Research Seminar Global Analysis

Prof. Dr. M. Lesch, Prof. Dr. W. Müller

Summer semester 2013

Talks

23.04.2013 Daniel Grieser (Oldenburg)
Title: The spectrum of triangles.
Abstract: The object of the talk is the spectrum of the Laplace operator with Dirichlet boundary conditions on Euclidean triangles. I will discuss two results. The first result is specific to triangles, while the second is really about a more general type of problem, which is most easily exemplified by the case of triangles.

The first result (joint work with S. Maronna) is a new proof of the fact that a triangle is (among the set of all triangles) uniquely determined by the spectrum. The only previously known proof of this uses wave invariants. The study of these is technically difficult. Our new proof uses heat invariants and is technically simpler, and also involves a curious and interesting -- and apparently new -- geometric fact about triangles.

The second result (joint work with R. Melrose) that I will discuss aims at the description of the full asymptotic behavior of the eigenvalues when the triangle degenerates into a line. This may happen in various ways. More precisely, there are two parameters describing the degeneration, and we give a complete asymptotic expansion in terms of both parameters. This involves a rather intricate and unexpected blow-up of the parameter space, which will be explained in the talk.
30.04.2013 Dan Burghelea (Ohio State)
Title: What a Morse function adds to the spectral geometry of a closed Riemannian manifold and some implications.
Abstract: I propose to investigate a small finite piece of the spectral package of a closed Riemannian manifold, eigenvalues and eigenforms of the Laplacians specified by a Morse function via Witten deformation, in the hope that this package can provide information usually derived using the entire infinite spectrum of the Laplacians and may be something new.
07.05.2013 Klaus Kroencke (Potsdam)
Title: Linear and Dynamical Stability of Einstein Metrics
Abstract: We study certain normalized variants of the Ricci flow near compact Einstein manifolds with non-vanishing Ricci curvature. We put conditions on an Einstein metric g under which any solution of the Ricci flow, starting close to g, exists for all time and converges to an Einstein metric as t goes to infinity.
14.05.2013 Julie Rowlett (MPI Bonn)
Title: Zeta Regularized Determinants of Polygons
Abstract: Explicit formulae for spectral quantities, even for domains as simple as Euclidean polygons, are a rarity. A few formulae are well known, such as the small-time asymptotic expansion of the heat trace. In this talk I will discuss joint work-in-progress with Clara Aldana and Werner Müller in which we are computing an explicit formula for the zeta-regularized determinant of the Euclidean Laplacian on polygonal domains with Dirichlet boundary condition.
28.05.2013 Nicolai Reshetikhin (Berkeley)
Title: On the Hamiltonian structure of classical gauge theories
Note the change of time and venue: at 15:45 in room Nr. 2.008
04.06.2013 Jasmin Matz (Bonn)
Title: An explicit bound for global coefficients in Arthur's trace formula for GL(n)
Abstract: We will discuss an explicit upper bound for the so-called global coefficients appearing on the geometric side of Arthur's trace formula in the case of GL(n). These coefficients are in general left unspecified, but a better understanding of them is essential for applications, and I will explain how the upper bound might be useful in proving a Weyl's law for Hecke operators on GL(n). t.b.a.
25.06.2013 Pierre Albin (Illinois)
Title: Compactness of relatively isospectral sets of surfaces
Abstract: Although one can not `hear the shape of a drum', it turns out that the set of isospectral metrics on a closed surface forms a compact set. I will discuss joint work with Clara Aldana and Frederic Rochon regarding the corresponding statement for non-compact surfaces.