Marked length spectrum rigidity and the geodesic stretch
In this talk I like to discuss recent progress on the marked length spectrum rigidity problem on closed manifolds of negative curvature. The marked length spectrum is a map which associates to each free homotopy class of a negatively curved manifold the length of the unique closed geodesic in this class. It has been conjectured by A. Katok in a MSRI problem session that this map should determine the metric up to isometry. In 1990 the conjecture has been proved by Otal for surfaces. However, his method cannot be extended to higher dimensions. In higher dimension the conjecture has been obtained in 1998 for isospectral deformations by Croke and Sharafutdinov using a certain Weitzenböck formula on the tangent bundle of a Riemannian manifold (Pestov’s identity). With the help of powerful methods from micro-local analysis Guillarmou and Lefeuvre where recently able to prove the conjecture locally, i.e. isospectral negatively metrics are isometric provided they are sufficiently close (to appear in Annals of Math.). Using the notion of geodesic stretch, which was introduced by Thurston for Teichmüller spaces, and methods from thermodynamical formalism, Guillarmou, Lefeuvre and myself improved this result even further by showing that asymptotic information of the length spectrum is sufficient for proving the local length spectrum rigidity.
A magnetic systolic inequality
In this talk, I will present the systolic inequality for magnetic systems on surfaces. First, following some work with Jungsoo Kang, I will describe how a local version of this inequality is a manifestation of a more general phenomenon in contact and odd-symplectic geometry. Second, I will discuss some work in progress with Luca Asselle to show the global systolic inequality for magnetic systems having an S^1-symmetry.
Zoll magnetic systems on surfaces
In this talk I will report on joint work with Gabriele Benedetti and Christian Lange on the structure of the space of Zoll magnetic systems on surfaces. First, I will show the construction of the first non-trivial examples of Zoll magnetic systems on the two-torus. In the second part of the talk, I will focus on rigidity phenomena and show e.g. that there are no non-trivial magnetic systems on the two-torus which are Zoll for every energy.
Alexander Caviedes Castro
Generalizing the Mukai conjecture
The Mukai conjecture is an inequality involving the second Betti number b2 and the index k of a Fano variety M, the index being the largest integer dividing the first Chern class of the tangent bundle, which is not zero in the Fano case. More precisely it asserts that
b2(k−1)<n+1 (∗),
where n denotes the complex dimension of M. It gives, in particular, an upper bound on the second Betti number, whenever the index is greater than one. The goal of this ongoing project, joint with Milena Pabiniak and Silvia Sabatini, is to see to which extent this inequality holds in the category of compact monotone symplectic manifolds endowed with a Hamiltonian S1-action. These spaces naturally carry almost complex structures which are compatible with the symplectic structure. We first provide a new criterion, in terms of the indices of certain line bundles, that implies (*) requiring only a choice of an almost complex structure (i.e., not necessarily integrable). In particular, this new approach generalizes the methods used so far to prove the Mukai conjecture in certain cases. We apply this to give an alternative proof of the Mukai conjecture for toric Fano varieties and for flag varieties.
Visualization of Flows and Features
In this talk, we examine recent advances in the visual analysis of discretized dynamical systems. Such analysis faces various challenges, ranging from high structural complexity to high dimensionality. We exemplify the utility of visualization research for the sciences and discuss possible directions of future work.
Symbolic dynamics for Reeb flows
I will explain how to construct symbolic dynamics for 3D Reeb flows with non-degenerate periodic orbits when a transversal foliation exists, e.g. via the Hofer-Wysocki-Zehnder holomorphic curve theory. The following dichotomy is a consequence: Either there is a global section or a horse-shoe. This is joint work with Umberto Hryniewicz and Gerhard Knieper. This will be for a general mathematical audience.
Proof and Progress in Polyfold Gromov-Witten Theory
Polyfold theory allows us to study and understand Gromov-Witten moduli spaces via classical diffeo-geometric methods. I will discuss some select exciting developments in polyfold GW-theory, which include proofs of the GW-axioms, of the strong Weinstein conjecture for certain contact manifolds, and of GW-invariants as intersection numbers.
Nonconvexity of momentum images and projective geometry
Let K be a connected, compact Lie group. I will discuss the following interrelated questions. What is the momentum image of an (irreducible) unitary representation of K? In particular, when is it convex and what causes nonvonvexity? What properties of the (unique) complex K-orbits the projective space, in the context of projective geometry, relate to properties of the momentum map? How many points from the complex orbits are necessary to write a given vector as a linear combination? What are the degrees of the generators of the ring of K-invariant polynomials on the representation space?
Shaped contact manifolds
A theorem by Eliashberg and Hofer determines whether a contact manifold with boundary diffeomorphic to the two-sphere is diffeomorphic to the three-ball depending on the existence of periodic reeb orbits. I will present a more general type of manifolds to which this theorem generalises.
Existence of closed geodesics on orbifolds
We discuss old and new results about the existence of closed geodesics on Riemannian orbifolds.
On the spectral characterization of Besse and Zoll Reeb flows
A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they further have the same minimal period. In recent joint projects with Suhr (for geodesic flows on 2-spheres) and Cristofaro Gardiner (for general Reeb flows on 3-manifolds) we showed that the Besse and Zoll properties can be read off from the action spectrum. In higher dimension the situation is still largely unexplored, and I will present a few results in this direction: for convex contact spheres, I will provide a characterization of Besse contact forms in terms of S^1-equivariant spectral invariants; for restricted contact type hypersurfaces of symplectic Euclidean spaces, I will give a sufficient condition for the Besse property via the Ekeland-Hofer capacities. This is joint work with Viktor Ginzburg and Basak Gürel.