SFB Seminar
Wintersemester 2005-2006
im AEI in Golm,
Max-Planck-Campus
Am Mühlenberg 1, 14476 Golm
Central Building, room number Z-050
15.11.2005
15:00 Uhr:
Monotone quantities for geometric evolution equations
16:30 Uhr:
Dr. Joachim Weber (HU)
Floer homology of cotangent bundles
Given a closed manifold, consider its cotangent bundle
equipped with the canonical symplectic structure. An almost complex
structure is provided by choosing a Riemannian metric on the manifold. In this situation one can define Floer homology associated to physical
Hamiltonians of the form kinetic plus potential energy and show that it
is naturally isomorphic to the singular homology of the free loop space
of the manifold. We review the three isomorphisms constructed by
Viterbo, Salamon-Weber, and Abbondandolo-Schwarz. Then we discuss some
applications, for instance existence of noncontractible periodic orbits
of a certain class of Hamiltonians.
6.12.2005
15:00 Uhr:
Leon Simon (guest of the AEI)
Singularities of Minimal Surfaces and Harmonic Maps
In the latter half of the 20'th century much was achieved in understanding regularity properties of minimal surfaces and harmonic maps, and major theorems were proved, for example in the direction of limiting the dimensional size of the singular set for various sub-classes of such surfaces and maps. In particular various "partial regularity theories'' based on ideas of De Giorgi, Reifenberg, Federer, Almgren and others were developed.
Nevertheless many very basic questions about the nature
of the singular set have remained completely open; for
example, does the singular set have some sort of stratified structure analogous to the stratification of the singular part of general real analytic varieties.
This talk will briefly survey the present state of knowledge and conclude with a discussion of recent work which aims to develop a framework for a special class of singular minimal surfaces which is on the one hand analytically manageable and on the other hand sufficiently rich in examples to provide insights into the general situation alluded to above.
16:30 Uhr:
Prof. Dr. Alexander Bobenko (TU)
Geometric variational principles on simplicial surfaces. Discrete Laplace-Beltrami operator and discrete Willmore flow.
A discrete Willmore energy is introduced for simplicial surfaces. This energy is invariant with respect to Moebius transformations. The associated geometric flow and discrete Willmore surfaces are studied. We define also a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its weights are positive. Our Laplace operator is similar to the one defined by Pinkall and Polthier (the so called "cotan formula'') except that it is based on the intrinsic Delaunay triangulations of the simplicial surfaces.
10.1.2005
14:00 Uhr:
Andrejewski lecture
Prof. Dr. Albrecht Klemm, University of Wisconsin, Madison
Mirror symmetry and the topological A- and B-model
Mirror symmetry on CY manifolds exchanges the symplectic structure on M, actually a complexified Kähler structure, with the complex structure on a mirror dual CY manifold W. The deformation theory of each of these structures can be described by a topological string theory called the topological A- and the B-model respectively. These models are cohomological theories defined by nilpotent operators QA and QB. We will show that QA exists on every symplectic manifold while QB exists only on CY manifolds and certain generalizations thereof. The latter fact is related to the Tian & Todorov theorem on the unobstructedness of complex structure deformations on CY spaces and generalizations by Hitchin. We will then discuss properties of cohomological theories theories notably the descend- and topological recursion relations. The solution of the topological B-model using these recursion relations and some classical methods of complex structure deformation theory on W is worked out and related by mirror symmetry to the Gromov-Witten theory captured by the topological A-model on M.
This lecture is a part of three Andreevski Lectures held by Prof. Dr. Klemm by invitation of the Humboldt University.
1st lecture:
09.01.2006, 16:00 Uhr "Gromov-Witten invariants"
3rd lecture:
12.01.2006, 16:00 Uhr:
“Methods of explicit calculation and physical applications.”
Both lectures were held in the main building of the Humboldt University, Room Nr. 3038/035 (2nd floor).
16:00 Uhr:
Prof. Dr. Jens Hoppe (Royal Institute of Technology, Stockholm), guest of the AEI
17:30 Uhr:
Dr. Simon Chiossi (HU)
"G2 structures on solvmanifolds"
‘Conformally G2’ manifolds are Riemannian manifolds with a G2 structure whose metric can be modified to a holonomy structure by a conformal change.
There is an interesting construction of homogeneous conformally G2 structures on solvmanifolds built from underlying SU(3) structures.
I will show how the corresponding non-homogeneous G2 metrics can be obtained also by evolving the SU(3) structure in time. (Possible reference: arXive math.DG/0510087)
31.1.2005
15:00 Uhr:
Dr. habil. Johanna Erdmenger, Max-Planck-Institut fuer Physik (Werner-Heisenberg-Institut)
AdS/CFT correspondence and differential geometry
The AdS/CFT correspondence (AdS: Anti-de Sitter space, CFT: Conformal field theory) provides a relation between quantum field theory and classical general relativity which is based on string theory. In particular, in its original form it relates field-theoretical properties of N=4 super Yang-Mills theory, such as the coefficients of the conformal anomaly, to geometrical properties of the space AdS_5 x S5. Recently, the correspondence has been generalised to quantum field theories in which supersymmetry is partially broken. In this case the five-sphere S5 is replaced by manifolds with a more involved metric. An example are quiver gauge theories which are related to Sasaki-Einstein manifolds. In the talk an outline of these results is given and methods for further studies of the generalised AdS/CFT correspondence are presented.
16:30 Uhr:
Prof. Dr. Jochen Bruening (HU)
Nonparabolic Dirac systems
14.2.2005
15:00 Uhr:
Q-curvature and Holography
16:30 Uhr:
Prof. Dr. Christian Baer (UP)
Deligne Cohomology, the B-field, and Variational Problems
Sommersemester 2005
19.04.2005 im Hörsaal ZIB:
15:15 Uhr:
Mirror-Konstruktion für Hyperflächen und vollständige Durchschnitte in torischen Varietäten
Wir berichten über Batyrevs klassische Konstruktion mittels der Dualisierung reflexiver Polyeder und deren Verallgemeinerung zu den sogenannten nef-Partitionen. Diese Konstruktionen liefern zunächst singulaere CY-Varietäten, und wir diskutieren die zwei Möglichkeiten ihrer Glättung: (Crepante) Auflösung und Deformationen.
17:00 Uhr:
Prof. Dr. Hermann Nicolai (AEI)
Supermembranen
10.05.2005 Arnimallee 3, SR 001
15:30 Uhr:
Prof. Dr. Stefan Theisen (AEI)
String Compactification
17:00 Uhr:
Prof. Dr. Bernold Fiedler (FU)
Sturm attractors: geometry and
global dynamics
21.06.2005: (Hörsaal ZIB)
15:00 Uhr:
Prof. Dr. Gerhard Huisken (AEI)
Geometrische Evolutionsgleichungen: Fragestellungen und Querverbindungen
17:00 Uhr:
Prof. Dr. Helga Baum (HU)
Holonomy of conformal structures, special conformal geometries and conform-parallel spinors
The holonomy theory of semi-Riemannian manifolds is well-known. Special metric holonomies are related to distinguished geometric structures (such as Kähler, Calabi-Yau, G2,... ) and allow us to decide how many parallel spinors exist. In the seminar I will dicuss the same question in conformal geometry. In the first part I will introduce some basics of conformal Cartan geometry which are necessary to define the notion of conformal holonomy. In the second part I will review recent results on the holonomy of Riemannian and of Lorentzian conformal structures. I will explain geometric properties of conformal structure with special holonomy, in particular the relation to Einstein manifolds and to the existence of conform-parallel spinors.
28.06.2005: (Hörsaal ZIB)
15:30 Uhr:
How to count holomorphic curves
I will recall how certain non-compactness phenomena for solution spaces of (partial) differential equations lead algebraic structures: chain complexes in Floer theory, quantum cup product in Gromov-Witten theory. Symplectic field theory describes the algebraic formalism to count a more general class of holomorphic curves: maps of a punctured Riemann surface into symplectic manifolds with a certain end structure with controlled behaviour at the punctures. The non-compactness is a mixture of breaking Floer trajectories, Gromov bubbling and Deligne-Mumford pinching. Finally, I will attempt to outline what the invariants should give for the cotangent bundle of a closed manifold.
17:00 Uhr:
Dr. Alan Rendall (AEI)
Cosmic acceleration and dark energy
It is less than ten years since the majority of astrophysicists became convinced that the expansion of our universe is accelerated. This requires some kind of repulsive force which overcomes the normal gravitational attraction. The cause of this has acquired a name, dark energy, but nobody knows what exactly dark energy is. Many models have been proposed and it is natural that one rich source of these is string theory. Questions about these models where mathematicians have a contribution to make are: what is the exact definition of the models, which of them are really different and what is the dynamics of
the solutions of the field equations? In this talk, after presenting the necessary background on cosmology, I will cover some of these
issues. In particular I will discuss $k$-essence models and the special case of the tachyon.
12.07.2005: (Hörsaal ZIB)
15:30 Uhr:
Dr. Ilka Agricola (HU)
Special Geometries, Holonomy and String Theory
17:00 Uhr:
Dr. Joerg Teschner (FU)
Strings, conformal field theory and the harmonic analysis on loop spaces
When studying corrections to general relativity as predicted by string theory one must in particular understand the consequences of the extended nature of the strings. This requires the methods of conformal field theory. We will explain in a concrete example how the conformal field theory assigned to a particular space generalizes the harmonic analysis on the same space.