Geometry and Analysis Seminar

We construct vertex algebras associated to divisors $S$ in toric Calabi-Yau threefolds $Y$, satisfying conjectures of Gaiotto-Rapcak and Feigin-Gukov, and in particular such that the characters of these algebras are given by a local analogue of the Vafa-Witten partition function of the underlying reduced subvariety $S^{red}$. These results are part of a broader program to establish a dictionary between the enumerative geometry of coherent sheaves on surfaces and Calabi-Yau threefolds, and the representation theory of vertex algebras and affine Yangian-type quantum groups.

After a brief introduction to the semiregularity maps of Severi, Kodaira and Spencer, and Bloch, I will focus on the Buchweitz-Flenner semiregularity map and on its importance for the deformation theory of coherent sheaves.
The subject of this talk is the construction of a lifting of each component of the Buchweitz-Flenner semiregularity map to an L-infinity morphism between DG-Lie algebras, which allows to interpret components of the semiregularity map as obstruction maps of morphisms of deformation functors.

As a consequence, we obtain that the semiregularity map annihilates all obstructions to deformations of a coherent sheaf on a complex projective manifold. Based on a joint work with R. Bandiera and M. Manetti.

We apply recent ideas in Floer homotopy theory to some questions in symplectic topology. We show that Floer homology can detect smooth structures of certain Lagrangians, as well as using this to find restrictions on symplectic mapping class groups. This is based on joint work-in-progress with Ivan Smith.

In this talk we give an introduction to the topic of Einstein metrics on spheres. In particular, we prove the existence of three non-round Einstein metrics with positive scalar curvature on $S^{10}.$ Previously, the only even-dimensional spheres known to admit non-round Einstein metrics were $S^6$ and $S^8.$ This talk is based on joint work with Jan Nienhaus.

The Lagrange multipliers method relates critical points on a submanifold with those on an enlarged space. In derived algebraic geometry, we are allowed to consider a more general type of functions called shifted functions and thus a shifted version of the Lagrange multipliers method. If we start with quasi-smooth derived stacks, the Borisov-Joyce-Oh-Thomas virtual Lagrangian cycle of the critical locus coincides with the cosection localized virtual fundamental cycle of the enlarged space. This immediately implies the quantum Lefschetz principle of Chang-Li and an analogous result for branched covers. Based on a joint work with Hyeonjun Park. 

Harmonic maps from surfaces to other manifolds is a fundamental object of geometric analysis with many applications, for example to minimal surfaces. In particular, there are many available methods of constructing them such, such as using complex geometry, min-max methods or flow techniques. By contrast, much less is known for harmonic maps from higher dimensional manifolds. In the present talk I will explain the role of dimension in this problem and outline the recent joint work with D. Stern, where we provide a min-max construction for higher-dimensional harmonic maps. If time permits, an application to eigenvalue optimisation problems will be discussed. Based on joint work with D. Stern.

Mathematicians have been interested in the problem of compactifying the Jacobian variety of curves since the mid XIX century. In this talk we will discuss how all 'reasonable' compactified Jacobians of nodal curves can be classified combinatorically. This suffices to obtain a combinatorial classification of all 'reasonable' compactified universal (over the moduli spaces of stable curves) Jacobians. This is a joint work with Orsola Tommasi.

I'll present a computation in the algebraic approach to Vafa-Witten invariants of projective surfaces, as introduced by Tanaka-Thomas. The invariants are defined by integration over moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components. The physical notion of S-duality translates to conjectural symmetries between these contributions.  One component, the "vertical" component, is a nested Hilbert scheme on a surface. I'll explain work in preparation with M. Kool and T. Laarakker in which we express invariants of this component in terms of a certain quiver variety, the instanton moduli space of torsion-free framed sheaves on $\mathbb{P}^2$. Using a recent identity of Kuhn-Leigh-Tanaka, we deduce constraints on Vafa-Witten invariants conjectured by Göttsche-Kool-Laarakker. One consequence is a formula for the contribution of the vertical component to refined Vafa-Witten invariants in rank 2.