## November – Dezember 2013

Vortrag im Rahmen des Seminars “Algebra und Geometrie”

Abstract: A major open question in convex algebraic geometry is whether all hyperbolicity cones are spectrahedral, i.e. the solution sets of linear matrix inequalities. We will use sum-of-squares decompositions of certain matrix polynomials to approach this problem. More precisely, we will prove that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that qh has a definite determinantal representation.

P R O G R A M M

Begrüßung durch den Dekan

Professor Dr. Michael Hellus (Universität Regensburg)

- Eine Anwendung von Matlis-Dualen lokaler Kohomologie-Moduln

Kaffeepause

Professor Dr. Bernd Fritzsche

- Würdigung der Tätigkeit von Professor Dr. Jürgen Stückrad an der Fakultät für Mathematik und Informatik

Professor Dr. Uwe Storch (Ruhr-Universität Bochum)

- Über reelle Nullstellensätze

Sektempfang

Physik-Kolloquium

Abstract: We review the remarkable connection that has emerged during the past 40 years between the theory of black holes in general relativity and the laws of thermodynamics. The issue of information loss occurring when a black hole evaporates due to its thermal emission will also be discussed.

Alle Teilnehmer sind ab 16.30 Uhr zu Kaffee vor dem Hörsaal eingeladen.

Vortrag im Rahmen des Seminars “Algebra und Geometrie”

Abstract: For a group G and a generating set S one defines a growth function f(n) as the number of elements in the group G which can be represented as words in S that have length less or equal to n. It is a famous result by M.Gromov that groups with a function f(n) bounded by a polynomial from above are virtually nilpotent. If there exists C>1 such that f(n)>C^n for all n>0 then the group G has exponential growth. We will discuss how the constant C can depend on the choice of the generating set S, mostly concentrating on free products and free products with amalgamation, showing that estimations of best possible constant C can be obtained by a mixture of algebra and geometry.

Vortrag im Rahmen des Seminars “Algebra und Geometrie”

Abstract: The Approximation Property of Haagerup and Kraus (AP) is an approximation property for groups that is weaker than weak amenability. The question whether there exist (exact) groups without it was open for a long time. In 2010, Lafforgue and de la Salle proved that SL(3,R) does not have the AP. Joint with Uffe Haagerup, we proved that also Sp(2,R) and its universal covering group do not satisfy the AP, which, together with the result of Lafforgue and de la Salle, gives rise to the fact that a connected simple Lie group has the AP if and only if its real rank equals zero or one. In this talk, I will discuss this work, and I will comment on a recent joint work with Mikael de la Salle, in which we prove, using similar methods, that connected higher rank simple Lie groups satisfy strong property (T) of Lafforgue, which in turn implies the fixed point property (F_X) of Bader, Furman, Gelander and Monod for such Lie groups and certain Banach spaces X.

Vortrag im Rahmen des Seminars “Algebra und Geometrie”

Abstract: Ultraproduct and central sequence algebra have been important tools for the analysis of II_1 factors. It is therefore natural to consider the ultraproduct/central sequence algebra for general von Neumann algebras. However, due to the lack of a trace, it is not clear how one should define ultraproducts. In fact there have been several competing definitions of ultraproducts (Ocneanu, Golodets, Groh-Raynaud) and central sequence algebras (Golodets, Ocneanu, Connes). In this talk I will discuss the relationship among them and show some structural results of the ultraproducts of type III factors. If there is enough time, I would like to mention about the related work with Haagerup and Winslow about Connes’ Embedding conjecture.

(Joint work with Uffe Haagerup)

Der Dekan lädt herzlich ein zu einem Mathematischen Kolloquium zum Gedenken an

Herrn Professor Dr. Hans-Joachim Roßberg (5.6.1927-13.10.2013) ein. Programm: Bernd Kirstein (Universität Leipzig): “Hans-Joachim Roßberg – ein Protagonist der Entwicklung der Stochastik in Leipzig”, Kaffeepause, Zoltan Sasvari (TU Dresden): “Über Faktorisierungssätze von Lévy, Cramér und Linnik”

Vortrag im Rahmen des Seminars “Algebra und Geometrie”

Abstract: The main goal is the investigation on the unit group of an order O in a rational group ring QG of a finite group G. In particular we are interested in the unit group of ZG. For many finite groups G a specific finite set B of generators of a subgroup of finite index in U(ZG) has been given. The only groups G excluded in this result are those for which the Wedderburn decomposition of the rational group algebra QG has a simple component that is either a non-commutative division algebra different from a totally definite quaternion algebra or a 2×2 matrix ring M2(D), where D is either Q, a quadratic imaginary extension of Q or a totally definite rational division algebra H(a, b, Q). In some of these cases, up to commensurability, the unit group acts discontinuously on a direct poduct of hyperbolic 2- or 3-spaces. The aim is to generalize Poincar ́e’s theorem on fundamental domains and group presentations to these cases. For the moment we have done a test-case: Let O is the ring of integers in Q( d) with d a square-free positive integer. ￼Moreover set O a principal ideal domain and consider SL2(O). It can be shown that SL2(O) acts discontinuously on H2 × H2. We have generalized Poincar ́e’s theorem to this case and hence get generators and relations of SL2(O) via a fundamental domain in H2 × H2.

Fragestunde für Doktoranden und Postdoktoranden.

Vortrag im Kolloquium des Felix-Klein-Collegs

Abstract: Universality of configuration spaces of classes of geometric objects is a prevalent phenomenon. After a short general introduction we give an outline of the proof of the following universality theorem for realization spaces of polyhedral maps.

Let P ⊆ R^3n be a semialgebraic set (real algebraic coefficients). Then there exists a map M with 5 + n + k vertices (on some orientable manifold) which contains only triangles and quadrangles such that the image of the realization space of M under the canonical projection p : R^3(n+k) → R^3n is equal to P∆, where ∆ = {(x_1,…,x_n) ∈ R^3n|x_i = x_j for some i ̸= j}. As a corollary we get that for each strict subfield L of the field of real algebraic numbers there is a map M which can be realized as a polyhedron in R^3 but not in L^3.

A realization of a map is an injective mapping from the set of vertices into R^3 such that the faces of M are mapped onto convex polygons and no self- intersections occur and polygons sharing an edge are not coplanar. The general idea for proving a universality theorem is to encode the polynomial equalities and inequalities by geometric or combinatorial configurations.

Vortrag im Rahmen des Seminars “Algebra und Geometrie”

Abstract: Typical questions in the area of conic integral geometry are the following: given a finite number of closed convex cones, each of them (independently) in a uniformly random orientation, what is the probability that they have a nontrivial intersection? Or, given a (standard normal distributed) random semidefinite program, what is the probability that its solution has rank k? These questions have surprisingly simple answers in terms of the intrinsic volumes of the cones. Moreover, the theory of conic intrinsic volumes, though at first glance being close to its well-studied Euclidean counterpart, has remarkable new features and interconnects several different mathematical disciplines, allowing the usage of different tools stemming from areas such as convex and differential geometry, statistics, functional analysis, or combinatorics. One central recent result is the concentration of the conic intrinsic volumes of a cone about its statistical dimension, which in an only slightly oversimplified way may be summarized by saying that “In high dimensions, a convex cone behaves statistically like a linear subspace.” This result has direct consequences for applications: it gives a rigorous explanation for the ubiquity of phase transition phenomena that appear in high-dimensional applications of convex optimization such as compressed sensing, and also allows the location of these as well as making predictions about quantitative aspects such as the transition widths. We will survey this fascinating new theory, highlight its various interconnections, and present some intriguing open questions and conjectures, among them a genuine conic analog of the Alexandrov-Fenchel inequalities and a new isoperimetric property of spherical balls.