Am 31. Januar 2025 findet am Institut für Mathematik das Kolloquium des SFB-TRR 358 statt. Die Vortragenden sind Prof. Thomas Schick (Göttingen) und Prof. Tobias Hartnick (KIT Karlsruhe).
Programm:
14:00 - 15:00 Prof. Thomas Schick (Göttingen)
15:00 - 15:30 Kaffee / Tee
15:30 - 16:30 Prof. Tobias Hartnick (KIT Karlsruhe)
16:30 Empfang / Buffet
Ort:
Die Vorträge finden im Hörsaal O1 statt, der sich im Erdgeschoss des Gebäudes O befindet. Die Kaffeepause und der Empfang werden im Raum O1.224 auf derselben Etage stattfinden.
Prof. Thomas Schick (Göttingen):
Titel: Topological T-duality
Abstract: The Fourier transform transports function from an abelian group to the dual group.
The Fourier-Mukay transform transports elements of the derived category from an abelian variety to its dual.
T-duality, the theme of the talk, transports de Rham cohomology classes from a smooth manifold to a “dual” manifold.
T-duality is a purely mathematical concept, inspired by mathematical physics.
In the talk, we introduce the underlying mathematical concepts of T-duality:
- When are two manifolds T-dual to each other?
- How is the T-duality tansform defined? (Fitting into the sequence of dualities listed above)
- Why is the T-duality transform an isomorphism (The fundamental result of this theory)?
Along the way, we discuss K-theory and twisted versions of it to obtain the most powerful formulation of T-duality and we will see how representation theory of compact groups enters to fully understand the equivariant situation.
We report on joint work with Ulrich Bunke and Tom Dove.
Prof. Tobias Hartnick (KIT Karlsruhe):
Titel: Transverse dynamical systems for discrete subsets of Lie groups
Abstract: What is an integral structure? Like the integers, it should be discrete and related to arithmetic. In this talk, I’d like to make the case that discreteness is the more fundamental property. I will illustrate this with a recent theorem of Hrushovski which implies that every sufficiently discrete large subset of a symmetric space of noncompact type is either a group or of arithmetic origin.
Discrete subsets which are also groups can be studied using a plethora of tools. In particular, one can study the topology and dynamics of the corresponding quotient space and consider various transforms associated with double fibrations over these quotient spaces.
Over the last 5-10 years, a similar tool kit has been developed for sufficiently discrete set without group structure. This has found applications from mathematical quasicrystals to point process theory and sphere packings. In these applications, it is typically not enough to consider periodic structures, and ideas from aperiodic order are needed. The natural replacement for quotient spaces in this context are so-called transverse dynamical systems.
After a general introduction to the subject I will focus on recent joint work with Michale Björklund (Chalmers) on double fibrations of transverse dynamical systems and discuss applications to the spectral theory of approximate lattices, including the construction of Schrödinger representations in the diffraction spectrum of Heisenberg quasicrystals and the modest beginnings of a theory of automorphic forms for discrete non-group patterns which does not distinguish the infinite places from the finite ones.
