25.10.24	Eva Miranda (UPC Barcelona & Köln) From Alan Turing to higher categories via symplectic and contact geometry Abstract: In 1936, Alan Turing laid the foundation for modern computation by proving the undecidability of the halting problem. In this talk, we will explore how concepts from symplectic and contact geometry can be applied to simulate Turing-complete dynamical systems, particularly through the study of particle movement in fluids. We begin by examining two geometric models that provide solutions to classical equations in fluid dynamics. On one side, we explore Beltrami fields - stationary solutions of the Euler equations - and their link to Reeb vector fields and contact structures [CMPP]. On the other, we investigate stationary solutions of the Navier-Stokes equations and their corresponding cosymplectic geometry [DGMP]. In both cases, cobordisms defined by the Reeb flow arise naturally, offering a framework to associate these fluid models with Turing-complete systems, which we refer to as "Fluid Computers". In the second part of the talk, we will discuss how these geometric constructions can be "glued" using 2-categories to create more advanced computational systems. Drawing on ideas from Topological Quantum Field Theory (TQFT), we will explore the quantization of Turing-complete dynamical systems, including those related to fluid dynamics. This leads to the introduction of Topological Kleene Field Theory (TKFT), which combines computation and field theory, using basic building blocks - called "flubits" - derived from 3D Euler/Navier-Stokes flows. This talk is based on joint work with Ángel González-Prieto and Daniel Peralta-Salas. [CMPP] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas, Constructing Turing complete Euler flows in dimension 3, Proc. Natl. Acad. Sci. USA, 118 (2021) [DGMP] S. Dyhr, Á. González-Prieto, E. Miranda, D. Peralta-Salas, Harmonic vector fields and cosymplectic manifolds, preliminary notes, 2024 [GMP] Á. González-Prieto, E. Miranda, D. Peralta-Salas, Computability and Topological Kleene Field Theories, preprint, 2024 This talk is based on joint work with Ángel González-Prieto and Daniel Peralta-Salas.
8.11.24	BACH Seminar in Bochum
15.11.24	Eva Miranda (UPC Barcelona & Köln) Desingularizing singular symplectic structures Abstract: The exploration of symplectic structures on manifolds with boundaries has naturally led to the identification of a "simple class" of Poisson manifolds. These manifolds are symplectic away from a critical hypersurface, but degenerate along this hypersurface. In the literature, they are referred to as b-symplectic or log-symplectic manifolds. They arise in the context of the space of geodesics of the Lorentz plane and serve as a natural phase space for problems in celestial mechanics such as the restricted 3-body problem. Geometrically, these manifolds can be described as open symplectic manifolds endowed with a cosymplectic structure on the open ends. The technique of "deblogging" or desingularization associates a family of symplectic structures to singular symplectic structures with even exponent (known as b2k-symplectic structures), and a family of folded symplectic structures for odd exponent (b2k+1-symplectic structures). This method has good convergence properties and generalizes to its odd-dimensional counterpart, contact geometry. In this way, the desingularization technique puts under the same umbrella various geometries, such as symplectic, folded-symplectic, contact, and Poisson geometry.
22. & 23.11.24	Geometric Dynamics Days in Köln
6.12.24	Joint Seminar on Complex Algebraic Geometry and Complex Analysis in Essen
13.12.24	Marc Kegel (HU Berlin) The search for exotic knot traces Abstract: Every knot leaves a trace in the 4-dimensional world. The trace of a knot is the smooth 4-manifold obtained by attaching a 2-handle to the 4-ball along a knot in the 3-sphere. We will introduce the relevant notions and present a strategy to disprove the smooth 4-dimensional Poincaré conjecture by finding knot traces with certain exotic properties. In the second part of the talk, we will discuss different methods to search for such exotic knot traces.
20.12.24	Marco Zambon (KU Leuven) Coisotropic branes in symplectic geometry Abstract: A brane in a symplectic manifold M is a coisotropic submanifold N together with a closed 2-form which is compatible in a specific sense. Despite being defined in terms of symplectic geometry, branes involve complex geometry in an essential way. We will make some remarks on the case N=M (space-filling branes), i.e., the case in which M carries a holomorphic symplectic form. For branes supported on lower-dimensional submanifolds, we then address the question of infinitesimal deformation of branes, and whether all coisotropic submanifolds nearby a given one are themselves branes. This talk is based on ongoing work with Charlotte Kirchhoff-Lukat.
24.1.25	Souheib Allout (RU Bochum) Compact Lorentz manifolds without closed geodesics Abstract: A classical theorem in Riemannian geometry asserts that every compact Riemannian manifold admits at least one closed geodesic. One then asks whether the same holds in the non-definite case. In this talk, we will show how to construct compact Lorentz manifolds that do not admit closed geodesics. This is a joint work with A. Zeghib and A. Belkacem.

H. Geiges, 7.10.24