Vorträge inklusive Abstracts im Wintersemester 2015/16
Die folgenden Vorträge haben im Sommersemester 2015 im Oberseminar Reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium Reelle Geometrie und Algebra stattgefunden.
Montag 19.10.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Donnerstag 22.10.2015 um 17:00 Uhr, Allgemeines Mathematisches Kolloquium
Pantelis Eleftheriou (Universität Konstanz)
On o-minimal Hilbert's fifth problem
Abstract: In model theory, we are interested in the study of "definable" objects in certain tame settings. This talk deals with the study of definable groups in the o-minimal setting. These groups were introduced by Pillay in 1988 who showed that every such group is also a topological group. Ever since, a number of theorems have been proved manifesting a strong relation between definable groups and real Lie groups. The climax of this resemblance is Pillay's Principle (2007), which is an o-minimal analogue of Hilbert's fifth problem, and the topic of this talk.
Freitag 23.10.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Mihai Putinar (University of California at Santa Barbara)
(Gast von Claus Scheiderer)
Schoenberg positivity theorem in fixed dimension
Abstract: A celebrated 1942 result of Schoenberg characterizes all entry-wise functions which preserve positivity of matrices of any size. Roger Horn PhD dissertation (late 60ies) improved the result by obtaining necessary conditions on a polynomial to preserve positivity on matrices of a fixed size. I will present a characterization of polynomials which preserve positivity when applied entry-wise on matrices of a fixed dimension. All put in historical context and motivated by recent demands of the statistics of large data and optimization theory. A sketch of the proof will take a detour through the representation theory of the symmetric group.
Joint work with Alexander Belton, Dominique Guillot and Apoorva Khare.
Montag 26.10.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 30.10.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Cordian Riener (Fields Institute, Toronto)
(Gast von Markus Schweighofer)
Topological complexity of symmetric semi-algebraic sets
Abstract: This talk will present results on the structure of cohomology modules of symmetric (as well as multi-symmetric) real varieties and semi-algebraic sets. We will show bounds on the number of irreducible representations of the symmetric group occurring in the isotypic decomposition of the cohomology modules of the sets in question, which are polynomial in the number of variables, as well as polynomial bounds on the corresponding multiplicities. As an application we will give some improvements on bounds for the Betti numbers of images under projections of (not necessarily symmetric) bounded real algebraic sets. (Based on joint work with Saugata Basu.)
Montag 02.11.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Arno Fehm (Universität Konstanz)
On sums of two squares
Abstract: The integers that are a sum of two squares are characterized by the classical theorem of Fermat as those with a prime factorization in which every prime congruent to 3 mod 4 appears with even multiplicity. A theorem of Landau gives the number of such integers up to x as roughly 0.764*x/sqrt(log(x)). I will report on joint work with Efrat Bank and Lior Bary-Soroker in which we study related problems in polynomial rings over large finite fields. This talk does not involve any model theory and should be accessible to a broad audience.
Freitag 06.11.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Erwan Brugallé (École polytechnique)
(Gast von Markus Schweighofer)
Pseudoholomorphic Harnack curves
Abstract: Simple Harnack curves have been introduced and classified by Mikhalkin in the early 2000's. These curves constitute extremal objects in real algebraic geometry, and appear in several areas of mathematics. After recalling their definition, I will give an alternative proof of the classification of topological types of simple Harnack curves. In particular, this proof extends Mikhalkin's result to real pseudoholomorphic curves.
Montag 09.11.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Gabriel Lehéricy (Universität Konstanz)
Total quasi-orders and C-structures on abelian groups
Abstract: Quasi orders on algebraic structures can serve as a generalization of orders and valuations. For example, it has been shown by Fakhruddin in [1] that if (K,\precsim) is a field \neq F2 endowed with a total quasi order compatible with the field operations, then \precsim is either a field order or is induced by a uniquely determined valuation via a\precsim b\Leftrightarrow v(a)\geq v(b).
The subject of this talk is the theory of totally quasi-ordered abelian groups (q.o.a.g). A q.o.a.g is an abelian group (G,+) endowed with a quasi-order \precsim compatible with +; ordered groups and groups endowed with a quasi-order defined by a valuation are two examples of such structures, but they are not the only ones. In my talk I will present the general structure of a q.o.a.g, describe the definable sets of some of them and explore the possiblity of defining a notion of "quasi-order minimality" in analogue of o-minimality, which will lead me to consider C-structures.
The notion of C-minimal structure is a generalization of the notion of o-minimality that has been developped by Steinhorn, Macpherson in [2] and Delon in [3]; a particular attention has been given to the study of C-groups. I will show that quasi-orders and C-structures are naturally related, since any quasi-order on an abelian group induces a C-structure compatible with +, and any C-structure canonically induces a quasi-order.
References:
[1] Syed M.Fakhruddin, Quasi-ordered fields, Journal of Pure and Applied Algebra 45 (1987) 207-210
[2] Dugald Macpherson and Charles Steinhorn, On variants of o-minimality, Annals of Pure and Applied Logic 79 (1996) 165-209
[3] Françoise Delon, C-minimal structures without the density assumption, In Raf Cluckers, Johannes Nicaise et Julien Sebag, éditeurs: Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry. Cambridge University Press, Berlin, 2011
Freitag 13.11.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Victor Vinnikov (Ben-Gurion University of the Negev, Be'er Sheva)
(Gast von Markus Schweighofer)
On an application of free noncommutative function theory to determinantal representations
Abstract: In a recent joint work with Anatolii Grinshpan, Dmitry Kaliuzhnyi-Verbovetskyi, and Hugo Woerdeman, we have established that any polynomial over ${\mathbb C}$ that is strictly stable, i.e., has no zeroes in the closed unit polydisc in the complex affine space, admits a determinantal representation that witnesses this stability property (the result is in fact somewhat more general in that it applies to arbitrary products of Cartan domains of type I). The proof involves several ingredients: (a matrix-valued version of) a Hermitian Positivstellensatz due to Putinar; a realization technique from multidmensional system theory combined with a "lurking contraction argument" from multivariable operator theory; and a lifting of the problem from the polynomial ring to the free associative algebra that allows us to get rid of the possible extra factors in a determinantal representation. I will concentrate mostly on the last step, as an illustration of some powerful tools that are available in the free noncommutative setting and of their possible applications to usual commutative problems.
Montag 16.11.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 20.11.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Michele Serra (Universität Konstanz)
On Hilbertian fields and varieties of Hilbert type
Abstract: Hilbertian fields are fields for which Hilbert's irreducibility theorem holds. This theorem was proved by Hilbert, in 1892, over the rationals in his attempt to attack the inverse Galois problem, which concerns whether or not every finite group occurs as a Galois group over the field of rational numbers. Closely related to the notion of Hilbertian field is the one of variety of Hilbert type. I will introduce both these notions, explain how they are related and give some examples. I will also show some applications, with particular attention to the role they play in attacking the inverse Galois problem. In the last part I will address some open questions on varieties of Hilbert type and present some ideas to answer them.
Montag 23.11.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Sebastian Krapp (Universität Konstanz)
Ritt's Factorisation Theorem
Abstract: An exponential polynomial over an exponential field K (i.e. a field equipped with an exponential function) is a an expression of the form f(z) = a1 eb1z + ... + an ebn z where ai and bi are constants from K. Under standard addition and multiplication, the set of exponential polynomials over K forms a ring.
One can easily see that a factorisation into irreducible exponential polynomials does not exist in general. However, for the case that K is the complex exponential field, a unique factorisation into irreducibles and so-called simple exponential polynomials exists. This factorisation
was established by Ritt [2] in 1927. Ritt's Factorisation Theorem is particularly useful in the study of zeros of exponential polynomials, as one only has to consider the two cases of irreducible and simple exponential polynomials.
Everest and van der Poorten proved in [1] that Ritt's factorisation can be carried out in more general contexts, including the real exponential field.
This talk will present the main arguments in the proof of the complex case of Ritt's Factorisation Theorem and explain how a similar factorisation can be deduced for the real case.
[1] G. R. Everest and A. J. van der Poorten, Factorisation in the ring of exponential polynomials, Proceedings of the American Mathematical Society 125 (1997), no. 5, 1293–1298.
[2] J. F. Ritt, A factorization theory for functions \sum_{i=1}^n ai ealphai z}, Transactions of the American Mathematical Society 29 (1927), no. 3, 584–596.
Donnerstag 26.11.2015 um 17:00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Mikhail Tyaglov (Shanghai Jiao Tong University)
Criterion of total positivity of generalized Hurwitz matrices
Abstract: als pdf-Datei hier
Freitag 27.11.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Mario Kummer (Universität Konstanz)
Real fibered morphisms
Abstract: A real fibered morphism between varieties over the real numbers has the property that a point is mapped to a real point if and only if it is a real point itself. We will present some remarkable
properties of this kind of morphisms and apply these results to hyperbolic varieties. This is joint work with Eli Shamovich.
Montag 30.11.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Merlin Carl (Universität Konstanz)
Computing beyond Constructibility: The Recognizability Strength of Ordinal Time Machines
Abstract: Transfinite machine models of computation provide an approach to an `effective mathematics of the uncountable'. However, their set-theoretical interest seems to be limited by the fact that even the strongest such model, Koepke's Ordinal Turing Machines with parameters (pOTMs), can only compute constructible sets.
Recognizability is a more liberal notion than computability in that it only requires the machine to be able to identify a certain object when it is given to it as an input, not to produce that object.
By invoking notions from algorithmic randomness and considering recognizability rather than computability, we connect transfinite computability to large cardinals and forcing axioms incompatible with the axiom of constructibility on the one hand and inner models for large cardinals on the other. In particular, under appropriate large cardinal assumptions, a real number is heriditarily recognizable by a pOTM if and only if it is an element of the mouse for one Woodin cardinal. This is joint work with Philipp Schlicht and Philip Welch.
Freitag 04.12.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Christoph Hanselka (Universität Konstanz)
Montag 07.12.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Patrick Helbig (Universität Konstanz)
Elementary equivalence of strongly complete profinite groups
Abstract: A profinite group is a compact, Hausdorff, totally disconnected topological group; it can be shown that profinite groups are precisely the Galois groups of finite and infinite Galois extensions. Jarden and Lubotzky considered the elementary theory of profinite groups as abstract groups and showed that if two profinite groups are elementarily equivalent (as abstract groups) and one of them is (topologically) finitely generated, then they are isomorphic (as profinite groups). The proof of this relies on results related to a theorem by Nikolov and Segal which states that all finitely generated profinite groups are strongly complete, i.e. every subgroup of finite index is open, and on the fact that finitely generated profinite groups are small, i.e. there are only finitely many open subgroups of index n for each positive integer n. Extending the approach used by Jarden and Lubotzky, I show that if two profinite groups are elementarily equivalent and one of them is strongly complete, then they are isomorphic.
Freitag 11.12.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Christian Kuehn (Technische Universität Wien)
(Gast von Maria Infusino)
Blowing-Up the Olsen Model & An Introduction to Moment Closure
Abstract: In this talk I shall explain two related problems arising in the theory of polynomial vector fields. One is the resolution of dynamical singularities in the Olsen model for the peroxidase-oxidase reaction and the second part if a brief introduction to moment closure and how it may link to geometric approximation conditions encountered for multiscale dynamical systems. (The first part if joint work with Peter Szmolyan, Vienna).
Montag 14.12.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 18.12.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Tobias Kuna (University of Reading)
(Gast von Maria Infusino)
The truncated discrete moment problem on the real line
Abstract: Denote by N the degree of the truncated moment problem (TMP), that is, the degree of the highest polynomial for which the moments are pre-given. This talk will deal with discrete TMP of any degree N, where discrete means that the measure solving the given TMP is required to be supported on a discrete subset of the real line. The concrete case when the required support of the measures to be constructed is $\mathbb{N}_0$ will be presented. An algorithm to derive N necessary and sufficient conditions to solve the TMP of degree N will be explained.The main step is a reduction to a truncated Stieltjes moment problem, whose solution can be for example characterized by the results of Curto and Fialkow. Only the cases N=1,2,3 of the discrete TMP have been solved before, at least in principle. The case N>3 was considered to be very difficult and beside great activity in the area, no results were known before. It will become clear why N>3 is essentially more challenging. The extension to the general truncated discrete moment problem will be also discussed. This work is in collaboration with Maria Infusino (Konstanz), Joel Lebowitz and Eugene Speer (Rutgers).
Montag 21.12.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Im Zeitraum vom 23.12.2014 bis zum 06.01.2015 finden auf Grund der allgemeinen Betriebsschließung der Universität Konstanz keine Vorträge in den Oberseminaren statt.
Freitag 08.01.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag 11.01.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 15.01.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag 18.01.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Sylvy Anscombe (University of Central Lancashire, Preston)
(Gast von Arno Fehm)
Henselianity in the language of rings
Abstract: In this talk I will report on joint work with Franziska Jahnke in which we try to clarify the relationships between the following properties of a field K:
- K admits a nontrivial henselian valuation
- all fields L elementarily equivalent to K admit a nontrivial henselian valuation
- K admits a definable nontrivial henselian valuation
- K admits a 0-definable nontrivial henselian valuation.
I will describe the implications that hold between these properties in the classes of fields of characteristic zero with fixed residue characteristic (of the canonical henselian valuation). As ever, open questions remain in positive characteristic; although we obtain some results using the theory of tame valued fields.
Freitag 22.01.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag 25.01.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Pablo Cubides (Université de Caen Normandie)
(Gast von Salma Kuhlmann)
On externally definable sets in algebraically closed valued fields
Abstract: Given a pair of structures M < N and a set X ⊆ Nk definable with parameters in N, its trace in M (that is, X ∩ Mk) is called an externally definable set of M over N. A pair of models M < N of a first-order theory T is said to be stable if all externally definable sets of M over N are definable with parameters in M. Marker and Steinhorn characterized stable pairs of models of o-minimal theories as pairs M < N where M is Dedekind complete in N. In this talk we provide a characterization of stable pairs of algebraically closed valued fields. To get a flavor of the topic, different examples will be discussed and a brief introduction to some model-theoretic aspects of stable pairs will be given. This is a joint work with Françoise Delon.
Freitag 29.01.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Thomas Unger (University College Dublin)
(Gast von Claus Scheiderer)
Sums of squares and positivity in algebras with involution
Abstract: (This is joint work with Vincent Astier.) Using the theory of signatures of hermitian forms over algebras with involution (as presented at the Oberseminar Reelle Geometrie und Algebra in January 2015) I will introduce a notion of positivity for symmetric elements and prove a noncommutative analogue of Artin's solution to Hilbert's 17th problem, characterizing totally positive elements in terms of weighted sums of hermitian squares. As a consequence I will obtain an earlier result of Procesi and Schacher and show how to reformulate their question about representation of elements as sums of hermitian squares so that it has a positive answer. (Their question in its original formulation was shown to have a negative answer in general in earlier work with Igor Klep). Time permitting, I will discuss positive cones on algebras with an involution. These extend orderings on the base field of the algebra and generalize positive semidefinite symmetric matrices over fields.
Montag 01.02.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 05.02.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Simone Naldi (Università degli Studi di Firenze)
(Gast von Daniel Plaumann)
Computer algebra algorithms for semidefinite programming
Abstract: Bounding the arithmetic complexity of an optimization problem is very important. Recent works focus on semidefinite programming (SDP) for its outstanding importance and universality in polynomial optimization. Interior-point algorithms can be used to compute floating point approximations of a solution of an SDP in polynomial time (in finite precision arithmetic). On the other hand, neither efficient exact algorithms for SDP are available, nor a complete understanding of its theoretical complexity has been achieved. In this talk I will present some new results in this direction, based on joint work with Didier Henrion and Mohab Safey El Din.
Montag 08.02.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Lorenzo Galeotti (Universität Hamburg)
(Gast von Salma Kuhlmann)
Generalised Weihrauch Degrees
Abstract: The theory of Weihrauch degrees is about representing classical theorems of analysis in the Baire space and comparing their strength. In this talk, we are exploring a version of this theory for generalised Baire spaces. The first part of the talk will be devoted to the presentation of the construction of an extension of the real line suitable for extending the theory of Weihrauch degrees to uncountable cardinals. In the second part of the talk we will be focusing on generalising notions from computable analysis. Finally we will show how this new framework can be used to characterize the strength of the version of the Intermediate Value Theorem we presented in the first half of the talk.
Freitag 12.02.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Grey Violet (Universität Konstanz)
Topology of root clustering problems
Abstract: It is known that the space of univariate monic hyperbolic polynomials of degree n is homeomorphic to ${\mathbb R}\times {\mathbb R}^{n-1}_+$.
Notion of hyperbolicity is one of the many possible notions concerning different distributions of a root of polynomial with respect to some set Ω, which for the hyperbolic case is the real line. Those notions are usually called stability conditions (or root clustering problems), among them one could note such conditions as Hurwitz stability for characteristic polynomial of linear ODE, Schur stability for linear difference equations, pole placement problems and many others.
Our goal is to describe topology of the sets of polynomials with fixed distribution of roots relative to the Ω, adjacencies between these sets and to provide an explanation for the special position of hyperbolicity, Hurwitz and Schur stabilities.
