiburg in 1994 in Bangert's group, to the research of the younger Burago, when he was\na long term guest in Freiburg and other aspects.\n
Herausforderungen beim Lernen von Bruchzahlen: Größenvorstellungen aufbauen, Konzeptwechsel unterstützen, Bias vermeiden
Tuesday, 23.4.24, 18:30-19:30, Hörsaal II, Albertstr. 23b
Beim Übergang von den ganzen Zahlen zu den Bruchzahlen müssen Lernende einen Konzeptwechsel (Conceptual Change) vollziehen, da manche der von den ganzen Zahlen vertrauten Eigenschaften ihre Allgemeingültigkeit verlieren. Dies führt zu typischen Schwierigkeiten und kognitivem Bias bei bestimmten Aufgabenstellungen. \nIm Vortrag werden Studien vorgestellt, in denen neben Aufgabenbearbeitungen auch Reaktionszeiten und Blickbewegungen gemessen wurden, um Denkprozesse genauer zu beschreiben. Ferner wird eine Interventionsstudie vorgestellt, in der untersucht wurde, inwieweit der Aufbau von Größenvorstellungen für Bruchzahlen gezielt gefördert werden kann und zu einer Reduktion von Bias führt. Implikationen für das Unterrichten von Bruchzahlen werden zur Diskussion gestellt\n
On the injectivity and non-injectivity of the \(l\)-adic cycle class maps
Friday, 26.4.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
We study the injectivity of the cycle class map with values in Jannsen's continuous étale cohomology, by using refinements that go through étale motivic cohomology and the ``tame'' version of Jannsen's cohomology. In particular, we use this to show that the Tate and the Beilinson conjectures imply that its kernel is torsion in positive characteristic, and to revisit recent counterexamples to injectivity.\n
Multiplication of BPS states in heterotic torus theories
Monday, 29.4.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
The space of states of an N = 2 superconformal field theory contains an infinite-dimensional subspace of Bogomol'nyi–Prasad–Sommerfield (BPS) states, defined as states with minimal energy given their charge. In particular, they arise in worldsheet theories of strings. In this setting, Harvey and Moore introduced a bilinear map on BPS states.\n\nThis talk presents a mathematically rigorous approach to this construction, which has been considered promising but not properly understood for almost 30 years now. The example used throughout is that of a heterotic string with all but four dimensions compactified on a torus. For this case, the BPS states were claimed to form a Borcherds–Kac–Moody algebra, as introduced in Borcherds' proof of the monstrous moonshine conjectures.\n\nThe first half of the talk, unfortunately, consists in pointing out problems with the proposed construction. The second half will provide more details on selected aspects, such as the existence of a finite-dimensional Lie algebra of massless BPS states.
Two-scale finite element approximation of a homogenized plate model
Tuesday, 30.4.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We study the discretization of a homogenized and dimension reduced model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Velčić in 2014. Thereby, a nonlinear bending energy is based on a homogenized quadratic form which acts on the second fundamental form associated with the elastic deformation. Convergence is proven for a multi-affine finite element discretization of the involved three-dimensional microscopic cell problems and a discrete Kirchhoff triangle discretization of the two-dimensional isometry-constrained macroscopic problem. Finally, the convergence properties are numerically verified in selected test cases and qualitatively compared with deformation experiments for microstructured sheets of paper.