The Research Seminar usually meets on Mondays 5-7 pm in E2.304 or on Fridays 2-4 pm in D1.338. 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 its justification. Nevertheless, there may be talks on topics which are of predominantly functional analytic, number theoretic, representation theoretic or geometric nature.
Program for the winter term 2008/2009
- Monday, October the 13th 2008; 5:15-6:15 pm in E2.304
David Blottière (Universität Paderborn): Introduction to root valuation data
Abstract: The purpose of this talk is to introduce some notions needed in Eva Viehmann's forthcoming talk. The structure of the absolute Galois group of the field of Laurent series with complex coefficients will be recalled, as well as some basic properties of linear algebraic groups (emphasizing on the example of the special linear group). The main part of this talk will deal with the concept of root valuation data whose definition will be explained and elementary properties discussed. - Monday, October the 20th 2008; 5:15-6:45 pm in E2.304
Dr. Eva Viehmann (Universität Bonn): Coverings of root valuation strata
Abstract: Affine Springer fibres play an important role in the representation theory of linear algebraic groups over local fields. For a given root valuation datum we define and study a new kind of associated generalized affine Springer fibres. They are nonempty exactly over the closure of the given root valuation stratum and define interesting coverings of the given stratum. - Monday, October the 27th 2008; 5:15-6:15 pm in E2.304
Dr. Martin Laubinger (Universität Münster): Frölicher groups with Lie algebra
Abstract: A Frölicher space is a set together with a smooth structure which is determined by its smooth curves. The category of Frölicher spaces is closed under various category theoretical constructions. Also, there is a natural tangent functor. Given a group in the category of Frölicher spaces, we investigate the question whether the tangent space at identity can be equipped with a Lie bracket. - Monday, November the 3rd 2008; 5:15-6:45 pm in E2.304
Dr. Sven Meinhardt (Universität Bonn): Quotient categories and birational geometry
Abstract: My talk deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. It turns out that this category has homological dimension c. As an application of this, I will describe all exact equivalences between the derived categories of these quotient categories in the particular case c=1, which is closely related to classification problems in birational geometry. If there is some time left, I will say a few words about the space of stability conditions on these quotient categories in the case c=1.
- Monday, November the 10th 2008; 5:15-6:15 pm in E2.304
Prof. Dr. Cornelia Vizman (West University of Timisoara, Romania): A higher dimensional version of the vortex filament equation
Abstract: A model for the motion of vortex filaments in an ideal fluid is the evolution equation for closed space curves $\gamma$:$\dot\gamma=\gamma'\times\gamma''$.Using the fact that it descends to a Hamiltonian equation on the space of unparameterized knots with the Marsden-Weinstein symplectic form, we give a generalization to higher dimensions.This is the flow in the space of codimension two submanifolds (a non-linear Grassmannian) along the mean curvature vector field rotated by 90 degrees. We study the special case of surfaces in R^4 and solutions consisting of orbits of an isometric action. - Monday, November the 17th 2008; 5:15-6:45 pm in E2.304
Prof. Dr. Jan Schröer (Universität Bonn): Euler characteristics of varieties of composition series
Abstract: (Joint work with C. Geiss and B. Leclerc) Given a finite-dimensional module $M$ over an algebra $A$, one can study the projective variety of composition series of $M$ such that the simple subfactors of the composition series occur in a given order.
So to each composition series type, there is a map which associates to a module the Euler characteristic of the corresponding variety of composition series. These maps generate (as a vector space) the complex Ringel-Hall algebra $C(A)$ associated to $A$.
Our aim is to obtain a better understanding of such Euler characteristics. We focus on some subcategories of modules over preprojective algebras. Especially for many rigid modules we obtain algorithms which compute the Euler characteristics explicitly. These results have application to the theory of cluster algebras.
- Monday, November the 24th 2008, 5:15-6:45 pm in E2.304
Dr. Alexander Alldridge (Universität Paderborn): Invariant Differential Operators on Riemannian Symmetric Superspaces
Abstract: Supermanifolds are generalisations of manifolds whose definition is motivated by the necessity to construct, in physics, a classical model incorporating supersymmetry.
We consider (reductive) Riemannian symmetric superspaces. These are generalisations of Riemannian symmetric spaces to the realm of supermanifolds. They occur in various contexts, such as Howe dual pairs, and the universality classes of random matrix theory.
In particular, we are interested in invariant differential operators on such spaces. We show that the algebra of invariant differential operators on a symmetric superspace is supercommutative. In the Riemannian case, it is commutative, and isomorphic to a certain subalgebra of the algebra of Weyl-group invariants on a Cartan subspace ("Chevalley's Restriction Theorem"). This paves the way for a study of spherical functions in this context.
The above results are part of joint work in progress, together with Joachim Hilgert and Martin R. Zirnbauer (Cologne). - Monday, December the 1st 2008, 5:15-6:45 pm in E2.304
Prof. Dr. Daniel Grieser (Universität Oldenburg): Classical and new ideas in the theory of pseudodifferential operators
Abstract: Pseudodifferential operators (PsiDO) were invented in the 1960s to obtain detailed information on solutions of partial differential equations (PDEs). They have since become an effective and systematic tool in areas ranging from global analysis to mathematical physics. Classical PsiDO theory is well suited for the study of PDEs on compact manifolds, but on non-compact spaces only gives local information. A focus of modern PsiDO research is to extend the theory to cover global problems on non-compact manifolds. Examples of such manifolds are Euclidean space, locally symmetric spaces or smooth parts of compact singular spaces, such as algebraic varieties. The problems include asymptotic behavior of solutions, scattering theory, Hodge theory and L^2 cohomology.
In the talk I will recall the classical theory and explain some central ideas of the modern developments, which include recent joint work with E. Hunsicker. - Monday, December the 8th 2008, 5:15-6:15 pm in E2.304
Prof. Dr. Dieter Vossieck (Universidad Michoacana de San Nicolas de Hidalgo, Mexico): A convenient abelian hull for the exact category of commutative complex Lie groups
Abstract: For two commutative complex Lie groups G and G’, Hom(G,G’) admits a natural Lie group structure whereas Ext(G,G’) in general does not. We will explain how to accommodate Ext(G,G’) in a suitable abelian hull of the exact category of all commutative complex Lie groups. - Monday, December the 15th 2008, 5:15-6:15 pm in E2.304
Dipl.-Math. Stefan Wolf (Universität Paderborn): Geometry of Quiver Flag Varieties
Abstract: Fix a finite dimensional K-representation of a quiver Q, or, more generally, a finite dimensional module of a K-algebra A, for K some field. Then, one has a natural generalisation of the flag variety by additionally demanding that each subspace is also a submodule of the given module.
In general these schemes are neither reduced nor irreducible. I will explain how to calculate the tangent space to this variety and then show, that, under some additional conditions, it is smooth and irreducible. Then I will interpret this result as some coefficient of a Hall polynomial. - Monday, January the 5th 2009, 5:15-6:45 pm in E2.304
Prof. Dr. Helge Glöckner (Universität Paderborn): The structure theory of totally disconnected, locally compact groups, illustrated by the special case of Lie groups over local fields
Abstract: In the first half of the talk, I'll give an introduction to the structure theory of totally disconnected, locally compact topological groups, which started with a paper by George A. Willis in 1994. In particular, I'll interpret the basic concepts of this theory in the special case of Lie groups (notably, for algebraic groups) over local fields.
The second half of the talk is devoted to contraction groups (G,f), meaning that f is an automorphism of G such that f^n(x) tends to the neutral element 1 as n tends to infinity, for each x in G. I'll describe the classification of the simple totally disconnected contraction groups (obtained jointly with Willis), which implies that unipotent p-adic algebraic groups are among the basic building blocks of general locally compact contraction groups.
Finally, I'll discuss the structure of Lie groups over local fields which admit a contractive analytic automorphism. - Friday, January the 9th 2009, 2:15-13:15 pm in D1.338
Dr. Roland Knevel (Université du Luxembourg): Parametrized Lattices and Automorphic forms
Abstract: Automorphic forms over the upper half plane with respect to a lattice in SL(2, R) are a well-known example of geometric quantization. In my talk I would like to present parametrized (!) lattices as a generalization of ordinary lattices and an approach to the spaces of automorphic and cusp forms on the upper half plane H with respect to parametrized lattices of SL(2,R) using the concept of parametrized Riemann surfaces. Hereby 'parametrized' means: we are talking about whole families of discrete subgroups, Riemann surfaces etc. with finitely many nilpotent parameters, which is of course an example of local deformation. This concept originally comes from the theory of super manifolds where it is more essential, however even in this simple non-super situation parametrization is a nice study object producing striking phenomena.
The work I would like to present in my talk consists of 4 steps:
- show that the quotient of the upper half plane by a parametrized lattice is a parametrized Riemann surface with compact body after adding cusps,
- define the spaces of automorphic forms via growth conditions generalizing the usual non-parametrized case,
- show that any parametrized Riemann surface with compact body corresponds to a parametrized point of the moduli space,
- investigate the spaces over the parametrized Riemann surface corresponding to automorphic forms on H with respect to the parametrized lattice.
- Monday, January the 12th 2009, 5:15-6:45 pm in E2.304
Dr. Frederick Magata (Universität Münster): A general Weyl-type integration formula for isometric group actions
Absract: Many geometric properties of an isometric group action of a Lie group $G$ on a manifold $M$ can be reduced to the study of an associated action of a Lie group $W$ on a submanifold $\Sigma\subseteq M$. In some cases the group $W$ may be significally smaller than $G$. For instance, in the case of a compact Lie group equipped with a bi-invariant metric and which acts on itself via conjugation, $W$ is finite. In this talk we show that integration over a $G$-manifold can be expressed in terms of $W, \Sigma$, an orbit part $G/N$ and a volume scaling factor $\delta$. As a special case, the formula yields Weyl's classical integration formula for compact Lie groups. - Monday, January the 19th 2009, 5:15-6:15 pm in E2.304
Prof. Dr. Jochen Heinloth (University of Amsterdam): Moduli spaces related to twisted groups over curves
Abstract: In the study moduli spaces of bundles on curves, it is often useful to equip the bundles with aditional structures. In this talk I would like to explain that more spaces of bundles with aditional structures can be viewed as moduli spaces of bundles for non-constant groups on curves than one might expect at first.
This was motivated by a series of conjectures of Pappas and Rapoport. I would also like to explain why the general setup of their conjectures is actually very useful in the proofs. - Monday, January the 26th 2009, 5:15-6:45 pm in E2.304
Prof. Dr. Reinhold Meise (Universität Düsseldorf): Power series spaces and their significance in analysis
Abstract: After a short introduction of the basic facts about power series spaces, examples are presented which come up in connection with Schauder bases in certain spaces. To obtain other examples and to present deeper results, the linear topological invariants (DN) and $(\Omega)$ and their significance are discussed. Finally, it is indicated why power series spaces come up as kernels of certain convolution operators. - Monday, February the 2nd 2009, 5:15-6:15 pm in E2.304
Dipl.-Math. Anke Pohl (Universität Paderborn): Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds
Abstract: The main concern of symbolic dynamics is the construction and investigation of discretizations in space and time and symbolic
representations of flows on locally symmetric spaces, or more generally, on orbifolds.
We will consider the geodesic flow on orbifolds of the form $\Gamma\backslash H$, where $H$ is the hyperbolic plane and $\Gamma$ a geometrically finite subgroup of $\PSL(2,\R)$. Further we will require that $\infty$ is a cuspidal point of $\Gamma$ and that $\Gamma$ satisfies an additional weak (and easy to check) condition concerning the structure of the set of the isometric spheres of $\Gamma$. We will construct cross sections for the geodesic flow on $\Gamma\backslash H$ for which the associated discrete dynamical systems are conjugate to discrete dynamical systems on $\R$. For each of these cross sections there is a natural labeling in terms of certain elements of $\Gamma$. We will see that the arising coding sequences of unit tangent vectors belonging to the cross section can be reconstructed from the endpoints of the associated geodesics.
The boundary discrete dynamical systems (and the generating functions for the symbolic dynamics) are of continued fraction type. In turn, the transfer operators produced from them have a particularly simple structure.
If time permits, we will discuss steps towards a generalization to other locally symmetric good orbifolds of rank one. - Tuesday, February the 24th 2009, 4:00-5:00 pm in E2.304
Prof. Dr. Azzedine Lansari (Abou Bekr Belkaid University of Tlemcen, Algeria): Ideals of finite codimension of the Fréchet-Lie algebras of vector fields
Abstract: This is a report on joint work with Zineb Achouri concerning ideals of finite codimension in Fréchet-Lie algebras of rapidly decreasing vector fields and applications of the Nash-Moser inverse theorem in this area. - Monday, March the 2nd 2009, 5:15-6:15 pm in E2.304
Dr. Martin Laubinger (Universität Münster): Almost complex structures and coadjoint orbits of mapping groups
Abstract: There is a classical integrability condition for almost complex structures on principal fiber bundles. Isomorphism classes of integrable structures are in a bijective correspondence with certain orbits of the gauge group of the bundle.
In our Diplom thesis, we gave an alternative proof of the integrability condition for smoothly trivial bundles by computing the Newlander-Nijenhuis tensor of the almost complex structure.
We will sketch this proof and outline our current research, joint with Christoph Wockel, on non-trivial bundles and a generalization to bundles whose fibers are complex Banach-Lie groups. - Monday, March the 30th 2009, 5:15-6:15 pm in E2.304
Dr. Walther Paravicini (Universität Münster): The Bost conjecture
Abstract: There is a family of conjectures having as prominent members the Baum-Connes conjecture and its cousin, the Farrell-Jones conjecture.
In this talk I would like to present the little sister of the Baum-Connes conjecture, the so-called Bost conjecture, that asserts a way to compute the K-theory of the L^1-algebra L^1(G) of a locally compact group G.
We will discuss to what extent the Bost conjecture for a group G passes to subgroups of G. On the way, we will take a look at groupoids, objects which are surely of general interest for those working with groups.