WS 2019/20

Montag, 25. November 2019 um 15:15 Uhr, Oberseminar Math. Logik, Mengenlehre und Modelltheorie

Hazel Brickhill

Generalised Closed Unbounded and Stationary Sets

The notions of closed unbounded and stationary set are central to set theory. I will introduce a new generalisation of these notions, and describe some of their basic theory. Surprisingly for a new concept is set theory, generalised closed unbounded and stationary sets are very simple to define and accessible. Generalised stationary sets were first introduced by Bagaria, Magidor and Sakai in their paper "Reflection and indescribability in the constructible universe" and are closely related to the phenomena of stationary reflection and indescribability, but they can also be characterised in terms of derived topologies. I will present work from my PhD thesis that followed on from this paper, and if there is time some more recent work on generalising further to $\mathcal{P}_{\kappa}(\lambda)$.

Freitag, 29. November 2019 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Thorsten Mayer

Polyhedral Faces in Gram Spectrahedra of Binary Forms

Given a homogeneous polynomial $f$ with real coefficients, its Gram spectrahedron $\mathrm{Gram}(f)$ parametrizes the representations of $f$ as a sum of squares, modulo orthogonal equivalence. We study the facial structure of this convex body in the case of binary forms. In this talk, we examine the relationship between rank and dimension of faces $F \subseteq \mathrm{Gram}(f)$ and we show which pairs $(\mathrm{rk}(F), \dim(F))$ can occur. In addition, we will use representations of binary forms as Hermitian squares to find polyhedral faces of Gram spectrahedra.

Freitag, 13.Dezember 2019 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Julian Vill

The Gram spectrahedron $\mathrm{Gram}(f)$ of a real form $f$ parametrizes its sos representations up to orthogonal equivalence. Let $F\subset\mathrm{Gram}(f)$ be a face of rank $r$. We determine an upper bound for the dimension of faces while fixing the rank $r$ (as well as the number of variables and the degree of $f$). This defines an interval of possible dimensions for every $r$. For faces of almost maximal rank we will then show that in this interval only a very small number of values is attained, although both bounds are tight.

Montag, 16. Dezember 2019 um 15:15 Uhr, Oberseminar Math. Logik, Mengenlehre und Modelltheorie

Michele Serra

Hahn fields are fields of generalised power series over a field with exponents in an abelian group. I will study the group of automorphisms of such fields, aiming at generalising a result of Schilling on automorphisms of Laurent series to those automorphisms of general Hahn fields that are strongly linear (i.e., they commute with infinite sums).
To study the automorphism group, an important role is played by the lifting property i.e., the possibility of lifting an automorphism of the exponent group to an automorphism of the Hahn field. I will introduce the stronger notion of canonical lifting property, which allows a deeper understanding of the structure of the automorphism group; a large class of Hahn fields satisfying this property will be described.

Freitag, 10. Januar 2020 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

David de Laat (Delft University of Technology)

(Gast von Markus Schweighofer)

Exact semidefinite programming bounds for packing problems

Semidefinite programs are usually solved using interior point methods that work in floating point arithmetic. This means the solution given by the solver is not an exact solution to the problem and it may be non-obvious how to round it to an exact solution. When applying semidefinite programming in extremal geometry, we often do require an exact solution however. In this talk I will explain our algorithm to round the output of semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We use this to prove that some optimal configurations in discrete geometry are unique up to isometry.

Joint work with Maria Dostert and Philippe Moustrou.

Freitag, 17. Januar 2020 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Mario Kummer (TU Berlin)

(Gast von Claus Scheiderer)

Tropical Abelian Varieties

A lattice is a discrete subgroup of $\mathbb{R}^n$ whose span is full-dimensional. We will examine lattices from the point of view of tropical geometry where they serve as analogues to abelian varieties. This talk is based on a joint work with Lynn Chua and Bernd Sturmfels.

Freitag, 17. Januar 2020 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Arthur Forey (ETH Zürich)

(Gast von Salma Kuhlmann)

Bounded motivic integral and motivic Milnor fiber

Building on ideas of Hrushovski and Loeser, I will present a new motivic integration morphism, the bounded integral, that interpolates Hrushovski and Kazhdan's integrals with and without volume forms. It has applications to the complex and real motivic Milnor fibers. This is joint work with Yimu Yin.

This talk is also part of the 5th meeting of the DRMTA (Donau–Rhein Modelltheorie und Anwendungen).