The Research Seminar usually meets on Fridays 2-4 pm, in D1.320, or Mondays 5-7 pm in D1.320. The topics treated come from the present research interests of the members of our group. A common feature is the existence of many symmetries in the objects studied which give the name of the seminar a certain justification. Nevertheless, there may be talks on topics which are of predominantly functional analytic, number theoretic or differential geometric nature.
Program for the summer term 2008 (April through September 2008):
Mon, April 7, 2008; 5:15 pm in D1.320
I. Roy (Metz/Paderborn): Foliated eta and rho invariants
Abstract:
The Eta invariant is a spectral invariant for elliptic differential operators and measures the "spectral asymmetry". However, these invariants are very unstable under perturbations of the operator. Atiyah Patodi and Singer found that the difference of two eta invariants is a much more stable quantity, called the (APS)rho-invariant. We'll give the background and motivation for these 'secondary' invariants, and outline their important properties. For the more general case of a foliation, we'll define the eta invariant on the holonomy groupoid and a possible extension of the rho-invariant on the monodromy groupoid for the foliated flat bundle case using the traces on the Von Neumann algebras associated to the foliation.
Mon, April 14, 2008; 5:15 pm in D1.320
A. Baklouti (Sfax): On the deformation space of Clifford-Klein forms
Abstract:
Let G be a Lie group, H a closed subgroup of G and Γ a finitely generated discrete subgroup of G. In this talk, I will deal with the deformation and the moduli space of Clifford-Klein forms which are not necessarily compact in the context of connected exponential Lie groups. Let me recall that Γ does not in general act properly discontinuously on G/H when H is not compact, so the problem of deformation consists in seeking how to deform Γ by means of homomorphisms from Γ to G (thus to consider the set Hom(Γ,G) of all these homomorphisms) in a way such that the deformed discrete subgroup acts properly on G/H. The problem of describing deformations was first advocated by T. Kobayashi for the general non-Riemannian setting.
Kolloquium des Instituts für Mathematik Tue, April 15, 2008; 6:00 pm in D2
S. Iyengar (Lincoln): Finite group actions and commutative algebra
Abstract:
The problem that gave rise to the research to be reported in this lecture is the following: Given a (finite) group G and a finite dimensional topological space X, can G act freely on X? In the early 1980's Gunnar Carlsson, Bill Browder, Steve Halperin, and others found a number of interesting algebraic obstructions to free actions.
In my talk, I will present certain aspects of recent work in commutative algebra that is motivated by, and perhaps clarifies, some of their results. This is based on joint work with Avramov, Buchweitz, and Miller.
Mon, April 21, 2008; 5:15 pm in D1.320
D. Blottière (Paderborn): An introduction to polylogarithms
Abstract:
The sheaf theory of polylogarithms on the complex space minus 0 provides a geometrical construction of values of the Riemann zeta function at positive integers. The aim of this talk is to explain this theory (due to A.A. Beilinson, P. Deligne, D. Ramakrishnan,...), its relations with the artihmetic and some of its extensions.
Mon, April 28, 2008; 5:15 pm in D1.320
E. Mohamed Moutuou (Metz/Paderborn): The Brauer Group of a Lie Groupoid
Abstract:
Given a Lie groupoid (G, G°), we define the Brauer group Br(G) to be the set of Morita equivalence classes of Diximier-Douady bundles over (G, G°). We show that each element of this group corresponds to a unique element in the set of Morita equivalence classes of S1-central extentions of (G, G°), which is denoted by Ext(G,S1), and vise-versa. Finally, by the well-known result of groupoid cohomology which states that Ext(G,A) = H2(G, A), we have an isomorphism Br(G) = H2(G, S1). In particular, when the groupoid is given by a compact manifold X, we obtain Br(X) = H3(X, Z).
Kolloquium des Instituts für Mathematik Tue, April 29, 2008; 6:00 pm in D2
M. Kreck (Bonn): Codes und 3-Mannigfaltigkeiten
Abstract:
Zunächst wird erklärt, was ein Code ist und welche Bedeutung er bei der Korrektur von Übertragung mit Fehlern hat. Dann wird erklärt, wie man aus einer 3-dimensionalen Mannigfaltigkeit mit Involution Codes bekommt. Es stellt sich heraus, dass es sich um sogenannte selbst-duale Codes handelt. Diese haben eine hochinteressante Verbindung zur Arithmetik (Gitter), welche erläutert wird. Schließlich geht es um die Frage, welche selbst-dualen Codes durch 3-Mannigfaltigkeiten realisiert werden.
Tue, May 13, 2008; 6:00 pm in D2
G. Olshanski (Moscow and Bielefeld): Representation theory and random point processes
Abstract:
I will describe connections between harmonic analysis on groups and certain problems in probability theory and asymptotic combinatorics.Mon, May 19, 2008; 5:15 pm in D1.320
Ch. Wockel (Bonn): Gauge groups for lifting gerbes and principal 2-bundles
Abstract:
Gauge (transformation) groups are the internal symmetry groups of principal
bundles and thus of their associated gauge theories. Classical gauge theory
has proven to be a very successful concept, providing a link between
theoretical physics (gauge field theory) and (differential) topology. In the
past years, the so called "higher gauge theory" has developed quickly,
trying to carry this deep connection further.
In the talk, we present
a global point of view to principal 2-bundles (or categorified principal
bundles), the fundamental objects of higher gauge theory. The analysis of
its symmetry groups shall be of particular interest, and we show that they
are, in fact, infinite-dimensional Lie 2-groups. Moreover, we shall work out
lifting gerbes as an example, linking these higher gauge groups to the ones
in the classical setting.
Mon, May 26, 2008; 5:15 pm in D1.320
Abstract:
An old result of Myers and Steenrod states that the group of
isometries of a Riemannian manifold is a Lie transformation group. We use the
approach developed by Nomizu and Kobayashi, based on the frame bundle and
on 1-structures, to investigate similar problems in the context of
manifolds modeled on Hilbert or Banach spaces. Results are obtained in this
way also about other types of groups, e.g. the affine
transformation groups. The enlargability problem for Lie algebras, the
lack of coordinate systems of the second kind, the non-separability of the
algebra of operators and the lack of closed complements for Banach subspaces
are important obstructions, resulting in a series of open problems.
Tue, May 27, 2008; 6:00 pm in D2
Kolloquium des Instituts für Mathematik
K.-H. Neeb (Darmstadt): Konvexität in der Darstellungstheorie von Lie-Gruppen
Abstract:
Die unitäre Darstellungstheorie von Lie-Gruppen verallgemeinert den Spektralsatz für selbstadjungierte Operatoren, der die Darstellungstheorie der additiven Gruppe der reellen Zahlen beschreibt. Um Ordnung in den Zoo der Darstellungen zu bringen, unterscheidet man Typen von Darstellungen gemäß der Halbbeschränktheit von Spekten ihrer infinitesimalen Erzeuger. Diese Methode führt für endlichdimensionale Gruppen zu einer sehr reizvollen Theorie mit zahlreichen Verbindungen zu symplektischer Geometrie und komplexer Analysis. Eine dieser Verbindungen konkretisiert sich in der Impulsabbildung einer unitären Darstellung, die zwischen dem projektiven Hilbertraum und dem Dual der Lie-Algebra vermittelt. Auch für unendlichdimensionale Lie-Gruppen lassen sich viele Klassen interessanter Darstellungen in diesem Rahmen einheitlich behandeln.
Mon, June 2, 2008; 5:15 pm in D1.320
Operator
Abstract:
Both Dirichlet and Neumann boundary conditions for the Laplace equation are of fundamental importance in Mathematics and Physics. Given a compact connected Riemannian manifold M with boundary ∂M the Dirichlet-to-Neumann operator Λg maps Dirichlet boundary data f to the corresponding Neumann boundary data Λg f = ∂ν u|∂Mu denotes the unique solution to the Dirichlet problem Δg u=0 in M, u|∂M = f. The main statement is that this operator is a first order elliptic pseudodifferential operator on the boundary ∂M.
We will first give a brief overview of how to define the Dirichlet-to-Neumann operator as a map Λg:H1/2(∂M) ---> H-1/2(∂M)
between Sobolev spaces. In order to show that it is actually a
pseudodifferential operator we introduce tangential pseudodifferential
operators. This allows us to derive a microlocal factorization of the
Laplacian near boundary points. Together with a regularity statement
for the heat equation this will finally give the main result.
Mon, June 9, 2008; 5:15 pm in D1.320
E. Mohamed Moutuou (Metz/Paderborn): A Short Introduction To Twisted K-theoryAbstract:
Given a Lie groupoid (G,G°) and a Dixmier-Douady bundle A over G, the twisted K-theory of G by A, denoted by K(G;A), is defined as the K-theory of the reduced crossed-product of the the C*-bundle A by the action of G. I will give a relation between Br(G) and K(G;A) and finally I will mention very quickly the analogue of Thom isomorphism in the twisted case.
Kolloquium des Instituts für Mathematik Tue, June 10, 2008; 6:00 pm in D2
J.-Ph. Anker (Orleans): Evolution equations in negative curvature
Abstract:
We are interested in classical linear and nonlinear evolution equations, such as the heat equation, the Schrödinger equation, the wave equation, ... on manifolds with nonpositive curvature. In this talk we shall restrict to the simplest cases, namely real hyperbolic spaces and homogeneous trees. After recalling the setting, we will present and discuss a few results and properties, and compare them with the classical Euclidean case.
Mon, June 16, 2008; 5:15 pm in D1.320
M. Schröder (Paderborn): Non-euclidean Pseudodifferential Analysis on Symmetric Spaces
Abstract:
Let X be a general symmetric space of the noncompact type. We use the Fourier transform for symmetric spaces to introduce a calculus of non-euclidean pseudodifferential operators. I will give group-theoretically backgrounds, then a brief overview of the geometry of symmetric spaces, present first results about pseudodifferential operators and conclude with open problems of the calculus, such as the question about stationary phase asymptotics.
Mon, June 23, 2008; 5:15 pm in D1.320
M. Vaeth (Würzburg/Gießen/Berlin): Three faces of coincidence point theory
Abstract:
Besides the usual definition of a mapping degree there is a related but purely homotopic notion of so-called essential (or epi) maps. Both concepts generalize in a natural way to coincidence point theories. However, since decades also other types of coincidence point degrees are known whose relation with the above generalizations is not clear - in a sense, they present "dual" theories. The talk gives a survey on the known theories (i.e. the three known faces of coincidence point theory) and also discusses the relatively new and thus unknown "fourth" face. It is also sketched what can be expected by combining the theories.
Fri, July 18, 2008; 2:15 pm in D1.320
D. Mayer (Clausthal):The transfer operator approach to the Hecke triangle groups
Abstract:
We use the Hurwitz-Nakada continued fractions to derive a symbolic dynamics for the geodesic flow on the Hecke surfaces Mq = Gq\H for Gq the Hecke triangle groups generated by Tqz = z+βq where βq
= 2cos( π/q), q = 3, 4, .. and Sz = −1/z . For this we construct a
Poincare section with a Poincare map which is closely related to the
natural extension Fq of the generating interval map fq : Iq --> Iq for the H-N fractions. The closed orbits of the geodesic flow can then be characterized by the periodic points of the map fq.
Standard procedures allow us to define a tranfer operator for the flow,
whose Fredholm determinant is closely related but not identical to the
Selberg zeta function for Gq. We can finally derive a functional
equation for the eigenfunctions of this operator
which corresponds at least for the case of the modular group G3 to the well known Lewis equation.
Mon, July 21, 2008; 16:00 -- 18:00 pm in D1.320 (double feature)
R. Kerr (Glasgow): Products of Toeplitz operators on a vector-valued Bergman space
Abstract:
We give a necessary and a sufficient condition for the boundedness of
the Toeplitz product $T_FT_{G^*}$ on the vector valued Bergman
space $L_a^2(\mathbb{C}^n)$, where $F$ and $G$ are matrix symbols with
scalar valued Bergman space entries. The results generalize existing results
in the scalar valued Bergman space case. We also characterize boundedness
and invertibility of Toeplitz products $T_FT_{G^*}$ in terms of the
Berezin transform, generalizing results found by Zheng and Stroethoff for
the scalar valued Bergman space.
B. Sehba (Glasgow): Test functions, Paraproducts, Multiplication operators and norm
estimates
Abstract:
We combine test functions and Carleson type measures to characterize boundedness of multiplication operators in the continuous case and the paraproducts in the dyadic setting. We then obtain equivalent definitions of some function spaces and lower estimates of the multiplication operators and paraproducts on the corresponding spaces.
Thu, August 7, 2008; 14:15 -- 15:00 in D1.320
C. Köhler (Paderborn): The Gabriel-Roiter measure and covering theory
Abstract:
The Gabriel-Roiter measure is a combinatorial invariant which is defined on the category of finite length modules. In my talk I will introduce this measure and present a method to compute it for a certain class of algebras, the so-called string algebras. To this end, I will explain how covering theory helps to simplify the computation of the Gabriel-Roiter measure.
IRTG-Summerschool: Large-N limits
August 18-29, 2008 in L'Etang de Hasselfurth near Bitche, France
Main Speakers:
- J. Faraut (Université Paris VI):
Finite and infinite dimensional spherical analysis - S.G. Rajeev (University of Rochester):
Spectral statistics of large dimensional matrices S. Zelditch (Johns Hopkins University):
Large-N limits in geometry and physics
Fri, September 12, 2008; 2:15 pm in D1.320
Abstract:
If we allow for super symplectic forms to be non-homogeneous, some surprising results appear: the Poisson algebra is no longer the set of smooth functions on the manifold, but consists of pairs of functions and prequantization in terms of complex line bundles and in terms of principal finer bundles are no longer equivalent procedures. However, non-homogeneous symplectic forms are rule more than exception in the class of coadjoint orbits. In this tak I will explain these differences.
Program of previous semesters
winter 2007/2008, summer 2007 , winter 2006/2007, summer 2006, winter 2005/2006, summer 2005, winter 2004/2005, summer 2004
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