Topology @ TUM group seminar

Many interesting TQFTs can be realized as sigma models. When coupled with a sheaf theory, sigma models can be used in algebraic geometry to construct TQFTs valued in (higher) categories. With the advent of higher categorical algebra developed by Lurie and others, this perspective has been employed to great success in geometric representation theory, as exemplified, for instance, by the work of Ben-Zvi, Gaitsgory, Nadler, and others in the various forms of the geometric Langlands program. In this talk, I will demonstrate how the study of categorified link invariants can also benefit from this point of view, using the theory of Soergel bimodules as an example. In particular, I will describe the value of the corresponding TQFT on the circle and relate it to the category of character sheaves, yielding, as a consequence, a relation between the HOMFLY-PT link homology theory and coherent sheaves on the Hilbert schemes of points on $\mathbb{C}^2$ as conjectured Gorsky--Negut--Rasmussen. This is joint work with Penghui Li. The talk will be informal, requiring no background in link homology, algebraic geometry, or representation theory.

To the singularities of a hypersurface one may associate increasingly sophisticated invariants: Milnor numbers, sheaves of vanishing cycles and categories of Matrix factorizations. It is an interesting problem to globalize these locally defined invariants. The Milnor numbers associated to the moduli spaces of suitable sheaves on a Calabi-Yau 3-fold leads to the highly influential Donaldson-Thomas invariants. Brav-Bussi-Dupont-Joyce-Szendroi have globalized the sheaf of vanishing cycles to construct a categorification of Donaldson-Thomas invariants (obstructed by some orientation data). I will take about globalizing higher categorical invariants. This depends on the study of (-1)-shifted symplectic structures on derived schemes, but I will not expect the audience to know what those words mean. This is work in progress, joint with B. Hennion and M. Robalo.