Strict minimality and algebraic relations between solutions in Poizat’s family of equations.
Freitag, 5.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In the last twenty years, techniques from model theory and differential algebra have been successfully applied to study integrability and algebraic dependence/independence of solutions in certain families of algebraic differential equations. I will discuss some of these techniques on a concrete example: the family of (non-linear) differential equations of the form y’’/y’ = f(y) where f(y) is a rational function. From a model-theoretic perspective, the study of this family was initiated by Poizat in the 80’s. \n\nThis is joint work with James Freitag, David Marker and Ronnie Nagloo.
Rigidity of Hyperelliptic Manifolds
Freitag, 12.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
A rigid compact complex manifold is one which admits no non-trivial deformations in a neighborhood of the base point. I will start by discussing the notions of rigidity and hyperelliptic manifolds (these are torus quotients with certain properties). After that, I will explain how to classify rigid hyperelliptic manifolds in complex dimension four (which is the minimal dimension in which rigid hyperelliptic manifolds exist). The classification has differential and complex geometric as well as algebraic flavors. This is joint work with Christian Gleißner (Universität Bayreuth).\n\n\n
A geometric approach to the charge statistic
Freitag, 19.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The charge is a statistic defined on the set of Young tableaux by Lascoux and Schützenberger in 1978. They showed that the generating functions for this statistic are the q-analogue of the weight multiplicities in type A.\nIn this talk, I will explain a different and geometrically motivated approach to the charge statistic. The q-weight multiplicities can in fact be obtained as Kazhdan-Lusztig polynomials for the affine Grassmannian. In this setting, we will recover the charge statistic by studying hyperbolic localization for a family of cocharacters.
Schinzel's Hypothesis with probability 1 and rational points on varieties in families
Freitag, 26.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In a joint work with Efthymios Sofos we prove that Schinzel's Hypothesis (H) holds for 100% of polynomials of any fixed degree. I will pass in silence the proof of this analytic result, but will explain how to deduce from this that among varieties in specific families over Q, a positive proportion have rational points. The main examples are varieties given by generalised Châtelet equations (a norm form equals a polynomial) and diagonal conic bundles of any fixed degree over the projective line.\n