WS 2018/19

Freitag, 26. Oktober 2018 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Igor Klep (University of Auckland)

(Gast von Markus Schweighofer)

Positive trace polynomials and the Procesi-Schacher conjecture

Positivstellensatz is a fundamental result in real algebra providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this talk Positivstellensätze for trace polynomials positive on semialgebraic sets of $n\times n$ matrices are provided. A Krivine-Stengle-type Positivstellensatz is proved characterizing trace polynomials nonnegative on a general semialgebraic set $K$ using weighted sums of hermitian squares with denominators. The weights in these certificates are obtained from generators of $K$ and traces of hermitian squares. For compact semialgebraic sets $K$ Schmüdgen- and Putinar-type Positivstellensätze are obtained: every trace polynomial positive on $K$ has a sum of hermitian squares decomposition with weights and without denominators. The methods employed are inspired by invariant theory, classical real algebraic geometry and functional analysis.
Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a Hilbert's 17th problem for the universal central simple algebra of degree n: is every totally positive element a sum of hermitian squares? They gave an affirmative answer for $n=2$. In this talk we will present a negative answer for $n=3$. Consequently, including traces of hermitian squares as weights in the Positivstellensätze is indispensable.
This is based on joint work with Špela Špenko and Jurij Volčič.

Freitag, 9. November 2018 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Rola Alseidi (Universität Konstanz)

On spectral properties of nonsingular matrices that are strictly sign-regular for some order

A matrix is called strictly sign-regular of order $k$ (denoted by $SSR_k$) if all its $k\times k$ minors are non-zero and have the same sign. For example, totally positive matrices, i.e., matrices with all minors positive, are $SSR_k$ for all $k$. Another important subclass are those that are $SSR_k$ for all odd $k$. Such matrices have interesting sign variation diminishing properties, and it has been recently shown that they play an important role in the analysis of certain nonlinear cooperative dynamical systems.

In this talk, the spectral properties of  nonsingular matrices that are $SSR_k$ for a specific value $k$ are studied. One of the results is that the product of the first $k$ eigenvalues is real and of the same sign as the $k\times k$ minors, and that linear combinations of certain eigenvectors   have specific sign patterns. It is then shown how known results for matrices that are $SSR_k$ for several values of $k$ can be derived from these spectral properties. Using these theoretical results,  the notion of a totally positive discrete-time  system (TPDTS) is introduced. This may be regarded as the discrete-time  analogue of the important notion of a totally positive differential system. It is shown that TPDTSs entrain to periodic excitations.

This work is joint with Jürgen Garloff and Michael Margaliot.

Montag, 12. November 2018 um 15:15 Uhr, Oberseminar Math. Logik, Mengenlehre und Modelltheorie

Michele Serra (Universität Konstanz)

Hahn groups, Hahn fields and their group of automorphisms

Hahn fields are fields of generalised power series. In some particular cases, their automorphism groups have been studied successfully, for example, Schilling described the (internal) automorphism group of the field of Laurent series, using methods from valuation theory. The construction of a Hahn field can be generalised to that of a Hahn group. The importance of Hahn groups (resp. fields), besides applications, comes from the fact that they are, in some sense, universal among ordered divisible abelian groups (resp. real closed fields). Many aspects of the theory of Hahn groups parallel those of Hahn fields. We highlight these analogies in order to transfer Schilling's ideas to the study of automorphisms of more general Hahn fields and Hahn groups.

Montag, 26. November 2018 um 15:15 Uhr, Oberseminar Math. Logik, Mengenlehre und Modelltheorie

Lothar Sebastian Krapp (Universität Konstanz)

Ordered fields dense in their real closure

Let $\mathcal{L}_{\mathrm{r}} = \{+,-,\cdot,0,1\}$ be the language of rings and $\mathcal{L}_{\mathrm{or}} = \mathcal{L}_{\mathrm{r}} \cup \{<\}$ the language of ordered rings.
The study of definable valuations in certain fields is motivated by the general analysis of definable subsets of fields as well as by decidability questions. For instance, there is a vast collection of results giving conditions on $\mathcal{L}_{\mathrm{r}}$-definability of henselian valuations in a given field, many of which are from recent years (cf. [1]).   
In the setting of ordered fields, convex valuations are well-understood. In [2], the authors provide a construction procedure of $\mathcal{L}_{\mathrm{or}}$-definable non-trivial convex valuations for ordered fields which are not dense in their real closure. This result also motivates the study of ordered fields which are not dense in their real closure.   
In my talk, I will apply the construction procedure of $\mathcal{L}_{\mathrm{or}}$-definable non-trivial convex valuations to certain ordered Hahn fields.   
This talk is on joint work with S. Kuhlmann and G. Lehéricy.

[1] A. Fehm and F. Jahnke, 'Recent progress on definability of Henselian valuations', Ordered Algebraic Structures and Related Topics, Contemp. Math. 697 (Amer. Math. Soc., Providence, RI, 2017), 135--143, doi:10.1090/conm/697/14049.

[2] F. Jahnke, P. Simon and E. Walsberg, 'Dp-minimal valued fields', J. Symb. Log. 82 (2017) 151--165,  doi:10.1017/jsl.2016.15.

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

Christoph Schulze (Universität Konstanz)

Facial structure of local psd-cones

We consider the set $\mathcal{P}^{Loc}$ of polynomials $f\in \mathbb{R}[\underline{x}]=\mathbb{R}[x_1,\dots,x_n]$ which are non-negative in some neighbourhood of the origin. These sets are convex cones, which naturally appear in the study of cones of globally non-negative forms of some degree.
We will state some basic properties of these cones and study their facial structure in the case $n=2$. In this case, we present the facial structure of faces of codimension up to $27$ modulo automorphisms of $\mathbb{R}[x_1,x_2]$. Furthermore, we will give an (countably infinite) graph which may be seen as a proper description of the facial structure of $\mathcal{P}^{Loc}$ ($n=2$) for faces of finite codimension.

Montag, 3. Dezember 2018 um 15:15 Uhr, Oberseminar Math. Logik, Mengenlehre und Modelltheorie

Martin Goldstern (TU Wien)

Cardinals between $\aleph_1$ and continuum

Georg Cantor's "continuum hypothesis (CH)" (1877) states that all infinite subsets of the real line are
-  either countable (equinumerous with the set of natural numbers)
-  or "of size continuum" (equinumerous with the set of all reals),
or in other words: the equivalence relation of equinumerosity (=equal cardinality) divides the infinite subsets of the reals into only 2 classes.

Based on Paul Cohen's forcing method (invented or discovered in 1963), set theorists have constructed many set-theoretical universes where CH fails, often in strong way.  For example, it is well known that  the number of equinumerosity classes may be uncountable, or even equal to the continuum itself.

While proofs that an explicitly given set S of reals is uncountable often exhibit a perfect subset of S, thus showing that S is in fact equinumerous with the real line,  there are also many cardinal numbers in the interval from $\aleph_1$ (the first uncountable cardinal) to continuum which appear as the answers to natural questions about the minimal size of a "pathological" set, such as
-  "How many points must a non-measurable set have?"
-   or "How many meager sets are needed to cover the real line?".

In my talk I will present some methods that can be used to construct set-theoretic universes where many nicely definable cardinals between $\aleph_1$ and continuum  (such as the cardinals in Cichońs diagram) have different values. In particular, I will talk a bit about using the method of Boolean ultrapowers in such constructions.

Freitag, 7. Dezember 2018 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Kaie Kubjas (Aalto University/ Sorbonne Universitè)

(Gast von Maria Infusino)

Mathematics of 3D Genome Reconstruction in Diploid Organisms

The 3D organization of the genome plays an important role for gene regulation. Chromosome conformation capture techniques allow one to measure the number of contacts between genomic loci that are nearby in the 3D space. In this talk, we study the problem of reconstructing the 3D organization of the genome from whole genome contact frequencies in diploid organisms, i.e. organisms that contain two indistinguishable copies of each genomic locus. In particular, we study the identifiability of the 3D organization of the genome and optimization methods for reconstructing it. Since every possible 3D organization is a solution to a system of polynomial equations, the identifiability question reduces to a question in algebraic geometry. For reconstruction, we use semidefinite programming methods.

This talk is based on joint work with Anastasiya Belyaeva, Lawrence Sun and Caroline Uhler.

Montag, 10. Dezember 2018 um 15:15 Uhr, Oberseminar Math. Logik, Mengenlehre und Modelltheorie

Simon Müller (Universität Konstanz)

On the Tree Structure of Orderings and Valuations on arbitrary unital Rings

We first introduce the notion of quasi-orderings on arbitrary unital rings, which axiomatically subsumes the classes of all orderings and valuations. This enables us to uniformly define a coarser relation $\geq$ on the set of all quasi-orderings, generalizing the respective notion known from valuation theory. Our main result states that, given a prime ideal $\mathfrak{q}$ of a unital ring $R,$ the set of all quasi-orderings on $R$ with support $\mathfrak{q}$ is a tree w.r.t. $\geq$, i.e. a partially ordered set admitting a maximum, such that for any such quasi-ordering $\preceq$ the set of its coarsenings is linearly ordered. We conclude the talk by discussing to what extend the tree structure theorem can be expanded to quasi-orderings with different supports.

Freitag, 14. Dezember 2018 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Tobias Fritz (Perimeter Institute for Theoretical Physics, Waterloo, Canada)

(Gast von Markus Schweighofer)

A generalization of Strassen's Positivstellensatz and applications to random walks

Strassen's Positivstellensatz is concerned with preordered semirings satisfying a strong Archimedeanicity assumption, and characterizes the preorder induced on large powers as the preorder induced by homomorphisms to the nonnegative reals. I will present a generalization
of this result, replacing the Archimedeanicity by polynomial growth and deriving further equivalent characterizations of the homomorphism-induced preorder. Some of the ideas that enter the proof may be useful in real algebraic geometry more generally. When applied to the semiring of measures under convolution, this new Positivstellensatz implies that iid random walks with bounded step size are classified asymptotically by their moment-generating functions.

Freitag, 11. Januar 2019 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Hamza Fawzi (University of Cambridge)

(Gast von Claus Scheiderer)

Positive semidefinite rank

The positive semidefinite (psd) rank is a measure of complexity for convex bodies: a convex body has psd rank $d$ if it can be expressed as the image of an affine slice of the $d\times d$ positive semidefinite cone. In this talk I will explain some connections of the psd rank with some matrix factorization problems as well as with the algebraic degree of semidefinite programming, and will show how these connections can be used to get lower bounds on the psd rank.

Freitag, 18. Januar 2019 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Christoph Schulze (Universität Konstanz)

Multiplicative cones and cones of non-negative polynomials

The set of polynomials in $n$ variables being non-negative on some subset $S\subseteq \mathbb{R}^n$ is a convex cone which additionally is closed under multiplication. Other examples of such cones are cones of sums of squares or more generally a preorder in some commutative integral domain containing $\mathbb{R}$. In this talk, we will study the convex structure of such cones as well as similar questions in a graded setting.
As an application of the introduced notions, we will sketch the proof of a statement in the graded setting concerning the case of suitable basic-closed semi-algebraic $S\subseteq \mathbb{P}^n(\mathbb{R)}$: it describes how faces of low codimension of the positivity cone corresponding to $S$ (in some even degree $2d$) are connected to faces of the corresponding local positivity cones (in degree $2d$).

Freitag, 25. Januar 2019 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Patrick Michalski (Universität Konstanz)

Projective limit techniques for the infinite dimensional moment problem

Consider the following general version of the classical moment problem for a linear functional $L$ on a unital commutative $\mathbb{R}$-algebra $A$: When can $L$ be represented as an integral w.r.t. a Radon measure on the character space $X(A)$ of $A$ equipped with the Borel $\sigma$-algebra generated by the weak topology $\tau_A$?
In this talk, we show that this problem is solvable if for each finitely generated subalgebra $S$ of $A$ there exists a (unique) representing Radon measure for $L\restriction_S$. The crucial step here is to construct $X(A)$ as a projective limit of the character spaces of finitely generated subalgebras of $A$. Our result allows us to generalize to infinitely (even uncountably) generated $\mathbb{R}$-algebras some of the classical theorems for the moment problem, e.g. the ones by Nussbaum and Putinar.

This is joint work in progress with Maria Infusino, Salma Kuhlmann and Tobias Kuna.

Freitag, 1. Februar 2019 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Antonio Lerario (SISSA, Triest)

Pathological examples in real geometry

It is well known that, when working in the definable setting, it is possible to control in a quantitative way the complexity of the various objects of study. For example: if we intersect a semialgebraic curve $X$ with an algebraic curve $Y$ of degree $d$ in the plane, the number of points of intersection (if finite) is bounded by $O(d)$. If we only assume that $X$ is definable, it is still possible to bound the number of points of intersection as function of the degree of $Y$ - however such bound is not explicit. In fact Gwozdziewicz, Kurdyka and Parusinski have showed that it is possible to produce a definable curve $X$ and a sequence of curves $Y_d$ of degree $d$ such that the number of points in their intersection is exponential in $d$.
Generalizing this construction, I will show that, up to extracting subsequences, the intersection in $n$-dimensional projective space of a definable hypersurface $X$ with a hypersurface $Y$ of degree $d$ can be as complicated as we want (meaning that, up to subsequences, the sequence $X\cap Y_d$ can be any sequence of smooth manifolds!). I call these "pathological examples".  On the other hand, I will show that these examples are essentially unaccessible, meaning that the volume of the set of polynomials defining hypersurfaces $Y_d$ with this behavior is extremely small (the more complicated the pathology, the smaller the volume).

Freitag, 8. Februar 2019 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra

Dijana Kreso (TU Graz)

Diophantine equations and polynomial decomposition

In my talk I will give an overview of my research in the field of Diophantine equations. Of my particular interest are Diophantine equations of separated variables type, $f(x)=g(y)$ in polynomials $f$ and $g$, and some related problems about the factorization of polynomials with respect to functional composition. I will then focus on the case of lacunary polynomials (polynomials with a bounded number of terms) and separated variables equations with lacunary polynomials. Finally, I will discuss some related Diophantine problems, such as the classical Diophantine equation asking for integers with all digits equal to one with respect to two distinct bases, and present some recent results of mine obtained jointly with Bennett and Gherga.