On arithmetic properties of discrete Toda flows
Freitag, 6.6.25, 10:30-11:30, Seminarraum 404
We will explain a new linearization of the discrete periodic Toda flow (a well known discrete integrable system) in terms of Mumford's description (via quadratic forms and Gauß composition law) of Jacobians of hyperelliptic curves. A subtle integrality property appearing in this set-up opens the way to use p-adic methods for the study of the Toda flow. We will give an elementary talk explaining all the main actors and also point to some intriguing connections with number theory.
A braided monoidal 2-category via Soergel bimodules
Montag, 16.6.25, 14:15-15:15, Seminarraum 232
Braided monoidal categories are good sources for topological invariants. To get more refined invariants, or even a full TQFT, one might want to lift this a categorical level higher and construct a braided monoidal 2-category. The talk will start by reviewing some well-known facts with an important example of a braided monoidal category. Then I will try to explain why the construction of a braided monoidal 2-category, and even the definition of such an object, is not obvious. The second part of the talk will then indicate an actual construction using complexes of Soergel bimodules.
Ramified periods and field of definition
Freitag, 20.6.25, 10:30-12:00, Seminarraum 404
In a joint work with Dragos Fratila and Alberto Vezzani, we construct hyperelliptic curves of large genus, defined over quadratic fields that are isomorphic to their Galois conjugates but do not descend to Q. The obstruction to descent is new and we call it “ramified periods”. These are p-adic numbers that arise from the comparison between de Rham cohomology and crystalline cohomology (hence the term periods). These numbers can reveal interesting information if p ramifies in the quadratic field.
Random complex planar curves of large degree are almost hyperbolic
Freitag, 27.6.25, 10:30-12:00, Seminarraum 404
A transfer principle in asymptotic analysis
Freitag, 25.7.25, 10:30-12:00, Seminarraum 404
Hardy fields form a natural domain for a “tame” part of asymptotic analysis. They may be viewed as one-dimensional relatives of o-minimal structures, and have applications to dynamical systems and ergodic theory. In this talk I will explain a theorem which permits the transfer of statements concerning algebraic differential equations between Hardy fields and related structures, akin to the “Tarski Principle” at the basis of semi-algebraic geometry, and sketch some applications, including to some classical linear differential equations. (Joint work with L. van den Dries and J. van der Hoeven.)