### Geometric Analysis Seminar

### Oberseminar Geometrische Analysis

Dienstag 16 Uhr c.t., **Raum 903, **RMayer-Straße 10

Aktuelle Vorträge

**Wintersemester 2019/20 (14.10.2019 - 14.02.2020)**

**22.10.2019 Ricardo Arconada **(Goethe-Universität)**Titel: **Donnelly's theorem

**(Goethe-Universität)**

05.11.2019 Jonas Knörr

05.11.2019 Jonas Knörr

**Titel: Smooth valuations on convex functionsAbstract: **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)

**In this talk, we give an introduction to the theory of D-modules.**

Titel: D-modules and applications

Abstract:

Titel: D-modules and applications

Abstract:

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.

**(TU Wien)**

28.01.2020 Franz Schuster

28.01.2020 Franz Schuster

Titel: “Affine” isoperimetric inequalities in real space forms

Titel: “Affine” isoperimetric inequalities in real space forms

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

**(Alfréd Rényi Institute of Mathematics, Budapest)**

28.4.2020 Karoly Böröczky

28.4.2020 Karoly Böröczky

**tba**

Titel:Titel:

12.05.2020 Antonio Lerario (SISSA Trieste)

**Titel: tba26.05.2020 Georg Hofstätter (Wien)Titel: tba**

**02.06.2020 Oscar Ortega (TU Wien)Titel: tba09.06.2020 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 BachelorarbeitTitel: **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

**Manifolds all of whose geodesics are closed have been studied a lot,**

Abstract:

Abstract:

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

Titel:

**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.

**Multiply-Periodic Hypersurfaces with constant nonlocal mean Curvature**

27.11.2018 Dr. Ignace Minlend (AIMS Mbour/Senegal)

Titel:

27.11.2018 Dr. Ignace Minlend (AIMS Mbour/Senegal)

Titel:

**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 fillingsAbstract: **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.

**Twistors and conics**

17.04.2018 Maciej Dunajski

Title:

17.04.2018 Maciej Dunajski

Title:

**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.

**Geodesics on singular spaces**

08.05.2018 Daniel Grieser (Universität Oldenburg)

Titel:

08.05.2018 Daniel Grieser (Universität Oldenburg)

Titel:

**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.

**Perelmans Pseudolocality Theorem (Vortrag zur Bachelorarbeit)**

05.06.2018 Lucas Schäfer

19.06. 2018 Roger El Andary (Frankfurt am Main)

Titel:

05.06.2018 Lucas Schäfer

19.06. 2018 Roger El Andary (Frankfurt am Main)

*in Raum 711 groß*Titel:

**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.

**Complexification of real analytic Kahler manifolds with applications to Teichmuller theory**

24.10.2017 Andy Sanders (Universität Heidelberg)

Title:

24.10.2017 Andy Sanders (Universität Heidelberg)

Title:

**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.**

Abstract:

Abstract:

**Harmonic maps, integrable systems and twistor spaces**

21.11.2017 Sebastian Heller (Universität Hamburg)

Title:

21.11.2017 Sebastian Heller (Universität Hamburg)

Title:

**In this talk I discuss harmonic maps of compact Riemann surfaces into certain**

Abstract:

Abstract:

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) Titel: **Das Chern-Gauss-Bonnet Theorem (Bachelorarbeit)

05.12.2017 Friederike Dittberner (Universität Konstanz)

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.

**Scalar curvature via local extent**

19.12.2017 Giona Veronelli (Universite Paris 13)

Title:

19.12.2017 Giona Veronelli (Universite Paris 13)

Title:

**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.**

Abstract:

Abstract:

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

16.01.2018 Thomas Hack (TU Wien)

Title:

16.01.2018 Thomas Hack (TU Wien)

Title:

**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.**

Abstract:

Abstract:

**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

23.05.2017

**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.**

Abstract:

Abstract:

**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)

**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.**

Abstract:

Abstract:

27.06.2017 Ruth Kellerhals (Université de Fribourg)

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)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 110Tristan Daus (Frankfurt), BachelorvortragTitel:** Inverse flow in AdS-Schwarzschild manifold

(proof of new Minkowski inequalities)

**Wintersemester 2016/1715.11.2016 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

Titel:

**A connection on the tangent bundle of a smooth manifold M can be understood**

Abstract:

Abstract:

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)

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.

13.05.2014 Manuel Ritoré (Univ. Granada)

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 $M\times\{x\}$, $x\in R^k$.

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

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

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

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.

**Title: Valuations on Riemannian manifolds**

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

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

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.