A spectral result for massive electromagnetic Dirac Hamiltonians
Monday, 2.12.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
From the physical perspective of relativistic wave mechanics, Dirac operators on Riemannian manifolds can be interpreted as infinitesimal generators of time translation, i.e. as Hamiltonians. Most mathematical research on Dirac operators focuses on what amounts to the study of free, massless fields. In this talk however, we will consider a Dirac Hamiltonian that is coupled to an electromagnetic field and a spatially non-constant mass term. After motivating and setting up the necessary notions, we will proceed to show that, given a suitably behaved "potential well", the spectrum of such a Dirac Hamiltonian must be discrete. The result being presented is a generalization of a theorem by N. Charalambous and N. Große in 2023.
TBA
Thursday, 5.12.24, 10:15-11:15, Raum 414, Ernst-Zermelo-Str. 1
Sequential topological complexities and sectional categories of subgroup inclusions
Thursday, 12.12.24, 10:00-11:00, Raum 414, Ernst-Zermelo-Str. 1
The sequential topological complexities (TCs) of a space are integer-valued homotopy invariants that are motivated by the motion planning problem from robotics and express the complexity of motion planning if the robots are supposed to make predetermined intermediate stops along their ways. After outlining their definitions, I will discuss the sequential TCs of aspherical spaces in the first part of my talk and describe how they can be investigated by purely algebraic means. We will also take a look at a generalization of this algebraic setting, namely sectional categories of subgroup inclusions. In the second part of my talk, I will present a general lower bound on their values and derive consequences for sequential TCs and parametrized topological complexities of epimorphisms. We will investigate the methodology of the proof of this lower bound, in which all key steps are carried out using elementary homological algebra. This is joint work with Arturo Espinosa Baro, Michael Farber and John Oprea.
Moduli spaces of 3-manifolds with boundary are finite
Monday, 16.12.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
In joint work with Rachael Boyd and Corey Bregman we study the classifying\nspace \(B\bmathrm{Diff}(M)\) of the diffeomorphism group of a connected, compact, orientable\n3-manifold \(M\). I will recall the construction of this \(B \bmathrm{Diff}(M)\), also known as\nthe "moduli space of \(M\)", and explain how it parametrises smooth families of\nmanifolds diffeomorphic to \(M\).\n\nMilnor's prime decomposition and Thurston's geometrisation conjecture allow us\nto cut \(M\) into "geometric pieces", which each admit complete, locally\nhomogeneous Riemannian metric. For such geometric manifolds \((N,g)\) recent work\nusing Ricci flow shows that a certain space of metrics is contractible and thus\nthat the generalised Smale conjecture (often) holds: the diffeomorphism group\n\(\bmathrm{Diff}(N)\) is homotopy equivalent to the isometry group \(\bmathrm{Isom}(N,g)\).\n\nThe purpose of this talk is to explain a technique for computing the moduli\nspace \(B\bmathrm{Diff}(M)\) in terms of the moduli spaces of the pieces. We use this to\nprove that if \(M\) has non-empty boundary, then \(B \bmathrm{Diff}(M\btext{ rel boundary})\) has the\nhomotopy type of a finite CW complex, as was conjectured by Kontsevich.\n\n\n