Dienstag 14 Uhr c.t.**, **RM10 - 903

**Aktuelle Vorträge**

**23.04.2024 Mario Santilli (Universita dell'Acquila)**

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

**Abstract**: We discuss the extension of well known rigidity results for constant mean curvature hypersurfaces of Alexandrov

(in hyperbolic space) and Brendle (in warped product spaces) to sets of finite perimeter with constant distributional mean curvature. Joint work with Francesco Maggi.

**07.05.2024 Oscar Ortega Moreno (TU Wien)**

**Title**: Moment inequalities for Gaussian vectors

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

**14.05.2024 Elisabeth Werner (Case Western
Reserve University Cleveland)**

**Title:** Weighted floating functions and weighted functional affine surface areas

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

**28.05.2024 ****Nikita Cernomazov (Frankfurt)**

**Title**: Constrained curve-flows and their self-shrinkers

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

**18.06.2024 ****Vadim Lebovici (University of
Oxford)**Title: Hybrid transforms of constructible functions

Abstract: A function on a real analytic manifold is constructibleif it is integer-valued and locally constant on a subanalytic

stratification of the manifold. Constructible functions are the measurable functions of a theory of integration with respect to the

Euler characteristic, which allows for the definition of topological integral transforms used in shape analysis.

After a short motivation of constructible functions from the point of view of valuations on manifolds, of persistent homology and of applied

geometry, I will present the so-called /hybrid/ transforms combining the Lebesgue integral and integration with respect to the Euler

characteristic. From a theoretical point of view, these transforms benefit from properties of regularity and invertibility. From an applied

point of view, they are efficiently computable and provide flexibility for extracting relevant information in data analysis.

This presentation is based on the article [arXiv:2111.07829] and the

article with Olympio Hacquard [arXiv:2303.14040].

25.06.2024 Peter Bürgisser (TU Berlin)

Title: Kähler package for algebra of Grassmann zonoids

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

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

**09.07.2024 Olivia Vicanek-Martinez (Uni Tübingen)**

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

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

**16.07.2024 ****Stefan Le Coz (Toulouse)**

Titel: Blow-up on a star-graph

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

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**ARCHIV**

**Wintersemester 2023/24**

**25.10.2023 Liangjun Weng (Roma Tor Vergata)**

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

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

**01.11.2023 Prachi Sahjwani (Cardiff University)**

**Title**: Stability of geometric inequalities in various spaces

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

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

**15.11.2023 Amelie Menges (TU Dortmund)**

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

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

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

**29.11.2023 Simon Ellmeyer (Wien)**

**Ti****tel:
**Complex
Lp-Intersection Bodies and related Busemann–Petty type problems

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

This is joint work with Georg C. Hofstätter

**13.12.2023 Bachelor-Vortrag: Sebastian Schmidt (in German)**

**Title**: Zur Stabilität in der Heintze-Karcher Ungleichung

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

**10.01.2024 Ben Lambert (Leeds University)**

**Title**: Alexandrov immersed mean curvature flow

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

**17.01.2024 Kostiantyn Drach (Universitat de Barcelona)**

**Title**: Reverse isoperimetric inequality under curvature constraints

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

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

**31.01.2024 Leo Brauner (TU Wien)
**

**Title: **Lefschetz operators on Minkowski valuations**Abstract: **Minkowski
valuations are finitely additive operators on the space of convex
bodies. They form a rich class of geometric maps, including the
difference body, projection body, and mean section body maps. The
Lefschetz operators allow us to move between valuations of different
degrees of homogeneity. In this talk, we discuss the action of the
Lefschetz operators on continuous Minkowski valuations that are
compatible with rigid motions.

This is joint work with Georg C. Hofstätter and Oscar Ortega-Moreno.

**Sommersemester 2023**

**12.04.2023 Xuwen Zhang (Xiamen/Frankfurt)**

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

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

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

**26.04.2023 Dmitry Faifman (Tel Aviv)**

**Title:** Extensions of valuations.

**Abstract. **A valuation is a finitely additive measure on convex bodies or
some other family of test sets, typically with some analytic
restrictions or invariance properties.

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

**03.05.2023 Joe Hoisington (MPI Bonn)**

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

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

**10.05.2023 FRANKFURTER SEMINAR **

**17.05.2023 Frederick Herget (Frankfurt)**

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

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

**31.05.2023 Armin Schikorra (Pittsburgh)**

**Titel:** Regularity results for n-Laplace systems with antisymmetric potential**Abstract: **n-Laplace
systems with antisymmetric potential are known to govern geometric
equations such as n-harmonic maps between manifolds and

generalized
prescribed H-surface equations. Due to the nonlinearity of the leading
order n-Laplace and the criticality of the equation they

are very difficult to treat.

I
will discuss some progress we obtained, combining stability methods by
Iwaniec and nonlinear potential theory for vectorial equations by

Kuusi-Mingione.

Joint work with Dorian Martino

**07.06.2023 FRANKFURTER SEMINAR **

**14.06.2023 Guofang Wang (Freiburg)**

**Title**: Capillary hypersurfaces

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

**28.06.2023 FRANKFURTER SEMINAR **

**05.07.2023 Elisabeth Werner (** Case Western University Cleveland, USA)

**Title**:
Extremal affine surface area in a functional setting

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

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

parallels results in the setting of convex bodies.

Based on joint work with Stephanie Egler.

**Wintersemester 2022/23**

**18.10.2022 Samuel Held (Frankfurt) - Vortrag zur Masterarbeit**

**08.11.2022 Thomas Wannerer (Jena)**

**title: **Affine Minkowski valuations

**abstract:** In convex geometry, the maps that assign to a convex body its
difference body or projection body

have the following properties:
They are (1) continuous; (2) finitely additive; (3) compatible
with the action

the special linear group. In this talk we explore
the question whether there exist other constructions

with these
properties. We have found the following dichotomy: There are no
new examples if one

assumes translation-invariance, but there are
plenty of new examples without this assumption.

Based on joint work with Jakob Henkel.

**15.11.2022 Chiara Meroni (MPI MIS Leipzig)**

of curves. This is a work in progress with Carlos Améndola and Darrick Lee. We generalize the class of curves for which

a certain integral formula works, using the technique of signatures. I will then give a geometric interpretation

of this volume formula in terms of lengths and areas, and conclude with the example of logarithmic curves, which draws

connections to polylogarithms and Feynman integrals.

**22.11.2022 Nikita Cernomazov ** **(Darmstadt)**

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

**Abstract: **The
discussion of self-similar solutions is a cornerstone in the study of
the famous Curve Shortening Flow ∂_t c = kN. Already in its infancy,
W.W. Mullins provided examples of curves that shrink due to uniform
scaling. Later, U. Abresch and J. Langer were able to classify all
homothetically shrinking solutions.

In
this talk, we consider CSF's less famous, but certainly not less
interesting cousin: the Area-Preserving Curve Shortening Flow ∂_t c =
(k-\bar{k})N with \bar{k} being the average curvature. First, we derive
characterizing equations for general self-similar solutions of APCSF.
Following this, we will rephrase the search for homothetically shrinking
solutions to a problem in Hamiltonian mechanics. Finally, we will prove
that there are homothetically shrinking solutions of APCSF and give
examples.

**6.12.2022
Ernst Kuwert ** **(Freiburg)**

**title:** Curvature varifolds with orthogonal boundary

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

**13.12.2022 Tobias König (Frankfurt)**

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

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

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

**10.01.2023 Julian Scheuer (Frankfurt)****Title**: Quermassintegral inequalities for convex free boundary hypersurfaces in the ball

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

**17.01.2023 Georg Hofstätter (Wien)title: ** Restrictions of Valuations

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

This is joint work in progress with D.
Faifman

**24.01.2023 Michele Stecconi (Universite du Luxembourg)**

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

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

** **

**31.01.2023
Markus Wolff (Tübingen)
title: **Ricci-Flow on surfaces along the standard light cone in the $3+1$ Minkowski spacetime

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

**15.02.2022 ** **Axel Fünfhaus, Tobias Weth** (Frankfurt)

Vorträge im interdisziplinären Seminar zur **Topologie und Festkörperphysik**

Titles: tba**14.09.2021**, Raum 901 **Dmitry Faifman** (Tel Aviv/Israel)

Title: **Between the Funk metric and convex geometry.**

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

**Sommersemester 2020 14.07.2020 Ludwig Hammer **(Videovortrag zur Masterarbeit)

**Titel:** **Regularität von Minimierern des TV-L2-Modells**

**Abstract: **Die Reduktion von Bildrauschen mit dem Verfahren nach L. Rudin, S. Osher und E. Fatemi ist empirisch wie analytisch gut erforscht. Es basiert auf der Minimierung eines konvexen Funktionals auf der Menge der Funktionen beschränkter Variation, das eng mit der Mean-Curvature-Gleichung zusammenhängt.

Dieser Vortrag widmet sich von analytischer Seite her der Frage, wie sich in Dimension bis 7 die Stetigkeit und lokale Hölderstetigkeit von den Ausgangsdaten auf das Ergebnis übertragen, folgend einer Technik von A. Chambolle, V. Caselles und M. Novaga. Die zentrale Beobachtung dabei ist, dass die Niveaumengen stetiger Funktionen sich nicht berühren, beziehungsweise im Falle von Hölderstetigkeit einen positiven Abstand haben. Die Niveaumengen von Minimierern des fraglichen Funktionals lassen sich mit Methoden der geometrischen Maßtheorie analog behandeln.

**2.6.2020 Rüdiger Krämer** (Videovortrag zur Bachelorarbeit)

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

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

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

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

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)

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.

28.01.2020 Franz Schuster

Titel: “Affine" isoperimetric inequalities in real space forms

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

28.4.2020 - FÄLLT AUS - Karoly Böröczky

Titel: U(n) equivariant tensor valuations

Abstract:

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: tba26.05.2020 Georg Hofstätter (Wien) - ABGESAGTTitel: tba**

**02.06.2020 Oscar Ortega (TU Wien) - ABGESAGTTitel: 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.

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.

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:

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.

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.

27.11.2018 Dr. Ignace Minlend (AIMS Mbour/Senegal)

Titel:

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.

**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

17.04.2018 Maciej Dunajski

Title:

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:

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)

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

24.10.2017 Andy Sanders (Universität Heidelberg)

Title:

Abstract:

21.11.2017 Sebastian Heller (Universität Hamburg)

Title:

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)

**12.12.2017 Nguyen Bac Dang (Ecole Polytechnique Paris-Saclay)Title: **A positive cone in the space of continuous translation invariant

valuations.

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:

Abstract:

16.01.2018 Thomas Hack (TU Wien)

Title:

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

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

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

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)

Abstract:

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

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)

Titel:

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)

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

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

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

10.05. 2016

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

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

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

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

Abstract

03.11.2015

Abstract

10.11.2015

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

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

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)

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)

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.

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.

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.

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