Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

 

Past events in this series


Tue, 16 Jun 2026
15:30
L4

Wall-crossing Package via Non-Abelian Localization

Ivan Karpov
(MIT)
Abstract
Recent and seminal work of Dominic Joyce and his coauthors has produced a new (and, indeed, the first) wall-crossing machinery in the context of certain quasi-smooth moduli stacks of abelian categories: quiver representations, sheaves on Fano threefolds, and so forth.
Henry Liu has later explained how its K-theoretic version should look like.
 
Most importantly, perhaps, this machinery defines reasonable virtual fundamental classes for moduli stacks that may contain strictly semistable objects.
Unfortunately, these results do not, without further modification, apply to stacks of objects in derived categories (as opposed to abelian ones) since they require certain additional data.
This data, the so-called 'framing functor', plays an important rôle in the original constructions, and is unavailable in the derived case.
 
I shall try to explain a modest extension of Joyce-Liu’s K-theoretic Monster Wall-Crossing Formalism which, in most cases, makes it possible to dispense with this additional data, and clarifies the relation to motivic wall-crossing.
Our proof of this extension is very different from Joyce’s own, and is based instead on Halpern-Leistner’s Non-Abelian Localization (NAL) Theorem, and on the use of Blanc's topological K-theory.
 
The applications include carrying out the Feyzbakhsh–Thomas programme for Fano threefolds with even canonical class, and proving (simultaneously with R. Anderson and D. Joyce, though under stricter assumptions on the underlying variety) rationality and functional equations for generating functions of Pandharipande–Thomas invariants.
 
Time permitting, I shall also try to sketch a very short proof of the wall-crossing formula for Calabi–Yau 4-folds (conjectured by Joyce and later investigated by Bojko) which follows the NAL strategy and uses the so-called Drinfeld–Gaitsgory degeneration. This argument explains also the relation between the NAL story and the hyperbolic localization package.
 
Everything is joint with M. Moreira, and is partly in progress.
Add to calendar