Oberseminar Geometrische Analysis

Termin und Ort:

Dienstag 14 Uhr c.t., RM10 - 903


Prof. Dr. A. Bernig

Prof. Dr. J. Scheuer

Prof. Dr. T. Weth

Aktuelle Vorträge

23.04.2024     Mario Santilli (Universita dell'Acquila)

Title: Rigidity and compactness of rectifiable boundaries with constant mean curvature in warped product spaces

Abstract: We discuss the extension of well known rigidity results for constant mean curvature hypersurfaces of Alexandrov
(in hyperbolic space) and Brendle (in warped product spaces) to sets of finite perimeter with constant distributional mean curvature. Joint work with Francesco Maggi.

07.05.2024    Oscar Ortega Moreno (TU Wien)

Title: Moment inequalities for Gaussian vectors

Abstract: The Gaussian product inequality is a long-standing conjecture relating the moments of an arbitrary centred normal random vector to the moments of a standard one. In connection to this problem we present some new extensions of moment inequalities for Gaussian vectors.

14.05.2024      Elisabeth Werner (Case Western Reserve University Cleveland)

Title: Weighted floating functions and weighted functional affine surface areas

Abstract: We introduce the new concept of weighted floating functions associated with log concave or s-concave functions. This leads to new notions of weighted functional affine surface areas. Their relation to more traditional versions of functional affine surface areas as well as the classical affine surface areas for convex bodies is discussed.

28.05.2024     Nikita Cernomazov (Frankfurt)

Title: Constrained curve-flows and their self-shrinkers

Abstract: Solitons provide an important pool of solutions for geometric flows as they often serve as 'limit shapes' in singularity analysis. For the well-studied curve shortening flow, all possible closed homothetic solutions have been identified as the so-called "Abresch-Langer curves." In this talk, we present a similar classification scheme for the self-shrinking solutions of a non-local, area-preserving variant of classical curve shortening. The proof of this classification also confirms a conjecture by J.-E. Chang relating to more general λ-curves.

18.06.2024     Vadim Lebovici (University of Oxford)

Title:     Hybrid transforms of constructible functions

Abstract:  A function on a real analytic manifold is constructibleif it is integer-valued and locally constant on a subanalytic
stratification of the manifold. Constructible functions are the measurable functions of a theory of integration with respect to the
Euler characteristic, which allows for the definition of topological integral transforms used in shape analysis.
After a short motivation of constructible functions from the point of view of valuations on manifolds, of persistent homology and of applied
geometry, I will present the so-called /hybrid/ transforms combining the Lebesgue integral and integration with respect to the Euler
characteristic. From a theoretical point of view, these transforms benefit from properties of regularity and invertibility. From an applied
point of view, they are efficiently computable and provide flexibility for extracting relevant information in data analysis.

This presentation is based on the article [arXiv:2111.07829] and the
article with Olympio Hacquard [arXiv:2303.14040].

25.06.2024     Peter Bürgisser (TU Berlin)

Title: Kähler package for algebra of Grassmann zonoids

Abstract: The Grassmann zonoid algebra of a Euclidean space V, introduced by Breiding, Bürgisser, Lerario and Mathis (2022), is a commutative graded Banach algebra, giving a new perspective on classical objects in convex geometry. Its elements of degree k are the real Radon measures on the real Grassmannian G(k,V) of k-planes; alternatively, they can be seen as formal differences of certain zonoids in the kth exterior power of V. The multiplication in the zonoid algebra captures the mixed volume of zonoids, and the Alexandrov-Fenchel inequality for zonoids expresses the Hodge-Riemann property of the zonoid multiplication in degree one. The reduced Grassmann zonoid algebra A(V) is the commutative graded Banach algebra obtained by factoring out the kernels of the Radon transforms for Grassmannians. Nonnegative measures with full support on projective space G(1,V) define a positive cone in the degree one part of A(V).

We outline a proof that the reduced Grassmann zonoid algebra A(V) satisfies the Kähler package with respect to the above positive cone (Poincare duality, mixed hard Lefschetz property, mixed Hodge-Riemann relations). The algebra A(V) is closely related to the graded algebra of even, translation-invariant, smooth valuations (with Alesker's product), for which the properties of the Kähler package recently were proven by Bernig, Kotrbaty and Wannerer (2023).

09.07.2024     Olivia Vicanek-Martinez (Uni Tübingen)

Title: A geometric choice of asymptotically flat coordinates of initial data sets via foliations

Abstract: Asymptotically flat initial data sets model instants of time of an isolated system in GR. While the mass in this setting is well-known, a definition of a center of mass has proved very difficult and ambiguous. A geometric definition came from the abstract center of an asymptotic foliation by closed hypersurfaces of constant spacetime mean curvature, whose existence is ensured in this setting (Cederbaum-Sakovich, 2021). We construct coordinates from this purely geometric foliation exploiting the properties of the induced Laplacian of the foliation leaves via a delicate analysis. These coordinates are well-adapted to the center of mass, and moreover asymptotically flat, showing that the existence of the foliation actually characterizes asymptotic flatness. This is joint work with A. Piubello.

16.07.2024      Stefan Le Coz (Toulouse)

Titel:  Blow-up on a star-graph 

Abstract : We consider a metric star graph endowed with a nonlinear Schrödinger equation with critical nonlinearity. Depending on the mass of the initial datum, the corresponding solution might be global or blow-up in finite time. At the mass-threshold, we construct a solution with arbitrary energy, which blows up in finite time at the vertex of the star graph. The blow-up profile and blow-up speed are characterized explicitly. This is a joint work with François Genoud and Julien Royer.



Wintersemester 2023/24

25.10.2023     Liangjun Weng (Roma Tor Vergata)

Title: The relative isoperimetric inequality for a minimal submanifold with the free boundary in Euclidean space

Abstract: Given a minimal submanifold with a free boundary outside a convex set in Euclidean space, in this talk, I will discuss the isoperimetric problem for such submanifolds. Firstly, by adapting the ideas of the restricted normal cones in Choe-Ghomi-Ritoré [2006, JDG] to obtain an optimal area estimate for our generalized restricted normal cones. This optimal area estimate, together with the Alexandroff–Bakelman–Pucci (ABP) method, provides an alternative proof of the relative isoperimetric inequality obtained by Choe-Ghomi-Ritoré in [2007, CVPDE]. Furthermore, using this area estimate and the idea in Brendle [2021, JAMS], we solve the relative isoperimetric inequality for minimal submanifolds with free boundary up to codimension 2 [Choe, Clay Math. Proc. 2, Open Problem 12.6]. This talk is based on a joint project with Profs. Lei LIU and Guofang WANG.

01.11.2023     Prachi Sahjwani (Cardiff University)

Title: Stability of geometric inequalities in various spaces

Abstract: In this talk, I will discuss the stability of two inequalities: "Alexandrov-Fenchel inequalities in hyperbolic space" and "Minkowski's inequality in a general warped product space." I will give a brief overview of both inequalities and their respective stability problems. To understand what I mean by stability, I will first discuss it for Isoperimetric inequality, which is a special case of Alexandrov-Fenchel inequalities. I will also briefly discuss the proofs in both these cases. This is joint work with Julian Scheuer. This work is based on Wang/Xia's work on Alexandrov-Fenchel inequalities and Brendle/Hung/Wang's and Scheuer's work on Minkowski's inequality.

08.11.2023     FRANKFURTER SEMINAR (RM10 - 711): Annette Huber-Klawitter (Freiburg)

15.11.2023     Amelie Menges (TU Dortmund)

Title: 'Comparing the sets of volume polynomials and Lorentzian polynomials'

abstract: In 1901, Minkowski showed that the volume of a linear combination of convex bodies with non-negative coefficients gives rise to a homogeneous polynomial, the so-called volume polynomial. Starting with the Alexandrov-Fenchel inequality, there has been a lot of progress throughout the years in the study of the coefficients of volume polynomials. For example, Brändén und Huh (2019) showed that every volume polynomial is a Lorentzian polynomial, but the converse is generally not true. In this talk, using our knowledge of Lorentzian polynomials and operations preserving them, we consider the set of volume polynomials of degree d in n variables and give a complete classification of the cases when the two sets are equal.

22.11.2023     FRANKFURTER SEMINAR (RM10 - 711): Angkana Rüland (Bonn)

29.11.2023     Simon Ellmeyer (Wien)

Titel: Complex Lp-Intersection Bodies and related Busemann–Petty type problems

Abstract: Interpolating between the classical notions of intersection and polar centroid bodies, (real) Lp-intersection bodies, for −1 < p < 1, play an important role in the dual Lp-Brunn–Minkowski theory. Inspired by the recent construction of complex centroid bodies, a complex version of Lp-intersection bodies, with range extended to p > −2, is introduced, interpolating between polar complex centroid and complex intersection bodies. Moreover, related Busemann–Petty type problems are established.
This is joint work with Georg C. Hofstätter

13.12.2023     Bachelor-Vortrag: Sebastian Schmidt (in German)

Title: Zur Stabilität in der Heintze-Karcher Ungleichung

Abstract: Ich werde mithilfe der Heintze-Karcher-Ungleichung eine Stabilitätsaussage für C2−Hyperflächen mit positiver mittlerer Krümmung entwickeln. Das Ziel ist herauszufinden, wie sich aufgrund der Differenz beider Seiten jener Heintze-Karcher-Ungleichung die Ähnlichkeit der Fläche zu einer Sphäre verändert. Hierbei ist die Aussage, dass die Heintze-Karcher-Ungleichung strikt wird genau dann, wenn die Fläche eine Sphäre ist, grundlegend. Mithilfe einer Fehlerabschätzung durch die Ln+1-Norm der spurfreien Weingartenabbildung wird aus der üblichen Heintze-Karcher-Ungleichung die Stabilitätsaussage entwickelt. Diese liefert durch die Differenz der Terme aus der Heintze-Karcher-Ungleichung eine Abschätzung des Hausdorffabstands zwischen dem eingeschlossenen Gebiet und einem Ball.

10.01.2024     Ben Lambert (Leeds University)

Title: Alexandrov immersed mean curvature flow

Abstract: In this talk I will introduce Alexandrov immersed mean curvature flow and extend Andrew's non-collapsing estimate to include Alexandrov immersed surfaces. This estimate implies an all-important gradient estimate for the flow and allows mean curvature flow with surgery to be extended beyond flows of embedded surfaces to the Alexandrov immersed case. This is joint work with Elena Mäder-Baumdicker. 

17.01.2024     Kostiantyn Drach (Universitat de Barcelona)

Title: Reverse isoperimetric inequality under curvature constraints

Abstract: What is the smallest volume a convex body K in R^n can have for a given surface area? This question is in the reverse direction to the classical isoperimetric problem and, as such, has an obvious answer: the infimum of possible volumes is zero. One way to make this question highly non-trivial is to assume that K is uniformly convex in the following sense. We say that K is \lambda-convex if the principal curvatures at every point of its boundary are bounded below by a given constant \lambda>0 (considered in the barrier sense if the boundary is not smooth). By compactness, any smooth strictly convex body in R^n is \lambda-convex for some \lambda>0. Another example of a \lambda-convex body is a finite intersection of balls of radius 1/\lambda (sometimes referred to as ball-polyhedra). Until recently, the reverse isoperimetric problem for \lambda-convex bodies was solved only in dimension 2. In a recent joint work with Kateryna Tatarko, we resolved the problem also in R^3. We showed that the lens, i.e., the intersection of two balls of radius 1/\lambda, has the smallest volume among all \lambda-convex bodies of a given surface area. For n>3, the question is still widely open. I will outline the proof of our result and put it in a more general context of reversing classical inequalities under curvature constraints in various ambient spaces.

24.01.2024     FRANKFURTER SEMINAR (RM10 - 711): Sabine Jansen (LMU München)

31.01.2024     Leo Brauner (TU Wien)

Title:   Lefschetz operators on Minkowski valuations

Minkowski valuations are finitely additive operators on the space of convex bodies. They form a rich class of geometric maps, including the difference body, projection body, and mean section body maps. The Lefschetz operators allow us to move between valuations of different degrees of homogeneity. In this talk, we discuss the action of the Lefschetz operators on continuous Minkowski valuations that are compatible with rigid motions.
This is joint work with Georg C. Hofstätter and Oscar Ortega-Moreno.

Sommersemester 2023

12.04.2023     Xuwen Zhang (Xiamen/Frankfurt)

Title: Alexandrov-type theorem for singular capillary CMC hypersurfaces in the half-space

Abstract: Alexandrov theorem is a fundamental geometric result, describing the rigidity of closed embedded CMC hypersurfaces. A recent beautiful work [Matias Gonzalo Delgadino and Francesco Maggi: “Alexandrov's theorem revisited", Anal. PDE 12.6 (2019), pp. 1613–1642.] extends this classical theorem to the sets of finite perimeter setting. In this talk, we consider the rigidity of capillary CMC hypersurfaces in the upper half-space, and extends this result to the sets of finite perimeter setting under certain regularity assumptions.

19.04.2023     FRANKFURTER SEMINAR (RM10 - 711): Jean-Francois Le Gall (Paris-Saclay)

26.04.2023     Dmitry Faifman (Tel Aviv)

Title: Extensions of valuations. 

Abstract. A valuation is a finitely additive measure on convex bodies or some other family of test sets, typically with some analytic restrictions or invariance properties.
We will consider smooth, translation-invariant valuations in linear space, and study a question inspired by the Whitney extension theorem for smooth functions: given a restriction-compatible collection of valuations on a family of subspaces, do they extend to a valuation on the whole space? We consider both finite families, and submanifolds of the grassmannian. As a corollary, we will deduce a Nash theorem for valuations, and also show that all valuations on smooth manifolds are given by Crofton formulas. This is a joint work in progress with G. Hofstätter, and also a standalone sequel of his January talk.

03.05.2023     Joe Hoisington (MPI Bonn)

Titel:               Calibrations and energy-minimizing mappings of rank-1 symmetric spaces

Abstract:       We will prove lower bounds for energy functionals of mappings from real, complex and quaternionic projective spaces to Riemannian manifolds. For real and complex projective spaces these results are sharp, and we will characterize the family of mappings which minimize energy in these results. We will also discuss some connections between these results and several questions in systolic geometry.

10.05.2023     FRANKFURTER SEMINAR (RM10 - 711): Renzo Cavalieri (Colorado State)

17.05.2023     Frederick Herget (Frankfurt)

Title: Inverse mean curvature flow of complete hypersurfaces in hyperbolic space

Abstract: The inverse mean curvature flow (IMCF) was famously introduced by Robert Geroch to show the Riemannian Penrose inequality, which succeeded with Huisken's and Ilmannen's proof. It is the case of an inverse curvature flow. We present results that complete (and not necessarily compact) hypersurfaces under IMCF either converge to horospheres, other hypersurfaces or degenerate in finite time. With this we extend the results of Claus Gerhardt and Brian Allen.

31.05.2023     Armin Schikorra (Pittsburgh)

Titel: Regularity results for n-Laplace systems with antisymmetric potential

Abstract: n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and
generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they
are very difficult to treat.
I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by
Joint work with Dorian Martino

07.06.2023     FRANKFURTER SEMINAR (RM10 - 711): Carla Cederbaum (Tübingen)

14.06.2023     Guofang Wang (Freiburg)

Title: Capillary hypersurfaces

Abstract: In this talk we will discuss optimal geometric inequalities for capillary hypersurfaces and related problems.

28.06.2023     FRANKFURTER SEMINAR (RM10 - 711): Michael Joswig (TU Berlin)

05.07.2023     Elisabeth Werner ( Case Western University Cleveland, USA)

Title: Extremal affine surface area in a functional setting

Abstract: We introduce extremal affine surface areas in a functional setting. We show their

main properties, in particular we estimate the size of these quantities. This

parallels results in the setting of convex bodies.

Based on joint work with Stephanie Egler.

Wintersemester 2022/23

18.10.2022        Samuel Held (Frankfurt) - Vortrag zur Masterarbeit

08.11.2022        Thomas Wannerer (Jena)

title: Affine Minkowski valuations

abstract: In convex geometry, the maps that assign to a convex body its difference body or projection body
have the following properties: They are (1) continuous; (2) finitely additive; (3) compatible with the action
the special linear group. In this talk we explore the question whether there exist other constructions
with these properties. We have found the following dichotomy: There are no new examples if one
assumes translation-invariance, but there are plenty of new examples without this assumption.

Based on joint work with Jakob Henkel.

15.11.2022        Chiara Meroni (MPI MIS Leipzig)

Title: Convex hulls of curves: volumes and signatures

Abstract: How to compute the volume of the convex hull of a curve? I will try to answer this question, for special families
of curves. This is a work in progress with Carlos Améndola and Darrick Lee. We generalize the class of curves for which
a certain integral formula works, using the technique of signatures. I will then give a geometric interpretation
of this volume formula in terms of lengths and areas, and conclude with the example of logarithmic curves, which draws
connections to polylogarithms and Feynman integrals.

22.11.2022         Nikita Cernomazov  (Darmstadt)

Title: Are There Homothetically Shrinking Solutions to the Area-Preserving Curve-Shortening Flow?

Abstract: The discussion of self-similar solutions is a cornerstone in the study of the famous Curve Shortening Flow ∂_t c = kN. Already in its infancy, W.W. Mullins provided examples of curves that shrink due to uniform scaling. Later, U. Abresch and J. Langer were able to classify all homothetically shrinking solutions.
In this talk, we consider CSF's less famous, but certainly not less interesting cousin: the Area-Preserving Curve Shortening Flow ∂_t c = (k-\bar{k})N with \bar{k} being the average curvature. First, we derive characterizing equations for general self-similar solutions of APCSF. Following this, we will rephrase the search for homothetically shrinking solutions to a problem in Hamiltonian mechanics. Finally, we will prove that there are homothetically shrinking solutions of APCSF and give examples.

6.12.2022         Ernst Kuwert  (Freiburg)

title: Curvature varifolds with orthogonal boundary

abstract: The talk is concerned with the existence of upper mass bounds for m-dimensional surfaces in terms of curvature integrals. We focus on the case of surfaces confined to a set Ω in R^n meeting ∂Ω orthogonally along their boundary (joint work with Marius Müller, Freiburg). In a previous paper with Victor Bangert (Freiburg) there is a related result for 2-dimensional surfaces in Riemannian manifolds.

13.12.2022        Tobias König (Frankfurt)

Title: Stability of the Sobolev inequality: best constants and minimizers.

Abstract: Since the ground-breaking inequality of Bianchi and Egnell (1991), which bounds the 'Sobolev deficit' of a function in terms of a constant c_{BE} > 0 times its squared distance to the manifold of optimizers, it has been an open problem to determine the optimal value of c_{BE} and, if it is achieved, its optimizer.

In this talk, I will present some recent partial progress on this problem. The main result is that c_{BE} admits an optimizer for every dimension d \geq 3. The proof relies on new strict upper bounds on c_{BE}, which exclude that the optimal value c_{BE} is attained by sequences which are asymptotically equal to one or two Talenti bubbles (i.e. optimizers of the Sobolev inequality).

10.01.2023       Julian Scheuer (Frankfurt)

Title: Quermassintegral inequalities for convex free boundary hypersurfaces in the ball

Abstract: For smooth and bounded domains of the Euclidean space, higher derivatives of enclosed volume with respect to unit outward variation are given in terms of the quermassintegrals, which are integrals over elementary symmetric polynomials of curvature. The associated quermassintegral inequalities are well-known classical results in convex geometry. In a joint project with Guofang Wang (Freiburg) and Chao Xia (Xiamen) we introduced quermassintegrals for convex hypersurfaces with free boundary (perpendicularity) on the unit sphere. The justification of our definition stems from two directions: Firstly, their unit outward variation gives rise to the same beautiful formula as in the classical case and secondly, we have a Gauss-Bonnet-Chern property of the highest order quermassintegral. In this talk we will present a curvature flow approach to prove new quermassintegral inequalities for such convex free boundary hypersurfaces and initiate the discussion of some open questions.

17.01.2023       Georg Hofstätter (Wien)

Restrictions of Valuations

abstract: It is a well-known consequence of two very influential theorems by Klain and Schneider that every translation-invariant, continuous and k-homogeneous

valuation on a Euclidean vector space is uniquely determined by its restrictions to all subspaces of dimension k. In this talk, we study the complementary question of when a given family of valuations defined on all subspaces of a fixed dimension can be realized as restrictions of one globally defined valuation.
This is joint work in progress with D. Faifman

24.01.2023       Michele Stecconi (Universite du Luxembourg)

Title: Expectation of a random submanifold: the zonoid section.

Abstract: I will present a joint work with Léo Mathis. Given a smooth compact Riemannian manifold, consider the random submanifold Z defined as the zero level of a nice enough random smooth function (e.g. random polynomials, random eigenfunctions,...) The zonoid section of Z is a certain a family of convex bodies indexed by the points of the ambient manifold. It represents the expectation of the random submanifold. This family depends only on pointwise data and determines the expected volume and the expected current of the random submanifold. Moreover, such convex bodies belong to a particular class: zonoids, on which there exists a multiplicative structure (a recent result by Breiding-Bürgisser-Lerario-Mathis). We will see that this structure corresponds to the intersection of random submanifolds, in a similar fashion as for the cohomology ring.

31.01.2023       Markus Wolff (Tübingen)

Ricci-Flow on surfaces along the standard light cone in the $3+1$ Minkowski spacetime

abstract: By identifying the conformal structure of the round $2$-sphere with the standard lightcone in the $3+1$ Minkowski space we gain a new perspective on $2d$ Ricci flow on topological spheres in the context of General Relativity. It turns out that in this setting Ricci flow is equivalent to a null mean curvature flow first proposed by Roesch-Scheuer along null hypersurfaces. Thus, we can fully characterize the singularity models for this proposed flow in the standard Minkowski lightcone, where the metrics of constant scalar curvature (up to scaling) each correspond to a member of the restricted Lorentz group $SO^+(3,1)$. This new viewpoint of conformally round $2d$ Ricci flow as an extrinsic flow along the lightcone leads to a new proof of Hamiltons classical result using only the maximum principle.

15.02.2022     Axel Fünfhaus, Tobias Weth (Frankfurt)

Vorträge im interdisziplinären Seminar zur  Topologie und Festkörperphysik

Titles: tba

14.09.2021, Raum 901    Dmitry Faifman (Tel Aviv/Israel)

Title: Between the Funk metric and convex geometry.

Abstract: The Funk metric in the interior of a convex set is a lesser-known cousin of the Hilbert metric. The latter generalizes the Beltrami-Klein model of hyperbolic geometry, and both have straight segments as geodesics, thus constituting solutions of Hilbert's 4th problem alongside normed spaces. Unlike the Hilbert metric, the Funk metric is not projectively invariant. I will explain how, nevertheless, the Funk metric gives rise to many projective invariants, which moreover enjoy a duality extending results of Holmes-Thompson and Alvarez Paiva on spheres of normed spaces and Gutkin-Tabachnikov on Minkowski billiards. I will also discuss how extremizing the volume of metric balls in Funk geometry yields extensions of the Blaschke-Santalo inequality and Mahler conjecture.

Sommersemester 2020

14.07.2020     Ludwig Hammer
(Videovortrag zur Masterarbeit)

Titel: Regularität von Minimierern des TV-L2-Modells

Abstract: Die Reduktion von Bildrauschen mit dem Verfahren nach L. Rudin, S. Osher und E. Fatemi ist empirisch wie analytisch gut erforscht. Es basiert auf der Minimierung eines konvexen Funktionals auf der Menge der Funktionen beschränkter Variation, das eng mit der Mean-Curvature-Gleichung zusammenhängt.
Dieser Vortrag widmet sich von analytischer Seite her der Frage, wie sich in Dimension bis 7 die Stetigkeit und lokale Hölderstetigkeit von den Ausgangsdaten auf das Ergebnis übertragen, folgend einer Technik von A. Chambolle, V. Caselles und M. Novaga. Die zentrale Beobachtung dabei ist, dass die Niveaumengen stetiger Funktionen sich nicht berühren, beziehungsweise im Falle von Hölderstetigkeit einen positiven Abstand haben. Die Niveaumengen von Minimierern des fraglichen Funktionals lassen sich mit Methoden der geometrischen Maßtheorie analog behandeln.

2.6.2020  Rüdiger Krämer (Videovortrag zur Bachelorarbeit)

Titel: Positivität und Trivialität antiker Lösungen der semilinearen Wärmeleitungsgleichung auf Riemannschen Mannigfaltigkeiten

Abstract: Es wird die semilineare Wärmeleitungsgleichung ut = ∆u + u² auf Riemannschen Mannigfaltigkeiten betrachtet. Für antike Lösungen dieser Gleichung lässt sich unter bestimmten Voraussetzungen an die Mannigfaltigkeit und die Lösung selbst schließen, dass die Lösung positiv oder räumlich konstant sein muss. In diesem Vortrag sollen Fälle betrachtet werden für die Daniele Castorina und Carlo Mantegazza in einem Paper die Positivität beziehungsweise die räumliche Konstanz gezeigt haben.

Wintersemester 2019/20 (14.10.2019 - 14.02.2020)

22.10.2019     Ricardo Arconada (Goethe-Universität)

Donnelly's theorem

05.11.2019     Jonas Knörr 

Titel:                 Smooth valuations on convex functions

In recent years, valuations on functions arose as a natural generalization of
valuations on convex bodies, and various types of valuations on different spaces
of functions have been studied and classified.
I will present some results from an ongoing project examining the space of
dually epi-translation invariant valuations on convex functions. We will see how
these functionals are related to translation invariant valuations on convex bodies
and how one can exploit this relation to establish a notion of smoothness. It
turns out that the dense subspace of smooth valuations can be described using
integration of differential forms over the graph of the differential of a convex
function (or more generally, the differential cycle) and I will present a sketch of
proof for this result.
As an application, we will see that the subspace of smooth and rotation invariant
valuations admits a very simple description.           

12.11.2019      Anna-Laura Sattelberger
(MPI Leipzig)

Titel:                 D-modules and applications

In this talk, we give an introduction to the theory of D-modules.
This theory allows us to investigate systems of linear homogeneous partial
differential equations with polynomial coefficients by algebraic methods.
Many functions can be understood by their annihilating D-ideal. These functions
are said to be holonomic. Many functions in a mathematician's daily life indeed
are holonomic. We explain their use in concrete applications. Among others,
we explain how to compute the volume of a TV-screen with the help of them.

28.01.2020      Franz Schuster
(TU Wien)

Titel:                  “Affine" isoperimetric inequalities in real space forms


14.04.2020      - FÄLLT AUS - Frau Dan Ma
(Shanghai Normal University, currently at Karlsruher Institut für Technologie)

28.4.2020        - FÄLLT AUS -    Karoly Böröczky
(Alfréd Rényi Institute of Mathematics, Budapest)

Titel:                  U(n) equivariant tensor valuations
                      (joint with Gil Solanes and Matyas Domokos)

 For scalar valued translation invariant continuous valuations on convex
compact sets in R^{2n},  U(n) equivariant  valuations have been
characterized by Alesker,
and SU(n) invariant ones by Bernig. For vector valued valuations,
Wannerer calculated the dimension of  U(n) equivariant  valuations.
Contributing towards the theory of "Hermitian Integral Geometry"
initiated by Alesker, Bernig and Fu,  the talk presents a basis for
vector valued U(n) equivariant translation invariant continuous
valuations on convex compact sets in R^{2n}, and the dimensions of the
tensor valued valuations in all rank.         

12.05.2020     Antonio Lerario (SISSA Trieste) - ABGESAGT

Titel: tba

26.05.2020     Georg Hofstätter (Wien) - ABGESAGT

Titel: tba

02.06.2020      Oscar Ortega (TU Wien) - ABGESAGT

Titel: tba

09.06.2020     Daniel Rosen (Bochum)

Titel: tba

Sommersemester 2019 (15.04.2019 - 19.07.2019)

09.04.2019            Anton Galaev (University  of Hradec Králové, Tschechische Republik)

Titel:                          Classification problem for holonomy groups of pseudo-Riemannian manifolds

Abstract:                     The holonomy group of a pseudo-Riemannian manifold gives reach information
about the geometry of the manifold. A classical and important result is the classification of the connected
holonomy groups of Riemannian manifolds. In the lecture will be represented recent results about
classification for the holonomy groups of Lorentzian manifolds and some results about holonomy groups
of pseudo-Riemannian manifolds.

16.04.2019               Nico Lombardi (Florenz)

Titel:                     Real-valued valuations defined on the space of quasi-concave functions


30.04.2019                Nicolas Hilger (Frankfurt)
                                     Vortrag zur Bachelorarbeit

Das Vergleichsprinzip von Talenti

Abstract:               In seiner Arbeit vergleicht Talenti die Lösungen der Poisson-Gleichung und ihrer
Schwarz-symmetrisierten Form. Er trifft die Aussage, dass die Lösung der symmetrisierten
Gleichung punktweise größer ist als die Lösung der Ausgangsgleichung. Im Vortrag werden
wir diese Aussage mit Hilfe von zwei Resultaten zu Funktionen von beschränkter Variation
beweisen. Wir werden dann das Vergleichsprinzip verwenden, um die Sobolev-Konstanten
zweiter Ordnung eines Gebietes und seiner Schwarz-Symmetrisierung gegeneinander abzuschätzen.

07.05.2019               Olaf Mordhorst (Frankfurt am Main)

Titel: Fraktionelle Sobolev-Normen und Funktionen beschränkter Variation auf Mannigfaltigkeiten

Im euklidischen Raum konnten Bourgain, Brezis & Mironescu zeigen, dass die Sobolev-Norm
einer Funktion für p>1 der Grenzwert der fraktionellen Sobolev-Normen ist. Für den Fall p=1 zeigte Davila,
dass die fraktionellen Sobolev-Normen gegen die BV-Norm konvergieren. Wir wollen in diesem Vortrag die
Begriffe für kompakte riemannsche Mannigfaltigkeiten einführen und die entsprechenden Grenzwertsätze
vorstellen. Insbesondere betrachten wir auch den Fall von Mengen endlichen Perimeters.
Die Ergebnisse dieses Vortrages  sind in Zusammenarbeit mit Andreas Kreuml entstanden.

14.05.2019             Olivier Guichard (Université de Strasbourg)

Titel:                      Compactifications of some families of locally symmetric spaces

  We will report on a joint work with Fanny Kassel and Anna Wienhard.
The talk will address Anosov subgroups, a class of discrete subgroups of
Lie groups that should indeed be called generalized convex cocompact
The hyperbolic manifolds arising from (classical) convex cocompact
subgroups have well understood compactifications as well as many other
pleasant topological and geometrical properties.
We will explain how to obtain compactifications for the symmetric spaces
associated with Anosov subgroups and draw some consequences of the
explicit construction of the compactifications (topological tameness,
stability under deformations). Examples and counter-examples will
illustrate this discussion.

21.05.2019            Alberto Abbondandolo (Ruhr Universität Bochum)

Titel:                          On short closed geodesics, shadows of balls and polar bodies

Abstract: How long is the shortest closed geodesic on a Riemannian sphere? How large is the shadow
of a symplectic ball? How large is the volume of the polar of a centrally symmetric convex body?
I will discuss how these seemingly different problems can be addressed within the setting of Reeb dynamics.


28.05.2019                Lucas Schäfer (Frankfurt)
                             Vortrag zur Masterarbeit

Titel:                      Algebraische Operationen auf Bewertungen.

04.06.2019               Knut Smoczyk (Universität Hannover)

Titel:                          Type-II singularities of the Lagrangian mean curvature flow.

Abstract:                  We give a classification of type-II singularities of the Lagrangian mean curvature
                            flow of almost calibrated Lagrangian submanifolds in Calabi-Yau manifolds. We then prove that the
                            singularity of the Whitney sphere is of that type.

02.07.2019                Fabian Mußnig (TU Wien)

Titel:                 SL(n) invariant valuations on convex functions

Valuations on convex bodies have been of interest ever since they appeared in Dehn's solution of Hilbert's Third Problem in 1901. Two of the most fundamental valuations are the Euler characteristic and the n-dimensional volume and the first characterization of these operators as continuous, SL(n) and translation invariant valuations was obtained by Blaschke in the 1930s. Since then, many generalizations and improvements of his result were found.
More recently, valuations on function spaces have been studied. We will present SL(n) invariant valuations on convex functions and corresponding characterization results. In particular, we will highlight similarities and differences with the theory of valuations on convex bodies. Some of the presented results were obtained in joint work with Andrea Colesanti and Monika Ludwig.


Wintersemester 2018/19

30.10.2018     Christian Lange (Universität Köln)

Orbifolds all of whose geodesics are closed

Manifolds all of whose geodesics are closed have been studied a lot,
although there are only few examples known. The situation is quite different if
one allows in addition for orbifold singularities. In this case also new
phenomena occur, e.g. the geodesic length spectrum can be much more complicated.
In the talk we discuss examples, rigidity results and open questions.

13.11.2018     Frederick Herget (Frankfurt)

Inverse mean curvature flow for non-compact hypersurfaces in Hyperbolic space

Abstract:     The inverse mean curvature flow is an intrinsic geometric flow that was introduced to
prove the mass estimate for black holes given by the so called Penrose-inequality. Primarily studied
for the evolution of compact surfaces in Euclidean space, the case of non-compact hypersurfaces and
non-Euclidean background manifolds attracts increasing interest.
In the talk I will discuss primarily the case of the IMCF for initial non-compact hypersurfaces in
Hyperbolic space – not without also referring to the compact case – and the special role Horospheres play in it.

20.11.2018     Keegan Flood (University of Auckland)

Titel:               Scalar Curvature and Projective Compactification

In this talk we will use projective tractor calculus to describe the geometry
of solutions to the PDE governing the metrizability of projective manifolds. As a consequence
we will see that under suitable scalar curvature assumptions the "boundary at infinity" of a
projectively compact pseudo-Riemannian metric inherits a well-behaved geometric structure from
that of the interior. We will examine the non-vanishing scalar curvature case which yields a conforma
l structure on boundary, then the scalar-flat case which yields a projective structure on the boundary.

27.11.2018     Dr. Ignace Minlend (AIMS Mbour/Senegal)

Multiply-Periodic Hypersurfaces with constant nonlocal mean Curvature

Abstract: Hypersurfaces with Constant Nonlocal Mean Curvature (CNMC) can be
modelled as interfaces of coexistence in phase transition (of two liquids of different
density for instance) when long range interactions are allowed. Indeed, they are crit-
ical points of the fractional perimeter under a volume constraint.
In this talk, we use local inversion arguments to prove existence of smooth branches
of multiply-periodic hypersurfaces bifurcating from suitable parallel hyperplanes.

04.12.2018     Lucas Schäfer (Frankfurt), Vortrag zur Masterarbeit

15.01.2019     Kai Zehmisch (Universität Giessen)

Titel:                Diffeomorphism type of symplectic fillings

In 1991 Eliashberg-Floer-McDuff proved that compact symplectic manifolds
of dimension at least 6 that bound the standard contact sphere symplectically are
diffeomorphic to the ball provided there are no symplectic 2-spheres. This fundamental
result raised the question whether the boundary of a symplectic manifold determines the
interior. In my talk I will explain how holomorphic curves can be used to answer this open
question. For example, symplectically aspherical fillings of simply-connected, subcritically
fillable contact manifolds are unique up to diffeomorphism.

Sommersemester 2018 (09.04.2018 - 13.07.2018)

10.04. 2018     Prof. Dr. María de los Ángeles Sandoval-Romero

: Geometric Functional Spaces with Applications

Abstract: In this talk I will present two types of what we like to call Geometric Functional Spaces: The first of them is the space of differential forms with Sobolev class, which is a very classical construction and illustrative of how tools from Functional Analysis and Riemannian Geometry are combined. (See [1] for details.) The second one will be the space of differential forms with Besov and Triebel-Lizorkin class. In Functional Analysis the Besov and Triebel-Lizorkin spaces constitute a very wide class of function spaces that contain, in some sense, most of the possible spaces with a notion of differentiability. So, with our generalization we are constructing functional spaces with all the advantages of smoothness in the analytic and geometric sense. As an important application I will discuss the Hodge Decomposition. Finally, related to this result, in the context of electrodynamics I will discuss the absence of magnetic monopoles and the existence of magnetic potentials. These results are part of a joint work with Miguel Ballesteros and Francisco Torres of UNAM in Mexico City. [1]Schwarz, G. Hodge Decomposition-A method for Solving Boundary Value Problems. Springer. 1995.

17.04.2018     Maciej Dunajski

Twistors and conics

Abstract: I will describe the range of the Radon transform on the space of conics in CP2 ,
and show that for any function F in this range, the zero locus of F is a four-manifold
admitting a scalar-flat Kahler metric which can be constructed explicitly.
This is a joint work with Paul Tod.

08.05.2018     Daniel Grieser (Universität Oldenburg)

Geodesics on singular spaces

Abstract: The geodesics emanating from a point p in a Riemannian manifold together define the exponential
map based at p. We consider the question whether there is an exponential map based at a singular point.
We give an affirmative answer for special classes of singularities including conical or a cuspidal singularities.
However, the exponential map exhibits surprising properties in some cases, like not being injective in any
neighborhood of p. Important tools in the study of this question are blow-ups, Hamiltonian systems with
degenerate symplectic form and normally hyperbolic dynamical systems.

05.06.2018     Lucas Schäfer

19.06. 2018     Roger El Andary (Frankfurt am Main)

in Raum 711 groß

Perelmans Pseudolocality Theorem (Vortrag zur Bachelorarbeit)

Wintersemester 2017/18 (16.10.17 - 09.02.18)

17.10.2017     Gabriel Paternain (University of Cambridge)

Lens rigidity for a particle in a Yang-Mills field

Abstract: We consider the motion of a classical colored spinless particle under the influence of an external Yang-Mills potential A on a compact manifold with boundary of dimension $\geq 3$. We show that under suitable convexity assumptions, we can recover the potential A, up to gauge transformations, from the lens data of the system, namely, scattering data plus travel times between boundary points. This is joint work with Gunther Uhlmann and Hanming Zhou.

24.10.2017     Andy Sanders (Universität Heidelberg)

Complexification of real analytic Kahler manifolds with applications to Teichmuller theory

It is a classical theorem of Whitney that every real analytic manifold admits a totally real embedding into a complex manifold . Given a real analytic Kahler manifold , I will explain a number of natural differential geometric structures on which canonically extend the Kahler geometry of . In particular, admits a canonical complex symplectic structure, and a pair of transverse, holomorphic foliations. After discussing these general phenomena, I will discuss some applications to the geometry of Teichmuller space equipped with the Weil-Petersson Kahler structure, which elucidate a number of classical results in Teichmuller theory and the theory of quasi-Fuchsian groups. In particular, I will show that there is a canonical pseudo-Riemannian metric of neutral signature on the deformation space of quasi-Fuchsian groups which extends the Weil-Petersson metric on Teichmuller space.

21.11.2017     Sebastian Heller (Universität Hamburg)

Harmonic maps, integrable systems and twistor spaces

In this talk I discuss harmonic maps of compact Riemann surfaces into certain
symmetric spaces from the integrable systems point of view. The starting point is Deligne's
description of the twistor space of the hyper-K ̈ahler moduli space of solutions of Hitchin's self-
duality equation. It is a holomorphic fibration over the complex projective line. I will explain
which classes of (equivariant) harmonic maps (e.g., minimal surfaces in the 3-sphere and AdS_3 )
can be obtained as special sections of the twistor space, and how informations about solutions
and their moduli can be obtained from that point of view. The talk is partially based on joint
work with I. Biswas and M. Röser

28.11.2017     Nawal Sadawi (Frankfurt)

Das Chern-Gauss-Bonnet Theorem (Bachelorarbeit)

05.12.2017     Friederike Dittberner (Universität Konstanz)

Titel: Area preserving curve shortening flow
Abstract: This talk is about the enclosed area preserving curve shortening flow for non-convex embedded curves in the plane. I will show that initial curves with a lower bound of $-\pi$ on the local total curvature stay embedded under the flow and develop no singularities in finite time. Moreover, the curves become convex in finite time and converge exponentially and smoothly to a round circle. 

12.12.2017     Nguyen Bac Dang (Ecole Polytechnique Paris-Saclay)

A positive cone in the space of continuous translation invariant

Abstract: I will discuss a joint work with Jian Xiao.
In this talk, I will exploit some ideas coming from complex geometry to
define a cone in the space of continuous translation invariant
valuations. This "positive" cone allows us to define a topology for
which the convolution of valuations extends continuously.

19.12.2017     Giona Veronelli (Universite Paris 13)

Scalar curvature via local extent

In the first part we will present a metric characterization of the scalar curvature of an n-dimensional smooth Riemannian manifold, based on the asymptotic control of the maximal distance between (n+1) points in infinitesimally small neighborhoods of a given point. Since this characterization is purely in terms of the distance function, it could be used to introduce a notion of scalar curvature (bounds) on a non-smooth metric space. In the second part we will discuss this issue. We will focus in particular on Alexandrov spaces and surfaces with bounded integral curvature.

16.01.2018     Thomas Hack (TU Wien)

Spherical centroid bodies (joint work with F.  Besau, P. Pivovarov and F. E. Schuster)

Going back to C. Dupin and W. Blaschke, the notion of Euclidean centroid bodies, along with their associated isoperimetric inequalities, forms a classical part of the theory of convex bodies. In this talk, we give a new definition of centroid bodies in spherical space, explore its basic properties, and discuss isoperimetric problems associated with them.

Sommersemester 2017 (10. April bis 14. Juli 2017)

25.04.2017   Franziska Borer (ETH Zurich)

Titel:  Uniqueness of Weak Solutions for the Normalised Ricci Flow on Closed Surfaces

Abstract: "We show uniqueness of classical solutions of the normalised two-dimensional Hamilton–Ricci flow on closed,
smooth manifolds for H^2-data among solutions satisfying (essentially) only a uniform bound for the Liouville energy
and a natural space-time L^2-bound for the time derivative of the solution. The result is surprising when compared
with results for the harmonic map heat flow, where non-uniqueness through reverse bubbling may occur."

02.05. 2017     Katharina Neusser (Charles University Prag)

Titel: C-projective structures of degree of mobility at least two

Abstract: In recent years there has been renewed interest in c-projective geometry, which is a natural analogue
of real projective geometry in the setting of complex manifolds, and in its applications in Kähler geometry.
While a projective structure on a manifold is given by a class of affine connections that have the same
(unparametrised) geodesics, a c-projective structure on a complex manifold is given by a class of affine complex
connections that have the same ``J-planar'' curves. In this talk we will be mainly concerned with c-projective
structures admitting compatible Kähler metrics (i.e. their Levi-Civita connections induce the c-projective structure),
and will present some work on the geometric and topological consequences of having at least two compatible
Kähler metrics. An application of these considerations is a proof of the Yano--Obata conjecture for complete
Kähler manifolds---a metric c-projective analogue of the conformal Lichnerowicz conjecture. This talk is based
on joint work with D. Calderbank, M. Eastwood and V. Matveev.

        Karin Melnick (University of Maryland)

Topology of automorphism groups of parabolic geometries

Abstract: It is well known that the automorphism group of a rigid geometric structure is a Lie group. In fact, as
there are multiple notions of rigid geometric structures, the property that the local automorphisms form a Lie
pseudogroup is sometimes taken as an informal definition of rigidity for a geometric structure. In which topology
is this the case? The classical theorems of Myers and Steenrod say that C^0 convergence of isometries of a smooth
Riemannian metric implies C^\infty convergence; in particular, the compact-open and C^\infty topologies coincide
on the isometry group. I will present joint results with C. Frances in which we prove the same result for local
automorphisms of smooth parabolic geometries, a rich class of geometric structures including conformal
and projective structures. As a consequence, the automorphism group admits the structure of a Lie group
in the compact-open topology.

    Charles Frances (Université de Strasbourg)

Title: Dynamics and topology for 3-dimensional Lorentz manifolds

It is a well known phenomenon that in contrast to what happens for Riemannian manifolds, compact Lorentz structures might have a noncompact group of isometries. Such a property of the isometry group generally has strong consequences both on the geometry, and on the topology of the manifold. The aim of the talk is to present new results and methods on the subject, with an emphasis on closed $3$-dimensional manifolds.

13.06.2017     Sophia Jahns (Universität Tuebingen)

Titel: Trapped Light in Stationary Spacetimes

Abstract: Light can circle a massive object (like a black hole or a neutron star) at a „fixed distance“, or, more generally, circle the object without falling in or escaping to infinity. This phenomenon is called trapping of light and well understood in static, asymptotically flat (AF) spacetimes. If we drop the requirement of staticity, similar behavior of light is known, but there is no definiton of trapping available.

After a short introduction to General Relativity, I present some known results about trapping of light in static AF spacetimes. Using the Kerr spacetime as a model, I then show how trapping can be better understood in the framework of phase space and work towards a definition for photon regions in stationary AF spacetimes.

20.06.2017     Joel Kübler (Frankfurt)

Travelling-Wave-Lösungen nichtlinearer Klein-Gordon-Gleichungen auf der Sphäre (Vortrag Master-Arbeit)

Wir betrachten spezielle Lösungen einer nichtlinearen Klein-Gordon-Gleichung auf kompakten Mannigfaltigkeiten, die eine Verallgemeinerung  von euklidischen Traveling-Waves darstellen. Mithilfe variationeller Methoden zeigen wir die Existenz solcher Lösungen und erläutern deren Eigenschaften. Insbesondere konzentrieren wir uns auf die 2-Sphäre und untersuchen dort unter abgeschwächten Voraussetzungen Lösungen, die anschaulich um eine Achse rotieren. Dies führt schließlich auf Einbettungsresultate für geeignete Hilberträume, die in enger Beziehung zu fraktionalen Sobolev-Räumen stehen.

27.06.2017     Ruth Kellerhals (Université de Fribourg)

Titel: "Higher logarithmic integrals and non-euclidean volume"

Abstract: Starting with a simple concrete integral expression I shall explain its connection to spherical volume and the respective computational difficulties. Then, I shall pass to related higher logarithmic integrals and present recent work about hyperbolic volume in 5 dimensions.

04.07.2017    Farid Madani (Frankfurt)

Lokal konform Kählersche Geometrie und konforme Kählersche Metriken.

Abstract: Nach einer kurzen Einführung in die lokal konform Kählersche Geometrie, werden konforme nichthomothetische
Kählersche Metriken auf einer kompakten Mannigfaltigkeit klassifiziert. Der Vortrag basiert auf einer gemeinsamen Arbeit mit
A. Moroianu und M. Pilca.

Sondertermin am Mittwoch, den 26.07.2017 um 16 Uhr, Raum 110

Bachelorvortrag von Nikolai Krasnosselski

Titel: Krümmungsfluss unter Potenzen der Gausskrümmung

Sondertermin 17.8.2017, Raum 110

Tristan Daus (Frankfurt), Bachelorvortrag

Inverse flow in AdS-Schwarzschild manifold
          (proof of new Minkowski inequalities)

Wintersemester 2016/17

15.11.2016     Franz Schuster (TU Wien/Österreich)

: Affine vs. Euclidean isoperimetric inequalities

Abstract: In this talk we explain how every even, zonal measure on the Euclidean unit
sphere gives rise to an isoperimetric inequality for sets of finite perimeter which
directly implies the classical Euclidean isoperimetric inequality. The strongest
member of this large family of inequalities is shown to be the only affine invariant
one among them – the Petty projection inequality. As application, a family of
sharp Sobolev inequalities for functions of bounded variation is obtained, each of
which is stronger than the classical Sobolev inequality.
(joint work with Christoph Haberl)

6.12.2016      Kasri Khani-Alemouti (Frankfurt)

Titel: Symmetrische Räume

13.12.2016     Thomas Mettler (Frankfurt)

Minimal Lagrangian connections

A connection on the tangent bundle of a smooth manifold M can be understood
as a map into an affine bundle over M, whose total space carries a pseudo-Riemannian metric
as well as a symplectic form, both of which can be constructed in a canonical fashion from the
projective equivalence class of the connection. This viewpoint gives rise to the notion of a
minimal Lagrangian connection. I will discuss the classification of minimal Lagrangian
connections on compact oriented surfaces of non-vanishing Euler characteristic and show
how minimal Lagrangian connections naturally generalise the notion of an Einstein metric.

10.01.2017     François Fillastre (Université de Cergy-Pontoise)

Titel: A remark about the space of flat metrics with conical singularities on a compact surface

Abstract: W.P. Thurston showed that the area form naturally endows the space of
flat metrics on the sphere with prescribed n cone singularities of
positive curvature with a structure complex hyperbolic structure of
dimension n-3. Using classical polyhedral geometry in Euclidean space,
we note that this space is decomposed by (real) hyperbolic convex
polyhedra of dimensions (n-3) and between 0 and (n-1)/2.
By a result of W.~Veech, there is a fibration of the  space of flat
metrics on a compact surface with prescribed cone singularities of
negative curvature, and the area form naturally endows each leaves with
a structure of (complex) pseudo-sphere. Here the signature (that may be
degenerated) depends on the choice of the angles.
Using polyhedral surfaces in Minkowski space, we show that this space is
decomposed by spherical convex polyhedra.

17.01.2017     Luca Martinazzi (Universität Basel)

Title: The fractional Liouville equation in dimension 1 - Geometry, compactness and quantization

Abstract: I will introduce the fractional Liouville equation on the circle S^1 and its geometric
interpretation in terms of conformal immersions of the unit disk into the complex plane. Using
this interpretation we can show that the solutions of the fractional Liouville equation have very
precise compactness properties (including quantization and half-quantization) with a clear
geometric counterpart. I will also compare these result to analogue ones for the classical Liouville
equation in dimension 2, used to prescribe the Gaussian and Q-curvature. This is a joint work
with Francesca Da Lio and Tristan Riviere.

Sommersemester 2016

12.04.2016     Ignace Aristide Minlend (AIMS Mbour/Senegal)

Titel: Existence of self-cheeger sets on Riemannian Manifolds


19.04.2016     Micha Wasem (ETH Zürich)

Titel: Convex Integration, Isometric Extensions and Approximations of Curves

Abstract: In this talk, I will present some applications of convex integration — the tool used in order
to prove the celebrated Nash-Kuiper theorem. I will show how convex integration can be used to
describe parallel parking explicitly, what it has to do with an optimality question related to a coin
trick and how it leads to the construction of knots with prescribed curvature.

26.04.2016     Roland Hildebrand (WIAS Berlin)

Titel: Canonical barriers on regular convex cones

Abstract: Conic optimization is concerned with the minimization of linear objective functions over
affine sections of regular convex cones. One type of solution methods are the so-called interior point
methods, which need a barrier function defined on the interior of the cone. The barrier function is a
smooth, locally strongly convex, logarithmically homogeneous, self-concordant function
which tends to infinity if the argument tends to the boundary of the cone. The speed of the interior
point algorithm depends on a scalar parameter of the barrier. In this talk we present a universal barrier,
i.e., one which is defined for every regular convex cone, which has a parameter at most equal to the
dimension of the cone. The construction is closely linked to the Calabi conjecture on hyperbolic affine hyperspheres.

03.05. 2016    Jonas Knörr (Frankfurt)

Titel: The hard Lefschetz theorem

10.05. 2016    Daniele Alessandrini (Universität Heidelberg)

Titel: Geometric structures on manifolds and Higgs bundles

Abstract:  Higgs bundles can be used to construct geometric structures
on manifolds. I will explain how to use them to construct the closed
Anti-de Sitter 3-manifolds, and some real and complex projective
structures with holonomy in the Hitchin components. The construction
uses the solutions of Hitchin equations, a system of global elliptic
PDEs on a Riemann surface. This is joint work with Qiongling Li.

17.05.2016     Wojciech Kryński (IMPAN Warschau)

Titel: On geometry of GL(2,R)-structures. Abstract: The GL(2,R)-structures appear as natural generalizations of 3-dimensional
conformal geometry and are immanently connected to geometric theory of ODEs and certain integrable PDEs. We shall present
the basic concepts in the GL(2,R)-geometry and its applications as higher-dimensional counterparts of the Einstein-Weyl structures.
We shall also present new results on connections between the GL(2,R)-structures and complex geometry.

24.05.2016     Dmitry Faifman (University of Toronto/Kanada)

Titel: Some kinematic formulas for O(p,q).

Abstract: A central object in integral geometry are the kinematic formulas. Given a group G of motions of the plane, one wishes to write a closed expression for the average of the values of a certain G-invariant valuation - an integro-geometric quantity - of the intersection of two convex bodies (or manifolds) over their various relative positions under the affine action of G. I will first recall the theory behind such formulas for compact groups G, then present an attempt at obtaining kinematic formulas when G is the non-compact O(p,q).

07.06.2016     Saikat Mazumdar (Nancy)

Titel: Higher order Elliptic problems with Critical Sobolev Growth on a compact Riemannian Manifold:
         Best constants and existence.

Abstract: We investigate the existence of solutions to a nonlinear elliptic problem involving the critical Sobolev
exponent for a Polyharmomic operator on a Riemannian manifold   M. We first show that the best constant of the
Sobolev embedding on a manifold can be chosen as close as one wants to the Euclidean one, and as a
consequence derive the existence of minimizers when the energy functional goes below a quantified threshold.
Next, higher energy solutions are obtained by Coron's topological method, provided that the minimizing solution
does not exist and the manifold satisfies a certain topological assumption. To perform the topological argument,
we obtain a decomposition of Palais-Smale sequences as a sum of bubbles and adapt Lions's concentration-compactness lemma.

21.06.2016     Lukas Poerschke (Frankfurt)

Komplexe Raumformen

Wintersemester 2015/16

13.10.2015     Elhadji Abdoulaye Thiam (AIMS Senegal, z. Zt. Frankfurt)

Titel: Hardy-Sobolev inequality with cylindrical weight on Riemannian manifolds


03.11.2015     Ignace Aristide Minlend (AIMS Senegal, z. Zt. Frankfurt)

Titel: Construction of solutions to Serrin's overdetermined problem on the 2-sphere


10.11.2015    Bernardo González Merino (TU München)

Titel: On the Minkowski measure of symmetry"

Abstract:  The Minkowski measure of symmetry s(K) of a convex body K, is the smallest positive dilatation
of K containing a translate of -K. In this talk we will explain some of its basic properties in detail.
Afterwards, we will show how s(.) can be used to strengthen,  smoothen, and join different geometric
inequalities, as well as its connections to other concepts such as diametrical completeness, Jung's
inequality, or Banach-Mazur distance

24.11.2015     Alexander Lytchak (Universität Köln)
Beginn 16.30 Uhr

Titel: Classical Plateau problem in non-classical spaces

19.01.2016     Nicolas Tholozan  (University of Luxembourg)

Titel:  Entropy of Hilbert geometries


02.02.2016     Ana Peón-Nieto (Universität Heidelberg)



09.02.2016     Stefan Rosemann (Universität Jena)

Titel: Complex projective transformations on (pseudo-)Kähler manifolds

Abstract: The complex projective transformations of a (pseudo-)Kähler metric are defined by the property that they preserve the set of so-called J-planar curves. These curves satisfy that the acceleration is complex proportional to the velocity and can be viewed as natural generalizations of geodesics to the complex setting. In may talk I will discuss the proof of the following statement: for a closed (pseudo-)Kähler manifold all infinitesimal complex-projective transformations preserve the canonical connection of the metric unless the metric is the Fubini-Study metric. This statement is due to recent joint work with V. Matveev and A. Bolsinov and generalizes previously obtained results in the positive definite case.

Sommersemester 2015

26.05.2015     Dr. Farid Madani (GU Frankfurt am Main)

Titel: S^1-Yamabe invariant on 3-manifolds.

Abstract: After a short overview on the (non-equivariant) Yamabe invariant, we introduce the equivariant one.
We show that the S^1-Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the
(non-equivariant) Yamabe invariant of the 3-sphere. Moreover, we give a topological upper bound for the
S^1-Yamabe invariant of any closed oriented 3-manifold endowed with a circle action. This is joint work
with Bernd Ammann and Mihaela Pilca.

23.06.2015     Olaf Müller (Universität Regensburg)

Conformal techniques, bounded geometry and the Yamabe flow

Abstract: After giving an overview over some recently developed conformal methods in Riemannian and Lorentzian geometry,
we focus on a result of a joint work with Marc Nardmann (Dortmund) stating that every conformal class contains a metric of
bounded geometry. Finally, we sketch implications of the result in the theory of the Yamabe flow on noncompact manifolds.

Wintersemester 2014/15

28.10.2014     Gil Solanes (UAB)

Titel: Invariant valuations in complex and quaternionic spaces.

11.11.2014     Florian Besau (TU Wien)

Titel: The spherical convex floating body


Donnerstag, 04.12.2014, Raum 404, 15.15 Uhr       Prof. Dr. Franz Schuster (TU Wien)

Titel: Crofton formulas for Minkowski valuations and the Christoffel problem

Abstract: The classical Crofton formula for rigid motion invariant valuations is one of the starting points for many developments in modern integral geometry.
In this talk we present a new Crofton formula for translation invariant and SO(n) equivariant Minkowski valuations which leads to a surprising connection to C. Berg's solution of the Christoffel problem for area measures of order one of convex bodies.

09.12.2014     Dr. Thomas Mettler(ETH Zürich)

Title: Projective surfaces, holomorphic curves and the SL(3,R)-Hitchin component

Abstract: A projective structure P on a surface M is an equivalence class of affine torsion-free connections on M where two connections are called projectively equivalent if they share the same geodesics up to parametrisation. An oriented projective surface (M,P) defines a complex surface Z together with a projection to M whose fibres are holomorphically embedded disks. Moreover, a conformal connection in the projective equivalence class corresponds to a section whose image is a holomorphic curve in Z. Findig a section of Z->M whose image is “as close as possible" to a holomorphic curve turns out to be related to the parametrisation of the SL(3,R)-Hitchin component in terms of holomorphic cubic differentials.

Sommersemester 2014

29.04.2014     Prof. Uwe Semmelmann (Univ. Stuttgart)

Title: Almost complex structures on quaternion-Kähler manifolds and homogeneous spaces

Abstract: In meinem Vortrag möchte ich zeigen, wie man die Nicht-Existenz von
fast-komplexen Strukturen auf verschiedenen Klassen von Mannigfaltigkeiten
zeigen kann. Die Beweise beruhen auf einer Anwendung des Atiyah Singer
Indexsatzes für gewisse getwistete Dirac Operatoren.

13.05.2014     Manuel Ritoré (Univ. Granada)

Title: Large isoperimetric regions in the product of a compact
manifold with Euclidean space

Abstrakt: Given a compact Riemannian manifold $M$ without boundary, we
show that large isoperimetric regions in the Riemannian product $M\times
R^k$ of $M$ with the $k$-dimensional Euclidean space $R^k$ are tubular
neighborhoods of $M\times\{x\}$, $x\in R^k$.

03.06.2014 Prof. Dr. Heiko von der Mosel (RWTH Aachen)

Title: On minimal surfaces in Finsler spaces (joint work with P. Overath)

Abstract: In contrast to classic minimal surface theory relatively little seems to be
known about minimal surfaces in Finsler manifolds. We explore a connection
between the Busemann-Hausdorff volume in Finsler spaces and Cartan functionals
to prove new results in that direction, such as Bernstein theorems, a
uniqueness result, and removability of singularities for Finsler-minimal
graphs, isoperimetric inequalities and enclosure theorems for minimal
immersions in Finsler space, and we treat the Plateau problem in Finsler

24. Juni 2014   Prof. Joseph Fu (University of Georgia, Athens)

Title: Valuations on Riemannian manifolds

Abstract: A smooth valuation on a general manifold M is a finitely additive set function m, defined on a restricted class of “nice" subsets A, determined by a pair of differential forms, one living on M and the other on its cosphere bundle. The value m(A) is then expressed as the sum of the integral over A of the first and the integral of the second over the manifold of outward conormals to A. S. Alesker has shown that smooth valuations admit a natural multiplication.

 Historically the first such expression predated this formal definition, in the form of Chern's generalized Gauss-Bonnet theorem giving  the Euler characteristic of a Riemannian manifold with boundary in terms of integrals of differential forms arising from the curvature tensor. We introduce a universal family of valuations that arise in similar fashion, and note that under Alesker multiplication it is naturally a module over the polynomial algebra R[t]. We determine the multiplication table for this module and give an application to the integral geometry of complex space forms.

8. Juli 2014     Prof. Dmitry Faifman (Tel Aviv)

Title: Concentration of measure and the flat torus.

Abstract. The phenomenon of concentration of measure goes back to Paul
Levy, and was used extensively by Gromov, Milman, Pisier and many others
in the study of the local theory of Banach spaces, also known as
asymptotic geometry. We will survey some theorems describing the source
of concentration of measure, and give examples. Then we will present a
result from our recent short note (joint with Klartag and Milman), where
no concentration of measure is present, but some consequences of
concentration appear nevertheless.

29. Juli 2014      Christian Beck (Frankfurt)

Titel:                  Freies Randwertproblem fur Niveaus mit mehreren Phasen


Wintersemester 2013/14

29.10.2013   Lukas Parapatits, TU Wien

TITLE: Minkowski Valuations and the Special Linear Group

ABSTRACT: The systematic study of the space of valuations, i.e. finitely additive maps on convex bodies, that are compatible with some subgroup of linear transformations has its origins in the work of Hadwiger. Through the seminal work of Ludwig, convex-body-valued valuations that intertwine the special linear group have become the focus of recent research. In this talk, I will give a survey on characterizations of these Minkowski valuations.

17.12.2013   Sven Jarohs, Frankfurt

Titel: Overdetermined problems involving the fractional Laplacian.

Abstract: In 1971 Serrin proved that if there is a positive solution to the Poissonproblem with constant nonnegative right-hand side and Dirichlet boundary conditions in a domain Omega, such that the outernormal derivative of u along the boundary of Omega is constant, then Omega must be a ball. Several works since then have been devoted to the study of this kind of overdetermined problems. I will show how this result and its generalization can be extended to problems involving the fractional Laplacian. After a short introduction to the fractional Laplacian, I will present a version of Hopf's Lemma for weak continuous solutions for linear problems involving that operator. This especially induces the strong maximum principle. Based on this I will give the main geometric ideas of the proof and reason why the nonlocal structure enables us to a more general result than in the local case. My talk is based on a joint work with Moustapha Fall.

28.01.2014     Dr. Astrid Berg (TU Wien)

Title: Log-Concavity Properties of Minkowski Valuations (joint work with Lukas Parapatits, Franz E. Schuster, Manuel Weberndorfer

Abstract: The famous Brunn-Minkowski inequality expresses the log-concavity of the volume functional. We use a recent result of Wannerer and Schuster to extend this classical inequality and prove log-concavity properties of intrinsic volumes of rigid motion compatible Minkowski valuations. We also obtain new Orlicz-Brunn-Minkowski inequalities for such valuations. These inequalities extend previous results by Alesker, Bernig, Lutwak, Parapatits and Schuster.