Vorträge inklusive Abstracts im Sommersemester 2014
Die folgenden Vorträge haben im Sommersemester 2014 im Oberseminar reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium Reelle Geometrie und Algebra stattgefunden.
Donnerstag, 03.04.2014 um 17.00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Bruce Reznick (University of Illinois at Urbana- Champaign, USA)
(Gast von Salma Kuhlmann)
Hilbert's 17th Problem during the Late 20th Century
Abstract: For thirty years after Motzkin and Robinson produced the first concrete examples of real psd forms which are not a sum of squares, most of the published work on this subject was done by Man-Duen Choi, T. Y. Lam and their collaborators, including the speaker. This talk will present several vignettes of research along these lines, with a particular focus on symmetric forms, and is intended to be comprehensible to graduate students.
Freitag, 04.04.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Bruce Reznick (University of Illinois at Urbana- Champaign, USA)
(Gast von Salma Kuhlmann)
Ternary forms with lots of zeros
Abstract: We are concerned with real psd ternary forms p(x,y,z) of degree 2d, especially for d=3,4. The Robinson form is a ternary sextic with exactly 10 zeros (viewed projectively.) Choi, Lam and Reznick proved in 1980 that a psd ternary sextic with more than 10 zeros has infinitely many zeros and is sos and used the Robinson form to construct psd ternary forms of degree 6k with exactly 10k^2 zeros. We will give a psd-not-sos ternary octic with exactly 17 zeros. This is ongoing joint work with Greg Blekherman.
Donnerstag, 17.04.2014 um 11.45 Uhr, Oberseminar Reelle Geometrie und Algebra
Bill Helton (UC San Diego)
(Gast von Markus Schweighofer)
Real algebraic geometry in free noncommutative variables
Abstract: There is getting to be a reasonable theory of Noncommutative (free) Real Algebraic Geometry.
The talk will compare what is known and to some extent compare the methods in the free and the classical cases.
Freitag, 25.04.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 28.04.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Sudesh Kaur Khanduja (Indian Institute of Science Education and Research, Mohali, India)
(Gast von Salma Kuhlmann)
Irreducible Polynomials
Abstract: The irreducible polynomials have a long history. In 1797, Gauss proved that the only irreducible polynomials with real coefficients are linear or quadratic polynomials. However, in view of Eisenstien Irreducibility criterion proved in 1850, for each number n\geq 1, there are infinitely many irreducible polynomials of degree n over rationals. We discuss some generalizations of this criterion discovered in recent years using the theory of valuations, which yield irreducibility criteria by Akira and Tverberg.
Freitag, 02.05.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 05.05.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Arno Fehm (Konstanz)
Fields with almost small absolute Galois group
Abstract: Already Serre's "Cohomologie Galoisienne" contains an exercise regarding the following condition on a field F: For every finite field extension E of F and every n, the index of the n-th powers (E*)^n in the multiplicative group E* is finite. Model theorists recently got interested in this condition, as it is satisfied by every superrosy field and also by every strongly^2 dependent field, and occurs in a conjecture of Shelah-Hasson on NIP fields. I will explain how it relates to the better known condition that F is bounded (i.e. F has only finitely many extensions of degree n, for any n - in other words, the absolute Galois group of F is a small profinite group) and why it is not preserved under elementary equivalence. The main arguments are group theoretic, and most of the talk is accessible without knowledge of model theory. Joint work with Franziska Jahnke.
Freitag, 09.05.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 12.05.2014 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 16.05.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Sebastian Gruler (Konstanz)
Lower degree bounds for the Real Nullstellensatz in the case of binomial ideals
Abstract: The Real Nullstellensatz gives a representation 1+$\sum h_i^2=\sum g_if_i$, if there is no common real root of the $f_i$`s. Compared to upper bounds, there is little known about lower bounds on degrees of the polynomials $g_i$. I present the work of Gregoriev (2001), who gave the first lower bounds for the Real Nullstellensatz, in the special case that the $f_i$`s are binomials.
Montag, 19.05.2014 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 23.05.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 26.05.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Kein Vortrag
Freitag, 30.05.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Evelyne Hubert (INRIA, Mediterranee)
(Gast von Salma Kuhlmann)
Algebraic Constructions for Rational Invariants
Abstract: Computational efforts in invariant theory have been mainly focused on
polynomial invariants of linear actions.Yet rational invariants have interesting geometric properties and they can be computed for a wider class of
relevant actions.The talk will first review a construction that applies to any rational
action of an algebraic group and computes generating sets of rational invariants endowed with rewrite rules. It is based on Gröbner bases and the concept of section to the orbits.I will then focus on diagonal actions of tori and finite groups and show
how the computations rely solely on linear algebra over the integers. A rational section comes as a side product.The constructions induce a symmetry reduction scheme of polynomial systems.We introduce a construction based on the section that provides a similar scheme for any group actions and generalizes a construction for finite groups that I found in a paper of J. Cimpric, S. Kuhlmann, C. Scheiderer
This talk is partly based on joint works with Irina Kogan (North
Carolina State University) or George Labahn (University of Waterloo).
Montag, 02.06.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Arthur Paul Pedersen (Center for Adaptive Rationality, Max Planck Institute for Human Development, Berlin)
(Gast von Salma Kuhlmann)
Numerical Representations of Ordered Vector Spaces, with Applications to the Foundations of Probability
Abstract.To motivate my interest in ordered (and more generally valued) vector spaces, I shall first discuss problems in the foundations of probability and expected utility which have prompted a number of authors to consider accounts of probability and expected utility admitting non-Archimedean representations. I shall thereupon introduce a normative theory of subjective probability and expected utility that rests upon qualitative criteria regulating preference judgments (or comparative judgments of expectation). The theory abandons the commitment to the technically convenient but rationally non-obligatory dogma of real-valued representability---the presumption that each agent in any given context is committed to a system of preference judgments (or judgments of probability and expectation) representable by a real-valued indicator. Like other normative theories (e.g., Savage’s theory of personal probability), the theory I advance presumes that an agent is committed to a system of *qualitative, comparative* preference judgments (or judgments of probability and expectation).
I shall then explain a key lemma supporting a full numerical representation of preference (or comparative expectations) in terms of subjective expected utilities formed from possibly non-Archimedean probabilities and utilities. Using Hahn's Embedding Theorem, I shall describe how a simple construction transforming (embedding) a Hahn lexical vector space into a "small" Hahn lexical field can be used to show that expected utilities take a very simple numerical form in terms of power series in a single infinitesimal with addition and multiplication naturally defined by means of the familiar operations of addition and multiplication of power series and with a natural lexicographic ordering. Other accounts of non-Archimedean probability or expectation are insufficiently general and philosophically inadequate. Finally, I shall discuss outstanding mathematical questions to be addressed during my visit.
If time permits, I shall explain a qualitative criterion of coherence reminiscent of de Finetti’s numerical criterion of coherence for his theory of probability (or prevision). The qualitative criterion of coherence is formulated for a truly arbitrary collection of gambles (random quantities), free of structural constraints. Furthermore, the criterion does not require an agent’s comparative judgments to be reflexive, complete, or even transitive. Despite these weak structural constraints, the qualitative criterion of coherence satisfies an analogue of de Finetti’s Fundamental Theorem of Prevision, ensuring that each coherent system of comparative judgments can be extended to a coherent weakly ordered system of comparative judgments over any space of gambles (random quantities). This result furnishes a basis for an account admitting a representation of uncertainty in terms of numerically indeterminate, and possibly non-Archimedean, probabilities and expected utilities in the style of, for example, Levi, Smith, Walley, and Williams.
Freitag, 06.06.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Charu Goel (Konstanz)
Extension of Hilbert's 1888 Theorem to Even Symmetric forms
Abstract: In 1888: Hilbert proved that: $\mathcal{P}_{n,m}$ = $\Sigma_{n,m}$ iff $n=2, m=2$, or $(n,m)=(3,4)$, where $\mathcal{P}_{n,m}$ and $\Sigma_{n,m}$ are respectively the cones of positive semidefinite (psd) and sum of squares (sos) forms (real homogenous polynomials) of degree $m$ in $n$ variables. Thus in all other cases $\Sigma_{n,m}\subsetneq\mathcal{P}_{n,m}$ (I will call them non-Hilbert cases).
I will start by presenting an analogue of Hilbert's 1888 Theorem for Symmetric forms given by Choi and Lam in 1976. They claimed that $S\mathcal{P}_{n,m}$ = $S\Sigma_{n,m}$ iff $n=2, m=2$, or $(n,m)=(3,4)$, where $S\mathcal{P}_{n,m}$ and $S\Sigma_{n,m}$ are respectively the cones of symmetric psd and symmetric sos forms of degree $m$ in $n$ variables. I will give a construction of explicit forms $f \in S\mathcal{P}_{n,4} \setminus S\Sigma_{n,4}$ for $n \geq 5$, thereby completing their claim.
Next, I will consider even symmetric forms and give a degree jumping principle that can be used to find psd not sos even symmetric $n-$ary forms of degree $2d+4r (r \in \mathbb{Z}_{+}; r \geq 2)$ and 2d+2n from psd not sos even symmetric $n-$ary forms of degree $2d$. Then I will give a construction of explicit even symmetric psd not sos $n-$ary octic forms (for $n \geq 5$) after stating
known results by Choi, Lam, Reznick and Harris relating even symmetric psd and sos forms in some of the non-Hilbert cases.Finally I will show how these methods leads to an extension of Hilbert's 1888 Theorem for even symmetric forms.
Montag, 09.06.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Feiertag, kein Vortrag
Freitag, 13.06.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Kein Vortrag
Montag, 16.06.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Derya Çıray (Konstanz)
Counting rational points using mild parametrization
Abstract:This talk is about a method used to estimate the density of rational points (of bounded height) on subsets of the reals.
The Theorem of Pila and Wilkie (2006) states that the transcendental part of a set X definable in an o minimal expansion of the real ordered field, has a bound
$C(X,\epsilon)(H)^{\epsilon}$
on the number of its rational points with height smaller or equal to H. It is known that one can not get a better bound for all o-minimal expansions of reals, but Wilkie (2006) has conjectured that this bound can be improved to
$ C_1(X)(logH)^{C_{2}(X)} $
for sets definable in real exponential field.
A set has a mild parametrization, if it can be covered by a finite number of smooth functions with certain bounds on their derivatives (mild functions). I will explain how mild parametrization is used to achieve good bounds (of the kind that Wilkie conjectured) on the density of rational points for certain sets, in particular of Pfaffian curves. Wilkie's conjecture and how mild parametrization could be used to prove it, will also be discussed.
Freitag, 20.06.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Kein Vortrag
Montag, 23.06.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Marcus Tressl (University of Manchester, UK)(Gast von Alexander Prestel)
Title: Externally definable sets in real closed fields.
Abstract: For a real closed field R, a subset of R^n is called externally definable if it is the intersection of R^n with a semi-algebraic subset of some S^n, where S is a real closed field containing R. For example, convex valuation rings of R are externally definable.
Over the real field, every externally definable set is semi-algebraic by a Theorem of Bröcker.
I will explain how this statement can be phrased for non-archimedean R, what it means geometrically and prove it for fields that are close to being maximally valued. As a by-product we get an algebraic characterization of definable types in real closed valued fields. This is joint work with Francoise Delon.
Dienstag, 24.06.2014 um 13.30 Uhr, Oberseminar Modelltheorie (Room: D406)
Vagios Vlachos (University of Athens)
(Gast von Pantelis E.Eleftheriou)
Title:Prime Models of o-Minimal Theories
Abstract: Consider a first-order structure S=(M,< ,...) such that < is interpreted by a dense linear order on M. S is called o-minimal if every first-order definable subset of M is a finite union of singletons and intervals. A first-order theory T will be called o-minimal if every model of T is an o-minimal structure. The notion of o-minimality was first presented in the ‘80s (van den Dries, Pillay and Steinhorn) and since then has been studied thoroughly by many authors.
In this talk we will show the existence and uniqueness of prime models of o-minimal theories [PS86]. The recipe of the proof is the same with Shelah's analogous result for ω-stable theories which uses Ressayre's result for constructible models [Mar02].
References
[Mar02] David Marker. Model theory: an introduction. Grad. Texts in Math. 217. Springer, 2002.
[PS86] Anand Pillay and Charles Steinhorn. “Definable sets in ordered structures. I”. In: Trans. Amer. Math. Soc. 295.2 (1986), pp. 565–592.
Freitag, 27.06.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Patrick Michalski
(Gast von Salma Kuhlmann)
"Determinantal representation of homogeneous polynomials in few variables and those expressible as a product of linear polynomials" .
Abstract:In this talk I will present some main aspects of my bachelor thesis supervised by Salma Kuhlmann. Let K be an algebraically closed field, then a homogeneous polynomial p\in K[x_0,...,,x_n] of degree r has a determinantal representation if there exist rXr matrices A_0,..., A_n with entries in K such that p = \det(x_0A_0+\cdots +x_nA_n). L. E. Dickson (1921) has shown, that a general homogeneous polynomial in at most three variables has always such representation. I will give his prove and if time remains I will show, that any homogeneous polynomial with a determinantal representation A_0,...,A_n satisfying A_0,...,A_n\in K[A] for some matrix A is a product of linear polynomials.
Montag, 30.06.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Nadja Hempel (Université Lyon 1, France)
(Gast von Cédric Milliet)
"Artin-Schreier extensions in generalizations of NIP theories"
The study of algebraic extensions of fields defined in a first order
theory has been of special interest. It is known that NIP fields of
positive characteristic are Artin-Schreier closed. Chernikov, Kaplan
and Simon showed that any field of positive characteristic defined in
an NTP_2 theory admits only finitely many Artin-Schreier extensions.
I will introduce the notion of NIP_n theories due to Shelah and extend
the result of NIP fields to this wider class. Using this, I will also
present some applications to valued fields defined in this setting and
show that non separable closed pseudo-algebraically closed (PAC)
fields lie outside of the hierachie of NIP_n theories, a result due to
Duvet in the NIP context.
Donnerstag, 03.07.2014 um 17.00 Uhr, Schwerpunktkolloquium (Room: A 702)
Alexandre Borovik (University of Manchester)
(Gast von Margaret Thomas)
Heads-real, tails-complex: probabilistic recognition of black box groups.
Abstract: (joint work with Sukru Yalcinkaya)
The so-called "black box" methods of computational group theory usually
start with a very innocent question.
Assume that we are given a few matrices $g_1,\dots, g_l$ of sise
$n\times n$ over the finite field of ordere $q$. They generate a finite
group $X$. What can we say about $X$? Can we determine its order? Its
isomorphism class?
The catch is that even in relatively simple cases $X$ is of astronomic
size and no deterministic approach to these problems is presently known,
only probabilistic methods. Probabilistic sampling is the only approach.
My talk will discuss an asymptotic case: what can be said about a Lie
group if it is given to us as a "black box" that produces random
elements and performs, at our requests, operations over them? To avoid
technical details, I will concentrate on two smallest (but also the most
important) groups, the group of Euclidean rotations $SO_3(\mathbb{R})$
and the proper orthochronous Lorentz group $PSL_2(\mathbb{C})$. Can we
tell one from another? These asymptotic considerations and simple
quantum mechanical analogies provide some important breakthroughs in
black box recognition of finite groups.
Freitag, 04.07.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
M. Dickmann (Institut de Mathematiques de Jussieu)
(Gast von Prf. Alexander Prestel)
A few things I know about spectral spaces (and
want to share with you)
Abstract: I'll begin with a general presentation of the theroy of spectral spaces from a purely topological viewpoint (i.e not deriviting from Stone duality), emphasizing its main structural invariants:
- The specialization partial order
- The lattices of quasi-compact open sets and of closed constructible sets
- The associated constructible and inverse topologies
Then I'll briefly describe a couple of the less known (and delicate) topological constructions in this theory:
- Directed inductive limits
- Quotiens modulo arbitrary binary relations
Finally, I'll review some important special classes of spectral spaces dwelling in more detail with the topological structure of the Zariski and the real spectra of rings.
Note: All the material to be presented (and MUCH MORE) appears in the forthcoming book Spectral spaces by N.Schwartz, M.Tressl and myself.
Montag, 07.07.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Kein Vortrag
Freitag, 11.07.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Kein Vortrag
Montag, 14.07.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Kein Vortrag
Freitag, 18.07.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Markus Schweighofer (konstanz)
Inclusion of spectrahedra, free spectrahedra and coin tossing (joint with Bill Helton, Igor Klep and Scott McCullough)
Abstract: Spectrahedra are generalizations of (convex) polyhedra sharing many of the good algorithmic features with polyhedra but allowing for roundness in their shapes. Given two spectrahedra in form of a linear matrix inequality, it is in general hard to decide whether one contains the other. To the linear matrix inequalities one can however associate not only scalar solutions but also matrix solutions. This gives rise to so called free spectrahedra. Inclusion of free spectrahedra is in many cases easy to decide, e.g., by using a generalization of the Gram matrix method and a linear Positivstellensatz for symmetric linear matrix polynomials due to Helton, Klep and McCullough . The natural question arises how the inclusion of two free spectrahedra relates to the inclusion of the corresponding classical spectrahedra. Surprisingly, this question is related to very subtle properties of Binomial distributions some of which are known and non-trivial and some of which are most probably unknown. At this stage of the work, unfortunately we have to keep secret the unknown properties.
Montag, 21.07.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Katharina Dupont (Konstanz)
Freitag, 25.07.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Daniel Plaumann (konstanz)
Algebraic Loci for Polynomial Zeros
Abstract: This talk will be about generalisations of Grace's theorem concerning the location of zeros of polynomials in the complex plane. (Joint work in progress with Mihai Putinar)
