Schwerpunkt reelle
Geometrie und Algebra

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Vorträge im Sommersemester 2011

Die folgenden Vorträge haben im Sommersemester 2011 im Oberseminar reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium reelle Geometrie und Algebra stattgefunden.

 

Donnerstag, 14.04.2011 um 16.00 Uhr, Oberseminar Modelltheorie

Bijan Afshordel (Freiburg)

(Gast von Arno Fehm)

Beschränkte PAC-Unterstrukturen von stabilen
Strukturen

$\subsection*{Abstract:}$

$PAC-Körper gehen auf James Ax Ende der 1960er Jahre zurück. Später führten Hrushovski den Begriff der PAC-Unterstruktur von streng minimalen Strukturen und danach Pillay und Polkowska von stabilen Strukturen ein.$

$In meinem Vortrag möchte ich den Begriff einer PAC-Unterstruktur einer beliebigen Struktur einführen. Desweiteren gebe ich elementare Invarianten a la Cherlin, van den Dries und Macintyre für beschränkte PAC-Unterstrukturen von stabilen Strukturen an und diskutiere Amalgamationseigenschaften.$

 

Freitag, 15.04.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra

Arne Grenzebach (Bremen)

(Gast von Salma Kuhlmann)

Der Satz von Cayley-Bacharach

$\subsection*{Abstract:}$

$In meinem Vortrag werde ich den Satz von Cayley-Bacharach in einer schematheoretischen Version vorstellen. Dazu werde ich auf den Begriff des Schemas, insbesondere des nulldimensionalen Schemas näher eingehen.$

$Abschließend werde ich die klassischen Sätze von Chasles, Pascal und Pappus aus dem Satz von Cayley-Bacharach herleiten.$


Dienstag, 19.04.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Martin Hils (Paris 7)

(Gast von Itay Kaplan)

Bad fields and uniformity results in Kummer theory

$\subsection*{Abstract:}$

$A bad field is an algebraically closed field $K$ together with a proper infinite subgroup $U$ of the multiplicative group such that $(K,+,x,U)$ is a structure of finite Morley rank. In characteristic $0$, bad fields exist (joint work with Baudisch, Martin Pizarro and Wagner) and may be obtained by 'collapsing' an infinite rank analogue due to Poizat. The construction uses is via Hrushovski's amalgamation method, and some non-trivial results from algebraic geometry are needed to make this method work in the particular context:$

$(i) Ax's differential version of Schanuel's Conjecture (or rather a consequence thereof due to Zilber, called weak CIT), in order to get a definable control of the predimension;$

$(ii) a uniformity result in Kummer theory, in order to definably control multiplicities.$

$In the talk, we will start with an overview of the construction of the bad field. We will then focus on (ii), presenting a definability result in Kummer theory in the case of the multiplicative group. Gabber suggested a different proof which generalises to the case of semi-abelian varieties. We will explain a further generalisation to the context of definable abelian groups of finite Morley rank, and which allows a purely model-theoretic proof. This part is very recent and joint work with M. Bays and M. Gavrilovich.$

 

 

Freitag, 29.04.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra

Daniel Plaumann (Konstanz)

Der Satz von Cayley-Bacharach II

$\subsection*{Abstract:}$

$Der klassische Satz von Cayley-Bacharach (ursprünglich von Chasles) besagt: Sind neun Punkte in der Ebene der Schnitt zweier Kubiken und enthält eine weitere Kubik acht von ihnen, so enthält sie bereits alle neun. In diesem Vortrag werden die Geschichte des Satzes und mehrere Fassungen in aufsteigender Allgemeinheit diskutiert. Wir folgen dabei der Darstellung von Eisenbud, Green und Harris (Bulletin AMS 33 No. 3, 1996).$

 

Dienstag, 03.05.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Christoph Hanselka (Konstanz)

Sums of 2m-th Powers (Part II)

$\subsection*{Abstract:}$

$Using the Ax-Kochen-Ershov principle for Henselian valued fields, I will talk about a suitable elementary description of the class of Laurent series in order to prove a result by Prestel on the existence of a degree bound in a criterion for the representability of polynomials over the reals as sums of 2m-th powers of rational functions.$


Freitag, 06.05.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra

Annalisa Conversano (Konstanz)

The commutator subgroup of a semialgebraic group

$\subsection*{Abstract:}$

$We give an example of a semialgebraic group whose commutator subgroup is not semialgebraic, and we show how semialgebraicity of the commutator subgroup is related to semialgebraicity of Levi subgroups.$


Dienstag, 10.05.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Annalisa Conversano (Konstanz)

Connected components of groups and o-minimality
(joint papers with Anand Pillay)

$\subsection*{Abstract:}$

$In the talk we first introduce the definably connected component $G^{0}$, the type-definable connected component $G^{00}$ and the Aut-invariant connected component $G^{000}$ of a group $G$ definable in a saturated structure $M$. Then we specialize to the case where $M$ is o-minimal giving a description of the quotients $G/G^{00}$, $G/G^{000}$, $G^{00}/G^{000}$ as topological groups endowed with the logic topology, in terms of a basic decomposition of $G$.$


Freitag, 13.05.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra

Tobias Kaiser (Passau)

(Gast von Salma Kuhlmann)

Zahme Maße

$\subsection*{Abstract:}$

$Wir sind interessiert an einer Maß- und Integrationstheorie, die mit o-Minimalität verträglich ist. Dazu führen wir folgende Definition ein:$

$Gegeben sei eine o-minimale Struktur auf dem Körper der reellen Zahlen und ein  Maß $\mu$, das auf den Borelmengen des $\mathbb{R}^n$ definiert ist. Wir nennen $\mu$ $\mathcal{M}$-zahm, falls es eine o-minimale Expansion von $\mathcal{M}$ gibt, so dass für jede parametrisierte Familie von Funktionen auf $\mathbb{R}^n$, die definierbar in $\mathcal{M}$ ist, die entsprechende Familie der Integrale bzgl. $\mu$ in dieser o-minimalen Expansion definierbar ist.$

$Im ersten Teil des Vortrages bringen wir die Definitionen und motivieren sie durch existierende und viele neue Beispiele. Im zweiten Teil betrachten wir das Lebesgue-Maß in diesem Licht. Im ersten Teil behandeln wir definierbare Versionen wichtiger Sätze wie des Satzes von Radon-Nikodym und des Rieszschen Darstellungssatzes. Diese Ergebnisse erlauben uns, zahme Maße explizit zu beschreiben.$


Dienstag, 17.05.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Karen Lange (Notre Dame)

(Gast von Salma Kuhlmann)

Limit computable integer parts

$\subsection*{Abstract:}$

$An {\em integer part} $I$ of a real closed field $R$ is a discrete ordered subring  containing $1$ such that  for all $r\in R$ there exists  a unique $i\in I$ with $i\leq r < i+1$.  Mourgues and Ressayre showed  that every real closed field $R$ has an integer part.   For a countable real closed field $R$, we showed that the integer part obtained by the procedure of Mourgues and Ressayre is $\Delta^0_{\omega^\omega}(R)$.   We would like to know whether there exists a construction that yields a computationally simpler integer part, perhaps one that is {\em limit computable}, i.e.,  $\Delta_2^0(R)$.$

$All integer parts are {\em Z-rings}, discretely ordered rings that have  the euclidean algorithm for dividing by integers.  By a result of Wilkie, any $Z$-ring can be extended to an integer part for {\em some} real closed field.  We show that we can compute a maximal $Z$-ring $I$ for any real closed field $R$ that is $\Delta^0_2(R)$, and we then examine whether this $I$ must serve as an integer part for $R$.$

$We also show that certain subclasses of $\Delta^0_2(R)$ are not sufficient to give integer parts for arbitrary $R$.$

$This is joint work with Paola D'Aquino and Julia Knight.$

$\begin{thebibliography}{999}

$\bibitem{AKL} P. D'Aquino, J. Knight, and K. Lange, ``Limit computable integer parts,"  to appear in {\em Archive for  Math. Logic}.

$\bibitem{KL} J. Knight and K. Lange, ``Countable real closed fields and developments of bounded length," to appear in {\em Proceedings of London Math. Soc.}

$\bibitem{MR}  M.\ H.\ Mourgues and J.-P.\ Ressayre, ``Every real closed field has an integer part,'' \emph{J.\ Symb.\ Logic}, vol.\ 58 (1993), pp. 641-647.

$\bibitem{W}  Alex Wilkie, ``Some results and problems on weak systems of Arithmetic", in \emph{Logic Colloquium '77}, North Holland.

 

Freitag, 20.05.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra

Roland Speicher (U. des Saarlandes, Saarbrücken)

(Gast von Salma Kuhlmann)

Was sind und was sollen Quantenpermutationen
oder
Quantum Groups Made Easy

$\subsection*{Abstract:}$

$Quantum groups describe symmetries in a noncommutative context. I will discuss a special "easy" class of such quantum symmetries. Basic examples are quantum permutations and quantum rotations. Those strengthen the corresponding classical symmetries. I will explain the representation theory of these quantum groups and what it means to be invariant under them.$

 

D ienstag, 24.05.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Karen Lange (Notre Dame)

(Gast von Salma Kuhlmann)

Generalized power series and  real closed fields

$\subsection*{Abstract:}$

$Mourgues and Ressayre showed that every real closed field $R$ has an integer part by constructing a special embedding of  $R$ into a field  $k\langle\langle G\rangle\rangle$ of generalized power series.   Let $k$ be the residue field of $R$, and let $G$ be the value group of $R$.  The field $k\langle\langle G\rangle\rangle$  consists of elements of the form  $\Sigma_{g\in S}a_gg$ where $a_g\in k$ and the support of the power series $S\subseteq G$ is well ordered.   Julia Knight and I previouslyanalyzed an algorithmic version of their construction for countable $R$ and showed that  generalized power series in the image of $R$ are of length less than $\omega^{\omega^\omega}$ and that $R$ has an integer part that is $\Delta_{\omega^\omega}(R)$.   Ressayre showed that every real closed exponential field has an integer part $I$ that is closed under $2^x$ for positive elements of $I$  using the same approach as in M. H. Mourgues and J.-P. Ressayre, ``Every real closed field has an integer part''. However, he had to  choose more carefully the value group $G$ and the embedding of $R$ into $k\langle\langle G\rangle\rangle$.  We demonstrate that  these alterations cause Ressayre's construction in the exponential case to be much more complex than Mourgues and Ressayre's original construction.$

$This is joint work with Paola D'Aquino, Julia Knight, and Salma Kuhlmann.$

$\begin{thebibliography}{999}$

$\bibitem{KL} J. Knight and K. Lange, ``Countable real closed fields and developments of bounded length," to appear in {\em Proc. of London Math. Soc.}$

$\bibitem{MR}  M.\ H.\ Mourgues and J.-P.\ Ressayre, ``Every real closed field has an integer part,'' \emph{J.\ Symb.\ Logic}, vol.\ 58 (1993), pp. 641-647.$

$\bibitem{R} J.-P.\ Ressayre, ``Integer parts of real closed exponential fields," in {\em Arithmetic, Proof Theory, and Computational Complexity}, Oxford Logic Guides, vol.\ 23 (1993), pp. 278-288.  $

 

Donnerstag, 26.05.2011 um 17.00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Hans Mittelmann (Arizona State University)
(Gast von Cordian Riener)

Computing Strong Bounds in Combinatorial Optimization

$\subsection*{Abstract:}$

$As is well-known semidefinite relaxations of discrete optimization problems can yield excellent bounds on their solutions. We present three examples from our collaborative research. The first addresses the quadratic assignment problem and a formulation is developed which yields the strongest lower bounds known for larger dimensions. Utilizing the latest iterative SDP solver and ideas from verified computing a realistic problem from communications is solved for dimensions up to 512.$

$A strategy based on the Lovasz theta function is generalized to compute upper bounds on the spherical kissing number utilizing SDP relaxations. Multiple precision SDP solvers are needed and improvements on known results for all kissing numbers in dimensions up to 23 are obtained. Generalizing ideas of Lex Schrijver improved upper bounds for general binary codes are obtained in many cases.$

 

Freitag, 27.05.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
Ken Dykema (Texas A&M)
(Gast von Markus Schweighofer)

Sofic dimension

$\subsection*{Abstract:}$

$An introduction to sofic groups will be followed by a description of sofic dimension, which is an invariant (recently obtained in joint work with David Kerr and Mikael Pichot) that is defined in terms of the asymptotics of the number of sofic approximations to a group. A formula for amalgamated free products holds, and similar results exist for group actions and measurable equivalence relations.$

 

 

Donnerstag, 09.06.2011 um 16.15, Lectures on the geometry of plane quartic curves

Daniel Plaumann und Cynthia Vinzant

Announcement:

We plan a short informal series of lectures on the geometry of plane quartic curves, focussing on determinantal and sums-of-squares representations. The first session will be this Thursday, June 9, 16:15-18:00 in F426 with one talk by each of us. We will start with Dixon's construction of determinantal representations of plane curves, and the combinatorics of determinantal representations and the 28 bitangents of a plane quartic.

 

Freitag, 10.06.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra

Roland Abuaf (Institut Fourier, Grenoble)

(Gast von Rainer Sinn)

Theorems on tangencies in convex and projective geometry

$\subsection*{Abstract:}$

$In their recent work on the convex hull of a real algebraic variety, Ranestad and Sturmfels provide an interesting description of the algebraic boundary of such a variety. Their formula relate the irreducible components of this boundary to some stratas of the projective dual of the associated complex projective variety. One technical point in their proof is to give a sharp bound for the dimension of the set of hyperplanes whose tangency locus with the variety spans a $P^k$. While this is easily done if the variety has only finitely many hyperplanes which are tangent along infinitely many points, the general "scheme-theoretic" case seems to be completely out of reach.$

$In this talk, I will introduce some connexions between these bounds and classical theorems in convex geometry. I will also explain how one can prove a weak version of the result conjectured by Ranestad and Sturmfels. These "topological" bounds are already powerful. Indeed, I will derive from them a new proof of Severi's famous theorem on non-linearly normal surfaces in $P^5$.$


Dienstag, 14.06.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Itay Kaplan (Konstanz)

Groups in dependent theories

$\subsection*{Abstract:}$

$In this talk I will first introduce dependent theories, and then discuss some results about group in them. I hope to prove the existence of the type definable connected component theorem due to Shelah.$



Freitag, 17.06.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra

Michael Wibmer (RWTH Aachen)

(Gast von Arno Fehm)

Algebraic difference equations and a theorem of Chevalley

$\subsection*{Abstract:}$

$In difference algebra one studies systems of algebraic difference equations in the same way as one studies systems of algebraic equations in commutative algebra and algebraic geometry. In this talk I will first introduce the basic concepts of difference algebra and explain some fundamental properties of difference varieties. Then I will present a difference analog of a theorem of Chevalley which states that the image of a morphism of varieties is a constructible set. From an algebraic point of view Chevalley's theorem can be formulated as a lifting theorem for prime ideals.$


Dienstag, 21.06.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Martin Bays (Paris 7)

(Gast von Margaret Thomas)

Categoricity of Exponential Maps

$\subsection*{Abstract:}$

$Exponential maps of semi-abelian varieties, of which the familiar complex exponential map and the Weierstrass p-functions are special cases, are some of the most natural non-algebraic complex analytic functions in existence. I will discuss the model theory of some associated structures, with an emphasis on categoricity/classification (higher amalgamation) concerns. Zilber, Shelah, Kummer, Faltings and others will make their appearances, as model theoretic and number theoretic techniques mesh.$

 

Dienstag, 28.06.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Will Anscombe (Oxford)

(Gast von Arno Fehm)

Aspects of definability in henselian fields of positive characteristic

$\subsection*{Abstract:}$

$We investigate definability in fields of power series over finite fields, motivated by open problems in the model theory of these fields. Inevitably, many of the difficulties are due to the positive characteristic. Using Prestel-Ziegler's t-henselian fields, we will look at existential definability; first looking locally and then drawing conclusions about existentially definable sets, subsets, and substructures. Specialising the context to power series fields $ K((t))$ but not restricting to existential formulas, we study subsets definable with parameters from $K$. We will use the well-known automorphism group together with a Hensel-like property to find that the orbits of elements are existentially definable (although using the parameter $t$). Finally, we use these results to find an existential definition of the valuation ring which uses parameters only from $K$.$

 

Donnerstag, 30.06. um 17.00 Uhr, Kolloquium

Bruce Reznick (Urbana-Champaign)

(Gast von Claus Scheiderer)

The secret lives of polynomial identities

$\subsection*{Abstract:}$

$Polynomial identities can reflect deeper mathematical phenomena. In this talk, I will discuss some of the stories behind the following four identities (and their relatives). The stories involve algebra, analysis, number theory, combinatorics, geometry and numerical analysis. Fourteenth powers of polynomials will show up. \begin{equation} \begin{gathered} 1024x^{10} + 1024 y^{10} + (x+\sqrt 3\ y)^{10} + (x- \sqrt 3\ y)^{10} \\ + (\sqrt 3\  x+ y)^{10} + (\sqrt 3\ x- y)^{10}= 1512(x^2 + y^2)^5,\end{gathered}\end{equation} \begin{equation} x^3 + y^3 = \left( \frac{x(x^3+2y^3)}{x^3-y^3} \right)^3 +\left( \frac{y(y^3+2x^3)}{y^3-x^3} \right)^3,\end{equation} \begin{equation} \begin{gathered} (x^2 + \sqrt{2}\ xy - y^2)^5 + (i x^2 - \sqrt{2}\ xy +i y^2)^5 +  \\ (-x^2 + \sqrt{2}\ xy + y^2)^5 +  (-ix^2 - \sqrt{2}\ xy -i y^2)^5 = 0, \end{gathered} \end{equation}\begin{equation}\sum_{1 \le i < j \le 4}\bigl( (x_i+x_j)^4 + (x_i-x_j)^4 \bigr)= 6(x_1^2 + x_2^2+x_3^2 + x_4^2)^2.\end{equation}Equation (1) has roots in 19th century mathematics; (2) is due to  Vi\'ete (1590's); (3) was independently found by Desboves (1880) and Elkies (1995); (4) was used by Liouville (1859) to prove that every positive integer is a sum of at most 53 4th powers of integers.$


Freitag, 01.07.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra

Cynthia Vinzant (Berkeley)

(Gast von Daniel Plaumann)

The central curve of a linear program

$\subsection*{Abstract:}$

$The central curve of a linear program is an algebraic curve specified by the associated hyperplane arrangement and cost vector. This curve is the union of the various central paths for minimizing or maximizing the cost function over any region in this hyperplane arrangement. Here we will discuss the algebraic properties of this curve and its beautiful global geometry, both of which are controlled by the corresponding matroid of the hyperplane arrangement.$


Dienstag, 05.07.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Katharina Dupont (Konstanz)

Definable Valuations

$\subsection*{Abstract:}$

$Given a field $F$ we are interested if there exists a formula $\varphi(x)$ in the language of rings $L_{\textrm{ring}}=(0,1;+,-,\cdot)$, possibly with parameters, such that $\left\{x\in F\mid \varphi(x)\right\}$ is a non-trivial valuation ring on $F$. $


$We will give examples of fields on which there is a definable valuation and of fields for which we can proof that there is no definable valuation. For some of the positive examples we will show that the valuation is definable without parameters. $

$The talk will be mainly based on the preprint "Definable Valuations" of Jochen Koenigsmann.$

 

Freitag, 08.07.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra

Sebastian Wenzel (Konstanz)

Sums of squares and strip problem

$\subsection*{Abstract:}$

$Recently, M.Marshall answered a long outstanding question in algebraic geometry by showing that if $f\in\mathbb{R}[x,y]$ and $f \geq 0$ on the strip $[0,1]\times\mathbb{R}$, then $f$ has a representation $f = g + x(1-x) h$, where $g,h \in\mathbb{R}[x,y]$ are sums of squares.$

$In this talk, I will give a short review of the background of Marshalls result, outline the proof of the theorem and talk about possible generalisations of this result.$

 

 

 

Dienstag, 12.07.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Mikael Matusinski

Surreal numbers as exp-log series

$\subsection*{Abstract:}$

$In his monograph [Gon86], H. Gonshor showed that Conway’s real closed field of surreal numbers carries an exponential and logarithmic map. Subsequently, L. van den Dries and P. Ehrlich showed in [vdDE01] that it is a model of the theory of Rexp. Our aim is to show that surreal numbers form a field of exp-log series and transseries. This would help us, applying previous results of ours [KM11][Sch01], to endow the field of surreal numbers with a natural derivation.$

$After giving a quick overview of these results, we will present in this talk our first step in this direction : the description of what we call the exponential equivalence classes of surreal numbers. This notion takes place naturally between the Conway’s omega map and the generalized epsilon numbers.$

$This is a joint work with J. van der Hoeven and S. Kuhlmann.$

$\subsubsection*{References:}$

$[Gon86] H. Gonshor, An introduction to the theory of surreal numbers, London Mathematical Society Lecture Note Series, vol. 110, Cambridge University Press, Cambridge, 1986.$

$[KM11] S. Kuhlmann and M. Matusinski, Hardy type derivations on fields of exponential logarithmic series., to appear in J. Algebra, arxiv 1010.0896 (2011).$

$[Sch01] M.C. Schmeling, Corps de transséries, Ph.D. thesis, Université Paris-VII, 2001.$

$[vdDE01] Lou van den Dries and Philip Ehrlich, Fields of surreal numbers and exponentiation, Fund. Math. 167 (2001), no. 2, 173–188.$

 

Donnerstag, 14.07.2011 um 16.00 Uhr im Schwerpunktskolloquium Reelle Geometrie und Algebra

Alex Bartel (POSTECH in Pohang)

(Gast von Arno Fehm)

Yet another occurrence of group representations in number theory

$\subsection*{Abstract:}$

$Representation theory enters number theory in several ways. The business with l-adic Galois representations has gained some notoriety, at latest with Wiles's proof of Fermat's Last Theorem. In this talk, I will present two rather different kinds of representation theory that number theorists have to deal with. One has to do with the fact that Galois groups naturally act on many finitely generated abelian groups, which give rise to integral Galois representations. The other comes from so-called Brauer relations between induced representations and the resulting identities between L-functions. The number theoretic applications I will present will include results on the structure of Mordell-Weil groups of elliptic curves and a mysterious connection between the Galois module structure of units in number fields and class numbers.$


Freitag, 15.07.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra

Jochen Koenigsmann (Oxford)

(Gast von Salma Kuhlmann)

On fields with the absolute Galois group of Q

$\subsection*{Abstract:}$

$We will resume the theme of Alex Bartel's talk from the previous day, humming the tune of the importance of the absolute Galois group of $\mathbb{Q}$ as instanced by Andrew Wiles' proof of Fermat's Last Theorem. We will show that fields with the absolute Galois group of $\mathbb{Q}$ share most arithmetic properties with $\mathbb{Q}$.$