Projekte

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Uwe Abresch (Bochum):
The Isometric Problem in Symmetric Spaces of Non-Compact-Type
This is a project about the interplay of geometry and analysis. One aspect is to prove qualitative properties like convexity, smoothness, or uniqueness up to congruence. A second aspect is to determine the solutions explicitly when the rank is ≥2.

Uwe Abresch, Thomas Püttmann (Bochum):
The geometry of exotic spheres and exotic projective spaces
Exotic spheres and exotic projective spaces are fundamental examples in topology and geometry. These spaces are manifolds that are homotopy equivalent but not diffeomorphic to one of the standard spheres. It is a classical problem to find simple geometric models for these spaces and to investigate what kind of Riemannian metrics they admit. In particular, one is interested in the question which of the exotic spheres or exotic projective spaces admit a metric with nonnegative or even positive sectional curvature. We propose to extend our recent approach to exotic projective spaces in dimensions 5, 6, 13, and 14. In particular, we want to investigate the geometric structures related to the various models for the exotic spheres in dimensions 7 and 15.

Uwe Abresch, Thomas Püttmann (Bochum):
Stratifications of spaces with nonnegative sectional curvature and their relation to global structures and invariants
The subject of this project is a quest for interesting exceptional geometric situations where suitable stratifications allow a complete solution or at least an essential reduction of a global problem for a specific manifold. The global problems we focus on are the precise determination of geometric invariants, the construction of metrics with certain curvature properties, and the presentation of homotopy groups.

Peter Albers:
Functoriality in Floer homology
Exotic The object of this project is the theory of Floer homology in symplectic geometry. The principal problem to be analyzed is the question of functoriality for Floer homology with respect to symplectic immersions of positive codimension. We are using an approach based on a Lagrangian boundary value problem for Floer half-cylinders. The main problems are to extend this concept in broad generality which requires a careful analysis of holomorphic disks and to develop a suitable concept of obstruction classes and algebraic correction terms.

Christian Bär (Potsdam):
Dirac Operators on Lorentzian manifolds and their quantization
In the attempts of unifying quantum theory and general relativity theory one was able to quantize certain types of fields on curved spacetimes (i.e. timeoriented connected Lorentzian manifolds) which were most relevant from the physical point of view. The aim of the proposed project is to develop a unified and systematic approach to this kind of quantization. This requires a good understanding of the analysis of normally hyperbolic operators and of generalized Dirac operators. Certain geometric structures on Lorentzian manifolds such as foliations by smooth spacelike Cauchy hypersurfaces are important.

Christian Bär (Potsdam):
Noncommutative Geometry and Geometric Structures
Special geometric structures such as special holonomy should incorporated into noncommutative geometry. Interesting examples should be found. A workable concept of noncommutative Lorentzian manifold should be developped.

Werner Ballmann (Bonn), Werner Müller (Bonn),Dorothee Schüth (HU Berlin):
Spectral Theory of Dirac and Laplace Operators
In this project we investigate problems of spectral theory of Dirac and Laplace type operators on Riemannian manifolds. On compact manifolds we study questions of spectral rigidity versus isospectral defomability, relations to the length spectrum, and properties of metrics and potentials which are not spectrally determined. We also study higher spectral invariants (eta invariants, analytic torsion). Another aim is to understand the relation between the continuous spectrum and the geometry at infinity for certain classes of noncompact Riemannian manifolds. One of the tools used here is geometric scattering theory. The class of locally symmetric manifolds of finite volume is of special interest. Spectral theory on such manifolds has important applications in the theory of automorphic forms and in number theory.

Oliver Baues (Karlsruhe):
Affine manifolds and complex geometry
The project aims to develop the interactions of the differential geometry of affine manifolds with the theory of complex and symplectic manifolds. Part of this interaction arises from the Strominger-Yau-Zaslow idea which shows that integral affine manifolds play a role in the study of the mirror-phenomenon for complex manifolds. In this realm, we study geometric properties and construction methods for integral affine manifolds, aiming to contribute towards answering long standing questions on affine manifolds.

Helga Baum (HU Berlin), Felipe Leitner (Stuttgart), Thomas Leistner (Adelaide):
Geometry of Lorentzian manifolds with special holonomy
Manifolds with special geometries can be described by their holonomy representation. The irreducible holonomy representations of (simply-connected) Riemannian and pseudo-Riemannian manifolds are well known and geometric implications are intensively studied. In the pseudo-Riemannian case a new type of holonomy representations appears, the weakly irreducible but non-irreducible ones, which are - contrary to the irreducible case - not completely classified and geometrically less understood. In the same way as the holonomy of a metric, the holonomy of a conformal structure is defined (using the unique normal conformal Cartan connection). In the first period of the project the complete classification of Lorentzian holonomy groups was achieved (Th. Leistner). The aim in the second part of the project is to study the geometric structure and to construct geometric models of Lorentzian manifolds with special holonomy. Furthermore, we want to study systematically the holonomy of conformal structures - in particular in the Lorentzian case - and their geometric implications in conformal geometry.

Andreas Bernig (Zürich):
Differential Geometry of Singular Spaces
The aim of this project is to study Riemannian manifolds with curvature restrictions such as manifolds with positive scalar curvature or Einstein manifolds using tools from Geometric Measure Theory and Subanalytic Geometry. The normal cycle construction of Zaehle and Fu makes it possible to study curvature tensors of singular spaces and yields a topology in which scalar curvature and Einstein tensor are continuous. The collapsing of sequences of Riemannian manifolds will be studied using this topology. One of the potential applications is Gromov's conjecture about manifolds with positive scalar curvature.

Alexander I. Bobenko, Ulrich Pinkall (TU Berlin):
Constrained Willmore Surfaces
We want to study a class of surfaces in 3-space, the so-called "constrained Willmore surfaces". These are defined as the critical points of the Willmore functional &int H² (H the mean curvature) when only those variations are allowed that preserve the conformal type of the surface. Our main interest is the construction and classification of new compact examples, mainly topological spheres and tori.

Ulrich Bunke (Göttingen):
Geometrische Indextheorie für Mannigfaltigkeiten mit Ecken
One of the aims of the project is to develop the appropriate language of index theory for manifolds with corners. Families are considered in order to exhibit the geometric nature of the objects involved. The model cases are the interpretation of the eta-invariant of a manifold with corners as an element in a determinant line associated to the boundary, or as an element of the line, which arises as the difference of the two trivializations of the "two-line" associated to the corner of codimension two, which are given by the codimension-one faces.

Ulrich Bunke, Thomas Schick (Göttingen):
Geometric and Twisted Topology
In the geometric topology part we will construct and investigate smooth extensions of cohomology theories. This topic is in one part motivated by recent developments in field theory with non-trivial topological content, but also by the natural questions arrising from the already existing mathematical constructions (uniqueness, products, lifts of natural transformations, chracterization of the associated flat theory, equivariant extensions). The twisted topology part is devoted to the foundations of parametrized topology and the calculation of explicite examples in the case of equivariant K-theory.

Kai Cieliebak (LMU München):
The Symplectic vortex equations and applications
The symplectic vortex equations are equations on a symplectic manifold with a Hamiltonian group action recently introduced by Cieliebak, Gaio, Mundet and Salamon. Over the past years we developed the solution theory of these equations. In this project we will apply the symplectic vortex equations to questions in global differential geometry. The main application is to enumerative geometry, extending work of Kontsevich-Manin on Gromov-Witten invariants and Givental on mirror symmetry. Other applications concern Witten's conjecture on the Verlinde algebra, and the relation between different gauge theoretical invariants of smooth four-manifolds.

Kai Cieliebak (LMU München), Klaus Mohnke (HU Berlin)
Punctured Holomorphic Curves in Symplectic Geometry
The aim of this project is to systematically apply punctured holomorphic curves to questions in symplectic geometry. This is particularly promising for Lagrangian embeddings, where we expect new results on Lagrangian intersections, intersections of Lagrangian submanifolds with balls, Maslov class and symplectic area class rigidity, and unknottedness in dimension four. Moreover, we will lay the foundations for further applications by studying punctured holomorphic curves in cotangent bundles and their relations to closed geodesics and harmonic maps.

Anand Dessai (Münster):
Topology of positively curved manifolds with symmetry
The main aim of this project is to explore relations between curvature properties and the theory of elliptic genera. Also the classification of low dimensional manifolds of positive sectional curvature shall be pursued under weak symmetry assumptions. Another aim of this project lies in the study of the geometry of string manifolds. Here the main focus is on the question whether the kernel of the Witten genus can be represented by string manifolds of positive Ricci curvature.

Josef Dorfmeister (TU München):
Surfaces of Constant Mean Curvature with prescribed Fundamental Group
The main goal of this proposal is to use the generalized Weierstraß representation to construct a multitude of surfaces of constant mean curvature (CMC) in R³ of prescribed toplogical type. In particular, we will investigate 1. CMC tori with finitely many embedded ends attached, and 2. periodic surfaces, i.e. surfaces invariant under certain groups of translations in R³. In addition we plan to start the construction of compact CMC surfaces of genus g>1.

Felix Finster (Regensburg):
Curvature problems in Lorentzian manifolds
This project comprises two main research subjects
1) Lorentzian manifolds without spacelike codimension-one foliations
We want to investigate Lorentzian manifolds which do not admit any spacelike codimension-one foliation. A central question is to which extent this phenomenon can occur for metrics which satisfy one of the important partial differential relations in General Relativity: a positive energy condition or the vacuum Einstein equations.
2) Asymptotically flat manifolds of small mass
Our goal is to derive estimates which bound curvature from above by the total energy of the underlying Lorentzian manifold. Since in general relativity, space is described by a spacelike hypersurface of a Lorentzian manifold, the physical question under consideration is how and to which extent the energy of a physical system controls the gravitational field.

Urs Frauenfelder (LMU München):
Hamiltonian chords of quantized action
The objective of this proposal is to investigate Hamiltonian chords of quantized action. More precisely, we want to establish their existence in a quantitative way e.g. finding lower bounds on their number and giving estimates period and energy. The main tool is our newly defined Floer homology for non-compactly supported Hamiltonian functions in negative line bundles.

Thomas Foertsch (Zürich):
Large Products of hyperbolic metric spaces
Given two Gromov hyperbolic metric spaces (X_i,d_i), i=1,2, a certain subset Y of their product X₁ × X₂ endowed with a certain metric d, depending on d₁ and d₂ only, yields once again a Gromov hyperbolic metric space. (Y,d) is called the hyperbolic product of (X₁,d₁) and (X₂,d₂). The aim of the program is to, on the one hand, investigate this construction method further and to, on the other hand, seek for an analogue construction method for CAT(κ)-spaces.

Thomas Friedrich (HU Berlin):
Special Geometries and Fermionic Field Equations
Non integrable special geometries were introduced in Riemannian geometry for dimensions smaller than 8 by A. Gray in the 70ies, who subsequently studied them with his collaborators. More recently, non integrable geometries became interesting in the context of string theory. They are solutions to Strominger's equations. The goal of the project is to investigate special geometries (holonomy concept, "canonical" connections with torsion and their Dirac operators) and intend to make a contribution of modern differential geometry to recent developments in string theory.

Hans-Jörg Geiges (Köln):
Contact circles and surgery
This project studies certain families of contact structures, so-called contact circles and contact spheres, on 3-dimensional manifolds. The aim is to understand the relation of these structures to the Teichmüller theory of complex structures on surfaces, the dynamics of special flows on 3-manifolds, and constructions of hyperkähler metrics arising in physics such as the Gibbons-Hawking ansatz. Other strands of the project are concerned with surgery presentations of contact 3-manifolds and the existence of contact structures on higher-dimensional manifolds.

Sebastian Goette (Regensburg):
Higher Torsion Invariants and Applications to Smooth Maps, Bundles and Foliations
The Bismut-Lott index theorem for flat vector bundles relates the Kamber-Tondeur classes of a flat vector bundle on the total space of smooth a fibre bundle with compact fibres to the Kamber-Tondeur classes of the fibre-wise cohomology as a vector bundle on the base. By Dwyer, Weiss and Williams, this index theorem fails for merely topological fibre bundles in its present form. We want to use this fact as a starting point to generalise the Bismut-Lott theorem to larger classes of smooth maps, thus obtaining cohomological invariants of singular fibre bundles. Associated to the Bismut-Lott index theorem is a secondary invariant for smooth a fibre bundles, the Bismut-Lott higher analytic torsion. We want to relate this invariant to Igusa's topologically defined higher Franz-Reidemeister torsion using generalised fibre-wise Morse functions. As an application, we want to compute both invariants in many cases, and we want to use it to detect families of fibre-bundles that are pairwise topologically, but not differentially, isomorphic. Heitsch and Lazarov generalized the higher analytic torsion to foliations. We want to use leaf-wise Morse theory to compute this torsion. As an application, we want to exhibit families of foliations that are pairwise topologically, but not differentially, isomorphic.

Karsten Große-Brauckmann (Darmstadt):
Surfaces with prescribed curvature in theory and application
The moduli spaces of complete constant mean curvature surfaces in Euclidean space which are Alexandrow embedded, coplanar, have genus 0, and finitely many ends have been characterized joint with Kusner and Sullivan. We want to investigate one of the following questions, raised by these results: Are the moduli space smooth manifolds? What is a suitable description of the case of infinitely many ends? Do the restrictions on the asymptotic Delaunay parameters generalize at all to the non-coplanar case?

Daniel Grieser (Oldenburg):
Geometry and analysis of semi-algebraic sets
We study singular semi-algebraic and sub-analytic sets from a geometric and an analytic point of view. On the geometric side we study the inner metric properties of such sets, and in particular the behavior of differential geometric quantities at and near the singular locus: geodesics, the local volume growth function, the intrinsic distance function. On the analytic side we focus on regularity questions for harmonic forms, heat kernel asymptotics and L2 Hodge theory. This is closely related to the geometric questions since a proper understanding of the distance function is central to the study of the resolvent of the Laplace-Beltrami operator. Key points of our approach are the use of resolutions of singularities, adapted to the metric structure, and the development of a pseudodifferential calculus adapted to the degenerations of the metric on such resolutions.

Ursula Hamenstädt (Bonn):
Symplectic invariants of geodesic flows in negative curvature
The goal of the project is to develop a deformation theory for closed Riemannian manifolds of negative sectional curvature via modern invariants from symplectic geometry. The main guideline is classical Teichmüller theory and the theory of Kleinian groups.

Bernhard Hanke (LMU München), Thomas Schick (Göttingen):
Positive scalar curvature at the intersection of global analysis, topology and coarse geometry
This project is located around the investigation of Riemannian metrics of positive scalar curvature on smooth manifolds. Existence, obstructions and the space of positive scalar curvature metrics are particular questions in this context. Special attention will be paid to the interaction of methods from differential geometry, global analysis, algebraic topology and coarse geometry. The positive scalar curvature question will also be regarded as a motivation to formulate and investigate problems that are of interest in these specific areas independently from the positive scalar curvature question.

Ernst Heintze (Augsburg):
Submanifolds and group actions
The following problems will be studied:
1) A classification of polar (proper Fredholm) actions on Hilbert spaces. The main problem here is whether these always arise from isotropy representations of symmetric spaces of affine Kac-Moody type.
2) A classification of polar actions on symmetric spaces.
3) Are singular Riemannian submersions on Rⁿ ? up to few exceptions ? induced by isometric group actions? This might lead to a generalization of results of Thorbergsson, Gromoll-Grove and Gromoll-Walschap.
4) The investigation of symmetric spaces of affine Kac-Moody type G^/K^, where G^ is an affine Kac-Moody group and K^ the fixed point group of an involution. These spaces seem to be a perfect generalization of finite dimensional symmetric spaces with many similar properties.

Ines Kath (Leipzig):
The structure of pseudo-Riemannian symmetric spaces and holonomy groups
This project deals with two related classification problems in pseudo-Riemannian geometry: The first one is that of the classification of symmetric spaces which play a distinguished role among pseudo-Riemannian manifolds because of their special curvature properties. The second, more general problem is that of the classification of holonomy groups of pseudo-Riemannian manifolds. Using a recently developed structure theory for pseudo-Riemannian symmetric spaces we want to obtain explicit classification results for special classes of such spaces and we want to develop a similar structure theory for holonomy groups.

Gerhard Knieper (Bochum), Jens Heber (Kiel):
Harmonic Spaces in Riemannian Geometry
The project aims at structure or classification results for noncompact Riemannian harmonic spaces (i.e. with distance spheres of constant mean curvature), combining differential geometric, dynamical and algebraic methods. While nonsymmetric (homogeneous) examples are known to exist (Damek-Ricci), additional geometric assumptions (compact quotients, K<0) lead to rigidity (Besson-Courtois-Gallot)

Dieter Kotschick (LMU München):
Geometric formality
We shall investigate Riemannian metrics on compact oriented manifolds for which all wedge products of harmonic forms are harmonic. In small dimensions, we aim to classify these metrics. In arbitrary dimensions, forms harmonic for such a metric have very special properties and define interesting geometric structures, like foliations and symplectic structures. We investigate homogeneous examples, and those close to homogeneous. Further we study constructions of such metrics via symplectic geometry. Within symplectic geometry, we study the analogous property for symplectically harmonic forms in the sense of Brylinski instead of the harmonic forms in the sense of Hodge theory.

Dieter Kotschick (LMU München):
Asymptotic invariants of manifolds
We investigate several asymptotic invariants of closed manifolds which are related to the minimal volume entropy, with special emphasis on the isoperimetric constant, the minimal eigenvalue, and the spherical volume. We study the behaviour of these invariants under homotopy equivalence, under homeomorphism and under bordism. We prove inequalities between these invariants and we consider applications to geometric problems in which asymptotic invariants appear naturally.

Linus Kramer (TU Darmstadt), Stephan Stolz (Notre Dame):
Classification of isoparametric hypersurfaces and of manifolds which are like projective spaces
Isoparametric hypersurfaces are hypersurfaces in spheres with constant principal curvatures. The known examples either come from isotropy representations of symmetric spaces or from representations of Clifford algebras. The isoparametric hypersurfaces arising from symmetric spaces are the principal orbits of the isotropy representation of the symmetric space. In contrast to this, most of the examples arising from Clifford algebras do not admit a transitive isometry group. The classification of isoparametric hypersurfaces is one of the major open problems in submanifold geometry. The aim of the project is the classification of isoparametric hypersurfaces with 4 distinct principal curvatures.

Wolfgang Kuehnel (Stuttgart), Hans-Bert Rademacher (Leipzig):
Conformal Geometry of Generalized Brinkmann Spaces
Conformal Symmetries have been studied in General Relativity during many decades, and many concepts in differential geometry and mathematical physics are conformally invariant. In dimension four vacuum spacetimes with non-trivial conformal symmetries are pp-metrics first introduced by Brinkmann. We plan to extend the class of pp-metrics and study the conformal geometry and the spin geometry of these metrics.

Bernhard Leeb (LMU München):
Polygons in Symmetric Spaces and Buildings
We will study the geometry of a particularly important class of metric spaces with curvature bounded above, namely of nonpositively curved spaces of higher rank, i.e. of symmetric spaces of noncompact type and Euclidean buildings. Concrete questions we are interested in are e.g. to determine the restrictions on the side lengths of closed polygons, and to describe the natural geometric structures on moduli spaces of polygons with fixed side lengths. Besides its own geometric interest, the project is motivated by applications to various classical problems in algebra such as the eigenvalues of a sum problem for self-adjoint operators going back to Hermann Weyl, or the decomposition of tensor products in representation theory.

Joachim Lohkamp (Münster):
Scalar Curvature Contents
There are two by now nearly classical approaches to (obstruction theory for) positive scalar curvature. Spin Geometry (initiated by Lichnerowicz, Gromov, Lawson, Witten) and the inductive argument via minimal hypersurfaces introduced by Schoen and Yau. Whereas the first approach is restricted to spin manifolds, the second one covers arbitrary topologies. But there is one serious drawback: Due to the appearance of singularities the method could be applied only in dimension less or equal 7. The aim of this project is to overcome this dimensional obstacle by some local stabilisation process and to extend this approach to more general situations.

Vladimir S. Matveev (Freiburg):
Global theory of geodesically equivalent metrics
I am trying to understand under what conditions a diffeomorphism of a closed manifold preserving (unparameterised) geodesics must be a homothety. During the first phase of the project I understood completely the Riemannian case. My current goal is to study the pseudo-Riemannian case. More precisely, I hope to solve the Beltrami Problem for closed pseudo-Riemannian 2-manifolds, to prove or disprove in the pseudo-Riemannian case the Projective Lichnerowicz-Obata Conjecture, and to find first obstructions preventing a closed manifold from possessing different pseudo-Riemannian metrics sharing the same geodesics.

Lorenz Schwachhöfer (Dortmund):
Riemannian metrics with lower curvature bounds
Among the fundamental questions in differential geometry is the determination of obstructions to the existence or the description of new examples of manifolds with given lower curvature bounds. In particular, it is of interest to investigate obstructions to or examples of metrics with positive, nonnegative or almost nonnegative sectional or Ricci curvature. The manifolds which are of interest here are those which admit a group action of low cohomogeneity, in particular those of cohomogeneity at most two.

Lorenz Schwachhöfer (Dortmund):
Symplectic connections and symplectic realizations
A connection on a manifold is the description of the parallel transport of tangent vectors along differentiable paths. If the manifold is symplectic, then we call the connection symplectic if parallel transport preserves the symplectic form. We investigate symplectic connections whose curvature satisfies certain conditions, namely either the vanishing of some part of the curvature, or restrictions on its holonomy group. There is a canonical method to construct such connections locally. This method is based on the local existence of symplectic realizations of certain Poisson structures related to quaternionic symmetric spaces. The aim of this project is to determine in which case there can be a global realization, and hence when the local obstruction methods can be extended globally. In particular, it is important to decide if the Poisson structures involved admit a realization by a symplectic groupoid.

Matthias Schwarz (Leipzig):
Analysis of Floer Homology, its Natural Ring Structure and its S¹-equivariant version, in Relation with the Free Loop Space and Symplectic Invariants
The target of this project is Floer homology for the free loop space of a symplectic manifold. One of the main issues is to extend the understanding of Floer homology together with its pair-of-pants structure beyond the range of closed symplectic manifolds. For example a first step is the ring isomorphism with the Free Loop Space homology and its Loop Product in the case of the cotangent bundle. Further aims are the S¹-equivariant theory and the relation to String homology, as well as the connection with contact homology.

Uwe Semmelmann (Hamburg):
Nearly and almost Kähler geometry

Bernd Siebert (Freiburg):
Mirror symmetry, affine geometry and Gromov-Hausdorff limits
"Mirror symmetry" is a phenomenon discovered by physicists around 1990. It relates different geometries (symplectic and holomorphic) on pairs of different spaces. This project concerns differential-geometric aspects of my joint program with Mark Gross (UCSD) for a comprehensive explanation of this phenomenon by a notion of dual degeneration limits. The program also points to a dictionary of independent interest between symplectic and holomorphic geometry, respectively, on one side and integral affine geometry on the other side.

Knut Smoczyk, Matthias Schwarz (Leipzig):
Analysis of singularities of the Lagrangian mean curvature flow with pseudo-holomorphic curves
The prime aim of the project concerned with the Lagrangian mean curvature flow is to analyze and understand the process of formation of singularities as well as the possible limit solutions. The particular aim of this project in the intersection of geometric analysis and symplectic geometry is to use the now rich class of available methods from symplectic topology, that is specifically pseudo-holomorphic curve techniques, combined with the analytical methods from the theory of parabolic equations of heat flow and Monge-Ampère type.

Gudlaugur Thorbergsson (Köln):
Lie transformation groups in Riemannian Geometry
The area in which we plan to work can be described as "Lie transformation groups in Riemannian Geometry". The emphasis will be on isometric actions of compact Lie groups on Euclidean spaces and compact symmetric spaces, but more general Riemannian manifolds will also be considered. The goal is to understand the geometry and topology of the orbits of some of the most interesting classes of actions on Riemannian manifolds. The methods will be both geometric and algebraic.

Wilderich Tuschmann (Kiel):
Manifolds with Nonnegative and Almost Curvature
Manifolds with nonnegative and almost nonnegative curvature have turned out to play a crucial role in investigating the interplay of lower curvature bounds and global shape through geometric limiting processes and collapsing techniques. The present research project aims at establishing new topological obstructions to the existence of almost nonnegatively and nonnegatively curved metrics on closed manifolds and finding further geometrical properties of these spaces.

Hartmut Weiß (LMU München):
Deformations of 3-dimensional cone-manifold structures

Burkhard Wilking, Wilderich Tuschmann (Münster):
Representations whose orbit spaces have boundary and non-collapsing phenomena
Representations of compact Lie groups play a crucial role all over mathematics and physics. Although representations are classified by their highest weight, one often faces the hard problem to relate a priori knowledge on the geometry of a representation to its highest weight. We plan to classify representations whose orbit spaces have boundary. Since these representations occur naturally in different contexts of mathematics, a classification can be the starting point for solving various other problems. In a second part of the project we are concerned with non-collapsing phenomena. A non-collapsing phenomenon is present if a certain class of Riemannian manifolds has a uniform lower bound on the volume. Once one has established such a result, the understanding of this class of Riemannian manifolds as a whole improves significantly. Many recent progress has been made in the field. We propose that the recent progress should allow to attack the Klingenberg Sakai conjecture, which asserts that for each manifold a non-collapsing phenomenon is present in the moduli space of positively pinched metrics.

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