# Seminars & Colloquia Calendar

## Nested Hilbert schemes, local Donaldson-Thomas theory, and Vafa-Witten / Seiberg-Witten correspondence

#### Artan Sheshmani (Harvard)

Location: ** Serin E372**

Date & time: Thursday, 08 February 2018 at 1:30PM - 2:30PM

Abstract: We report on the recent rigorous and general construction of the deformation-obstruction theories and virtual fundamental classes of

nested (flag) Hilbert scheme of one dimensional subschemes of a smooth projective algebraic surface. The nested Hilbert scheme is a moduli space, which parametrizes a nested chain of configurations of curves and points in the ambient surface. This construction will provide one with a

general framework to compute a large class of already known invariants, such as Poincare invariants of Okonek et al, or the reduced

local invariants of Kool and Thomas in the context of their local surface theory. We show how to compute the generating series of

deformation invariants associated to the nested Hilbert schemes, and via exploiting the properties of vertex operators, prove that in

some cases they are given by modular forms. We finally establish a connection between the Vafa-Witten invariants of local-surface

threefolds (recently analyzed also by Tanaka and Thomas) and such nested Hilbert schemes. This construction

(via applying Mochizuki’s wall- crossing techniques) enables one to obtain a relation between the generating series of

Seiberg-Witten invariants of the surface, the Vafa-Witten invariants, and some modular forms. This is joint work with Gholampour and Yau

appeared in arXiv:1701.08902 and arXiv:1701.08899.

Chiara Damiolini, Ian Coley and Franco Rota -Charles Weibel Organizer's Page

Brooke Logan

Wujun Zhang Organizer's webpage

P. Gupta, X.Huang and J. Song Organizer's webpage

Swastik Kopparty, Sepehr Assadi Seminar webpage

Jeffry Kahn, Bhargav Narayanan, Jinyoung Park Organizer's webpage

Brooke Ogrodnik, Website

Robert Dougherty-Bliss and Doron Zeilberger --> homepage

Paul Feehan, Daniel Ketover, Natasa Sesum Organizer's webpage

Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward Organizer's webpage

Jason Saied Seminar webpage

Brian Pinsky, Rashmika Goswami website

Quentin Dubroff Organizer's webpage

James Holland; Organizer website

Edna Jones Organizer's webpage

Brooke Ogrodnik website

Yanyan Li, Zheng-Chao Han, Jian Song, Natasa Sesum Organizer's Webpage

Organizer: Luochen Zhao

Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song Organizer's Page

Lisa Carbone, Yi-Zhi Huang, James Lepowsky, Siddhartha Sahi Organizer's webpage

Simon Thomas website

Kasper Larsen, Daniel Ocone and Kim Weston Organizer's page

Joel Lebowitz, Michael Kiessling

Yanyan Li, Haim Brezis Organizer's Webpage

Stephen D. Miller, John C. Miller, Alex V. Kontorovich, Alex Walker seminar website

Stephen D. Miller

Brooke Ogrodnik, Website

Organizers: Yanyan Li, Z.C. Han, Jian Song, Natasa Sesum

Yael Davidov Seminar webpage

Kristen Hendricks, Xiaochun Rong, Hongbin Sun, Chenxi Wu Organizer's page

Fioralba Cakoni Seminar webpage

Ebru Toprak, Organizer

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