Veranstaltungsübersicht -- Wintersemester 2023/2024

Montag,
2023-10-23
16:00 Uhr
Raum 404, Ernst-Zermelo-Str. 1
Stefan Waldmann: Convergence of Star Products on $T^*G$

Star products can be seen as a generalization of a symbol calculus for differential operators. In fact, for cotangent bundles, the global symbol calculus yields a star product of a particular kind. While formal star products have been studied in detail with deep and exciting existence and classification theorems, convergence of the formal star products is still a widely open question. Beside several (classes of) examples, not much is known. In this talk I will focus on a particular class of examples, the cotangent bundles of Lie groups, where a nice convergence scheme has been established. I will try to avoid the technical details as much as possible and focus instead on the principal ideas of the construction. The results are joint work with Micheal Heins and Oliver Roth.

Montag,
2023-10-30
16:15-17:45 Uhr
Raum 125, Ernst-Zermelo-Str. 1
Yan Junrong (Beijing): Witten deformation for non-Morse functions and gluing formulas

Witten deformation is a versatile tool with numerous applications in mathematical physics and geometry. In this talk, we will focus on the analysis of Witten deformation for a family of non-Morse functions, leading to a new proof of the gluing formula for analytic torsions. Then we could see that the gluing formula for analytic torsion can be reformulated as the Bismut-Zhang theorem for non-Morse functions. Furthermore, this approach can be extended to analytic torsion forms, which also provides a new proof of the gluing formula for analytic torsion forms.

Montag,
2023-11-13
16:15-17:45 Uhr
Raum 404, Ernst-Zermelo-Str. 1
Maximilian Stegemeyer (Freiburg): String topology of the space of paths with endpoints in a submanifold

String topology is the study of algebraic structures on the homology of the free loop space of a closed manifold. The most famous operation is the Chas-Sullivan product which is a graded commutative and unital product on the homology of the free loop space. In this talk we study the space of paths in a manifold whose endpoints lie in a chosen submanifold. It turns out that the homology of this space also admits a product which is defined similarly to the one of Chas and Sullivan. Moreover, the homology of this path space is a module over the Chas-Sullivan ring. We will see that in some situations both structures together form an algebra - i.e. the product on homology of the path space with endpoints in a submanifold is an algebra over the Chas-Sullivan ring - but that this property does not hold in general.

Montag,
2023-11-20
16:15-17:45 Uhr
Raum 404, Ernst-Zermelo-Str. 1
Philipp Reiser: Positive intermediate Ricci curvature, surgery and Gromov's Betti number bound

Positive intermediate Ricci curvature is a family of interpolating curvature conditions between positive sectional and positive Ricci curvature. While many results that hold for positive sectional or positive Ricci curvature have been extended to these intermediate conditions, only relatively few examples are known so far. In this talk I will present several extensions of construction techniques from positive Ricci curvature to these curvature conditions, such as surgery, gluing and bundle techniques. As an application we obtain a large class of new examples of manifolds with a metric of positive intermediate Ricci curvature, including all homotopy spheres that bound a parallelisable manifold, and show that Gromov's Betti number bound for manifolds of non-negative sectional curvature does not hold from positive Ricci curvature up to roughly halfway towards positive sectional curvature. This is joint work with David Wraith.

Mittwoch,
2023-11-29
16:15 Uhr
Hörsaal II, Albertstr. 23b
Alberto Cattaneo: Poisson structures from corners of field theories

The BV formalism and its shifted versions in field theory have a nice compatibility with boundary structures. Namely, one such structure in the bulk induces a shifted (possibly degenerated) version on its boundary, which can be interpreted as a Poisson structure (up to homotopy). I will present the results for some field theories, in particular, 4D BF theory and 4D gravity.

Montag,
2023-12-04
16:15 Uhr
Raum 404, Ernst-Zermelo-Str. 1
Pascal Schade: Topological Censorship
Montag,
2023-12-11
16:15-17:45 Uhr
Raum 404, Ernst-Zermelo-Str. 1
Julian Seipel: Prescription of Dirac Eigenvalues, Partial Eigenbundles and Surgery

The prescription of eigenvalues of the Dirac operator on a closed spin manifold requires, besides the usual analytical methods à la Uhlenbeck and Dahl, also surgery methods to transport spectral data along a bordism. In this talk, I will give the necessary basics as well as an overview of the prescription of double eigenvalues on spin manifolds.

Montag,
2023-12-18
16:15 Uhr
Raum 404, Ernst-Zermelo-Str. 1
Philippe Kupper (KIT): String Topology of Compact Symmetric Spaces

On the homology of the free loop space of a closed manifold M there exists the so-called Chas-Sullivan product. It is a product defined via the concatenation of loops and can, for example, be used to study closed geodesics of Riemannian or Finsler metrics on M. In this talk I will outline how one can use the geometry of symmetric spaces to partially compute the Chas-Sullivan product. In particular, we will see that the powers of certain non-nilpotent homology classes correspond to the iteration of closed geodesics in a symmetric metric. Some triviality results on the Goresky-Hingston cohomology product will also be mentioned. This talk is based on joint work with Maximilian Stegemeyer.

Montag,
2024-01-08
16:00 Uhr
Ort noch nicht bekannt
Thomas Tony (University of Münster): Scalar curvature comparison geometry and the higher mapping degree

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 $f\colon M\to N$ of non-zero $\hat{A}$-degree between two closed connected oriented Riemannian manifolds.

In this talk, I will extend this classical result to maps between not necessarily orientable manifolds and replace the topological condition on the $\hat{A}$-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 $\mathrm{C}^\ast$-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.

Montag,
2024-01-15
16:15 Uhr
Raum 125, Ernst-Zermelo-Str. 1
Thorsten Hertl: Moduli Spaces of Positive Curvature Metrics

Besides the space of positive scalar curvature metrics $\mathrm{Riem}^+(M)$, various moduli spaces have gained a lot of attention. Among those, the observer moduli space arguably has the best behaviour from a homotopy-theoretical perspective because the subgroup of \textit{observer diffeomorphisms} acts freely on the space of Riemannian metrics if the underlying manifold $M$ is connected.

In 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.

Montag,
2024-01-22
16:15-17:45 Uhr
Raum 125, Ernst-Zermelo-Str. 1
Georg Frenck: Surgery on fold maps

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.

Montag,
2024-01-29
16:15-17:45 Uhr
Raum 125, Ernst-Zermelo-Str. 1
Marcel Bigorajski: Generalized Seiberg-Witten equations and where to find them

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.

We 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.

Montag,
2024-02-05
16:15-17:45 Uhr
Raum 125, Ernst-Zermelo-Str. 1
Thorsten Hertl: Observations on G2 Moduli Spaces

In his two seminal articles, Dominic Joyce not only constructed the first examples of closed manifolds with G2-holonomy metrics, but also proved that the moduli space of all G2-metrics on a closed manifold is itself a finite-dimensional manifold. The statement is, however, only a local one, and the global topological properties of these moduli spaces have remained quite mysterious ever since. Indeed, up to now, we only know that they may be disconnected by the work of Crowley, Goette, and Nordström; the question whether all path components are contractible or not has not been answered yet.

In this talk, I will give a short introduction to G2 metrics and their moduli spaces and outline a construction of a non-trivial element in the second homotopy of the easier-acceesible observer moduli space of G2 metrics on one of Joyce's examples. If time permits, I will indicate why and how this non-trivial example might also descend to the (full) moduli space.

This talk is based on ongoing joint work with Sebastian Goette.


Abonnieren

Die Einträge dieser Veranstaltung können im iCalender Format abonniert werden. die URL dazu lautet:
http://wochenprogramm.mathematik.uni-freiburg.de/ical/WS2023-2024/OS-DiffGeo.ics