### Oberseminar Geometrische Analysis

Dienstag 16 Uhr c.t.

Prof. Dr. A. Bernig

Prof. Dr. T. Weth

Aktuelle Vorträge

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)

Titel:
Donnelly's theorem

05.11.2019     Jonas Knörr
(Goethe-Universität)

Titel:                 Smooth valuations on convex functions

Abstract:
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

Abstract:
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

Abstract

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)

Abstract:
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

Abstract

30.04.2019                Nicolas Hilger (Frankfurt)
Vortrag zur Bachelorarbeit

Titel:
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

Abstract:
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

Abstract:
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
subgroups.
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

Abstract:
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.

ARCHIV

Wintersemester 2018/19

30.10.2018     Christian Lange (Universität Köln)

Titel:
Orbifolds all of whose geodesics are closed

Abstract:
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)

Titel:
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

Abstract:
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)

Titel:
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

Abstract:
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

Titel
: 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

Title:
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)

Titel:
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

Titel:
Perelmans Pseudolocality Theorem (Vortrag zur Bachelorarbeit)

Wintersemester 2017/18 (16.10.17 - 09.02.18)

17.10.2017     Gabriel Paternain (University of Cambridge)

Title:
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)

Title:
Complexification of real analytic Kahler manifolds with applications to Teichmuller theory

Abstract:
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)

Title:
Harmonic maps, integrable systems and twistor spaces

Abstract:
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

Titel:
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)

Title:
A positive cone in the space of continuous translation invariant
valuations.

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)

Title:
Scalar curvature via local extent

Abstract:
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)

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

Abstract:
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.

16.05.2017
Karin Melnick (University of Maryland)

Titel:
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.

23.05.2017
Charles Frances (Université de Strasbourg)

Title: Dynamics and topology for 3-dimensional Lorentz manifolds

Abstract:
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)

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

Abstract:
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.

Titel:
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

Titel:
(proof of new Minkowski inequalities)

Wintersemester 2016/17

15.11.2016     Franz Schuster (TU Wien/Österreich)

Titel
: 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)

Titel:
Minimal Lagrangian connections

Abstract:
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

Abstract

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)

Titel:
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

Abstract

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

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

Abstract

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

Abstract

02.02.2016     Ana Peón-Nieto (Universität Heidelberg)

Titel: SU (p, p + 1)-HIGGS BUNDLES AND THE HITCHIN MAP

Abstract

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)

Titel:
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

Abstract

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.

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
$3$-space.

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

Abstrakt

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.