# 2022

## Upcoming

Time and place:
Sep. 29, 2022 2:15 PM–4:00 PM,
NHA B1120

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## Previous

Time and place:
Sep. 22, 2022 2:15 PM–4:00 PM,
NHA B1120

I will explain how motivic homotopy theory can be used to attack problems regarding finite projective modules over smooth affine k-algebras. I will recall in particular the foundational theorem of Morel and Asok-Hoyois-Wendt, and the construction of the Barge-Morel Euler class. Time permitting, I will explain recent progress on Murthy's splitting conjecture.

Time and place:
Sep. 15, 2022 2:15 PM–4:00 PM,
NHA B1119

Abstract (PDF)

Time and place:
Aug. 25, 2022 2:15 PM–4:00 PM,
NHA B1119

I will discuss the question in the title. This is joint work with Alex Degtyarev and Ilia Itenberg. This will be a talk involving very classical topics in algebraic geometry. I will try to make the talk accessible to students at master- and PhD level.

Time and place:
Jan. 20, 2022 2:15 PM–4:00 PM,
NHA B1120

Hilbert schemes of points for a surface are a well studied subject with many famous results like Göttsche’s formula for its Betti numbers. A natural generalization comes from studying Grothendieck’s Quot-schemes and the associated enumerative invariants. Unlike the former, punctual Quot-schemes are smooth only virtually admitting perfect obstruction theories and virtual fundamental classes. This has recently been used to study invariants counting zero-dimensional quotients of trivial vector bundles by multiple authors who used virtual localization and therefore could not treat the case of a general vector bundle. We rely on other techniques which use a general wall-crossing framework of D. Joyce to study these. Our methods rely on existence of a Lie algebra coming from vertex algebras constructed out of topological data. I will explain how these arise naturally in the Quot-scheme setting and how one can obtain explicit invariants and study their symmetries.