- 3/14/2022: A blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces and its applications

- 9/10/2018: On the Vafa-Witten theory on closed four-manifolds

### March 14, 2023

TITLE: A blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces and its applications

ABSTRACT: I’ll talk about a blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces and its applications at the level of the generating series of those invariants. For instance, we obtain blowup formulae for the generating series of the virtual Euler characteristics and virtual χ_{y}-genera of the moduli spaces, in which modular forms appear in the same way as in Vafa-Witten’s original paper in ’94, as Goettsche and Goettsche-Kool conjectured. These enable one to compute some of universal functions in the generating series of the instanton part of the Vafa-Witten invariants on a projective surface, provided they existed as Goettsche-Kool and Goettsche-Kool-Laarakker conjectured. This talk is based on joint work arXiv:2107.08155 with Nikolas Kuhn and arXiv:2205.12953 with Nikolas Kuhn and Oliver Leigh.

### September 10, 2018

TITLE: On the Vafa-Witten theory on closed four-manifolds

ABSTRACT: Vafa and Witten introduced a set of gauge-theoretic equations on closed four-manifolds around 1994 in the study of S-duality conjecture in N=4 super Yang-Mills theory in four dimensions. They predicted from supersymmetric reasoning that the partition function of the invariants defined through the moduli spaces of solutions to these equations would have modular properties. But little progress has been made other than their original work using results by Goettsche, Nakajima and Yoshioka.

However, it now looks worth trying to figure out some of their foreknowledge with more advanced technologies in analysis and algebraic geometry fascinatingly developed in these two decades. This talk discusses issues to construct the invariants out of the moduli spaces, and presents possible ways to sort them out by analytic and algebro-geometric methods; the latter is joint work with Richard Thomas.