9.10.24	Murat Sağlam The topology of graph links
16.10.24	Kein Seminar
23.10.24	Eva Miranda (UPC Barcelona & Köln) Blow-up in Euler and Navier-Stokes equations? Abstract: The Euler and Navier-Stokes equations describe the motion of incompressible fluids, with or without viscosity, through partial differential equations. Despite their widespread practical applications, the theoretical understanding of these equations remains incomplete. In three dimensions, it is still unproven whether smooth solutions always exist or if counterexamples can be found. The phenomenon of non-smoothness in the solutions is often referred to as blow-up or singularities. This challenge is known as the Navier-Stokes existence and smoothness problem, one of the Millennium Prize Problems posed by the Clay Mathematics Institute. In this talk, we will review the current state of the problem and examine recent attempts to disprove the conjecture, following a programme initiated by Terence Tao. We will focus on approaches that use geometrical data, such as Riemannian structures, which offer flexibility in tackling the problem. Additionally, we will discuss how the flexibility of contact geometry plays a role, as a class of stationary solutions to the Euler equation can be reparametrized as Reeb vector fields on a contact structure, leading to potential applications. Finally, we will introduce a novel geometric construction using Topological Quantum Field Theory, developed in collaboration with Ángel González-Prieto and Daniel Peralta-Salas, which may provide new methods for testing the blow-up phenomenon.
30.10.24	Eva Miranda (UPC Barcelona & Köln) Blow-up in Euler and Navier-Stokes equations? - Part II Abstract: In this second part of my talk I will focus on approaches that use geometrical data, such as Riemannian structures, which offer flexibility in tackling the problem. Additionally, I will discuss how the flexibility of contact geometry plays a role, as a class of stationary solutions to the Euler equation can be reparametrized as Reeb vector fields on a contact structure, leading to potential applications. I will also offer details of proofs of the talk in the Oberseminar on Friday 25 October. Finally, I will close up the session with a new conjecture concerning the blow-up of Navier-Stokes equations.
6.11.24	Ángel González-Prieto (Universidad Complutense de Madrid) Quantization of character varieties Abstract: Character varieties are complex manifolds that parametrize representations of the fundamental group of a given manifold. They play a central role in calibrating the existence of geometric structures, such as hyperbolic structures, and exhibit a fascinating hyperkähler structure arising from the non-abelian Hodge correspondence. In this talk, we will explore various aspects of character varieties from both geometric and algebraic perspectives. We will review how the natural symplectic structure present in character varieties led, through geometric quantization, to Witten's construction of a Topological Quantum Field Theory (TQFT) that computes the Jones polynomial of knots. Additionally, we will demonstrate that a different quantization approach, inspired by the Fourier-Mukai transform, can be employed to produce an alternative TQFT that analyzes the Hodge structure of character varieties. Time permitting, we will also discuss how this approach can be applied to construct new knot invariants of an arithmetic nature.
13.11.24	Kein Seminar
20.11.24	Murat Sağlam The topology of graph links II
27.11.24	Hansjörg Geiges Bott-integrability of overtwisted contact structures
4.12.24	Murat Sağlam The topology of graph links III
11.12.24	Murat Sağlam The topology of graph links IV
18.12.24	Murat Sağlam The topology of graph links V

H. Geiges, 7.10.24