Winter Retreat

CRC/TRR 191

March 27–31, 2023

Ludwig-Windthorst-Haus

Gerhard-Kues-Straße 16

49808 Lingen-Holthausen

Germany

Practical information

Arrival:

March, 27 – before dinner

Departure:

March, 31 – after lunch

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Calender

Speakers

Pierre-Alexandre Arlove

Time functions and Lorentzian distances on the contactomorphism group

As recently pointed out by Abbondandolo, Benedetti and Polterovich, both contactomorphisms groups and Lorentzian manifolds admit natural partial binary relations coming from cone structure. In the context of Lorentzian manifolds these relations have been intensively studied since they have important physical interpratation. I will present some notions of global Lorentzian geometry that can be borrowed to study the binary relation on the group of contactomorphisms. In particular I will focus on the notions of time functions and Lorentzian distances and state some of their properties in the context of contactomorphisms groups.

Luca Asselle

Magnetic geometry and applications

To a magnetic system on a compact manifold one can naturally associate a magnetic curvature tensor, which encodes both the geometric properties coming from the Riemannian structure and the perturbation due to the magnetic interaction, and which is expected to carry valuable information about the magnetic dynamics. In this talk I will give a quick overview on recent progresses on the topic and discuss possible future research directions, including multiplicity/stability of periodic orbits, (non-)minimality, applications to Reeb dynamics, marked length spectrum rigidity, spectral characterization of Zoll systems, as well as certain open questions in celestial mechanics.

Franziska Beckschulte

Vertical convexity

We discuss a boundary condition for a contact form on a compact manifold with boundary and indicate, how this can be used to relate the diffeomorphism typ to the systol of the contact form.

Maria Bertozzi

Quiver varieties and symplextic reduction

We will construct the moduli spaces of quiver representation using the symplectic reduction, see some examples and possible applications.

Johanna Bimmermann

Hofer-Zehnder capacity for the (co-)tangent bundle of the real & complex projective space

We will determine the Hofer-Zehnder capacity of disc subbundles of the (co-)tangent bundle of RP^n & CP^n. For the lower bound we will study suitable geodesic billiards, while the upper bound is obtained using that DRP^n resp. DCP^n compactifies (using a Lerman cut at the boundary) to CP^n resp. CP^n x CP^n and then looking at holomorphic spheres.

Oliver Brammen

The shifted wave equation on non flat harmonic manifolds

I will introduce the Abel transform and its dual on harmony manifolds and present an application involving the shifted wave equation on a non compact simply connected harmonic manifold with mean curvature of the horospheres 2\rho>0 be giving an explicit representation of the solution.

Xian Dai

Riemannian and Finsler metrics in higher Teichmüller spaces

In this talk, we will introduce Riemannian metrics, called pressure metrics, and Finsler metrics, called Thurston's asymmetric metrics in higher Teichmüller spaces. Both are generalization of classical metrics on Teichmüller spaces and are constructed using Thermodynamic formalism. The pressure metrics were first studied in works of Bridgeman, Canary, Labourie and Sambarino. The Thurston's asymmetric metrics were recently introduced in joint work with Leon Carvajales, Beatrice Pozzetti and Anna Wienhard.

James Farre

The mapping class group action on Euler number zero representations

We consider the action of the mapping class group of a closed orientable surface on the component of its PSL_2R character variety containing the identity. We show that there is an abundance of partially hyperbolic dynamics of the Torelli group action on the singular locus, consisting of diagonal representations. Using these dynamics, we produce a neighborhood of the singular locus where every representation has a simple closed curve which maps to an elliptic element. This answers a question of Bowdtich in a neighborhood of the singular locus of Euler number 0 representations. This is based on joint work in progress with Martin Bobb and Peter Smillie.

Zachary Greenberg

Cluster Algebras for Symplectic Geometers

We will introduce cluster algebras to symplectic geometers and explain connections to moduli spaces of hyperbolic structures and representations of surface groups into SP_2n(R).

Pengfei Huang

Quiver representations and Betti moduli spaces

The celebrated nonabelian Hodge correspondence on a noncompact curve, for any reductive group G, establishes a one-to-one correspondence among filtered (ir)regular G-Higgs bundles, filtered (ir)regular meromorphic G-connections, and filtered (Stokes) G-local systems. However, the absence of a moduli space for filtered (Stokes) G-local systems renders the theory incomplete. The talk aims to demonstrate an application of quiver representation theory in constructing the moduli spaces of filtered (Stokes) G-local systems. Based on a joint work in progress with Hao Sun.

Caner Nazaroglu

Mock Modularity and its Generalizations within Stringy Geometry

I will overview the theory of mock modular forms (of arbitrary depth) through its applications to Gromov-Witten invariants, Vafa-Witten invariants, and elliptic genera while placing these objects in the context of string theoretical dualities. I will also briefly mention recent developments and future prospects for generalizations of mock modular forms.

Jacobus Sander de Pooter

Uniquness of measures of maximal entropy for some geometric flows

The variational principle states for continuous transformations of compact Hausdorff spaces states, that the topological entropy is equal to the supremum of all measures theoretic entropies. Here the supremum runs over all Radon measures invariant under the transformation. One can wonder what sufficient conditions are for existence and unicity of a 'maximizing' invariant measure, and what it can tell you about the original transformation.
In this talk we will recall some results in this direction for geodesic flows with certain properties, and wonder how to extend this to the Reeb flow setting. This is part of ongoing research for my thesis.

Murat Saglam

On the existence of integrable Reeb flows

This talk is devoted to studying a notion of Bott integrability for Reeb flows on contact 3-manifolds.

Giorgia Testolina

Bifurcations of balanced configurations for the n-body problem in R^3

We study the existence of bifurcation points along trivial branches of planar S-balanced configurations and give a lower bound on the number of bifurcation points. This follows from an abstract bifurcation result which shows that, for a continuous family of C^2 functionals on a finite dimensional manifold, the non vanishing of the spectral flow for the Hessians along a trivial branch of critical points implies the existence of bifurcation points.

Simon Vialaret

Systolic inequalities for S^1-invariant contact forms

In contact geometry, a systolic inequality aims to give a uniform bound on the period of the shortest closed Reeb orbit for contact forms with fixed volume. It is now known that no such inequality does hold for general contact forms on a fixed contact manifold. We investigate whether and when such an inequality holds on the class of invariant contact forms on a S^1-principal bundle. As a partial result, we will state a systolic inequality for invariant forms on S^1-principal bundles over the 2-sphere.

Michael Vogel

Systolic inequality around Besse contact forms

Alberto Abbondandolo and Gabriele Benedetti proved local systolic inequality around Zoll contact forms in higher dimensions using a normal form theorem for contact forms close to a fixed Zoll contact form. I will present a generalization of this normal form theorem assuming Besse instead of Zoll and will give some applications including a similar local systolic inequality statement.

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