Geometric Analysis Seminar

Oberseminar Geometrische Analysis

Dienstag 16 Uhr c.t.,  Raum 404, RMayer-Straße 10


Prof. Dr. A. Bernig

Prof. Dr. E. Cabezas Rivas

Prof. Dr. Th. Mettler

Prof. Dr. T. Weth



Aktuelle Vorträge:   

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.

Sommersemester 2013

 14.05.2013   Prof. Dr. Arnd Scheel, University of Minnesota

Titel: Pattern selection in the wake of fronts

Abstract: Motivated by the formation of complex patterns in the wake of growth
processes in bacterial colonies, we'll study more generally patterns
formed in the wake of moving fronts. We discuss a number of mathematical
problems that arise when one tries to predict which wavenumbers and what
type of pattern is created by the growth process. We therefore describe
in some detail how to derive linear predictions and analyze nonlinear
mechanisms that lead to failure of these predictions. This is joint work
with Matt Holzer and Ryan Goh.


11.06.2013    Prof. Dr. Elisabeth Werner (Western Reserve University, Cleveland)


Title:          Divergence for s-concave and log concave functions

Abstract: In information theory, probability theory and statistics, an
f-divergence is a function that measures the difference between two (probability)
distributions. It is a generalization of commonly used divergences such as relative
entropy  (Kullback-Leibler divergence), Renyi divergences, total variation
distance etc. Here, we introduce f-divergence for s-concave and for log-concave functions.


25.06.2013   Prof. Dmitry Faifman (University of Tel Aviv, Israel)


Title: Generalized translation-invariant valuations and the Lorentz group.

The field of valuations on convex bodies has been developing rapidly in the
last decade, stemming from the solution of McMullen's conjecture by
Alesker, and the rich algebraic structure that was subsequently discovered
in the space of smooth valuations. Among other things, it allowed the
complete classification of valuations invariant under various compact
groups, notably the unitary group, in several works by Alesker, Bernig and
Fu, thus extending the classical orthogonal-invariant classification due to
Hadwiger. For non-compact groups, however, the situation is more obscure.
In particular, some technical difficulties of analytic nature appear, as
much of the algebraic structure only exists for smooth valuations, while
valuations arising in the non-compact setting are typically non-smooth. In
this talk, I will describe the space of generalized translation-invariant
valuations, and extend some of the now-standard operations on valuations to
the space of generalized valuations. Then I will present a complete
classification of the generalized valuations that are invariant under the
Lorentz group. The talk is based on a joint work with Semyon Alesker.


02.07.2013 Prof. Eva Vedel Jensen (Aarhus Universität, Dänemark)


Title: Recent developments in rotational integral geometry

Abstract: In this talk, I will give an overview of recent developments
in rotational integral geometry. The focus will be on rotational integral
formulae for Minkowski tensors. The overview will include a rotational
Crofton formula and a principal rotational formula for Minkowski tensors.
Applications to local stereology will shortly be mentioned. Finally, I will
present and discuss some of the open problems in rotational integral geometry.


09.07.2013   Tolga Yesil (Goethe-Universität Frankfurt am Main)


Titel: Der Zerlegungssatz von Gerard

Abstract: Der Zerlegungssatz von Gerard besagt, dass sich jede beschränkte
Folge des homogenen fraktionalen Sobolev-Hilbertraums als asymptotisch
orthogonale Superposition translatierender und dilatierender Elemente
dieses Raums approximieren lässt, wobei diese Approximation im Sinne der
Norm des Bildraums unter der korrospondierenden Sobolev-Einbettung, also
in einen gewissen Lebesgueraum, zu verstehen ist.
In diesem Vortrag soll die Konstruktion dieser "Profilzerlegung"
erläutert werden, indem, die wesentlichen Beweisschritte tangierend,
die Extraktion des ersten Profils demonstriert wird.
Dazu wir es nötig sein, ein kleines fourieranalytisches Repertoire -
genauer die sogenannte Littlewood-Paley Zerlegung - sowie eine auf
dieser Zerlegung fußende verbesserte Sobolev-Ungleichung
vorauszuschicken. Damit gewappnet werden wir die im Beweis verwendete(n)
Ausschöpfungsmethode(n) beschreiben und anwenden können.


18.09.2012 Roland Hildebrand (Grenoble)


Titel: Affine hypersurface immersions with parallel cubic form



13.11.2012    Franz Schuster (Wien)


Titel: Umgekehrte Isoperimetrische Ungleichungen


08.01.2013  Dennis Amelunxen (Cornell University, Ithaca, USA)


Titel: Spherical intrinsic volumes - Applications, asymptotics, inequalities


Sommersemester 2012


19.06.2012 Patrick Mentrup 

Titel: Konvexität der L_p-Schnittkörper

03.07.2012   Susanna Dann (Missouri)


Titel: The Busemann-Petty Problem in Complex Hyperbolic Space

Abstract: The Busemann-Petty problem asks whether origin-symmetric convex bodies in R^n
with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative
if n<5 and negative if n>=5. We study this problem in the complex hyperbolic n-space and prove
that the answer is affirmative for n<3 and negative for n>=3.


10.07.2012  Mikhailo Saienko (Frankfurt)

Titel:       SU(n)-invariante Bewertungen


Wintersemester 2011/12


13.12.2011   Dr. Mouhamed Mustafa Fall (Frankfurt)


Titel: Sharp local upper bound for the first non-zero Neumann eigenvalue in
             Riemanian manifolds.

Abstract:   We study the local Szegö-Weinberger isoperimetric profile in a
geodesic ball in a Riemannian manifold. This profile is obtained by maximizing
the first nontrivial Neumann eigenvalue  of the Laplace-Beltrami
operator among subdomains of this geodesic ball with
fixed volume. We derive a sharp asymptotic upper bound of this profile
in terms of the scalar curvature. As a  consequence, we deduce
an isoperimetric comparison principle relative to the
corresponding profile for 2-dimensional manifolds


08.11.2011   Dr. Christoph Scheven (Universität Erlangen-Nürnberg)


Titel: The flow for surfaces of prescribed mean curvature: Existence and asymptotics

Abstract: The geometric significance of the H-surface equations lies in the fact
that conformal solutions parametrize a surface of prescribed mean
curvature H. Classical results guarantee the existence of solutions
under certain smallness assumptions on H, the most general of which is
an isoperimetric condition. The talk will present a new result on the
existence of solutions to the associated heat flow under the same
isoperimetric condition that implies existence in the elliptic case.
Moreover, it is shown that the solution to the flow subconverges to a
solution of the stationary problem as the time tends to infinity.
Moreover, some possible applications of the techniques to the heat flow
of m-harmonic maps are discussed.

25.10.2011  Dr. Andzrej Muravnik


Titel:  Integral transforms of measures and properties of solutions of singular differential equations

Sommersemester 2011


12.07.2011    Dr. Christoph Haberl (Universität Salzburg)

Titel:  Valuations and affine Sobolev inequalities

Abstract: Basic operators in convex geometry can be characterized as valuations
which are compatible with the special linear group. We present such
characterization results and show how these results lead to new affine
versions of classical Sobolev inequalities.


12.07.2011  Dr. Huy Nguyen (University of Warwick)

Titel:           Geometric Rigidity of Surfaces with Bounded Total Curvature.
Abstract:      We consider surfaces conformally immersed in R^3 with L^2 bounds on the norm of the second fundamental form.
We will classify certain limit cases of these bounds, for example we will suitably generalize Osserman's classification
of complete non-compact minimal surfaces  with total curvature equal to 8\pi to the case of complete non-compact surfaces
with total bounded curvature.


05.07.2011   Dr. Gil Solanes ((Universitat Autònoma de Barcelona)

Titel: Integral geometry in complex projective space.
Abstract: The classical formulas of integral geometry by Blaschke and others have been recently extended to complex affine spaces by Bernig and Fu. This opens the door to the development of integral geometry in complex spaces of constant holomorphic curvature. I will present the first step in this project: the determination of Crofton formulas with respect to totally geodesic complex spaces. This is a joint work with J. Abardia and E.Gallego.


05.07.2011    Dr. Yann Bernard (Universität Freiburg)

Titel:          Asymptotic analysis of branched Willmore surfaces
Abstract:        We consider Willmore surfaces in R^m (for m\ge3) with an isolated singularity of finite density at the origin.
The aim is to gather as much information as possible on the local behavior of the immersion and of the second fundamental
form around the singularity. In particular, we derive local asymptotitcs for the mean curvature vector near the singular point.
We also provide specific conditions under which the singularity is removable. This is joint-work with Tristan Rivière.


28.06.2011   Professor Elisabeth Werner (Case Western Reserve University Cleveland)

TitelNon additivity of the Renyi entropy and Dvorezky's theorem

Abstract: We show that the analysis of the minimum output $p$-Renyi entropy of a typical quantum channel essentially amounts to applying Dvoretzky's Theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden--Winter and Hastings disproving the additivity conjecture for the minimal output $p$-Renyi entropy.


14.06.2011  Thomas Wannerer (TU Wien)

Titel: Rotation-equivariant Minkowski valuations

Abstract: Valuations are a classical concept in convex geometry. They are
real-valued functions on the space of convex bodies satisfying an
additivity property. In recent years, the theory of valuations has been
generalized in various directions and in this talk we discuss
convex-body-valued valuations. We establish a new description Minkowski
valuations, discuss relations between this and other representations,
and give applications


07.06.2011    PD Marco Kuehnel (Universität Freiburg)

Titel: Krümmungserhaltende Deformationen hermitescher Mannigfaltigkeiten und
Moduli von Vektorbuendeln auf nicht-Kähler-Flächen

Abstract: Auf der Suche nach dem Deformationsverhalten offener
hermitescher Mannigfaltigkeiten trifft man auf einen Ansatz, der das
genannte Deformationsproblem für Kählermannigfaltigkeiten in einem
Spezialfall löst. Für kompakte nicht-Kähler-Flächen erscheint er aber besonders
interessant und führt schliesslich zu einer fast vollständigen
Klassifikation solcher Flächen mit $0$-dimensionalen Modulräumen von
Rang-2-Vektorbündeln. Hierbei ist dies im Rahmen der Klassifikation
von class VII-Flächen von gewissem Interesse.


10.05.2011    Dr. Reto Müller (Scuola Normale Superiore di Pisa)

Titel: A compactness theorem for complete Ricci shrinkers

Abstract: We prove precompactness in an orbifold Cheeger-Gromov sense
of complete gradient Ricci shrinkers with a lower bound on their entropy and
a local integral Riemann bound. In particular, we do not need any pointwise
curvature assumptions, volume or diameter bounds. In dimension four, under
a technical assumption, we can replace the local integral Riemann bound by
an upper bound for the Euler characteristic. The proof relies on a Gauss-
Bonnet with cutoû argument. The results are joint work with Robert Haslhofer.


15.02.2011     Professor Semyon Alesker (Tel Aviv University)

Titel:              A Radon type transform on valuations

Abstract:        Valuation is a classical notion of convex geometry with numerous applications to integral geometry. A valuation is a finitely additive measure on the class of all convex compact sets in R^n satisfying a continuity assumption. In recent years it turned out that the theory of valuations partly generalizes far beyond convexity, to all smooth manifolds. This extention has even broader relations to integral geometry, some of which will be discussed in the talk. We will describe the rich structures on valuations on manifolds assuming no preliminary knowledge of the classical theory. We introduce a Radon type transform on valuations and discuss its properties in various special cases.


01.02.2011     Miles Simon (Universität Freiburg)

Titel:    Ricci Fluss von Metriken mit Kegel Singularitäten

Abstract:    Wir zeigen, dass man positiv gekrümmte singuläre Kegel-Metriken mit dem Ricci Fluss evolieren kann. Die Metrik wird sofort glatt, und die Lösung ist ein expandierender Soliton.


18.01.2011    Professor Mohameden Ould Ahmedou (Universität Giessen)

Titel: A Poincare uniformization type-theorem on compact four-manifolds

Abstract: In this talk we report on some progress for the study on the problem of existence of conformal metrics with constant $Q$-curvature on closed four-dimensional Riemannian manifolds. This problem amounts to solve a fourth-order nonlinear elliptic equation involving  the Paneitz operator. The corresponding equation has a variational structure, however the associated Euler-Lagrange functional is not bounded from below nor from above in many situations. Furthermore, it does not satisfy the Palais-Smale condition in general. Using an algebraic topological argument, combined with a refined analysis of the loss of compactness, we solve the problem in many cases where blow-up does occur. Precisely, we prove that if the kernel of the Paneitz operator consists only of constant functions, then the above problem is solvable in all the cases (except one) left open after the celebrated works of Chang-Yang (Annals 1995) and Djadli-Malchiodi (Annals 2008).


Dr. Armin Schikorra (RWTH Aachen)

Titel: Beispiele für Regularität elliptischer Systeme mit antisymmetrischen Potentialen

Abstract:    Es werden Beispiele aus einerseits der Theorie der fraktionalen polyharmonischen Abbildungen und andererseits aus der Theorie der degeneriert elliptischen $n$-harmonischen Abbildungen vorgestellt, welche als Fortsetzung von Riviere's berühmten Resultats von 2007 über die Regularität von kritischen Punkten von konform invarianten Variationsfunktionalen in zwei Dimensionen gesehen werden können.


11.01.2011     Dr. Theodora Bourni (Max-Planck-Institut für Gravitationsphysik, Golm)

Titel: Curvature estimates for surfaces with bounded mean curvature

Abstract: In this talk I will discuss some recent results concerning estimates for the norm of the second fundamental form, |A|, for surfaces with bounded mean curvature. In particular I will show that for an embedded geodesic disk with bounded L^2 norm of |A|, |A| is bounded at interior points, provided that the W^{1,p} norm of its mean curvature is sufficiently small, p>2. This is joint work with Giuseppe Tinaglia.


14.12.2010 Professor Jan Metzger (Universität Potsdam)

Titel: Isoperimetric surfaces in asymptotically flat manifolds

Abstract: In this talk I will present joint work with Michael Eichmair. We consider the isoperimetric problem in asymptotically flat manifolds which are close in C^0 to Schwarzschild. The main result is that for given large enough volume there exists a smooth connected isoperimetric surface enclosing this volume. We furthermore derive position estimates for this surface. A corollary of our analysis is that the constant mean curvature foliation constructed by  Huisken and Yau consists of isoperimetric surfaces.


7.12.2010 Professor Oliver Schnürer (Universität Konstanz)

Titel: Stability of entire graphs evolving under mean curvature flow and Gauss curvature flow

Abstract: We consider the evolution of hypersurfaces described as graphs over R^n. The evolution is determined by the normal velocity. Here we consider the mean curvature and powers of the Gauss curvature as normal velocities. In both cases, we obtain stability results.

Long time existence for graphical solutions evolving under mean curvature flow is well known. Hence we focus on techniques to obtain stability results. For equations involving the Gauss curvature, we first have to establish a long time existence result. The proofs of our stability results are then similar to those for mean curvature flow.

In the first part, we investigate the qualitative behaviour of entire solutions to mean curvature flow. In the second part we mainly focus on an existence result for Gauss curvature flow.


30.11.2010 Dr. Judit Abardia (Frankfurt)

Title: Projection bodies in complex vector spaces

Abstract: The space of Minkowski valuations on an $n$-dimensional complex vector space which are continuous, translation invariant and contravariant under the complex special linear group is explicitly described. Each valuation with these properties is shown to satisfy geometric inequalities of Brunn-Minkowski, Alexandrov-Fenchel and Minkowski type.


23.11.2010 Dr. Patrick Breuning (Universität Freiburg)

Title: Immersions with local Lipschitz representation

Abstract: We consider immersions admitting uniform graph representations over the affine tangent space. Assume that any graph is defined on a ball of radius r and satisfies a specific property such as a Lipschitz bound or a C^0-bound. We show that such graph functions coming from immersions satisfy much better properties than a single function. Particularly we show that a sufficiently small C^0-norm of any graph implies Lipschitz continuity with small Lipschitz constant. This can be used e.g. for showing compactness of such immersions.


16.11.2010 Dr. Gil Solanes (Universitat Autònoma de Barcelona)

Title:   Total curvature of complete surfaces in hyperbolic space.

Abstract: I will discuss on a Gauss-Bonnet formula for the integral of the extrinsic curvature of complete surfaces in hyperbolic space. The formula contains a contribution from infinity which is a Möbius invariant of the ideal boundary curve. It can be described as the renormalized volume of the set of spheres linked with the curve. This is related to Banchoff-Pohl's definition of the area enclosed by space curves.  Also, connections with some known knot energies will be discussed.


9.11.2010 Christian Beck (Frankfurt)

Titel: Symmetrie von Minimierern von Variationsproblemen mit C1-Regularität

Abstract: Es wird eine Methode vorgestellt, wie bei bestimmten Variationsproblemen, und zwar solchen, bei denen polarisierte Minimierer wieder Minimierer sind und ferner hinreichende Regularität aufweisen, bewiesen werden kann, dass eine gewisse Art von Symmetrie, nämlich lokal geblätterte Schwarz-Symmetrie, vorliegt. Anwenden werden wir die gewonnenen Fertigkeiten, um Aussagen über die Gestalt von Eigenfunktionen des p-Laplace-Operators unter Neumann-Randbedingungen zum ersten positiven Eigenwert auf einem beschränkten radialsymmetrischen Gebiet zu treffen.


14.09.2010, 16.15 Uhr   Dr. Alvaro Guevera (Institut für Informatik)

Title:   Convergence Results for a Self-dual Regularization of Convex Problems

Abstract: We study a one-parameter regularization technique for convex optimization problems, which has as its main feature its self-duality with respect to the usual convex conjugation. The technique, introduced by Goebel, can be defined for both convex and saddle functions. When applied to the latter, we show that if a saddle function has at least one saddle point, then the sequence of saddle points of the regularized saddle functions converges to the saddle point of minimal norm of the original one.
For convex problems with inequality and state constraints, we apply the regularization directly on the objective and constraint functions, and show that, under suitable conditions, the associated Lagrangians of the regularized problem hypo/epi-converge to the original Lagrangian, and that the associated value functions also epi-converge to the original one. Finally, we find explicit conditions ensuring that the regularized sequence satisfies Slater’s condition.


23.02.2010.  Jean Van Schaftingen (Université catholique de Louvain )

Titel:    Existence of optimal functions for Poincaré-Sobolev inequalities


26.01.2010:    Thomas Mettler (Université de Fribourg)

Titel:     On projective surfaces with compatible Weyl structure

Abstract: The existence problem for Riemannian metrics on a manifold with prescribed (unparametrized) geodesics is a natural problem in the theory of over determined systems of partial differential equations. Unfortunately even for surfaces its solution is somewhat unpleasant. However if one generalizes the problem and looks for Weyl structures on surfaces with prescribed geodesics, the problem becomes tractable with techniques from complex geometry as will be shown in this talk.


12.01.2010:    Prof. Dr. Tobias Kaiser (Universität Passau)

Titel:     Zahme Geometrie und Analysis

Abstract: O-minimale Strukturen stellen eine umfassende Verallgemeinerung der klassischen Geometrien algebraischer Prägung dar. Sie zeichnen sich durch exzellente Endlichkeits- und Zahmheitseigenschaften aus und erfassen wichtige Konzepte der Analysis. Nach einer kurzen Einführung in o-minimale Strukturen zeige ich deren Bezug zu und Anwendung bei dynamischen Systemen (Hilbert 16, Teil 2), komplexer Analysis (Riemannscher Abbildungssatz) und partiellen Differentialgleichungen (Dirichlet-Problem).


3.11. 09:  Dr. Anna Dall' Acqua (Universität Magdeburg):

Titel: The Dirichlet boundary value problem for Willmore surfaces of revolution

Abstract: The Willmore functional is the integral of the square of the mean curvature over the unknown surface. We consider the minimisation problem among all surfaces which obey suitable boundary conditions. The Willmore equation as the corresponding Euler-Lagrange equation may be considered as frame invariant counterpart of the clamped plate equation. This equation is of interest not only in mechanics and membrane physics but  also in differential geometry.

We consider the Willmore boundary value problem for surfaces of revolution with arbitrary symmetric Dirichlet boundary condition. Using direct methods of the calculus of variations, we prove existence and regularity of minimising solutions.


27.10. 09:  Dr. Gautier Berck (Université de Fribourg):

Titel: Different aspects of the intersection bodies

Abstract: Classically, the intersection bodies where first introduced by Busemann in his theory of areas in normed and Finsler spaces. The notion was later successively extended by Lutwak, Zhang and Koldobsky leading to a complete solution of the first Busemann-Petty problem.

In the first, more descriptive, part of the talk, we will focus on Busemann's notion of area, the convexity of the intersection bodies and related convex geometric problems. In a second part, we will show how the theory of distributions and Fourier transforms may be used to address some of these problems.