Department of Mathematics
University of Fribourg

Research Seminar on Topology (Oberseminar Topologie)
Mondays 17-18 in 0.05 (PER 23)

Organizers: Anand Dessai, Philipp Reiser

Fall 2024 / Spring 2025

Tuesday 27 August 2024, 15:00 Leonardo Cavenaghi (University of Miami/University of Campinas)
New look at Milnor spheres

In recent years, built upon mirror symmetry reasoning, Katzarkov, Kontsevich, Pantev, and Yu have developed the theory of Atoms as an ``A side and ``B side'' patching to study birational geometry. The main revolution comes from applying Gromov-Witten invariants to birational geometry problems. On the other hand, recent developments (in the works of Tschinkel and Kresch) led to the concept of birational maps and rationality equivalence for stacks. Moreover, a cohomology theory based on Gromov-Witten invariants properly captures the cohomological aspects of Deligne-Mumford stacks. In the orbifold case, this is known as Chen-Ruan cohomology. In this talk, we briefly introduce these subjects aimed at a broader audience of mathematicians. We focus on descriptions using group actions and their representations. To the differential geometer, the consequences of an ongoing collaboration with L. Grama, L. Katzarkov, and M. Kontsevich aim to relate differential topology invariants with algebraic ones.

Monday 30 September 2024 Philipp Reiser (University of Fribourg)
Tangent cones of non-collapsed Ricci limit spaces

A non-collapsed Ricci limit space is a Gromov-Hausdorff limit of a sequence of Riemannian manifolds with a uniform lower bound on both the Ricci curvature and the volume. When studying the structure of such spaces, a central role is played by its tangent cones, which generalise the notion of a tangent space of a manifold. A tangent cone is a metric space obtained by scaling up the metric at a given point, that is, by "zooming" into the space. It was shown by Cheeger-Colding that any tangent cone of a non-collapsed Ricci limit space is in fact a metric cone over a compact metric space. At the same time, they showed that, in general, the metric on this space depends on the specific choice of sequence of scalings of the metric. Subsequently, Colding-Naber constructed an example where two different homeomorphism types of tangent cones appear at the same point, showing that even the homeomorphism type is non-unique in general. In this talk, I will give a brief introduction to the theory of Ricci limit spaces and their tangent cones, which will require only basic knowledge of Riemannian geometry. I will then prove a generalisation of the Colding-Naber result that allows to construct examples of non-collapsed Ricci limit spaces with highly non-unique tangent cones.

Monday 7 October 2024 no seminar
Monday 14 October 2024 Luke Higgins (University of Fribourg)
Introducing Elliptic Genera I

Constructing metrics of positive curvature and nontrivial group actions - which we usually would like to carry additional properties - on a given manifold is of great interest in topology and geometry. Of course, such pleasant geometric properties may not exist; it is therefore also useful to develop an obstruction theory. In the 60s and 70s, certain topological objects were found to vanish under the presence of various geometric properties. These objects shared the same property of being able to be viewed as homomorphisms from the relevant cobordism ring to the rational numbers. The study of such objects, which are known as \textit{(multiplicative) genera}, had begun. The hope was that by generalising the known examples, one could refine their obstructions. Some things are known about these generalisations, and much more is to be learned. In this introductory talk - the first of two - we motivate the study of \textit{elliptic} genera, cover the prerequisite machinery and important tools, and then begin studying our first examples. After this, we will have seen enough material to tackle the aforementioned generalisations of these examples; this will be the focus of the second talk.

Monday 21 October 2024 Luke Higgins (University of Fribourg)
Introducing Elliptic Genera II

In the first talk, the signature and A-hat genus were introduced, and some tools from equivariant index theory were recalled. Using this machinery, one can compute equivariant indices. If the equivariant index being studied is rigid, this gives a way of computing the index - assuming the existence of a compact Lie group action. If one can find conditions of this action which, for example, cause the equivariant index to vanish, we have an obstruction: such an action cannot exist on manifolds for which this index is nonvanishing. We begin to study elliptic genera and find that the genera we studied previously are elliptic. Another elliptic genus - which can be thought of as the equivariant signature of the loop space of a manifold - is defined. The vanishing/simplification of this genus is then studied under the presence of different kinds of S1 actions.

Monday 28 October 2024 no seminar
Monday 11 November 2024 no seminar
Monday 18 November 2024 no seminar
Thursday 28 November 2024 Darya Sukhorebska (Karlsruhe Institute of Technology)
Topology of positively curved manifolds with symmetry in low dimensions

The classification of simply connected closed manifolds which admit metrics of positive sectional curvature is a long-standing open problem in Riemannian geometry. The difficulty of the problem is that there are not many known examples of manifolds admitting such metrics, and all known obstructions are also obstructions to metrics of non-negative curvature. During the 1990s Karsten Grove suggested to study the structure of positively curved manifolds under symmetry assumptions. The fundamental results in this area are due to Grove and Searle, Wilking, Fang and Rong. In this talk, we will discuss the topological properties of positively curved manifold with symmetry in low dimensions. Namely, we will talk about the Euler characteristic of 16-dimensional manifolds with 3-dimensional torus action. The result improves of the work of Amann and Kennard on the topology of 16-dimensional positively curved manifolds with symmetry rank 4. This talk is based on ongoing joint project with Burkhard Wilking.

Monday 2 December 2024 no seminar
Monday 9 December 2024 Philipp Reiser (University of Fribourg)
Positive Ricci curvature and connected sums

The connected sum operation is a simple but useful tool in geometric topology to connect two given manifolds. However, if both manifolds are equipped with Riemannian metrics of positive Ricci curvature, it is surprisingly difficult to determine whether this condition can be preserved under the connected sum. In this talk, I will review previous work by Perelman and Burdick on this problem, and then discuss a new construction for Riemannian metrics of positive Ricci curvature on connected sums of certain fibre bundles.

Monday 17 February 2025 François Ammann (University of Fribourg)
Dirac Operator and Vanishing Theorems I

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