Camilla Felisetti(Modena): Parabolic bundles and intersection cohomology of moduli of vector bundles
Abstract: Intersection cohomology is a topological notion adapted to the description of singular topological spaces, and the Decomposition Theorem for algebraic maps is a key tool in the subject. Motivated by the work of Mozgovoy and Reineke, in joint work with Andras Szenes and Olga Trapeznikova, we give a complete description of the intersection cohomology of the moduli space of vector bundles of any rank via a detailed analysis of the Decomposition Theorem applied to a certain map from parabolic bundles. We also give a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning.
Jakob Hedicke (CRM/Radboud): Integrable Reeb Flows on Contact 3-Manifolds
Abstract: A Reeb flow describes the motion of a Hamiltonian system on a constant energy hypersurface. Every Reeb flow is transverse to a certain distribution of hyperplanes in the tangent bundle of the hypersurface, called a contact structure.
Often Reeb flows have a quite complex dynamical behaviour, and there are many known cases of contact structures such that any Reeb flow of that contact structure is chaotic. Hence a natural question to ask is, in which cases non-chaotic Reeb flows exist. After a brief introduction to contact geometry we will discuss various examples of chaotic and non-chaotic Reeb flows. We then focus on the classification of contact structures on 3-manifolds that admit a Bott-integrable Reeb flow, i.e., a Reeb flow for which there exists a Morse-Bott function invariant under the flow. This talk is based on joint work with Hansjörg Geiges and Murat Sağlam.
Eva-Maria Hekkelman(Bonn): Connes' Integration Formula of 1915
Abstract: Alain Connes proved an integration formula in 1988, which has taken on philosophical importance in the field of noncommutative geometry. I will attempt to convince you that Szegő already proved a special case of Connes' theorem in 1915, at least on the circle, if you use your imagination. More useful is that Szegő's version can be generalised way beyond the circle case, leading to a noncommutative version of Szegő's limit theorem. I'll make an attempt to include an entry-level explanation of all of the words in this abstract. Based on recent joint work with Ed McDonald.
Joshua Jackson (Cambridge): Moduli of Representations of Quivers with Multiplicities
Abstract: After reviewing the classical theory of quiver moduli spaces via Mumford’s reductive GIT, I will report on joint work with Victoria Hoskins and Tanguy Vernet which constructs moduli spaces for representations of quivers with multiplicities. Along the way I will indicate the challenges posed by this generalisations, namely the non-reductivity of the groups we need to quotient by, and outline the theoretical developments (joint with Eloise Hamilton and Victoria Hoskins) that make this possible.
Karin Melnick (Luxembourg): Conformal vector fields on Lorentzian manifolds
Abstract: The exponential map of a semi-Riemannian metric provides a linearization of any Killing vector field in a neighborhood of a singularity, leading to local normal forms for such vector fields. In 2013, C. Frances and I proved that conformal vector fields of real-analytic Lorentzian manifolds are linearizable around a singularity or the metric is conformally flat, thus obtaining local normal forms for such vector fields. In this colloquium, I will talk about work with S. Dumitrescu, C. Frances, V. Pecastaing, and A. Zeghib in which we show that certain non-isometric linear normal forms in this context give rise to local gravitational pp-wave metrics in the conformal class; moreover, if they occur on a compact Lorentzian manifold, then it is conformally flat. This global conclusion is pertinent to the Lorentzian Lichnerowicz Conjecture.
Sergey Neshveyev (Oslo): The Mackey machine and the ideal structure of crossed product type algebras
Abstract: Representations of a group or an algebra on Hilbert spaces are encoded by several spaces, the most common of which are the unitary dual and the primitive dual, equipped with Zariski-type topologies. These spaces have been computed in many cases as sets. In particular, for semidirect products of groups or, more generally, crossed product algebras, there is a general strategy of doing such computations known as the Mackey machine or the little group method. The method, however, does not say anything about the topology, and generally a description of the topology is available only in some special cases. The talk will review the history of the problem and some recent progress on it.
Sophie Zegers (Delft): On the classification of quantum lens spaces
Abstract: In the study of noncommutative geometry, various of classical spaces have been given a quantum analogue. A class of examples are the quantum lens spaces described by Hong and Szymański as graph C*-algebras. The graph C*-algebraic description has made it possible to obtain important information about their structure and to work on classification. Moreover, every quantum lens space comes with a natural circle action, leading to an equivariant isomorphisms problem.
In this talk, I will give an introduction on how to classify quantum lens spaces and how to obtain a number theoretic invariant in low dimensions. Moreover, I will briefly present some results from my recent joint work with S\o ren Eilers on the equivariant isomorphism problem of low dimensional quantum lens spaces.
