The triply graded link homology - A new approach
Mittwoch, 14.10.20, 10:00-11:00, Hörsaal II, Albertstr. 23b
Vorkurs Mathematik
Montag, 19.10.20, 00:00-01:00, der Lernplattform "kosmic" (Vorlesung online, Tutorate in Präsenz in verschiedenen Räumen) und endet am 23.10
A general approach to stability of the soap bubble theorem and related problems
Mittwoch, 21.10.20, 08:30-09:30, Virtueller SR 226 (Euwe)
The soap bubble theorem says that a closed, embedded surface of the Euclidean space with constant mean curvature must be a round sphere. Especially in real-life problems it is of importance whether and to what extent this phenomenon is stable, i.e. when a surface with almost constant mean curvature is close to a sphere. This problem has been receiving lots of attention until today. The purpose of this talk is to discuss two different approaches to problems of this type. The first one is a new general approach based on stability of the so-called "Nabelpunktsatz". The second one is of variational nature and employs the theory of curvature flows.
Stability problems and singularities of the Ricci flow
Mittwoch, 21.10.20, 11:30-12:30, Virtueller SR 226 (Euwe)
The Ricci flow is a geometric evolution equation for Riemannian metrics on a manifold, which is of fundamental importance in modern Riemannian geometry. Generally, its solutions may develop singularities after finite time. Such a singularity always admits a blowup limit, which is a self-similar solution of the Ricci flow, also called a Ricci soliton. In this talk, I will give a review on stability results for Ricci solitons under the Ricci flow in different geometric situations, with specific attention given to recent results in the asymptotically locally Euclidean (ALE) case. At the end of the talk, I will discuss how this stability analysis could potentially be used to extend the Ricci flow beyond singularities of certain kind.
Geometric analysis of the Einstein Equations
Mittwoch, 21.10.20, 15:30-16:30, Virtueller SR 226 (Euwe)
The Einstein equations of general relativity constitute a hyperbolic system of non-linear geometric partial differential equations. The equations’ non-linear geometric nature causes the formation of singularities and interesting asymptotics of solutions (e.g. black holes as final states). In this talk I will discuss how such phenomena can be tracked and controlled through the application of analysis tools (such as bilinear/trilinear estimates, sharp trace estimates, Besov spaces, Cheeger-Gromov theory and linear analysis). Specifically, I will present my research results and projects on (1) low regularity estimates for the Einstein equations (assuming the curvature to be only \(L^2\)-integrable) and (2) the asymptotic analysis of solutions along null hypersurfaces.
Geometric variational problems.
Donnerstag, 22.10.20, 08:30-09:30, Virtueller SR 226 (Euwe)
Many questions about things in nature can be answered by modeling them as equilibrium configurations of geometric variational problems. From a mathematical viewpoint, there are several types of interesting questions, for example, existence of critical points or minimizers, regularity of such objects, and properties of minimizers. I will give an overview of my research on geometric variational problems with the main focus on the Willmore functional which is the integral over the squared mean curvature of a surface.
Scale-invariant tangent-point energies for knots and their relation to harmonic maps
Donnerstag, 22.10.20, 15:30-16:30, Zoom Meeting https://pitt.zoom.us/j/4050917817
A substantial part of my research is dedicated to the analysis of nonlocal/fractional differential equations appearing in Geometric Analysis, in particular in the Calculus of Variations in combination with geometric or topological constraints. As one example, I will report about progress in the theory of minimizing and critical knots for a class of scale-invariant knot energies, the so-called O'Hara and tangent-point energies. I explain the relation to the theory of harmonic maps that we discovered and our attempts to exploit it for existence and regularity results. This talk is based on joint works with S. Blatt, Ph. Reiter, and N. Vorderobermeier.
b
Donnerstag, 22.10.20, 17:00-18:00, Hörsaal II, Albertstr. 23b
c
Motion by curvature of networks: analysis of singularities and “restarting” theorems
Freitag, 23.10.20, 08:30-09:30, Virtueller SR 226 (Euwe)
A regular network is a finite union of sufficiently smooth curves whose end points meet in triple junctions. I will present the state-of-the-art of the problem of the motion by curvature of a regular network in the plane mainly focusing on singularity formation. Then I will discuss the need of a “restarting” theorem for networks with multiple junctions of order bigger than three and I will give an idea of a possible strategy to prove it. This is a research in collaboration with Rafe Mazzeo (Stanford University), Mariel Saez (P. Universidad Catolica de Chile) and Jorge Lira (University of Fortaleza).
Some tools for non-local PDEs from conformal geometry
Freitag, 23.10.20, 11:30-12:30, Virtueller SR 226 (Euwe)
In this talk I will explain some new tools developed in conformal geometry to solve non-local elliptic semi-linear equations. These tools originally arose to study geometric properties. However, since they are analytic tools, they help us not only to solve geometric problems, but also several non-local / non-linear PDE problems (through the understanding of the instrinsic geometry which is present in the PDEs). Conformal geometry has been traditionally developed to deal with the study of scalar curvature (the natural generalization of the Gauss curvature to higher dimension), but this new approach (from a non-local point of view) leads to the study of other generalizations of the Gauss curvature, such as the Q-curvature. Moreover,\nthese tools are useful to study di↵erent equations, functionals and extremal solutions for inequalities arising in non-local geometric analysis. Would it be possible to use them for studying the extrinsic non-local geometry as well?
Singular structures in geometric variational problems
Freitag, 23.10.20, 15:30-16:30, Virtueller SR 226 (Euwe)
Solutions to variational problems arising in Geometry and Physics may exhibit singularities. A fine analysis of the size and structure of singular sets is of pivotal importance, both from the purely theoretical perspective, and in view of the applications, in particular as a confirmation of the suitability of the variational model towards a correct description of the observed phenomena. In this talk, I will describe my work on a variety of aspects concerning the physical relevance, the analytic properties, and the evolution of the singular structures arising in the solutions to some geometric variational problems, with an emphasis on minimal surfaces and mean curvature flows.
Einführungswoche
Montag, 26.10.20, 09:30-10:30, Räumen des Mathematischen Instituts, Ernst-Zermelo-Straße 1
On Parabolic Harnack inequalities
Mittwoch, 28.10.20, 08:30-09:30, Virtueller SR 226 (Euwe)
Motivated by the study of heat kernels with Dirichlet boundary condition, I will present results on parabolic Harnack inequalities for non-symmetric uniformly elliptic operators. The setting is that of an abstract metric measure Dirichlet space with volume doubling and Poincare inequality.
Compact manifolds with negative part of the Ricci curvature in the Kato class
Mittwoch, 28.10.20, 11:30-12:30, Virtueller SR 226 (Euwe)
The Ricci curvature encodes much geometric and analytic information of the underlying Riemannian manifold. For classes of compact Riemannian manifolds with a prescribed uniform lower bound on the Ricci curvature and an upper bounded diameter, quantitative estimates on the eigenvalues of the Laplace-Beltrami operator, the associated heat kernel, or on the isoperimetric constants can be derived. The resulting estimates depend heavily on the prescribed lower bound of the Ricci curvature. In view of geometric ows where metrics are deformed and the Ricci curvature possibly develops singularities, the obtained estimates become valueless even if there is only a small region where the singularity appears. For this reason, people became interested in relaxing the uniform lower Ricci curvature bound assumptions to integral conditions on the negative part of the Ricci curvature. Besides the commonly imposed \(L^p\)-assumptions, a part of my research is focussed on the implications of the even more general Kato condition, which appears naturally, e.g., in the theory of the Ricci ow. In this talk, I will present geometric and analytic properties of classes of compact Riemannian manifolds whose negative part of the Ricci curvature satisfies such a Kato condition and relate the results to recent work on manifolds with \(L^p\)-Ricci curvature assumptions.