Geometry Seminar main page

Next upcoming talk: tba

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.