Oberseminar Analysis, Geometrie und Physik
Freie Universität Berlin - Fachbereich Mathematik und Informatik
Arnimallee 2-6, 14195 Berlin-Dahlem (Raum 108/109)

Sommersemester 2004, Dienstag 17.15 Uhr
Veranstalter:
-- Prof. Dr. Klaus Ecker (FU Berlin)
-- Prof. Dr. Gerhard Huisken (MPI Gravitationsphysik Golm/FU Berlin)
-- Julie Clutterbuck (FU Berlin), Gruppe Geometric Analysis.

13.04.2004: Robert Schrader (FU Berlin):
Quantenstreutheorie auf Graphen: Eine Lösung des Travelling Salesman Problems
Gemeinsam mit V. Kostrykin (Aachen) ist ein systematisches Studium von Laplace Operatoren auf Graphen durchgeführt worden. Verschiedene Laplace Operatoren werden durch jeweils verschiedene lineare Randbedingungen an den Vertices erhalten. Jeder Laplace Operator definiert eine Streutheorie und die bei jeder Energie unitäre Streu Matrix ist Lösung einer Matrixinversion, wo die linearen Randbedingungen und die Länge der inneren Linien explizit eingehen. Andererseits ist jedes seiner Matrix Elemente darstellbar als eine Summe über Wege von Termen, in denen jeweils auch die Länge des Weges in einfacher Weise eingeht. Eine Fourieranalyse erlaubt dann die Bestimmung des kürzesten Weges, der jeden Vertex mindestens einmal besucht.
Die Bestimmung des Graphen und seiner Längen aus den Streudaten ist bei generischen Randbedingungen ebenfalls möglich.
20.04.2004: Two City Geometry Seminar at FU Berlin/ZIB Carlo Sinestrari (Tor Vergata Roma) and Neil Trudinger (ANU Canberra)
27.04.2004: Friedrich Sauvigny (TU Cottbus):
Ein Existenzsatz für nichtlineare elliptische Differentialgleichungssysteme
In der Differentialgeometrie erscheinen häufig nichtlineare elliptische Systeme von Differentialgleichungen mit dem Laplaceoperator als Hauptteil. Mit einer topologischen Methode soll unter Dirichlet-Randbedingungen eine weite Klasse dieser Systeme gelöst werden. Hierzu sind a-priori-Abschätzungen bis zum Rand erforderlich. Die hier gelösten Systeme sind i.a. weder eindeutig lösbar, noch entstammen sie notwendig einem Variationsproblem.
Literatur:
Friedrich Sauvigny: Partialle Differentialgleichungen der Geometrie und der Physik.
Erster Teil: Grundlagen und Integraldarstellungen.
Zweiter Teil: Funktionalanalytische Lösungsmethoden.
Erscheint voraussichtlich im März 2004 bzw. im September 2005 im Springer-Verlag.
04.05.2004: Oliver Schnürer (FU Berlin):
Longtime behavior of fully nonlinear flows
We study the flow of convex hypersurfaces for the Neumann and second boundary value problems. Solutions are shown to converge to translating or homothetic solutions.
11.05.2004: John Sullivan (TU Berlin):
Ropelength Criticality for Knots and Links
The ropelength problem asks us to minimize the length of a knot or link in space, subject to a thickness constraint (keeping different strands of the knot at unit distance from each other). We derive a Balance Criterion giving necessary and sufficient conditions for a space curve to be ropelength-critical. Our approach is modeled on finite-dimensional rigidity theory for frameworks; we prove an infinite-dimensional generalization of the Kuhn-Tucker theorem. Some examples of ropelength minimizers exhibit surprising geometric behavior: for two simply clasped ropes, there is a slight gap between them when they are pulled tight, and each has infinite curvature at its tip.
18.05.2004: Richard Hamilton (Columbia, NY):
Ricci flow
Special location: Zuse Institute Berlin, Takustraße 7, 14195 Berlin-Dahlem, Hörsaal im Rundbau.
25.05.2004: Claus Gerhardt (Uni Heidelberg):
The inverse mean curvature flow in general relativity
We first consider the inverse mean curvature flow in cosmological spacetimes N and give necessary and sufficient conditions for its longtime existence.
By adding some structural conditions to N, which are still fairly general, we can prove convergence results for the leaves of an inverse mean curvature flow.
Moreover, we define a new spacetime Ñ by switching the light cone and using reflection to define a new time function, such that the two spacetimes N and Ñ can be pasted together to yield a smooth manifold having a metric singularity, which, when viewed from the region N is a big crunch, and when viewed from Ñ is a big bang.
The inverse mean curvature flows in N resp. Ñ correspond to each other via reflection. Furthermore, the properly rescaled flow in N has a natural smooth extension of class C3 across the singularity into Ñ. With respect to this natural, globally defined diffeomorphism we speak of a transition from big crunch to big bang.
The C3 regularity is the best possible as counter examples show. However, in genuine Robertson-Walker spaces - with compact or non-compact Riemannian cross-section - that satisfies the Einstein equation for a perfect fluid, one can prove that the transition flow described above is of class C∞.
Similar results can also be proved for branes in a Schwarzschild-AdS bulk, where in this case a smooth transition from big crunch to big bang can be achieved without using the inverse mean curvature flow because the branes are embedded in the bulk space.
However, for abstract spacetimes which are not embedded in an ambient space, the inverse mean curvature flow, properly rescaled, seems to be the right vehicle for a smooth transition through the singularity.
01.06.2004: Helga Baum (HU Berlin):
Conformally invariant spinor field equations and special conformal Lorentzian geometries
There are two conformally invariant spinor field equations on 1/2 spinors, the Dirac and the twistor equation. We are interested in solutions of the twistor equation (socalled conformal Killing spinors) on Lorentzian manifolds. In the talk we will show special conformal Lorentzian structures that admit conformal Killing spinors, we will classify all local conformal structures with singularity free solutions up to dimension 7 and we will discuss some problems concerning essential conformal transformations on Lorentzian manifolds which are related to conformal Killing spinors with singularities.
08.06.2004: Two City Geometry Seminar at MPI Leipzig
Tristan Rivière (ETH Zürich) and Alan Rendall (MPI Golm)
15.06.2004: Julie Clutterbuck (FU Berlin):
Gradient estimates obtained by counting zeroes
22.06.2004: Robert Gulliver (Uni Minnesota):
Singularities of Soap Film-Like surfaces and Total Curvature of Graphs
A graph in Rn is a union of smooth arcs, which begin and end in two of the vertices of the graph. The number of arcs meeting at a vertex is called the degree of the vertex. In a physical experiment, it is possible to dip a wire framework into soap solution and withdraw it: a surface may be formed in which three pieces of soap film meet with equal angles along edges, and where six pieces of soap film may meet at a point. Jean Taylor in 1976 used a certain class of rectifiable sets, called soap film-like surfaces, as a model for such physical surfaces, and proved the partial interior regularity. We extend recent work on minimal surfaces with Jordan curves as boundaries to this case, proving that the density of a soap film-like surface may be controlled by the total curvature of its boundary graph. The possible singularities are restricted by this density. The contribution to the total curvature at a vertex is defined as the supremum over p of the sum of the complements of the interior angles to a cone over the graph with vertex p in Rn. This is joint work with Sumio Yamada.
29.06.2004: No seminar this week
06.07.2004: Neshan Wickramasekera (MIT)
13.07.2004: Herbert Koch (Uni Dortmund)
The geometry of free boundaries
After reviewing the use of Von Mises variables I will explain how the free boundary for the Mumford Shah functional can be reduced to an elliptic boundary value problem, from which analyticity of flat solution with analytic data can be deduced. In the second part a similar approach to the porous medium equation will be explained. Here we are forced to study a sub-Riemannian geometry. The talk is based on joint work with G. Leoni, M. Morini and J. Vazquez.
13.07.2004: Richard Schoen (Stanford)
An approach to the sharp isoperimetric inequality for minimal surfaces in 3-space
12.08.2004: Alan McIntosh (Australian National University)
The square root problem of Kato: Survey, Solution and Sequel
More about the Kato square root problem here.