Geometry Seminar main page

Tue May 3rd ∙ SR B ∙ 1 pm

A well known conjecture stated in 1984 by Burns and Katok asserts that the marked length spectrum of a closed and negatively curved Riemannian manifold determines the metric up to isometry. In 2018 Guillarmou and Lefeuvre confirmed the conjecture locally, i.e. for sufficiently small open neighborhoods of negatively curved metrics.

In this talk I will report on a different approach which is based on the geodesic stretch and methods from thermodynamical formalism. This yields a different proof of the local rigidity and provides new stability estimate for the marked length spectrum. We also obtain metrics on the space of negatively curved isometry classes which generalizes well known metrics on Teichmüller space.

The talk is based on joint work with Colin Guillarmou and Thibault Lefeuvre.

Tue May 17th ∙ SR B ∙ 1 pm

In this talk we will construct thin surface groups in lattices of split real Lie groups. A thin subgroup of a lattice is an infinite index subgroup that is Zariski-dense. Although thin subgroups are not themselves lattices, they share many properties with them and have been an active field of research in the last decade. The construction relies on special representations of surface groups in split real Lie groups, so called Hitchin representations. Those are faithful representations that form a connected component of the character variety. Our goal is to prove the existence of Zariski-dense Hitchin representations that have image in a lattice. To do so, we have to investigate the arithmetic properties of lattices.

Tue June 7th ∙ SR B ∙ 1 pm

A tropical analogue of line bundles on metric graphs is, by now, well-understood and reflects the various compactifications of the Jacobian over semistable degenerations of compact Riemann surfaces. The goal of this talk is to propose an up-to-now still missing analogue of vector bundles of higher rank on metric graphs. After defining these objects I will, in particular, talk about a tropical analogue of the Weil-Riemann-Roch-Theorem and of the Narasimhan-Seshadri correspondence. Then I will outline a tropicalization procedure that lets us connect this a priori only combinatorial theory with the classical story. As it turns out, this will work best in the case of the Tate curve. Given time, I might indulge in some speculations concerning a new approach to degenerations of vector bundles using methods from logarithmic geometry that incorporates and expands on both the algebro-geometric and the tropical story. The non-speculative aspects of this talk are based on joint work with Margarida Melo, Sam Molcho, and Filippo Viviani and joint work in progress with Andreas Gross and Dmitry Zakharov.

Tue June 14th ∙ SR B ∙ 1 pm

An important problem in symplectic topology is to determine when symplectic embeddings exist, and more generally to classify the symplectic embeddings between two given domains. Modern work on this topic began with the Gromov nonsqueezing theorem, which asserts that the ball symplectically embeds into the cylinder if and only if the radius of the ball is larger than that of the cylinder. Many questions about symplectic embeddings remain open, even for simple examples such as ellipsoids and polydisks. To obtain nontrivial obstructions to the existence of symplectic embeddings, one often uses various symplectic capacities. We shall discuss some questions about capacities, in particular the equality of two type of symplectic capacities. This is joint work with V.Ramos.

Thu July 7th ∙ SR 2 ∙ 4 pm

A source of richness in Teichmüller theory is that Teichmüller spaces have descriptions both in terms of group representations and in terms of hyperbolic structures and complex structures. A program in higher-rank Teichmüller theory is to understand to what extent there are analogous geometric interpretations of Hitchin components. In this talk, we will give a natural description of the SL(3,R) Hitchin component in terms of higher complex structures as first described by Fock and Thomas. Along the way, we will describe higher complex structures in terms of jets and discuss intrinsic structural features of Fock-Thomas spaces.

Tue July 26th ∙ SR B ∙ 1 pm

The theory of generalized Lusztig's positivity (or $\Theta$-positivity) developed by O. Guichard and A. Wienhard generalizes the total positivity for split real Lie groups and the maximality for Hermitian Lie groups to a larger class of simple Lie groups (e.g. $\SO(p,q)$, $p \neq q$, some exceptional Lie groups). Lie groups $G$ with a positive structure are of particular interest in the higher Techmüller theory because the representation space $\Hom(\pi_1(S),G)/G$, where $S$ is an orientable surface of finite type, admits connected components that consist entirely of discrete and faithful representations (so-called higher rank Teichmüller spaces). In my talk, I explain how the spaces of positive framed representations of the fundamental group of a punctured surface into a Lie group with a positive structure can be parametrized, and how we can describe the topology of this spaces using this parametrization. This is a joint work with O. Guichard and A. Wienhard.

Thu September 23rd ∙ Hörsaal (or video life stream) ∙ 10:30 am (Note the unusual time!)

We will describe the topological behaviour of the conjugacy class action of the mapping class group of an orientable infinite-type surface.

Mon October 11th ∙ SR 3 ∙ 1 pm

Let H^n be the hyperbolic n-space and D be a geometrically finite discrete subgroup in Isom_+(H^n) with cusps. In the joint work with Wenyu Pan, we establish exponential mixing of the geodesic flow over the unit tangent bundle T1(D \ H^n). Previously, such results were proved by Stoyanov for convex cocompact discrete subgroups and Mohammadi-Oh and Edwards-Oh for D with large critical exponent. We obtain our result by constructing a nice coding for the geodesic flow and then prove a Dolgopyat-like spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding, which is partly inspired by the works of Lai-Sang Young and Burns-Masur-Matheus-Wilkinson. I will also discuss the application of obtaining a resonance-free region for the resolvent on D\H^n.

Tue October 26th ∙ SR B ∙ 1 pm

The Selberg trace formula provides a link between the length spectrum and the Laplacian spectrum of a hyperbolic surface. I will speak about a joint project with Maxime Fortier Bourque in which we are using this formula to probe extremal problems in hyperbolic geometry. These are questions of the form: what is the hyperbolic surface of a given genus with the largest kissing number or the largest spectral gap? Concretely, I will explain the general principle of our method, which is inspired by ideas from the world of Euclidean sphere packings. Moreover, I will explain why the Klein quartic, the most symmetric Riemann surface in genus 3, solves one of our extremal problems.

Tue November 9th ∙ SR B ∙ 1 pm

The models we shall discuss are motions of a particle in the plane moving under the influence of a conservative force field which in addition reflect elastically against certain smooth reflection "wall". The dynamics of such a system depends on the force field and the shape of the reflection wall. While one could believe that the dynamics should generally be complicated, some of these systems are actually integrable. In this talk we shall explain how conformal correspondence of natural mechanical sytems extends to correspondence between integrable mechanical billiards. This provides a link between some apparently different integrable mechanical billiards, and also allows us to identify certain new integrable mechanical billiards defined with the Kepler and the two-center problems. The talk is based on joint work with Airi Takeuchi from Karlsruhe Institute of Technology.

Tue November 23rd ∙ SR B ∙ 1 pm

We will start with recalling property (T) and its relation to the group Laplacian Δ. This leads to the equivalent formulation of property (T) as the positivity of Δ² - λ Δ in the group C*-algebra. By Positivstellensatz for star-algebras this positivity is implied (an is equivalent by a result of Ozawa) by a single sum of squares decomposition for Δ² - λ Δ. I will show how to exploit the group structure of SL(n,Z) to endow Δ² with more structure and how to use it to extend a single solution to the sum of squares problem for SL(3,Z) to all n > 3. If time permits I'll talk about the extensions of this result to Sp(2n, Z) and groups graded by root systems.

Tue December 7th ∙ SR B ∙ 1 pm

In this talk I shall discuss the persistence of Lagrangian periodic tori for symplectic twist maps of the d-dimensional annulus and deduce some rigidity results for integrable twist maps. This is a joint work with Marie-Claude Arnaud and Jessica Massetti.

Tue January 18th ∙ SR B ∙ 1 pm

We formulate a moduli problem encoding the internal states of a neural network, and use linear algebra, geometric invariant theory and symplectic reduction to describe the resulting moduli spaces. This is joint work in progress with M. Armenta, T. Bruestle and S. Hassoun.

Tue February 15th ∙ SR B ∙ 1 pm

Wang tilings is an important class of tilings of the plane where each tile (called Wang tile) is a unit square with a color given to each edge. Tilings by Wang tiles are valid when the colors of the common edge of every pair of adjacent tiles within the tiling are the same. In 1961, Wang conjectured that when a finite set of Wang tiles admit a valid tiling of the plane, it must admit one which is periodic. This statement was proved false few years later by Berger, a student of Wang, who also provided a set of 20426 (lowered down later to 104) Wang tiles that admit valid tilings the plane none of them being periodic. Such set of Wang tiles are called *aperiodic*. In the 1970s, Penrose found an aperiodic set of two rhombus with matching rules, and their apparition in aluminium-manganese alloys was first noticed by crystallographers in the 1980s (leading to a Nobel Prize in Chemistry in 2011).

A question which was open for a long time was to find the smallest set of aperiodic Wang tiles (Penrose rhombus are not Wang tiles). Before 2015, the smallest aperiodic sets of Wang tiles known were Kari's 14 tiles and Culik's 13 tiles both discovered in 1996. Then in 2015, Jeandel and Rao closed the question and proved the existence of an aperiodic set of 11 Wang tiles and that no set of Wang tiles of cardinality ≤ 10 is aperiodic. Their computation of all candidates of size up to 11 "took approximatively one year on several hundred cores".

In this talk, we will discuss the structure of Jeandel-Rao aperiodic tilings admitted by the 11 Wang tiles discovered by Jeandel and Rao with in-hand wooden laser-cut puzzle pieces brought for the occasion. The most important result being the existence of a polygonal partition of the two-dimensional torus and a $\mathbb{Z}^2$-action on the torus allowing to easily construct valid Jeandel-Rao tilings. This can also be seen as the projection of points from a higher dimensional lattice as it was already noticed by de Bruijn in 1981 in the context of Penrose tilings. Time will not allow to discuss other results Jeandel-Rao Wang tilings related to the higher dimensional extensions of Sturmian sequences, substitutive structure, Markov partition for $\mathbb{Z}^2$-rotations, Rauzy induction, but the curious reader can find them in the three following articles published in 2021: https://doi.org/10.1007/s00454-019-00153-3, https://doi.org/10.5802/ahl.73, https://doi.org/10.3934/jmd.2021017

Thu 20. July ∙ Online (Zoom) ∙ 15:00 (Note the unusual time!)

Ordinary differential equations in one complex variable are a classical source of interesting geometric structures, leading to the uniformization theory of Riemann surfaces, discrete groups in Lie groups, and much more. I will describe a family of such differential equations, of hypergeometric type, that yields discrete subgroups in Sp(4,R) which are Anosov, and have a number of additional properties related to uniformization. Part of the work is joint with Charles Fougeron.

Thu 15. July ∙ Hörsaal ∙ 11:00 (Note the unusual time!)

The Borsuk-Ulam theorem, that any Z/2-equivariant map from S^n to R^n must hit zero, is renowned for its wide range of appealing applications. We begin with a brief overview of some applications, focusing on those which are lesser-known. We discuss a classical generalization for products of spheres and its usefulness in attacking some new questions about embeddability of manifolds and simplicial complexes; for example: 1. If X embeds in R^m, can any such embedding be isotoped to its mirror image? 2. If X does not embed in R^m and Y does not embed in R^n, can this nonembeddability be "simultaneously witnessed" for any map X x Y to R^{m+n}? We will formalize and discuss Borsuk-Ulam type results for these questions.

Tue 13. July ∙ Zoom ∙ 13:00 (Note the usual time!)

On closed surfaces of positive genus, through classical work of Dehn, simple closed curves can be described using intersection numbers. Now what if you want to describe curves with self-intersections in a similar way? This talk will be on joint work with Binbin Xu about this question, and where we end up constructing and studying so-called k-equivalent curves. These are distinct curves that intersect all curves with k self-intersections the same number of times.

Tue 13. July ∙ Hörsaal, also online ∙ 11:00 (Note the unusual time!)

Sol is one of the 3-dimensional Thurston geometries. In this talk I will give an exact characterization of the cut locus of Sol, in terms of the arithmetic geometric mean of Gauss, and prove, in particular that the metric spheres in Sol are topological spheres, smooth away from at most 4 singular arcs. This is joint work with Matei Coiculescu.

Tue 6. July ∙ Zoom ∙ 13:00

We will explain how to define a notion of joint Ruelle resonant spectrum for abelian Anosov actions and how one can deduce results on SRB type measures and periodic tori. This is joint work with Bonthonneau, Hilgert and Weich.

Tue May 25 ∙ Zoom ∙ 13:00

Anti de Sitter (AdS) space is the Lorentzian cousin of the hyperbolic 3-space: it is a symmetric space with constant curvature -1. In this talk, we will consider surface group representations in the isometry group of AdS space, called quasi-Fuchsian representations. There is 2 classical objects associated to those representations and one of the goal is to understand their interplay: the limit set which is a quasi-circle in the boundary at infinity of AdS space and a convex set inside AdS which is preserved by the group action and bounded by two pleated surfaces. I will conclude the talk by a report on a work in common with Jean-marc Schlenker where we extend the "Teichmüller" situation to the "universal Teichmüller".

Tue Feb 23 ∙ Zoom ∙ 13:00

We describe dynamical systems arising from the classification of locally homogeneous geometric structures and flat connections. Their classification mimics that of Riemann surfaces by the Riemann moduli space, the quotient of Teichmueller space by the mapping class group. However, unlike Riemann surfaces, the mapping class group actions on character varieties may be chaotic, leading to dynamic complexity. A striking elementary example is Baues's theorem that the deformation space of complete affine structures on the 2-torus is the plane with the usual linear action of GL(2,Z) (the mapping class group of the torus). We discuss specific examples of these dynamics for some simple surfaces, where the relative character varieties appear as cubic surfaces in 3-space. Complicated dynamics seems to accompany complicated topology, and we interpret them in terms of hyperbolic structures on surfaces, with or without singularities.

Tue Feb 16 ∙ Zoom ∙ 13:00

An electron hopping along a chain of atoms is a basic system in physics. When the sequence of atoms is determined by the shape of a flat surface, the motion of the electron becomes related to the shape of a pleated hyperbolic surface. This relationship is mediated by a motley crew of mathematical objects, including cutting sequences, difference equations, flat SL(2,C) bundles, and a weak version of the Anosov property. I'll show how they fit together into an unlikely bridge from geometry to physics.

Tue Dec 08 ∙ Zoom ∙ 13:00

The class of moduli spaces appearing in nonabelian Hodge theory has been significantly enriched over the past 20 years or so, by considering solutions of the 2d self-duality equations with more involved behaviour at the boundary. In brief one can relax Simpson's tameness condition, and this leads to stable meromorphic connections/Higgs fields with arbitrary order poles (on parabolic vector bundles). Much of this was motivated by examples occurring in Seiberg-Witten theory, and in the classical integrable systems literature. For example the topological Atiyah-Bott/Goldman symplectic structures were extended to this context by the speaker (Adv. Math 2001), the Corlette/Donaldson correspondence with complex connections was extended by Sabbah (Ann. Inst. Fourier 1999), and the construction of the hyperkahler moduli spaces plus the extension of the Hitchin/Simpson correspondence with Higgs bundles was carried out by Biquard and the author (Compositio 2004). Some more recent work has extended the TQFT (quasi-Hamiltonian) approach to these holomorphic symplectic varieties from the generic case to the general case, and clarified the extra deformation parameters that occur, leading to the notion of ``wild Riemann surface''. In this talk I'll review some of the simplest examples of complex dimension two, and their link to affine Dynkin diagrams (leading to the notion of ``global Weyl group''). Then I'll explain a way to extend this link by attaching a diagram to {\em any} nonabelian Hodge space on the affine line. This is an attempt to organise the vast bestiary of examples of complete hyperkahler manifolds that occur. A key idea is that all the nonabelian Hodge spaces have concrete descriptions as moduli spaces of Stokes local systems (the wild character varieties), generalising the well-known explicit presentations of the (tame) character varieties, coming from a presentation of the fundamental group. This is joint work with D. Yamakawa (Compte Rendus Math. 2020).

Tue Nov 17 ∙ Zoom ∙ 15:00 (Note the different time)

Floer homology and hyperbolic geometry are fundamental tools in the study of three-dimensional topology. Despite this, it remains an outstanding problem to understand whether there is any relationship between them. I will discuss some results in this direction that use as stepping stone the spectral geometry of coexact 1-forms. This is joint work with M. Lipnowski.

Tue May 26 ∙ Zoom ∙ 14:00

Hitchin systems are algebraically completely integrable systems on the moduli spaces of Higgs bundles on a Riemann surface X, that play an important role in the mirror symmetry conjectures on Higgs bundle moduli spaces. The regular fibers are complex projective tori associated to a curtain spectral cover of X. The structure of the singular fibers is much more complicated. We will discuss a curtain class of (singular) fibers of the Hitchin system associated to the complex symplectic group distinguished by the singularities of the spectral cover. These singular Hitchin fibers turn out to be isomorphic to (singular) SL(2,C)-Hitchin fibers and thereby can be described by semi-abelian spectral data.

Tue May 12 ∙ Zoom ∙ 14:00

As discovered by Thurston, hyperbolic 3-manifolds are abundant among all 3-manifolds. In this talk I will describe an explicit construction of a hyperbolic metric for the specific class of random 3-manifolds, a family of 3-manifolds that share the same combinatorial description (introduced by Dunfield and Thurston). The existence of a hyperbolic structure on such objects is not new as it follows from the work of Maher, Hempel and the solution of the Geometrization conjecture by Perelman. The construction we will discuss, instead, only uses tools from the deformation theory of Kleinian groups and will also be explicit enough to allow the computation of geometric invariants such as volume and diameter.

Tue Feb 4 ∙ Seminarraum C ∙ 13:00

Despite spectacular progress in the past 4 decades, the panorama of (smooth) 4-dimensional manifolds is still quite untamed. Often, additional geometric structures on these manifolds have been used to construct key examples. We will review some of this context and explain the existence, on any orientable 4-manifold, of a so-called folded symplectic form: that is a closed 2-form with mild singularities along a separating hypersurface. This result may be viewed as a version for 4-manifolds of the fact that any orientable 2-manifold admits an area form.

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.