Vorträge inklusive Abstracts im Wintersemester 2014/15
Die folgenden Vorträge haben im Wintersemester 2014/15 im Oberseminar Reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium Reelle Geometrie und Algebra stattgefunden.
Montag, 20.10.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Kein Vortrag
Freitag, 24.10.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Jose Fernando Galván (Universidad Complutense de Madrid)
(Gast von Claus Scheiderer)
On the se
t of points at infinity of a polynomial image of R^n (joint work with Carlos Ueno)
Abstract: The study of polynomial and regular images of R^n has been developed during the last 25 years. In our opinion we are still very far from obtaining a complete solution to this problem. Between the most relevant results we point out: the full characterizations of the 1-dimensional polynomial and regular images of R^n, the connexion of the set of points at infinity of a polynomial image (which is the main objective of this seminar) or the results concerning the representation of convex polyhedra and their interiors (as regular images) or their complements and the corresponding closures (as polynomial images). The representation of the open quadrant {x>0,y>0} as a polynomial images of R^2 has been a crucial step for the representation of further families.
In this seminar we will sketch the proof of the connexion of the set of points at infinity of a semialgebraic set S that is the image of a polynomial map form R^n to R^m. The proof involves important classical results from resolution of the indeterminancy of rational maps, complex algebraic geometry and algebraic topology. Similar techniques (which have inspired us) were used before in some Jelonek's works developed in the 1990s, where he studied the set of points at which a polynomial map is not proper. The connexion result of the sets of points at infinity is no longer true for general regular maps from R^n to R^m, although it still holds for an ample family of regular maps that we call quasi-polynomial maps. We will also present some enlightening examples for the general regular case.
As far as we know the general problem of determining which semialgebraic subsets of R^m are either polynomial or regular images of R^n was initially proposed by J.M. Gamboa in the 1990 congress "Réelle Algebraische Geometrie" celebrated in Oberwolfach. The results mentioned above have been developed together with either J.M. Gamboa or C. Ueno.
Montag, 27.10.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Kein Vortrag
Freitag, 31.10.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Kai Kellner (Goethe-Universität Frankfurt a.M.)
(Gast von Markus Schweighofer)
Containment problems for spectrahedra and polyhedra
Abstract: Given two spectrahedra, deciding whether they are contained in each other is a co-NP-hard problem. Thus relaxation methods are of particular interest. We formulate the spectrahedron containment problem as a polynomial feasibility problem. Using the Positivstellensatz of Hol and Scherer, we derive a hierarchy of semidefinite feasibility problems to decide containment. The central question is then in which cases and under which conditions the hierarchy converges in finitely many steps. We exhibit several cases.
Another co-NP-hard containment problem is the problem whether a spectrahedron is contained in a V-polytope. We formulate the containment problem in terms of a bilinear feasibility problem. If the spectrahedron is an H-polytope, we show that under mild preconditions the Putinar based relaxation converges in finitely many steps. Note that the H-in-V problem itself is co-NP-complete.
Finally, for containment problems concerning projections of polyhedra and spectrahedra, we discuss how a straightforward formulation as a polynomial optimization problem can fail.
Montag, 03.11.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Moshe Jarden (Universität Tel Aviv)
(Gast von Arno Fehm)
Starker Approximationssatz für absolut integre Varietäten über galoisschen PSC-Erweiterungen vor globalen Körpern
Abstract: Seien K ein globaler Körper, V eine unendliche eigentliche Untermenge der Menge aller Primdivisoren von K, S eine endliche Untermenge von V und K~ (bzW.K_s) ein fester algebraischer (bzW.separabler) Abschluss von K. Sei Gal(K)=Gal(K_s/K) die absolute galoissche Gruppe von K. Für jedes p in V$ wählen wir einen henselschen Abschluss (bzW, einen reallen oder algebraischen Abschluss) K_p von K bei p, falls p non-archimedisch (bz W archimedisch). Dann ist K_{tot,S}=\inter_{p in S}\inter_{\tau in Gal(K)}K_p^\tau die maximale galoissche Erweiterung von K innerhalb K_s, wo alle p in S total zerfallen. Ist p nonarchimedisch, bezeichnen wir mit O_{K_p,p} den Bewertungsring von K_p und mit O_{K~,p} den ganzen Abschluss von O_{K_p,p} in K_s. Für \sig=(\sig_1,...,\sig_e) in Gal(K)^e sei K_{tot,S} [\sig] die maximale galoissche Erweiterung von K innerhalb K_{tot,S}, der von \sig_1,...,\sig_e festgehalten ist. Dann, für fast alle sig in Gal(K)^e (in bezug auf den haarschen Massese), genügt der Körper M=K_{tot,S}[\sig] den folgenden starken Approximationssatz: Sei V eine affine absolut integre Varietät über K und sei S C T eine endliche Untermenge von V, so dass die Menge V\T nur aus nonarchimedischen Primdivisoren besteht. Für jedes p in S sei \Ome_p eine nicht leere p-offene Untermenge von V_{\simp}(K_p), für jedes p in T\S sei \Ome_p eine nicht leere \Gal(K_p)-invariante p-offene Untermenge von V(K_s). Wir setzen vorauss dass V(O_{K_s,p})<>emptyset für jedes p in V\T ist. Dann, existiert z in V(M), so dass z^{\tau} in\Omega_p für alle p in T und \tau in\Gal(K) und |z^\tau|_{p}
Freitag, 07.11.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Maria Infusino (Universität Konstanz)
New directions in the infinite dimensional moment problem
Abstract: The purpose of this talk is to introduce an infinite dimensional version of the classical moment problem, namely the full moment problem on nuclear spaces, and exploring some questions related to various instances and generalizations of this problem. Given a nuclear space X, the question addressed is whether an infinite sequence of functions m_n, s.t. each m_n is an element of the n-th symmetric tensor product of the topological dual X', is actually the moment sequence of a finite non-negative Borel measure supported on a given subset K of X'.
I start reviewing the main result by Berezansky, Kondratiev and Sifrin about the existence and the uniqueness of a solution for the analogue of the Hamburger moment problem on nuclear spaces, i.e. when K=X'. Concerning the case when K is a proper subset of X', there are interesting results under further restrictions on the geometry of the support K or on the form of X. In particular, I present a joint work with T. Kuna and A. Rota about the case when K is a generic closed basic semi-algebraic subset of the space of generalized functions on R^d. Our approach combines the result of Berezansky et al. with some techniques recently developed for the moment problem on basic semi-algebraic sets of R^d. In this way, we get a complete characterization of the support of the realizing measure in terms of its moment functions.
It is then natural to ask if our result can be extended to any topological locally convex space (not necessarily nuclear). To investigate such a generalization, in a joint work in progress with M. Ghasemi, S. Kuhlmann and M. Marshall, we reformulate the problem in terms of characterization of all the functionals on the symmetric algebra on a locally convex space X that can be represented as integrals with respect to uniquely determined Radon measures on X'. Under the assumption that the given functional is continuous we have already got a result which characterizes the support of the representing measure, but we are still working on more general versions of this theorem.
Montag, 10.11.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Gabriel Lehericy (Universität Konstanz)
Model-theoretic approach to a Galois theory for fields endowed with an operator (Teil I)
Abstract: The purpose of this talk is to introduce the notion of internality between definable sets inside arbitrary structures. These general results will then be applied to the particular cases of differential and difference fields in the context of Galois theory.
Let $\langue$ be a first-order language, $T$ an $\langue$-theory with elimination of imaginaries,
$\model$ a model of $T$ and $Q,C$ two definable subsets of some power of $M$.
If there exists a definable set $X$ and a definable map $f$ from $Q\times X$ to some power of $C$, we say that $Q$ is $(X,f)$-internal to $C$. If $Q$ is $(X,f)$-internal to $C$, and if $\Delta$ is an arbitrary set of formulas in the sorts $(Q,C,X)$, we are particularly interested in the set $Aut_{\Delta}(Q,X,C/C)$ of bijections of the sorts $Q,X,C$ fixing $C$ pointwise and preserving all formulas of $\Delta$.
We shall see that every map of $Aut_{\{f\}}(Q,X,C/C)$ can be obtained by composing maps of the form $f(.,x)$ for some $x\in X$; this will allow us to show that if $f\in\Delta$ then $Aut_{\Delta}(Q,X,C/C)$ is a type-definable group contained in some power of $M$ and we will give explicit formulas to define this group.
These abstract results can be applied to the Galois theory of difference fields. Consider a difference field $(k,\sigma)$ and and equation $AY=Y$ where $A\in GL_n(k)$. The set of solutions of this equation generates an extension $(K,\Sigma)$ of $(k,\sigma)$ and we would like to study the algerbaic structure of the Galois group of this extension.
Set $Q:=\{\text{solutions of the equation}\}$, $C:=\{x\in M\mid \Sigma(x)=x\}$; it is easy to find $(X,f)$ such that $Q$ is $(X,f)$-internal to $C$.
The Galois group associated to the equation can be identified with the set $Aut_{\Delta}(Q,X,C/C)$, where $\Delta:=\{\text{quantifier-free formulas in the sorts $Q,C,X$ }\}$. Since the theory $ACFA$ of algebraically closed fields with automorphisms eliminates imaginaries, we
can apply our previous results by embedding $(K,\Sigma)$ into a model of $ACFA$, which will enable us to give a precise description of the Galois group.
Freitag, 14.11.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Tomas Bajbar (Karlsruher Institut für Technologie)
(Gast von Markus Schweighofer)
Coercive polynomials and their Newton polytopes
Abstract: Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. We analyze the coercivity on $\mathbb{R}^n$ of multivariate polynomials $f\in\mathbb{R}[x]$ in terms of their Newton polytopes. In fact, we introduce the broad class of so-called gem regular polynomials and characterize their coercivity via conditions imposed on the vertex set of their Newton polytopes. These conditions solely contain information about the geometry of the vertex set of the Newton polytope, as well as sign conditions on the corresponding polynomial coefficients. For all other polynomials, the so-called gem irregular polynomials, we introduce sufficient conditions for coercivity based on those from the regular case. For some special cases of gem irregular polynomials we establish necessary conditions for coercivity, too. Finally we address some stability issues.
Montag, 17.11.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Gabriel Lehericy (Universität Konstanz)
Model-theoretic approach to a Galois theory for fields endowed with an operator (Teil II)
Abstract: (siehe oben)
Freitag, 21.11.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 24.11.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Salma Kuhlmann (Universität Konstanz)
Real Closed Fields and Models of Peano Arithmetic
Abstract: We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic characterization of countable IPA-real closed fields. Expanding on [3], we conclude the talk by considering recursively saturated o-minimal expansions of real closed fields and their IPs.
References:
[1] D'Aquino, P. - Kuhlmann, S. - Lange, K. : A valuation theoretic characterization ofrecursively saturated real closed elds , to appear in the Journal of Symbolic Logic, arXiv: 1212.6842
[2] Carl, M. - D'Aquino, P. - Kuhlmann, S. : Value groups of real closed elds and
fragments of Peano Arithmetic, arXiv: 1205.2254 (2012)
[3] Conversano, A. - D'Aquino, P. - Kuhlmann, S : -Saturated o-minimal expansions of real closed elds, arXiv: 1112.4078 (2012)
[4] Kuhlmann, S. :Ordered Exponential Fields, The Fields Institute Monograph Series, vol 12. Amer. Math. Soc. (2000)
Freitag, 28.11.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Pantelis E.Eleftheriou (Universität Konstanz)
Semilinear groups
Abstract: Let M be an ordered vector space over an ordered division ring D. A set is called semilinear if it is a boolean combination of solutions to linear equalities and inequalities with coefficients from D. A group is called semilinear if its domain and the graph of its group operation are semilinear. We first present a positive result, that every semilinear group (satisfying suitable assumptions of compactness and connectedness) is semilinearly isomorphic to a torus. Then we present a negative result, that there is a semilinear group which is not semilinearly embeddable into the affine space.
Montag, 01.12.2014 um 13.30 Uhr, Oberseminar Modelltheorie
Pantelis E. Eleftheriou (Universität Konstanz)
Around definable compactness in weakly o-minimal structures
Abstract: We present results stemming from Gil Keren's master thesis on definable compactness for weakly o-minimal structures. We argue that the usual notion of definable compactness from o-minimal structures is not meaningful in a weakly o-minimal structure M, unless M is o-minimal. A byproduct of the argument is the construction of new weakly o-minimal structures that do not have definable choice, extending results from Chris Shaw's PhD thesis "Weakly o-minimal structrues and Skolem functions, University of Maryland, 2008".
Freitag, 05.12.2014 um 15.15 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 08.12.2014 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 12.12.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Salma Kuhlmann (Universität Konstanz)
Classification of cuts in real closed fields extensions
Abstract: We give necessary and sufficient conditions for a cut in a real closed subfield of a real closed field $R$ to be realized in $R$. To this end, we consider $R$ as a valued field $(R, v)$ (where $v$ is natural valuation on $R$). The conditions are threefold: conditions on the (divisible ordered abelian) value group, on the residue field, and on pseudo-convergence of certain pseudo Cauchy sequences.
As an application, we derive necessary and sufficient conditions for $R$ to be $\aleph_{\alpha}$-saturated. We discuss a generalization of this theorem to arbitrary o-minimal expansions of real closed fields.
Montag, 15.12.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Daniel Palacin (Universität Münster)
(Gast von Pantelis Eleftheriou)
A Fitting theorem for groups in simple theories
Abstract: A certain amount of model-theoretic ideas for groups in the stable context can be adapted to the more general framework of simple theories. For instance, groups defined in this context satisfy a chain condition on centralizers (up to finite index). In this talk we present some of the main tools and notions of groups in simple theories. Our goal is to show that in a group type-definable in a simple theory, the Fitting subgroup, i.e. the group generated by all normal nilpotent subgroups, is itself nilpotent.
Freitag, 19.12.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Victor Magron (Imperial College London)
(Gast von Markus Schweighofer)
New applications of moment-sos hierarchies
Abstract: Semidefinite programming is relevant to a wide range of mathematic fields, including combinatorial optimization, control theory, matrix completion. In 2001, Lasserre introduced a hierarchy of semidefinite relaxations for particular polynomial instances of the Generalized Moment Problem (GMP).
My talk emphasizes new applications of this moment-SOS hierarchy, investigated during my PhD and Postdoc research. In the context of formal proofs for nonlinear optimization, one can combine the moment-SOS hierarchy with maxplus approximation of semiconvex functions. Such a framework is mandatory for formal certification of nonlinear inequalities, occurring by thousands in the proof of Kepler Conjecture by Hales.
I also present how to approximate, as closely as desired, the Pareto curve associated with bicriteria polynomial optimization problems or the projections of semialgebraic sets. For each problem, one builds a hierarchy of semidefinite programs, so that the sequence of bounds converges in L1 norm. Finally, this hierarchy allows to analyze programs containing loop invariants with polynomial assignments.
Montag, 22.12.2014 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Im Zeitraum vom 23.12.2014 bis zum 08.01.2015 finden auf Grund der allgemeinen Betriebsschließung der Universität Konstanz keine Vorträge in den Oberseminaren statt.
Freitag, 09.01.2015 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Tom-Lukas Kriel (Universität Konstanz)
Putinars Positivstellensatz with degree bounds
Abstract: Putinars Positivstellensatz states the following: Let g_1,...,g_m be real polynomials defining an Archimedean quadratic module M_g and a semialgebraic set S_g={x \in R^n | g_1(x),...,g_m(x) \geq 0}. If f is a real polynomial being positive on S_g, then f \in M_g.
We take this result as a starting point and show the following generalization: If, additionally, one requires that deg(f) < N, ||f|| < N and f > N on S_g for a fixed N \in \N, then one can find a degree bound d only depending on N \in \N and g such that there exists a representation f=\sum_i s_i g_i with s_i \in \sum R[X]^2, where the degree of each summand is bounded by d.
We will present an elementary topological proof and a proof using the theory of pure states on rings. These things are part of my nearly completed master's thesis.
Montag, 12.01.2015 um 15.15 Uhr, Oberseminar Modelltheorie
David Masser (Universität Basel)
(Gast von Margaret Thomas)
Zero estimates with moving targets
Abstract: Zero estimates have a long classical history in diophantine approximation and transcendence, as well as more recent applications to counting rational points on analytic varieties. We give some examples showing that the sharpest conceivable results can be false, and in some cases the natural guess has even to be doubled. This is joint work with Dale Brownawell. A by-product is a special case of the (unproved) "Zilber Nullstellensatz".
Freitag, 16.01.2015 um 13.30 Uhr, Obersminar Reelle Geometrie und Algebra
Thomas Unger (University College Dublin)
(Gast von Markus Schweighofer)
Signatures of hermitian forms and applications
Abstract: I will survey my recent joint work with V. Astier on signatures of hermitian forms over central simple algebras with involution with respect to orderings on the base field. I will discuss some of the tools we developed (e.g., the Knebusch trace formula, “real splitting") and applications to the structure of the Witt group (e.g., “prime ideals”, total signature, stability index).
Montag, 19.01.2015 um 15.15 Uhr, Oberseminar Modelltheorie
Merlin Carl (Universität Konstanz)
Structures Associated with Real Closed Fields and the Axiom of Choice
Abstract: An integer part I of a real closed field K is a discretely ordered subring of K such that every element of K lies between two consecutive elements of I. Mourgues and Ressayre showed that every real closed field has an integer part. Their construction implicitly uses the axiom of choice.
We show that AC is actually necessary to obtain the result by constructing a transitive model of ZF which contains a real closed field without an integer part.
On the way, we demonstrate that a class of questions containing the question whether the axiom of choice is necessary for the proof of a certain ZFC-theorem is algorithmically undecidable. We further apply the methods to show that it is independent of ZF whether every real closed field has a value group section and a residue field section.
This also sheds some light on the possibility to effectivize constructions of integer parts and value group sections which was considered by D'Aquino, Kuhlmann, Knight and Lange.
Freitag, 23.01.2015 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 26.01.2015 um 15.15 Uhr, Oberseminar Modelltheorie
Paola D'Aquino (Seconda Università degli Studi di Napoli)
(Gast von Salma Kuhlmann)
Exponential polynomials
Abstract: I will present some recent work on the solution sets of
certain exponential polynomials.
Freitag, 30.01.2015 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Maria Lopez Quijorna (Universität Konstanz)
Truncated moment problem and polynomial optimization with the GNS-truncated construction
Abstract: In this talk, I will introduce the GNS-truncated construction and I will show some small results that we get in polynomial optimization and in the truncated moment problem with the help of this construction.
Montag, 02.02.2015 um 15.15 Uhr, Oberseminar Modelltheorie
Mario J. Edmundo (Universidade Aberta, Lissabon)
(Gast von Pantelis Eleftheriou)
On the o-minimal Hilbert's fifth problem
Abstract: The fundamental results about definable groups in o-minimal structures all suggested a deep connection between these groups and Lie groups. Pillay's conjecture explicitly formulates this connection in analogy to Hilbert's fifth problem for locally compact topological groups, namely, a definably compact group is, after taking a suitable the quotient by a "small" (type definable of bounded index) subgroup, a Lie group of the same dimension. In this talk we will report on the proof of this conjecture in the remaining open case, i.e. in arbitrary o-minimal structures. Most of the talk will be devoted to one of the required tools, the formalism of the six Grothendieck operations of o-minimal sheaves, which might be useful on it own.
(joint with: M. Mamino, L. Prelli, J. Ramakrishnan and G. Terzo)
Freitag, 06.02.2015 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Florian Pausinger (IST Austria)
(Gast von Salma Kuhlmann)
Introduction to persistent homology
Abstract: Persistent homology is an algebraic tool for quantifying topolog-
ical features of shapes and functions, which has recently found wide applica-
tions in data and shape analysis. In this talk, I aim to present the underlying
algebraic ideas and basic concepts of this very active field of research.
Montag, 09.02.2015 um 15.15 Uhr, Oberseminar Modelltheorie
Serge Randriambololona (Galatasaray Üniversitesi, Istanbul)
(Gast von Salma Kuhlmann)
Trichotomy in strongly minimal additive reducts of ACVF_0
Abstract: Zilber's conjecture of trichotomy asserts that the geometry (in the sense of the algebraic closure operator) of a strongly minimal structure 1) is either trivial or constrains the structure to essentially be either 2) a vector space or 3) an algebraic closed field. Though the conjecture was proven false in its general form, the trichotomy principle holds true in many particular instances. After having stated the conjecture in detail (and introduced the vocabulary required for its understanding), I will give an overview of the history of the results related to it and discuss the particular case of additive reducts of ACVF_0, as studied in a joint work with P. Kowalski (Univ. Wrocław).
Freitag, 13.02.2015 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Murray Marshall (University of Saskatchewan)
(Gast von Salma Kuhlmann)
A continuous moment problem for locally convex spaces
Abstract: It is explained how a locally convex topology $\tau$ on a real vector space $V$ extends naturally to a locally multiplicatively convex (lmc) topology $\overline{\tau}$ on the symmetric algebra $S(V)$. This allows application of results on lmc topological algebras obtained by M. Ghasemi, S. Kuhlmann and M. Marshall in an earlier paper to obtain representations of $\overline{\tau}$- continuous linear functionals $L: S(V)\rightarrow \mathbb{R}$ satisfying $L(\sum A^{2d}) \subseteq [0,\infty)$ (more generally, of $\overline{\tau}$-continuous linear functionals $L: S(V)\rightarrow \mathbb{R}$ satisfying $L(M) \subseteq [0,\infty)$ for some $2d$-power module $M$ of $A$) as integrals with respect to uniquely determined Radon measures $\mu$ supported on special sorts of closed balls in the dual space of $V$. The result is simultaneously more general and less general than a corresponding result of M. Infusino, T. Kuna and A. Rota. It is more general because $V$ can be any locally convex topological space (not just a nuclear space), the result holds for arbitrary $2d$-powers (not just squares), and no Carleman conditions are required. It is less general because it is necessary to assume that $L : S(V) \rightarrow \mathbb{R}$ is $\overline{\tau}$-continuous (not just that $L$ is continuous on the homogeneous parts of degree $k$ of $S(V)$, for each $k\ge 0$).
This is joint work in progress with M. Ghasemi, M. Infusino and S. Kuhlmann.
