Oberseminar Geometrie, Topologie und Analysis

Oberseminar Geometrie, Topologie und Analysis

H. Geiges, M. Lesch, G. Thorbergsson

Sommersemester 2003

Freitag, 10:30-12:00, Seminarraum 1

26.04.03
Samstag! Gemeinsames Seminar mit den Bonner Geometern,
Großer Hörsaal des Mathematischen Instituts Köln

14:30 Uhr Prof. Dr. Victor Bangert (Freiburg)
Systolic inequalities

16:00 Uhr Prof. Dr. Pierre Pansu (Paris Sud)
Filling ice-cream cones

18:00 Uhr Gemeinsames Nachtessen
(im Alten Wartesaal, Köln)
Zum Nachtessen bitte anmelden bei Frau Stiehl-Schöndorfer (e-mail: Stiehl@math.uni-koeln.de) bis Mittwoch, den 16.4.03, 12:00 Uhr

9.05.03 Dr. Konrad Polthier (TU Berlin)
Unstable periodic discrete minimal surfaces

Zusammenfassung: The talk shows new results on polyhedral surfaces of constant mean curvature which are used to study the index of unstable minimal surfaces by evaluating the spectra of their Jacobi operators. The numerical estimates confirm known results on the index of some smooth minimal surfaces like the trinoid and the Costa surface, and provide new estimates for a large number of triply periodic minimal surfaces. Also it gives additional information on the geometry of their area-reducing variations.

The investigation of the index requires the numerical computation of unstable discrete minimal surfaces with excellent numerical qualities, which is among the very challenging problems. The essential ingredients in the algorithm are the introduction of the new alignment energy and a duality of conforming and non-conforming polyhedral minimal surfaces.

23.05.03 PD Dr. David Green (Wuppertal)
Essential classes in the cohomology of finite groups

Abstract: Consider the cohomology ring of a finite group G with coefficients in a field of characteristic p. A cohomology class is called essential if its restriction to each proper subgroup of G is zero. In the past one tried to show that the group one was interested in had no essential classes: one of Quillen's proofs of the Adams conjecture involves showing that wreath product groups have no essential classes. More recently, people have begun discovering that the presence of essential classes can have positive consequences.
My talk will consist of three parts. In the first part I will attempt to provide an introduction to the relevant aspects of group cohomology. This will be followed by a survey of the applications of the ideal of essential classes, including results concerning the depth of the cohomology ring. (Depth is an invariant from commutative algebra.) In the final part of the talk I shall discuss one component of the proofs: comodule structure.
(This talk is planned to last 90 minutes.)

30.05.03 HD Dr. Markus Pflaum (Frankfurt)
Deformationsquantisierung symplektisch stratifizierter Orbispaces

6.06.03 Michael Bohn (Köln)
Seiberg-Witten Invarianten auf geschlossenen 3-Mannigfaltigkeiten
(bis 12:00 Uhr)

27.06.03 Prof. Dr. Gerhard Knieper (Bochum)
Die Komplexität des geodätischen Flusses für generische Metriken auf S²

4.07.03 Dr. Michael Gruber (z. Zt. Augsburg)
Geometry, topology and analysis of magnetic Schrödinger operators

10.07.03
Donnerstag! Gemeinsames Seminar mit den Bonner Geometern,
Großer Hörsaal des Mathematischen Instituts Bonn, Wegelerstraße 10

16:00 Uhr Prof. Dr. Gérard Besson (Grenoble)
Uniform growth of fundamental groups of manifolds and some geometric applications

17:30 Uhr Prof. Dr. Michael Kapovich (Salt Lake City, z.Zt. MPI Bonn)
Polygons in symmetric spaces and Douady-Earle barycenter

19:00 Uhr Gemeinsames Nachtessen
(im Opera, Bonn)

18.07.03 Prof. Dr. Claudio Gorodski (São Paulo)
Copolarity of isometric action

25.07.03 Prof. Dr. Maxim Braverman (Northeastern University)
Cohomology of symplectic reduction of a Kähler manifold

Abstract: Let X be a compact Kähler manifold acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use the Witten deformation of the Dolbeault complex of L to show that the cohomology of the sheaf of holomorphic sections of the induced bundle on the symplectic (or Mumford) quotient of (X,L) is equal to the G-invariant part on the cohomology of the sheaf of holomorphic sections of L. This result generalizes a theorem of Guillemin and Sternberg, which addressed the global sections. It also shows that the Morse-type inequalities of Tian and Zhang for symplectic reduction are, in fact, equalities.

H. Geiges, 4.2.02


26.04.03 Samstag!	Gemeinsames Seminar mit den Bonner Geometern, Großer Hörsaal des Mathematischen Instituts Köln
14:30 Uhr	Prof. Dr. Victor Bangert (Freiburg) Systolic inequalities
16:00 Uhr	Prof. Dr. Pierre Pansu (Paris Sud) Filling ice-cream cones
18:00 Uhr	Gemeinsames Nachtessen (im Alten Wartesaal, Köln) Zum Nachtessen bitte anmelden bei Frau Stiehl-Schöndorfer (e-mail: Stiehl@math.uni-koeln.de) bis Mittwoch, den 16.4.03, 12:00 Uhr
9.05.03	Dr. Konrad Polthier (TU Berlin) Unstable periodic discrete minimal surfaces Zusammenfassung: The talk shows new results on polyhedral surfaces of constant mean curvature which are used to study the index of unstable minimal surfaces by evaluating the spectra of their Jacobi operators. The numerical estimates confirm known results on the index of some smooth minimal surfaces like the trinoid and the Costa surface, and provide new estimates for a large number of triply periodic minimal surfaces. Also it gives additional information on the geometry of their area-reducing variations. The investigation of the index requires the numerical computation of unstable discrete minimal surfaces with excellent numerical qualities, which is among the very challenging problems. The essential ingredients in the algorithm are the introduction of the new alignment energy and a duality of conforming and non-conforming polyhedral minimal surfaces.
23.05.03	PD Dr. David Green (Wuppertal) Essential classes in the cohomology of finite groups Abstract: Consider the cohomology ring of a finite group G with coefficients in a field of characteristic p. A cohomology class is called essential if its restriction to each proper subgroup of G is zero. In the past one tried to show that the group one was interested in had no essential classes: one of Quillen's proofs of the Adams conjecture involves showing that wreath product groups have no essential classes. More recently, people have begun discovering that the presence of essential classes can have positive consequences. My talk will consist of three parts. In the first part I will attempt to provide an introduction to the relevant aspects of group cohomology. This will be followed by a survey of the applications of the ideal of essential classes, including results concerning the depth of the cohomology ring. (Depth is an invariant from commutative algebra.) In the final part of the talk I shall discuss one component of the proofs: comodule structure. (This talk is planned to last 90 minutes.)
30.05.03	HD Dr. Markus Pflaum (Frankfurt) Deformationsquantisierung symplektisch stratifizierter Orbispaces
6.06.03	Michael Bohn (Köln) Seiberg-Witten Invarianten auf geschlossenen 3-Mannigfaltigkeiten (bis 12:00 Uhr)
27.06.03	Prof. Dr. Gerhard Knieper (Bochum) Die Komplexität des geodätischen Flusses für generische Metriken auf S²
4.07.03	Dr. Michael Gruber (z. Zt. Augsburg) Geometry, topology and analysis of magnetic Schrödinger operators
10.07.03 Donnerstag!	Gemeinsames Seminar mit den Bonner Geometern, Großer Hörsaal des Mathematischen Instituts Bonn, Wegelerstraße 10
16:00 Uhr	Prof. Dr. Gérard Besson (Grenoble) Uniform growth of fundamental groups of manifolds and some geometric applications
17:30 Uhr	Prof. Dr. Michael Kapovich (Salt Lake City, z.Zt. MPI Bonn) Polygons in symmetric spaces and Douady-Earle barycenter
19:00 Uhr	Gemeinsames Nachtessen (im Opera, Bonn)
18.07.03	Prof. Dr. Claudio Gorodski (São Paulo) Copolarity of isometric action
25.07.03	Prof. Dr. Maxim Braverman (Northeastern University) Cohomology of symplectic reduction of a Kähler manifold Abstract: Let X be a compact Kähler manifold acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use the Witten deformation of the Dolbeault complex of L to show that the cohomology of the sheaf of holomorphic sections of the induced bundle on the symplectic (or Mumford) quotient of (X,L) is equal to the G-invariant part on the cohomology of the sheaf of holomorphic sections of L. This result generalizes a theorem of Guillemin and Sternberg, which addressed the global sections. It also shows that the Morse-type inequalities of Tian and Zhang for symplectic reduction are, in fact, equalities.