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January 7: William Graham (University of Georgia)
Title: A generalization of the Springer resolution Abstract: The Springer resolution of the nilpotent cone of a semisimple Lie algebra has important applications in representation theory, and in particular was used by Springer to give a geometric construction of the irreducible representations of Weyl groups. This talk concerns a generalization of the Springer resolution constructed with the use of toric varieties. We will discuss how this is connected in type A with Lusztig's generalized Springer correspondence, as well as an analogue of an affine paving of the fibers. Part of this talk is joint work with Martha Precup and Amber Russell. |
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January 14:
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January 21: Tudor Pădurariu (Columbia University)
Title: Relative stable pairs and a non-Calabi-Yau wall crossing Abstract: For complex smooth threefolds, there are enumerative theories of curves defined using sheaves, such as Donaldson-Thomas (DT) theory using ideal sheaves and Pandharipande-Thomas (PT) theory using stable pairs. These theories are conjecturally related among themselves and conjecturally related to other enumerative theories of curves, such as Gromov-Witten theory. The conjectural relation between DT and PT theories is known only for Calabi-Yau threefolds by work of Bridgeland, Toda, where one can use the powerful machinery of motivic Hall algebras due to Joyce and his collaborators. Bryan-Steinberg (BS) defined enumerative invariants for Calabi-Yau threefolds Y with certain contraction maps Y→X. I plan to explain how to extend their definition beyond the Calabi-Yau case and what is the conjectural relation to the other enumerative theories. This conjectural relation is known in the Calabi-Yau case by work of Bryan-Steinberg using the motivic Hall algebra. In contrast to the DT/ PT correspondence, we manage to establish the BS/ PT correspondence in some non-Calabi-Yau situations. |
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January 28: Sergej Monavari (Utrecht)
SPECIAL TIME: 10:00am pre talk, 10:30am main talk Title: Double nested Hilbert schemes and stable pair invariants Abstract:Hilbert schemes of points on a smooth projective curve are simply symmetric powers of the curve itself; they are smooth and we know essentially everything about them. We propose a variation by studying double nested Hilbert schemes of points, which parametrize flags of 0-dimensional subschemes satisfying certain nesting conditions dictated by Young diagrams. These moduli spaces are almost never smooth but admit a virtual structure à la Behrend-Fantechi. We explain how this virtual structure plays a key role in (re)proving the correspondence between Gromov-Witten invariants and stable pair invariants for local curves, and say something on their K-theoretic refinement. |
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February 4:
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February 11:
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February 18:
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February 25:
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March 4: Anton Mellit (University of Vienna)
SPECIAL TIME: 10:00am pre talk, 10:30am main talk Title: Integrals over Hilbert schemes and Macdonald polynomials Abstract: We apply results of Garsia-Haiman-Tesler on Macdonald polynomials to the problem of computation of integrals of tautological classes over the Hilbert schemes of surfaces, studied by Marian-Oprea-Pandharipande. Using localization, these results allow us to find new functional equations for the generating series of integrals. MOP paper considers two kind of integrals: the so-called Chern integrals resp. Verlinde integrals. The answer to the problem is encoded in series A1, A2, A3, A4, A5 resp. B1, B2, B3, B4. All the series except A4, A5, B3, B4 were computed in MOP and a conjecture motivated by mathematical physics was formulated relating A4 to B3 and A5 to B4. It was also conjectured that A4, A5, B3, B4 are algebraic functions. Solving our functional equations we prove the former conjecture and obtain explicit formulas for A4 and B3, thus proving a part of the latter conjecture. We also give a conjectural formula for A5 and B4. This is a joint work with Lothar Göttsche. |
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March 11:
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Organizers: Elham Izadi, James McKernan and Dragos Oprea
This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Past quarters: Fall 2013, Winter 2014, Spring 2014, Fall 2014, Fall 2017, Winter 2018, Spring 2018, Fall 2018, Winter 2019, Fall 2019, Winter 2020, Spring 2020, Fall 2020, Winter 2021, Spring 2021, Fall 2021.
The design of this webpage is copied shamelessly from the MIT Number Theory seminar site. Contact Samir Canning at srcannin@ucsd.edu about problems with the website or posters.