Geometry Seminar / Graduate Lectures

Pietro Giavedoni
Classification of real Riemann surfaces in the critical case.

The problem of classification of real Riemann surfaces can be formulated as the challenge to decide if a real Riemann surface - or equivalently, a projective, smooth and irreducible curve - is endowed with real points, based on any "real" period matrix of its. Solved only for genus two (by Comessatti), it has been open for more than one century nowadays. I will propose an exhaustive and effective solution valid for all genera.

Lorenzo Baldi (MPI-MIS Leipzig)
Nonnegative Polynomials and Moment Problems on Algebraic Curves

Given a real projective curve X, we study the dual convex cones of nonnegative forms and positive Borel measures supported on X.
We show how to recover the facial structure of the nonnegative cone from the totally real effective divisors on X and the Jacobian variety.
We focus in particular on the genus one case, where our description is explicit and complete.

For the dual cone of measures, we deduce that the Caratheodory number is determined by the topology of the real locus of the curve, using a technique inspired by Hilbert's proof on ternary quartics.
This number determines the complexity of quadrature rules supported on the genus one curve X, and can be interpreted as the maximal rank in a Waring-type decomposition with nonnegative coefficients.

Through the talk, we present some possible future research directions of algebraic and differential nature, to extend our results beyond the genus one case.

Based on a joint work with Greg Blekherman and Rainer Sinn.

Jonas Pinke (Bielefeld)
A category theoretical proof of Pontryagins Duality Theorem

Pontryagins famous duality theorem establishes a strong relationship between locally compact abelian hausdorff (LCA) groups and their dual groups, providing deep insights into the structure and behavior of these groups. In the talk we will give a proof of the duality theorem based on ideas from category theory, which is guided by the structure within the category LCA.The duality theorem will first be established for the subcategory of compactly generated abelian Lie-Groups.Through the study of "formal" limits and colimits the duality is expanded to a duality between the subcategories of discrete abelian and compact abelian LCA groups. From there, a study of exact sequences will result in the full duality theorem for the category LCA.

no seminar

 

09.04.2024 (Tue)
Z21/242
15:00

The inverse Gowers $U^3$-theorem and Conze-Lesigne factors

The Gowers uniformity $k$-norm on a finite abelian group measures the averages of complex functions on such groups over $k$-dimensional arithmetic cubes. The inverse question about these norms asks whether a large norm implies correlation with a function of algebraic origin.

The analogue of the Gowers uniformity norms for measure-preserving abelian actions is the Host-Kra-Gowers seminorms, which are intimately connected to the Host-Kra-Ziegler factors of such systems. The corresponding inverse question, in the dynamical setting, asks for a description of such factors in terms of systems of algebraic origin.

In this talk, we present a satisfactory answer to inverse question in the case of the Gowers $U^3$-norm in the combinatorial setting by completely classifying Host-Kra-Ziegler factors of order $2$, also known as Conze-Lesigne factors, for all countable measure-preserving abelian actions.