SFB Retreat 2021

Maria in der Aue, 30 August - 3 September 2021

30 August
14:00 - 15:00 Peter Albers und Christian Lang
Visualizing flows on the 3-torus with WebGL

Abstract: We present a web tool (developed with WebGL) for visualizing flows of vector fields on the 3-torus. We demonstrate various examples of magnetic geodesic flows on the 2-torus where T3 is the phase space. Finally, we show first steps of the implementation of finite time Lyapunov exponents (FTLE).
31 August
09:00 - 10:00 Erik de Amorim
The Maxwell-Bopp-Lande-Thomas-Podolsky-Einstein system for a static point source

Abstract: We discuss a dynamical systems approach to prove the existence of a Lorentzian spacetime that is the solution of the Einstein field equations coupled with an electric field obeying the equations of Maxwell-Bopp-Lande-Thomas-Podolsky for a static point charge. This is a law of electromagnetism proposed in the 1940's that generalizes the usual Maxwell equations and, in the Minkowski space of Special Relativity, fixes the problem of the infinite self-energy of the field generated by a particle. We show that it also fixes the same problem in our general relativistic context, which will permit future investigations such as the joint problem for the time evolution of point charges and their electromagnetic fields. All of the relevant terms will be explained assuming no knowledge of Physics from the audience.
11:00 - 12:00 Michela Egidi
Partner orbits for the geodesic flow on manifolds with negative curvature

Abstract: We study pairs of closed orbits of the geodesic flow on compact manifolds with variable negative curvature, showing that a long enough closed orbit with a self-intersection is "shadowed" by a closed orbits without self-intersections. These orbits stay close at all times and have small action difference depending on the angle of the intersection. Such pairs of orbits have been first considered by Sieber and Richter and their study is motivated by the idea that periodic orbits bunching may lead to universal spectral fluctuations for chaotic quantum systems.
1 September
09:00 - 10:00 Tilman Becker
Geodesible vector fields

Abstract: A non-vanishing vector field on a manifold is called geodesible, if there exists a Riemannian metric turning all of its integral curves into geodesics. After discussing several classes of examples, I will give a short introduction into open book decompositions, and explain how to use them to prove a theorem originally stated by Gluck (and first proven by Hajduk and Walczak): Every closed oriented odd-dimensional manifold admits a geodesible vector field. Further, I will show that every finite number of closed curves can be realised as orbits of such a vector field. If time allows, I will also discuss some connections to contact geometry.
11:00 - 12:00 Rima Chatterjee
Knots and links in overtwisted contact manifolds

Abstract: Knot theory associated to overtwisted contact manifolds is less explored. It turns out that the knots in overtwisted manifolds behave differently compared to tight manifolds. There are two types of knots in an overtwisted manifold- exceptional/non-loose and non-exceptional/loose. In this talk I'll give an overview of these knots and discuss how they differ from the knots in tight manifolds. Next, I'll discuss my work on classifying loose links, define an invariant named ''support genus'' and show this invaraint vanishes for this class of knots/links. If time permits, I'll also discuss the satellite operation on these knots.
14:00 - 15:00 Pengfei Huang
An investigation of Dolbeault moduli spaces via Simpson-Mochizuki correspondence

Abstract: The Simpson-Mochizuki correspondence, is a generalization from the Corlette-Simpson correspondence between Higgs bundles and flat bundles to a correspondence between Higgs bundles and λ-flat bundles (any λ ≠ 0). This is mainly due to Simpson's work on the Kobayashi-Hitchin-type theorem for Higgs bundles, and Mochizuki's work for λ-flat bundles. In this talk, firstly I will give a quick review of nonabelian Hodge theory and this correspondence as the background setting. Then I will present a generalization of Mochizuki's result to the case when the base being more general. Afterwards, I will talk about how to apply the correspondence to construct dynamical systems with two parameters on Dolbeault moduli space (moduli space of Higgs bundles). Based on a joint work with Dr. Zhi Hu.
2 September
9:00 - 10:00 Abror Pirnapasov
A Denvir-Mackay Theorem for Reeb flows

Abstract: Denvir and Mackay showed that if a Riemannian metric on the two-dimensional torus has a contractible closed geodesic, then the topological entropy of its geodesic flow is positive. In this talk, we give a version of the theorem for Reeb flows. This joint work with Marcelo Alves, Umberto Hryniewicz and Pedro Salomão.
3 September
09:00 - 10:00 George Marinescu
Random polynomials and random holomorphic sections

Abstract: We will focus on the interplay between complex geometry and probability theory. More precisely, we will show how to use complex geometry and geometric analysis techniques in order to study several problems concerning local and global statistical properties of zeros of random polynomials/sections.