Next Meeting
Loughborough University - 27th March 2025
The next meeting of the COW will take place at Loughborough University on Thursday 27th March 2025. The meeting will start at 1:30pm and finish at 5:30pm. All talks will take place in room SCH.0.01 in the Schofield Buiding.
Speakers:- Gavin Brown (Warwick)
- Nick Rekuski (Liverpool)
- Fenglong You (Nottingham)
For those who do not wish to attend the meeting in-person, we also plan to broadcast talks live using Microsoft Teams. Information about how to join the Teams meeting will be circulated to the COW mailing list the day before the meeting. If you would like to join the COW mailing list, instructions on how to do so may be found on the mailing lists page. If you would like to attend the meeting remotely but do not want to join the mailing list, please send an email to Alan Thompson (A.M.Thompson (at) lboro.ac.uk) requesting the meeting information.
Schedule
Time | Speaker | Title |
1:30pm | Fenglong You | Relative mirror symmetry and some applications |
3:00pm | Gavin Brown | Classification of smooth flops |
4:30pm | Nick Rekuski | Stability of syzygy bundles |
Funding and Travel Claims
The COW has some funding to cover travel expenses for UK-based PhD students and postdocs. The COW is currently funded by the Heilbronn Institute for Mathematical Research under the UKRI/EPSRC Additional Funding Programme for Mathematical Sciences and the London Mathematical Society under a scheme 3 grant. To ensure that we can fund as many participants as possible, we ask that participants purchase "advance" or "off-peak" train tickets where practical, and non-travel expenses (e.g. accommodation, food) cannot be covered. For those under the age of 30, we also recommend looking into getting a railcard, which can offer substantial savings on the cost of train travel around the UK. Information about how to submit a claim is available on the COW homepage here.
Abstracts
- Fenglong You (Nottingham) - Relative mirror symmetry and some applications
- We consider mirror symmetry for a log Calabi--Yau pair (X,D), where X is a Fano variety and D is an anticanonical divisor of X. The mirror of the pair (X,D) is a Landau--Ginzburg model (X^\vee, W), where W is a function called the superpotential. Following the mirror constructions in the Gross--Sibert program, W can be described in terms of Gromov--Witten invariants of (X,D). I will explain a relative mirror theorem for the pair (X,D). As applications, I will explain how to compute the superpotential W, the classical period of W and more.
- Gavin Brown (Warwick) - Classification of smooth flops
-
I give an overview of a project with Michael Wemyss to classify simple 3-fold flops. This amounts to understanding which smooth 3-fold neighbourhoods C inside X of a rational curve C with normal bundle O(-3)+O(1) can be contracted. (The other possible flopping normal bundles are the Atiyah flop and Reid's Pagoda flops.)
As X is a manifold, one can describe the situation by glueing together patches, and in fact it is enough to glue together two copies of the affine space C^3 by a simple formula. However, the formula has infinitely many free parameters, and most choices describe neighbourhoods that do not contract. Even just finding examples of good glue functions seems to be needle-in-a-haystack. Instead, we translate the problem to one of classifying noncommutative germs f(x,y), or equivalently complete local Jacobian algebras up to isomorphism, as the putative contraction algebras of C in X. In this context, the conditions to make the right choices of parameters seem to be much more controllable.
Working with such germs feels much like a noncommutative version of classical singularity theory of function germs in the style of Arnold (types ADE and all that), and we can solve enough of this problem to construct all flops and to provide the classification. The resulting ADE classification presents Atiyah-Reid as Type A, an infinite discrete collection for Type D, and then families with moduli for the four (!) classes of Type E.
- Nick Rekuski (Liverpool) - Stability of syzygy bundles
- It is notoriously difficult to construct stable bundles with given topological invariants. This difficultly is partially because we have few general constructions of stable vector bundles. Recent work has focused on a conjectural construction called syzygy bundles which are bundles arising as the kernel of the evaluation map on global sections. In this talk, we show syzygy bundles associated to globally-generated line bundles are stable---confirming this conjectural construction.
This page is maintained by Alan Thompson and was last updated on 12/03/25. Please email comments and corrections to A.M.Thompson (at) lboro.ac.uk.