Department of Mathematics
University of Fribourg
Organizers: Anand Dessai, Philipp Reiser
Tuesday 27 August 2024, 15:00 | Leonardo Cavenaghi (University of Miami/University of Campinas) | New look at Milnor spheresIn 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. |
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Monday 30 September 2024 | Philipp Reiser (University of Fribourg) | Tangent cones of non-collapsed Ricci limit spacesA 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. |
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Monday 7 October 2024 | Luke Higgins (University of Fribourg) | Introducing Elliptic Genera IConstructing 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. |
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Monday 14 October 2024 | Luke Higgins (University of Fribourg) | tba |
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Monday 21 October 2024 | ||||
Monday 28 October 2024 | ||||
Monday 11 November 2024 | ||||
Monday 18 November 2024 | ||||
Thursday 28 November 2024 | Darya Sukhorebska (Karlsruhe Institute of Technology) | |||
Monday 2 December 2024 | ||||
Monday 9 December 2024 |
Monday 2 October 2023 | Philipp Reiser (University of Fribourg) | Spin structures on vector bundles | ||
Monday 9 October 2023 | Sam Hagh Shenas Noshari (University of Fribourg) | Spin manifolds and spin bordism, part I | ||
Monday 16 October 2023 | Sam Hagh Shenas Noshari (University of Fribourg) | Spin manifolds and spin bordism, part II | ||
Tuesday 24 October 2023, 15:15 | Anand Dessai (University of Fribourg) | Clifford algebras | ||
Monday 30 October 2023 | Anand Dessai (University of Fribourg) | Clifford modules and spinor bundles | ||
Tuesday 7 November 2023, 15:15 | Patrick Ghanaat (University of Fribourg) | Connections on spinor bundles I | ||
Monday 13 November 2023 | Patrick Ghanaat (University of Fribourg) | Connections on spinor bundles II | ||
Monday 27 November 2023 | Philipp Reiser (University of Fribourg) | Dirac operators | ||
Thursday 8 December 2023, 15:15 | Sam Hagh Shenas Noshari (University of Fribourg) | Vanishing theorems and applications | ||
Monday 11 December 2023 | Anand Dessai (University of Fribourg) | Atiyah-Singer index theorem I | ||
Thursday 11 January 2024, 11:00 | Anand Dessai (University of Fribourg) | Atiyah-Singer index theorem II | ||
Monday 22 January 2024, 13:15 | Sam Hagh Shenas Noshari (University of Fribourg) | Enlargeable manifolds | ||
Monday 12 February 2024, 13:15 | Patrick Ghanaat (University of Fribourg) | Positive energy theorem | ||
Wednesday 21 February 2024, 13:15 | Anand Dessai (University of Fribourg) | Lefschetz fixed point formula | ||
Tuesday 27 February 2024, 17:15 in 2.52 | Luke Higgins (University of Fribourg) | Atiyah-Hirzebruch Â-vanishing theorem | ||
Wednesday (Geometry seminar) 6 March 2024, 10:20 | Masoumeh Zarei (University of Münster) | Susceptibility of positive curvature conditions under the Ricci flowSince the introduction of Ricci flow by Hamilton in 1982, it has been a fundamental question to understand the evolution of metrics and their curvature conditions under the flow. While positive scalar curvature and 2-positive curvature operator are preserved in all dimensions, there exist infinitely many dimensions where certain curvature conditions lying in between are not preserved. In this talk, I first recall some basics about homogeneous Ricci flow. Then, I present joint works with David González-Álvaro in which we examine metrics with different curvature conditions on various homogeneous spaces and discuss the evolution of their metrics under the Ricci flow. |
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Monday 11 March 2024 | Philipp Reiser (University of Fribourg) | Positive Ricci curvature on twisted suspensionsThe twisted suspension of a manifold can be seen as a smooth analogue of the classical suspension operation for topological spaces. Its construction is motivated from the spinning operation in knot theory and it is obtained by surgery on a fibre of a principal circle bundle over the given manifold. In this talk I will show that Riemannian metrics of positive Ricci curvature can be lifted along twisted suspensions. As application we obtain first examples of simply-connected manifolds of positive Ricci curvature with maximal symmetry rank in any dimension, and we obtain new examples of (rational) homology spheres with a Riemannian metric of positive Ricci curvature. |
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Monday 18 March 2024 | Florian Johne (University of Freiburg i. Br.) | Topology and geometry of metrics of positive intermediate curvatureThe notion of $m$-intermediate curvature interpolates between Ricci curvature and scalar curvature. In this talk we describe extensions of classical results by Bonnet-Myers and Schoen-Yau to the setting of $m$-intermediate curvature: A non-existence result for metrics of positive $m$-intermediate curvature on manifolds with topology ${N}^{n}={T}^{m}$×${S}^{n-m}$; a gluing result for manifolds with $m$-convex boundary; inheritance of spectral positivity along stable minimal hypersurfaces, and estimates for the $m$-diameter for uniform positive lower bounds. This talk is partially based on joint work with Simon Brendle and Sven Hirsch, and joint work with Aaron Chow and Jingbo Wan. |
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Monday 8 April 2024 | no seminar | |||
Monday 15 April 2024 | David González Álvaro (Univ. Politécnica Madrid) | Examples of ${\mathrm{Ric}}_{2}\mathrm{>0}$ manifoldsIn this talk we will discuss the construction of ${\mathrm{Ric}}_{2}$>0 metrics on closed manifolds of dimensions 10, 11, 12, 13 and 14. These include ${S}^{6}$×${S}^{7}$, ${S}^{7}$×${S}^{7}$ and infinitely many homeomorphism types in dimension 13. The main new ingredient is a notion for homogeneous bundles which generalizes the concept of fatness and, together with other assumptions, ensures the existence of ${\mathrm{Ric}}_{2}\mathrm{>0}$ metrics on the total space of the bundle. Further examples are obtained by considering biquotients of the corresponding homogeneous spaces. This is joint work with Jason DeVito, Miguel Domínguez-Vázquez and Alberto Rodríguez-Vázquez. |
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Tuesday (Colloquium) 16 April 2024, 17:15 | David González Álvaro (Univ. Politécnica Madrid) | Spaces with lower curvature boundsIn this talk we shall review a few of the many existing notions of curvature in Riemannian and metric geometry, together with some of the main achievements and current lines of research. We will put special focus on manifolds of positive sectional curvature and those of positive Ricci curvature, and also on certain curvature conditions which interpolate between them. |
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Monday 22 April 2024 | no seminar | |||
Monday 29 April 2024 | Francesca Tripaldi (SNS Pisa) | Sequences of properly embedded minimal surfacesThis talk centres around the study of sequences of minimal surfaces in a three manifold. So far, an in-depth characterisation of such sequences has been carried out in the case where the surfaces considered are discs, both in regards to the curvature blow-up points and the limit surfaces that can arise. In order to study the topology of the limit leaves of sequences of properly embedded minimal discs, Bernstein and Tinaglia first introduced the concept of the simple lift property, since such limit leaves satisfy this property. They proved that an embedded surface Σ ⊂ Ω with the simple lift property must have genus zero, if Ω is an orientable three-manifold satisfying certain geometric conditions. During my PhD, I generalised this result by considering an arbitrary orientable three-manifold Ω and proving that one is still able to restrict the topology of an arbitrary surface Σ ⊂ Ω with the simple lift property. In this talk, I will present which topological results can be extended in the case of sequences of minimal surfaces of arbitrary finite genus. |
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Monday 6 May 2024, 17:30 | Philipp Reiser (University of Fribourg) | Manifolds of positive Bakry-Émery Ricci curvatureThe Bakry-Émery Ricci tensor is a generalization of the classical Ricci tensor to the setting of weighted Riemannian manifolds, i.e. Riemannian manifolds whose Riemannian volume forms are weighted by a smooth function. While it was defined by Bakry and Émery in the context of diffusion processes, it also appears naturally in the study of Ricci flow and collapsed Ricci limit spaces. In analogy to important open problems in the Riemannian case, we will consider the question of which manifolds admit a weighted Riemannian metric of positive Bakry-Émery Ricci curvature. We will show that surgery techniques that are used to construct examples in the Riemannian case can be extended and improved in the weighted case. As applications we construct new examples of such manifolds in dimension 5. This is joint work with Francesca Tripaldi. |
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