Geometry Seminar main page

Tue Dec 17 ∙ Seminarraum C ∙ 13:00

I shall present in this talk a joint result with M. Abreu, J. Kang and L. Macarini on the existence of periodic Reeb orbits. More precisely, that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive S

Tue Oct 22 ∙ Seminarraum C ∙ 13:00

Basmajian's identity expresses the volume of the boundary of a compact hyperbolic manifold as a sum over the orthogeodesics on the manifold. In this talk, I will introduce Basmajian's identity for real hyperconvex Anosov representations (in the sense of Pozzetti-Sambarino-Wienhard) and show how to extend them in the PGL(n, C)-character variety.

Tue Oct 29 ∙ Raum 5.104 ∙ 13:30

I will discuss constructions and properties of real hyperbolic manifolds of finite volume, with a focus on dimensions 2 and 3 and families of arithmetic manifolds. The results I want to discuss mainly concern the relation of the volume to various topological invariants (genus, Betti numbers,...).

Tue Nov 5 ∙ Seminarraum C ∙ 13:30

I will discuss results about some topological constrains on Lagrangian cobordisms that can be obtained by assuming them to be either exact or monotone. The talk is partially based on joint work with Jean-Francois Barraud.

Tue Nov 19 ∙ Seminarraum C ∙ 13:30

A Kähler group is a finitely presented group, which arises as fundamental group of a compact Kähler manifold. In this talk I will address Delzant and Gromov's question of which subgroups of direct products of surface groups can be obtained as images of homomorphisms from Kähler groups, which are induced by a holomorphic map to a direct product of Riemann surfaces. I shall begin by giving a general introduction to the theory of Kähler groups and explain the special role that surface groups play in their study. I will then proceed to provide a complete solution to Delzant and Gromov's question for homomorphisms to direct products of three surface groups. It is based on understanding the images on fundamental groups of complex hypersurfaces in direct products of Riemann surfaces.

Tue Nov 26 ∙ Seminarraum C ∙ 13:30

I will report on recent joint work with Laura Fredrickson, Rafe Mazzeo and Jan Swoboda, in which we study the asymptotics of the Hitchin metric on moduli spaces of parabolic Higgs bundles. These give rise to real 4-dimensional and complete hyperkähler metrics when considered e.g. on the 4-punctured sphere. We will discuss the question (raised by Hitchin and others) if the 4-punctured sphere moduli space is an ALG gravitational instanton.

Tue Dec 3 ∙ Seminarraum C ∙ 13:00

We will discuss generalizations of the front and Lagrangian projections in general contact 3-manifolds presented as an open book with a Morse structure and explain how to compute the Euler class of the associated contact structure. This is based on joint work with S. Durst and J. Licata.

Tue Jan 28 ∙ Seminarraum C ∙ 13:00

The aim of the talk is to introduce a natural functional on the space of holomorphic sections of the twistor spaces of a class of hyper-Kähler manifolds, including the Hitchin moduli spaces.

In the first part, we recall basic concepts of hyper-Kähler manifolds and their twistor spaces. For hyper-Kähler spaces admitting
a circle action (and some additional conditions) we explain the construction (by Haydys and Hitchin) of a holomorphic line bundle
with a meromorphic connection on its twistor space. The energy of a section is given by the residue of the pull-back of the
meromorphic connection. It is shown that the energy on the spaces of twistor lines coincides with the moment map for the circle
action. In the second part of the talk we consider the particular case of the Higgs bundle moduli space. Its twistor space can
be identified with the Deligne-Hitchin moduli space of λ-connections. We prove a natural formula for the energy of a
section of the Deligne-Hitchin moduli spaces in terms of a Serre-type pairing. Finally, we compute the energy for a class of
examples. The talk is based on joint work with F. Beck and M. Räser, and on joint work with L. Heller.

Upcoming talks:

Tue July 16 ∙ Seminarraum C ∙ 13:00

Convex integration is one of the most important tools in the construction of solutions of partial differential relations. It was first introduced by J. Nash in his work on C

Tue July 2 ∙ Seminarraum C ∙ 13:00

Several new results about d-dimensional lattice polytopes contained in the hypercube [0,k]^d will be presented, along with a number of related open questions. The first set of results is about the largest possible diameter for the edge-graph of these polytopes as a function of d and k, where the special case of primitive zonotopes plays an important role. The second set of results is on the number of primitive zonotopes, that pop up from a number of seemingly unrelated fields. The third set of results deal with a graph structure that can be put on the set of all lattice polytopes. The connectedness properties of this structure will be presented and discussed.

Tue June 18 ∙ Seminarraum C ∙ 13:00

The theory of the moduli space of complex structures on a surface, the classical TeichmÃƒÂ¼ller space, is a rich interplay between Riemannian, symplectic and complex geometry. Its algebraic generalisation to other character varieties, especially Hitchin components, admits much less geometric insights for the moment. We present a new geometric structure on surfaces, mixing concepts from symplectic and algebraic geometry (punctual Hilbert schemes), which generalizes the complex structure and whose moduli space is conjecturally Hitchin's component. Joint work with Vladimir Fock.

Tue May 7 ∙ Seminarraum C ∙ 13:00

Masur-Veech volumes govern the Teichmuller geometry of moduli space, with several parallels and many differences to Weil- Petersson geometry. We provide recursive formulas for Masur- Veech volumes that allow efficient computation and precise large genus estimates. Counting torus covers and intersection theory provides two different recursions for these volumes.

Tue May 21 ∙ Seminarraum C ∙ 13:00

Hamiltonian homeomorphisms are those homeomorphisms of a symplectic manifold which can be written as uniform limits of Hamiltonian diffeomorphisms. One difficulty in studying Hamiltonian homeomorphisms (particularly in dimensions greater than two) has been that we possess fewer tools for studying them. For example, (filtered) Floer homology, which has been a very effective tool for studying Hamiltonian diffeomorphisms, is not well-defined for homeomorphisms. We will show in this talk that using barcodes and persistence homology one can indirectly define (filtered) Floer homology for Hamiltonian homeomorphisms. This talk is based on joint projects with Buhovsky-HumiliÃƒÂ©re and Le Roux-Viterbo.

Tue Apr 16 ∙ Seminarraum C ∙ 13:00

Joint work with Y. Groman. We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles, without any restrictions on the magnetic form.

Tue Jan 29 ∙ Seminarraum C ∙ 13:00

In this talk I will describe work in progress on the construction of explicit geometric models for moduli spaces of stable parabolic Higgs bundles in genus 0 and rank 2, and explain how the nature of such construction elucidates the wall-crossing behavior of the moduli spaces in question under variations of parabolic weights. This work is motivated by some results related to the cohomology of natural Kahler forms on the moduli spaces, which I will briefly describe as well.

Tue Jan 15 ∙ Seminarraum C ∙ 13:00

For a semisimple real Lie group G, a non-abelian Hodge type correspondence provides a 1-1 correspondence between fundamental group representations of a surface with boundary into G, and G-Higgs bundles equipped with a parabolic structure over the points of a divisor D of finitely many distinct points on the Riemann surface X. In this talk, we discuss topological properties of connected components of the parabolic G-Higgs bundle moduli space for the cases when G is split real and when G is of Hermitian type. A correspondence of parabolic Higgs bundles over X to orbifold Higgs bundles over a finite Galois covering Y of X ramified along the divisor D, opens the way to new topological invartiants for further investigation of these components. Joint work with Hao Sun and Lutian Zhao.

Tue Dec 11 ∙ Seminarraum C ∙ 13:00

Some of the dynamical properties of the geodesic flow φ of a closed Riemannian manifold M are independent of the metric on M. For "most" manifolds M, one can detect a strong instability of the dynamics of φ, notably through the positivity of the exponential growth rate of the number of closed orbits or through the positivity of the topological entropy of φ. These classical results considerably generalize to the world of symplectic and contact geometry. I will give an overview on the relevant results in this direction. Also I will present a joint construction with Alves of contact structures on spheres such that all its Reeb flows have positive topological entropy. If time permits I explain how the exponential growth of Rabinowitz Floer homology, besides providing new interesting examples of contact manifolds that only carry Reeb flows with positive entropy, has applications to the dynamics of geodesic flows.

Tue Dec 4 ∙ Seminarraum C ∙ 13:00

I will present effective methods to compute equivariant harmonic maps. The main setting will be equivariant maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive the convergence of the discrete heat flow to the energy minimizer, with explicit convergence rate. We also examine center of mass methods, after showing a generalized mean value characterization for harmonic maps. We feature a concrete illustration of these methods with Harmony, a computer software with a graphical user interface that we developed in C++, whose main functionality is to numerically compute and display equivariant harmonic maps.

Tue Nov 6 ∙ Seminarraum C ∙ 13:00

In the last decade CAT(0) cube complexes have gotten a lot of attention in geometric group theory and related areas. This comes from the fact that many groups are "cubulated", i.e. act proper cocompactly on a CAT(0) cube complex (e.g. RAAGs, RACGs, hyperbolic 3-manifold groups), and that such an action (called "cubulation") allows to derive interesting algebraic properties of the group; a prominent example of that is the "recent" proof of the virtual Haken conjecture. We want to point out that in general a group admits many non-isomorphic cubulations; which is why we want to consider the following rigidity question of cubulations (called "marked length spectrum rigidity"): Let G be a group that acts cocompactly on two irreducible CAT(0) cube complexes X,Y. Assume that the translation lengths for all g in G are the same for the action on X and on Y. Are X and Y then G-equivariantly isomorphic? In this talk we show that with the right choice of metric and under some natural assumptions (e.g. no free faces) this holds if adding a small assumption on X or Y. If X and Y show low-dimensional behaviour (e.g. square-complexes or particular cubulations of surface groups), then the statement is true in full generality. To proof this, we construct a notion of cross ratio on particular boundaries of the cube complexes - generalising a classical object on boundaries of negatively curved spaces - and show that the boundary equipped with the cross ratio determines the isomorphism type of the cube complex. This is joint work with E. Fioravanti and M. Incerti-Medici.

Tue Oct 23 ∙ Seminarraum C ∙ 13:00

Aubry-Mather theory studies distinguished sets which are invariant under the flow associated to a convex Hamiltonian H on a cotangent bundle. These sets arise as the support of so-called Mather measures. Symplectic topology studies properties imposed on the flow by the (symplectic) topology of the cotangent bundle. It is interesting to understand how these theories can play together. In this talk I will show how different techniques from symplectic topology give rise to incarnations of Mather's α-function and that its relation to invariant measures continues to hold: Mather measures exist. I will discuss applications to Hamiltonian system on closed symplectic manifolds, R^2n and twisted cotangent bundles.