WS 2017/18

Montag, 20. November 2017 um 14:00 Uhr, Oberseminar Reelle Geometrie und Algebra

Nicolai Vorobjov (University of Bath)

(Gast von Cordian Riener)

Triangulation of definable monotone families of compact sets

Let $K \subset \mathbb{R}$ be a compact definable set in an o-minimal structure over $\mathbb{R}$, e.g., a semi-algebraic or a real subanalytic set. A definable family $\{S_{\delta}|\>  0 < \delta \in \mathbb{R} \}$ of compact subsets of $K$, is called a monotone family if $S_\delta \subset S_\eta$ for all sufficiently small $\delta > \eta > 0$. The main result in the talk is that when $\dim K \le 2$ or $\dim K=n=3$ there exists a definable triangulation of $K$ such that for each (open) simplex $\Lambda$ of the triangulation and each small enough $\delta > 0$, the intersection $S_{\delta} \cap \Lambda$ is equivalent to one of five (respectively, nine) standard families in the standard simplex (the equivalence relation and a standard family will be formally defined). As a consequence, we prove the two-dimensional case of the topological conjecture on approximation of definable sets by compact families.

This is a joint work with Saugata Basu and Andrei Gabrielov (Purdue).

Freitag, 01. Dezember 2017 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Thorsten Jörgens (Goethe-Universität Frankfurt)

(Gast von Markus Schweighofer)

Conic Stability of Polynomials

A multivariate polynomial is called stable, if it has no root whose imaginary part lies in the positive orthant $(\mathbb{R}_+)^n$. In this talk, we extend this notion by replacing the positive orthant by a general proper cone $K \subset \mathbb{R}^n$ and we call a polynomial $K$-stable, if there is no root whose imaginary part lies in $K$. Building upon this, we generalize results known for stable polynomials, e.g., the multivariate version of Hermite-Kakeya-Obreschkoff by Borcea and Brändén. And we generalize a major criterion for stability of determinantal polynomials to stability with respect to the cone of positive semidefinite matrices.

Montag, 04. Dezember 2017 um 15:15 Uhr, Oberseminar Modelltheorie

Erick Garcia Ramirez (Universität Konstanz)

Expansions of real closed valued fields and applications

Let $T$ be an o-minimal theory containing the axioms for real closed fields. To a model $R$ of $T$ we attach a convex subring $V\subseteq R$, resulting in a valued field $(R,V)$. Under certain model-theoretic conditions on $V$, $(R,V)$ is a well-behaved structure in terms of model theory. In this talk I will describe some of the properties of definable sets and functions in $(R,V)$.  I will also point out some applications to geometry, including applications to real o-minimal geometry.The results I will report on this talk were part of my PhD work at the University of Leeds.

Montag, 18. Dezember 2017 um 15:15 Uhr, Oberseminar Modelltheorie

Alex Savatovsky (Universität Konstanz)

Expansions of the real field which do not define new C-infinity functions

We will give a rough sketch of the following theorem : Let $L$ be the
language of a field and $P$ a predicate. Let $R$ be an RCF and assume that $(R, P)$ is a
$d$-minimal structure. Under some further model theoretic assumptions, we have
that every definable $C$-infinity function on an open connected domain is the
restriction of some $L$-definable function to this domain.

Montag, 08. Januar 2018 um 17:00 Uhr, Oberseminar Reelle Geometrie und Algebra

Marie-Françoise Roy (Université de Rennes 1)

(Gast von Markus Schweighofer)

Quantitative aspect of two algebraic proofs of the fundamental theorem of algebra

The best known algebraic proof of the fundamental theorem of algebra is due to Laplace and uses the intermediate value axiom for polynomials of degree exponential in $d$ in the proof of existence of the complex roots of a polynomial of degree $d$.
Much more recently Michael Eisermann proposed another algebraic proof using Cauchy index and winding numbers, completing a project of Sturm in the 1830.
His proof does not provide bounds on the degree of the intermediate value axiom needed in the proof of existence of the complex roots of a polynomial of degree $d$. Using subresultants, we obtain a modification of his proof, using the intermediate value axiom for polynomials of degree polynomial in $d$ in the proof of existence of the complex roots of a polynomial of degree $d$.
This illustrates a quantitative difference between these two proofs.

Work in progress with Daniel Perrucci.

Freitag, 19. Januar 2018 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Maria Dostert (École Polytechnique Fédérale de Lausanne)

(Gast von Cordian Riener)

New bounds for the density of geometric packings

How much of the space can be filled with pairwise non-overlapping copies of a given solid? This is a fundamental problem in geometric optimization. Packing problems of non-spherical solids are mathematical challenges and have several applications such as in materials science. In this talk, I present new bounds for the optimal packing density of some non-spherical solids in dimension three.

This is a joint work with Cristóbal Guzmán, Fernando Mário de Oliveira Filho, and Frank Vallentin.

Freitag, 09.Februar 2018 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Péter Frenkel (Eötvös Loránd University, Budapest)

(Gast von Cordian Riener)

Convergence of graphs with intermediate density

The talk will focus on a notion of graph convergence that interpolates between the Benjamini--Schramm convergence of bounded degree graphs and the dense graph convergence developed by László Lovász and his coauthors. Many motivating examples of convergent graph sequences will be given.

It turns out that spectra of graphs, and also some important graph parameters such as numbers of colorings or matchings, behave well in convergent graph sequences. For graph sequences of large essential girth, asymptotics of coloring numbers can be explicitly calculated. We also treat numbers of matchings in approximately regular graphs. These results are based on an algebraic method, namely, the investigation of certain graph polynomials.

I will introduce tentative limit objects that I call graphonings because they are common generalizations of graphons and graphings. Special forms of these, called Hausdorff and Euclidean graphonings, involve geometric measure theory. One can construct Euclidean graphonings that provide limits of hypercubes and of finite projective planes, and, more generally, of a wide class of regular sequences of large essential girth. For any convergent sequence of large essential girth, weaker limit objects can be constructed: an involution invariant probability measure on the sub-Markov space of consistent measure sequences (this is unique), or an acyclic reversible sub-Markov kernel on a probability space (non-unique).

I will also pose some open problems.

Montag, 12.Februar 2018 um 15:15 Uhr, Oberseminar Modelltheorie

Nikolaas Verhulst (TU Dresden)

(Gast von Salma Kuhlmann, Simon Müller)

Non-commutative geometry via groupoid valuations

In commutative algebraic geometry, valuations play an important role as a kind of translation  mechanism between geometry and algebra. One way of defining non-commutative geometry would be by finding a good generalisation of the concept of a valuation to the non-commutative world and mimicking this mechanism. However, the right non-commutative analogue of a field is in some sense a simple artinian ring - but these may contain zero-divisors! It is for this reason that groupoid valuations and groupoid valuation rings were introduced. In this talk, we will show that some important basic results of classical valuation theory still go through for groupoid valuations, which will allow us, in the end, to draw a simple example of a non-commutative curve.