Algebraic and Symplectic Geometry Seminar

I will discuss some very well studied cohomology groups that turn out to be captured by the machinery of critical CoHAs, for example the compactly supported cohomology of singular quiver varieties and untwisted character varieties. I will explain the usefulness of this extra CoHA structure on these groups, starting with a new proof of the Kac conjecture, and discuss a conjectural form for the CoHA associated to untwisted character varieties that provides a new way to think about the conjectures of Hausel and Rodriguez-Villegas. Finally I will discuss an approach to purity for the compactly supported cohomology of quiver varieties and a related approach to a conjecture of Shiffmann and Vasserot, analogous to Kirwan surjectivity for the stack of commuting matrices.

The talk will focus on how the asymptotic behavior of the Riemann-Hilbert correspondence (and, conjecturally, the non-abelian Hodge correspondence) on a Riemann surface is controlled by certain harmonic maps from the Riemann surface to affine buildings. This is part of joint work with Katzarkov, Noll and Simpson, which revisits, from the perspective afforded by the theory of harmonic maps to buildings, the work of Gaiotto, Moore and Neitzke on spectral networks, WKB problems, BPS states and wall-crossing.

We will start with a recollection on factorization algebras and factorization homology. We will then explain what fully extended TFTs are, after Jacob Lurie. And finally we will see how factorization homology can be turned into a fully extended TFT. This is a joint work with my student Claudia Scheimbauer.

We will explain how the result of Pantev-Toën-Vaquié-Vezzosi, about shifted symplectic structures on mapping stacks, can be extended to relative mapping stacks and Lagrangian structures. We will also provide applications in ordinary symplectic geometry and topological field theories.

The Contou-Carrère symbol has been introduced in the 90's in the study of local analogues of autoduality of Jacobians of smooth projective curves. It is closely related to the tame symbol, the residue pairing, and the canonical central extension of loop groups. In this talk we will a discuss a K-theoretic interpretation of the Contou-Carrère symbol, which allows us to generalize this one-dimensional picture to higher dimensions. This will be achieved by studying the K-theory of Tate objects, giving rise to natural central extensions of higher loop groups by spectra. Using the K-theoretic viewpoint, we then go on to prove a reciprocity law for higher-dimensional Contou-Carrère symbols. This is joint work with O. Braunling and J. Wolfson.

In 2000 Eliashberg-Polterovich introduced the natural notion of orderability of contact manifolds; that is, the existence of a natural partial order on the group of contactomorphisms. I will explain how one can study orderability questions using the machinery of Rabinowitz Floer homology. We establish a link between orderable and hypertight contact manifolds, and show that the Weinstein Conjecture holds (i.e. there exists a closed Reeb orbit) whenever there exists a positive (not necessarily contractible) loop of contactomorphisms.

Joint work with Peter Albers and Urs Fuchs.

In this talk we aim to study periodic orbits on the energy levels of a symplectic magnetic flow on the two-sphere using methods from contact geometry. In particular we show that, if the energy is low enough, we either have two or infinitely many closed orbits. The second alternative holds if there exists a prime contractible periodic orbit. Finally we present some generalisations and work in progress for closed orientable surfaces of higher genus.

The concept of a matrix factorization was originally introduced by Eisenbud to study syzygies over local rings of singular hypersurfaces. More recently, interactions with mathematical physics, where matrix factorizations appear in quantum field theory, have provided various new insights. I will explain how matrix factorizations can be studied in the context of noncommutative algebraic geometry based on differential graded categories. We will see the relevance of the noncommutative analogue of de Rham cohomology in terms of classical singularity theory. Finally, I will outline how the Kapustin-Li formula for the noncommutative Serre duality pairing (originally computed via path integral methods) can be mathematically explained using a combination of homological perturbation theory and local duality.
Partly based on joint work with Daniel Murfet.