Yamabe Metriken mit Nullwertiger Skalarkrümmung
Monday, 24.10.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
Wir werden eine bestimmte\nEigenschaft auf nicht-kompakten Riemannschen Mannigfaltigkeiten definieren, welche die\nExistenz vollständiger Yamabe Metriken mit nullwertiger Skalarkrümmung impliziert.\nDarüber hinaus werden wir zeigen, dass diese Eigenschaft auf asymptotisch flachen\nMannigfaltigkeiten mit einer gewissen Abfallsrate für die Skalarkrümmung immer erfüllt\nist. Zum Abschluss dieses Vortrags werden wir einen Ausblick zur Gültigkeit dieser\nEigenschaft auf asymptotisch lokal flachen Mannigfaltigkeiten geben.\n
Whittaker Fourier type solutions to differential equations arising from string theory
Monday, 31.10.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
In this talk, we find the full Fourier expansion of some special functions describing the graviton scattering in the string theory. We give a connection of the boundary condition on such Fourier series with convolution formulas on the divisor functions. Using\nmanipulations with divergent series, we obtain a class of\nformulas evaluating an infinite sum of divisor functions, including a\nsurprising equality\n\[\n\bsum d(|n_1|) d(|n_2|) ( (n_2-n_1) \blog( | n_1/n_2 | ) + 2 ) =\n(2-\blog(4 \bpi^2 |n|) ) d(|n|),\n\]\nwhere \(\bsum\) denotes the sum over all possible non-zero integers\n\(n_1\) and \(n_2\) such that \(n_1+n_2=n\).\n\nThis is a joint work with Kim Klinger-Logan.
Concordances in Positive Scalar Curvature and Index Theory
Monday, 7.11.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
Scalar curvature is a local invariant of a Riemannian manifold. It measures\nasymptotically the volume growth of geodesic balls. Understanding the topological space of\nall positive scalar curvature metrics on a closed manifold has been an active field of study\nduring the last 30 years. So far, these spaces have been considered from an isotopy\nviewpoint. I will describe a new approach to study this space based on the notion of\nconcordance. To this end, I construct with the help of cubical set theory a comparison space\nthat only encodes concordance information and in which the space of positive scalar\ncurvature metrics canonically embeds. After the presentation of some of its properties, I will\nshow that the indexdifference factors over the comparison space using a new model of real\nK-theory that is based on pseudo Dirac operators.
Diracoperatoren mit magnetischer Verschlingung
Monday, 5.12.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
In der Quantenmechanik beschreibt der Aharonov-Bohm-Effekt, welche Auswirkungen ein magnetisches Vektorpotential auf interferierende Elektronenstrahlen hat, die sich außerhalb eines Magnetfeldes befinden. Bei der Verallgemeinerung diese Effekts gehen wir nun von Magnetfeldern in \( \bmathbb{S}^3 \) aus, die auf glatten, geschlossenen Kurven getragen sind. Der Vortrag befasst sich mit Dirac-Operatoren, die das Vektorpotential eines solchen Magnetfeldes beinhalten. Die Selbstadjungiertheit dieser Operatoren ist zu Anfang nur bei der Wahl einer Domain ersichtlich, die sich nicht in der Nähe des Magnetfeldes befindet. Es soll nun darum gehen, selbstadjungierte Erweiterungen zu finden, die das Verhalten nahe des Feldes beschreibt.
Hands on the Algebraic Index Theorem
Monday, 12.12.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
In my talk I want to give a summary of results from Fedosov/Tsygan/Nest on the so-called algebraic index theorem, which links symplectic deformation quantizations to topological invariants and reproduces the Atiyah-Singer Index Theorem for the canonical quantization of cotangent bundles. The talk includes a gentle introduction to deformation quantization.
tba
Monday, 19.12.22, 16:00-17:00, Hörsaal II, Albertstr. 23b
ODD Riemannian metrics
Monday, 19.12.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
Positive scalar curvature and 2-type: an analysis via the Gromov--Lawson--Rosenberg conjecture.
Monday, 23.1.23, 16:15-17:15, Hörsaal II, Albertstr. 23b
I will talk about a conjecture that claims the PSC space of a (spin) manifold only depends on its 2-type. In particular, focusing on fundamental groups that satisfy the Gromov--Lawson--Rosenberg conjecture one can obtain positive results in certain cases. The latter conjecture claims that the non-vanishing of a certain cobordism invariant represents a total obstruction to positive scalar curvature. Index theory and surgery theory are at the base of the whole argument.
Localization methods and the Witten genus
Monday, 30.1.23, 16:15-17:15, Hörsaal II, Albertstr. 23b
In this talk I will give a brief introduction to equivariant cohomology and the localization formula. Applying the formula to infinite dimensional siutations one recovers interesting invariants like the A-hat genus or the Witten genus. In this representation one finds a natural explanation for the modularity of the Witten genus.
TBA
Monday, 6.2.23, 16:15-17:15, Hörsaal II, Albertstr. 23b
Special submanifolds in Joyce’s generalised Kummer constructions
Monday, 6.2.23, 16:15-17:15, Hörsaal II, Albertstr. 23b
Associative and coassociative submanifolds are the natural subobjects of 7-dimensional G2-manifolds. Besides having minimal volume among all submanifolds realising a fixed homology class, they play a prominent role in higher-dimensional gauge theory. In this talk we will focus on G2-manifolds arising as desingularisations of flat orbifolds\nand explain a method of constructing coassociatives inside them. The novelty of this construction is that the volume of these submanifolds tends to zero as the ambient manifold\napproaches its orbifold-limit.
Hopf-Galois extension in noncommutative differential geometry
Monday, 13.3.23, 15:00-16:00, Raum 404, Ernst-Zermelo-Str. 1
In this seminar I give a gentle introduction to the theory of Hopf-Galois extensions and their role in noncommutative differential geometry. From a geometric point of view they correspond to principal bundles on noncommutative algebras with a Hopf algebra replacing the structure group. Principality of such a bundle can equivalently be phrased in terms of a noncommutative Atiyah sequence. We continue by discussing differential calculi on Hopf-Galois extensions, proving that in the faithfully flat case such a calculus amplifies to a graded Hopf-Galois extensions if and only if the corresponding Atiyah sequence is exact, as well. As an example we discuss the q-monopole fibration. The presentation is partially based on a collaboration with Aschieri, Fioresi and Latini.