Scalar curvature comparison geometry and the higher mapping degree
Montag, 8.1.24, 16:00-17:00, Ort noch nicht bekannt
Llarull proved in the late '90s that the round \(n\)-sphere is area-extremal in the sense that one can not increase the scalar curvature and the metric simultaneously. Goette and Semmelmann generalized Llarull's work and proved an extremality and rigidity statement for area-non-increasing spin maps \(f\bcolon M\bto N\) of non-zero \(\bhat{A}\)-degree between two closed connected oriented Riemannian manifolds.\n\nIn this talk, I will extend this classical result to maps between not necessarily orientable manifolds and replace the topological condition on the \(\bhat{A}\)-degree with a less restrictive condition involving the so-called higher mapping degree. For that purpose, I will first present an index formula connecting the higher mapping degree and the Euler characteristic of~\(N\) with the index of a certain Dirac operator linear over a \(\bmathrm{C}^\bast\)-algebra. Second, I will use this index formula to show that the topological assumptions, together with our extremal geometric situation, give rise to a family of almost constant sections that can be used to deduce the extremality and rigidity statements.\n
Moduli Spaces of Positive Curvature Metrics
Montag, 15.1.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
Besides the space of positive scalar curvature metrics \(\bmathrm{Riem}^+(M)\), various moduli spaces have gained a lot of attention. \nAmong those, the observer moduli space arguably has the best behaviour from a homotopy-theoretical perspective because the subgroup of \btextit{observer diffeomorphisms} acts freely on the space of Riemannian metrics if the underlying manifold \(M\) is connected.\n\nIn this talk, I will present how to construct non-trivial elements in the second homotopy of the observer moduli space of positive scalar curvature metrics for a large class for four-manifolds. I will further outline how to adapt this construction to produce the first non-trivial elements in higher homotopy groups of the observer moduli space of positive sectional curvature metrics on complex projective spaces.
Surgery on fold maps
Montag, 22.1.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
In this talk I will explore the notion of fold maps which are a natural generalization of Morse functions. Morse functions play a central role in the classification of manifolds and getting rid of their critical points is a crucial step in the proof of the h-cobordism theorem. I will describe a similar procedure for eliminating so called fold-singularities. This is similar in spirit but more flexible compared to the above-mentioned removal of critical points as it allows to perform surgery on the underlying manifold. If time permits I will also explain how this can be used to study fiber bundles and their characteristic classes.
Generalized Seiberg-Witten equations and where to find them
Montag, 29.1.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We explore the framework of Generalized Seiberg-Witten Equations, aimed at deriving fresh invariants for smooth four-manifolds. These generalizations replace the standard spinor bundle with a suitable hyperKähler manifold for the spinor fields. This departure opens up exciting new possibilities for studying the smooth structures of four-dimensional manifolds, while also including a lot of well-known invariants, the most prominent example the Anti-Self-Duality equations and the resulting Donaldson invariants.\n\nWe then present how to compute the solution spaces in on of the most simple cases, where the spinor takes values in a four dimensional hyperKähler manifold, and show how this leads to invariants for four dimensional symplectic and Kähler manifolds, while also giving a geometric interpretation.