12
Fachbereich
Institut für Mathematik
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Oberseminar Geometrische Analysis​​ – 2025

Geometrische Analysis

Dez 16 2025
14:00

Raum 903

Shujing Pan (Frankfurt): The quermassintegral inequalities for horo-convex domains in the sphere

In this talk, I will introduces a new notion of convexity on the unit sphere, called horo-convexity, inspired by its analogue in hyperbolic space. For horo-convex hypersurfaces, we prove the smooth convergence of the Guan–Li inverse curvature flow and, as a consequence, establish the full set of quermassintegral inequalities on the sphere. The talk will briefly outline the definition, the flow approach, and the main geometric results. This talk is based on joint work with Julian Scheuer

Geometrische Analysis

Dez 2 2025
14:00

Raum 903

Marcus Flook (Australian National University): Mean curvature flow analogues in Cauchy-Riemann (CR) Geometry

In this talk, I will introduce and motivate Cauchy-Riemann (CR) geometry by considering real hypersurfaces embedded in complex Euclidean space. Firstly, I will discuss progress on both Darboux- and Alexandrov-type theorems in this setting. Secondly, I will introduce flows of CR hypersurfaces that are analogous to the mean curvature flow. Alongside the standard degeneracy due to tangential diffeomorphisms, such flows have an additional degeneracy due to the CR structure which will be discussed. Finally, I will discuss joint research with Ben Andrews on new flows which preserve key components of the CR structure. This talk will be accessible to those with a background in Riemannian geometry.

Geometrische Analysis

Nov 18 2025
14:00

Georg Hofstätter (TU Wien): Localization of valuations and Alesker’s irreducibility theorem

We introduce a new localization technique for translation-invariant valuations on
convex bodies. We then apply it to show that smooth, translation-invariant valua-
tions are representable by integration over the normal cycle. With this representa-
tion, we provide a new proof of Alesker’s famous irreducibility theorem.
This is joint work with J. Knoerr.

Geometrische Analysis

Nov 4 2025
14:00

Tim Laux (Heidelberg): Equivalence of weak solution concepts for mean curvature flow

Abstract:  Mean curvature flow is a fundamental geometric evolution equation with natural applications in almost every field of science. To study the evolution past singularities, several notions of weak solutions have been introduced over the last decades. The viscosity solution on the one hand is based on a geometric comparison principle. On the other hand, many other concepts are variational in nature as they are inspired by the gradient flow structure of mean curvature flow. In this talk, I will show that these two viewpoints are equivalent in the following sense: (i) any generic level set of the viscosity solution is a variational solution; (ii) any foliation by variational solutions has to be equal to the unique viscosity solution. This also implies the generic uniqueness of variational solutions.


Geometrische Analysis

Okt 21 2025
14:00

Nicola De Nitti (Pisa/Italien): Optimal transport of measures via autonomous vector fields

We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to construct a suitable autonomous vector field that realizes the (unique) monotone transport map as the time-1 map of its flow. Generically, this vector field can be chosen to be Lipschitz continuous. We then use Sudakov's disintegration approach to deal with the multidimensional case by reducing it to a family of one-dimensional problems. This talk is based on a joint work with Xavier Fernández-Real.

Geometrische Analysis

Okt 20 2025
14:00

Shouda Wang(Princeton University/USA): Realization space of mixed volumes 



A classical theorem of Minkowski asserts that the volume of a linear combination of convex bodies is a homogeneous polynomial with nonnegative coefficients. Since then, numerous inequalities among these coefficients have been discovered, with interesting applications in algebraic geometry and combinatorics. In this talk, I will discuss the inverse problem to this question: given a homogeneous polynomial, can one decide whether it arises as a volume polynomial? I will present known cases as well as new results based on joint work with Huang, Huh, Michałek, and Wang.


 

Sep 9 2025
14:00

​Malik Türkön (UC Irvine): Fundamental Gap Estimates in Conformally Flat Manifolds

The fundamental gap is the difference between the first two eigenvalues of the Laplace operator. For the Dirichlet boundary condition, lower bounds on this gap have been established for convex domains in Euclidean space and the round sphere. Recently, it was shown that in hyperbolic space, there are no uniform lower bounds of the fundamental gap for convex regions of any given diameter. This raises the natural question of whether a fundamental gap estimate can be proven under a stronger notion of convexity, which is conjectured to be true by Nguyen, Stancu, and Wei. In this talk, we provide a positive answer to this question and derive estimates in more general conformally flat geometries. This is based on joint work with G. Khan and S. Saha.

 

Aug 26 2025
14:00

Mouhammed Moustapha Fall (Mbour/Senegal): tba

Geometrische Analysis

Jul 22 2025
14:00

Leo Brauner (Frankfurt): The mixed Christoffel-Minkowski problem

In this talk, I will first define mixed area measures: these are fundamental geometric measures encoding joint information about curvature of a family of convex bodies (including also the classical Gauss and mean curvature). Then we discuss the central Minkowski and Christoffel problem, as well as their intermediate and mixed versions, which are about classifying mixed area measures. In the special case where the convex bodies involved share an axial symmetry, we resolve the problem completely.
This is joint work with Georg C. Hofstätter and Oscar Ortega-Moreno

Geometrische Analysis

Jul 15 2025
14:00

Esther Cabezas Rivas (Universitat Valencia/Spanien): How to define an inverse mean curvature flow coming out of crystals?


We will start by recalling some basics about smooth versus weak solutions of inverse mean curvature flow (IMCF) from the classical works by Huisken and Ilmanen, where the level set solutions are applied to prove the Riemannian Penrose inequality. But this notion of weak solutions cannot work in milder regularity settings, like those needed to model crystal growth, which require an anisotropic framework (with no regularity nor ellipticity extra requirements), and where the objects to evolve are at most Lipschitz regular. In this a priori unfriendlier scenario, during the second part of the talk, we will explain how we still manage to find a good notion of weak IMCF coming out of a crystal. This is based on joint works with Salvador Moll and Marcos Solera.