# University of Nottingham

## 14-15 September 2023

Our inaugural conference will take place at the University of Nottingham on Thursday 14th and Friday 15th September 2023.

All talks will take place in the Keighton Auditorium, which is number 56 on the campus map (linked below) and next door to the maths department (number 20).

Schedule

The schedule for the event will be as follows; abstracts may be found below.

Thursday 14th September

1:00pm - Rohini Ramadas (Warwick) - Intersections of tropical psi classes in genus zero, with non-trivial valuations

2:30pm - Emily Norton (Kent) - Parabolic Kazhdan-Lusztig polynomials and oriented Temperley-Lieb algebras

3:45pm - EPSRC Discussion Session

4:30pm - Nelly Villamizar (Swansea) - Multivariate splines in algebraic geometry

Evening - Conference dinner (self-funded)

Friday 15th September

9:30am - Gwyn Bellamy (Glasgow) - Birational geometry of quiver varieties

11:00am - Dominic Joyce (Oxford) - The structure of invariants counting coherent sheaves on complex surfaces

12:00pm - Lunch

1:30pm - Ruadhaí Dervan (Glasgow) - Valuations and stability of projective varieties

3:00pm - Richard Thomas (Imperial) - The quantum Lefschetz principle

Registration

Registration has now closed. If you have not already registered and would still like to attend the event, please email Alan Thompson (a.m.thompson (at) lboro.ac.uk).

Accommodation and Travel

We can provide overnight accommodation on 14th September for a limited number of UK-based students and postdocs. We also have some funding to cover travel costs for UK-based students and postdocs.

If you would like to apply for accommodation and/or travel funding, please indicate this on the registration form. The deadline for applying for accommodation is Friday 28th July 2023. In case we receive more applications than we have funding to cover, accommodation and travel funding will be allocated on a first-come, first-served basis.

To claim reimbursement for travel expenses, please download and complete the following form (leaving the "Financial Codes" column blank). Note that this link will access the form in Excel (.xlsx) format; if possible, the form should be completed and returned in the same format.

Scan the receipts for all expenses being claimed and email them together with the completed form to Anne-Sophie Kaloghiros (Anne-Sophie.Kaloghiros (at) brunel.ac.uk). Please retain your original receipts until you have been reimbursed.

Please note that all expenses claims must be received within three weeks of the date on which the expense was incurred.

Abstracts

Rohini Ramadas (Warwick) - Intersections of tropical psi classes in genus zero, with non-trivial valuations

Psi classes are tautological divisor classes on moduli spaces of stable curves. In genus zero they determine birational morphisms to projective space. Tropical psi classes were defined by Mikhalkin as subfans of M_{0,n}-trop. Their stable intersections were computed by Kerber and Markwig, and shown to be equivalent to the corresponding algebro-geometric intersections.

I will present joint work with Sean Griffin, Jake Levinson and Rob Silversmith, in which we expand the notion of tropical psi classes by considering tropicalizations of carefully chosen effective representatives defined over non-trivially valued fields. We study their intersections in this modified setting, and establish several attractive features. I will contrast with the Kerber-Markwig tropical intersections, and describe how the two versions encode dual information.

Emily Norton (Kent) - Parabolic Kazhdan-Lusztig polynomials and oriented Temperley-Lieb algebras

Work by Brundan and Stroppel in the 2010s showed that parabolic Kazhdan-Lusztig polynomials in type A arise from diagrammatic algebras they called extended Khovanov arc algebras. These polynomials appear all over the place in representation theory. I will talk about another place they seem to crop up -- work in progress with Olivier Dudas. And I will discuss how to explain the diagrammatic rule for computing them using an oriented version of the Temperley-Lieb algebra -- work with Chris Bowman, Maud De Visscher, Niamh Farrell, and Amit Hazi.

Nelly Villamizar (Swansea) - Multivariate splines in algebraic geometry

Splines are piecewise polynomial functions defined over a real domain which are continuously differentiable to some order r. For a fixed integer d, the space of splines of degree at most d and smoothness r is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when r>1. It is particularly difficult to compute the dimension of splines on partitions in which a vertex is completely surrounded by tetrahedra – we call these domains vertex stars.

In the talk, I will present a lower bound formula on the dimension of splines on vertex stars and how that leads to prove a lower bound on the dimension of splines defined over general tetrahedral partitions in large degree. The proofs use apolarity, some results from rigidity theory, and the so-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. I will show some examples and open problems in the area.

Gwyn Bellamy (Glasgow) - Birational geometry of quiver varieties

Nakajima quiver varieties provide a large class of interesting algebraic varieties whose geometry can be studied combinatorially. These varieties are generally very singular, but their singularities can (in most cases) be resolved by variation of GIT. In these cases, I'll describe how one can explicitly describe the GIT chamber structure (equivalently, Mori chamber structure) that describes all possible crepant resolutions. I'll include several explicit examples to illustrated the combinatorics involved. This is based on joint work with Alastair Craw and Travis Schedler.

Dominic Joyce (Oxford) - The structure of invariants counting coherent sheaves on complex surfaces

Let X be a complex projective surface with geometric genus p_g > 0. We can form moduli spaces M_{(r,a,k)}^{st} \subset M_{(r,a,k)}^{ss} of Gieseker (semi)stable coherent sheaves on X with Chern character (r,a,k), where we take the rank r to be positive. In the case in which stable = semistable, there is a (reduced) perfect obstruction theory on M_{(r,a,k)}^{ss}, giving a virtual class [M_{(r,a,k)}^{ss}]_{virt} homology.

By integrating universal cohomology classes over this virtual class, one can define enumerative invariants counting semistable coherent sheaves on X. These have been studied by many authors, and include Donaldson invariants, K-theoretic Donaldson invariants, Segre and Verlinde invariants, part of Vafa-Witten invariants, and so on.

In my paper https://arxiv.org/abs/2111.04694, in a more general context, I extended the definition of the virtual class [M_{(r,a,k)}^{ss}]_{virt} to allow strictly semistables, proved wall-crossing formulae for these classes and associated "pair invariants", and gave an algorithm to compute the invariants [M_{(r,a,k)}^{ss}]_{virt} by induction on the rank r, starting from data in rank 1, which is the Seiberg-Witten invariants of X and fundamental classes of Hilbert schemes of points on X. This is an algebro-geometric version of the construction of Donaldson invariants from Seiberg-Witten invariants; it builds on work of Mochizuki 2008.

This talk will report on a project to implement this algorithm, and actually compute the invariants [M_{(r,a,k)}^{ss}]_{virt} for all ranks r > 0. I prove that the [M_{(r,a,k)}^{ss}]_{virt} for fixed r and all a,k with a fixed mod r can be encoded in a generating function involving the Seiberg-Witten invariants and universal functions in infinitely many variables. I will spend most of the talk explaining the structure of this generating function, and what we can say about the universal functions, the Galois theory and algebraic numbers involved, and so on. This proves several conjectures in the literature by Lothar Gottsche, Martijn Kool, and others, and tells us, for example, the structure of U(r) and SU(r) Donaldson invariants of surfaces with b^2_+ > 1 for any rank r \ge 2.

Ruadhaí Dervan (Glasgow) - Valuations and stability of projective varieties

The notion of K-stability of a projective variety is of interest primarily due to its links with moduli theory (through the construction of moduli spaces of projective varieties) and Kähler geometry (through the existence of special Kähler metrics). The surrounding theory is, by now, almost fully understood in the special case of Fano varieties. While K-stability has its roots in geometric invariant theory, the success in the Fano setting is instead primarily due to new ideas from birational geometry and the theory of valuations. I will explain a new valuative approach to the theory of K-stability for general polarised varieties (joint work with E. Legendre, along with subsequent work of Boucksom-Jonsson), and will then explain some recent concrete applications of these techniques (joint work with T. Papazachariou).

Richard Thomas (Imperial) - The quantum Lefschetz principle

“Quantum Lefschetz” is a pretentious name for understanding how moduli spaces -- and their virtual cycles and associated invariants -- change when we apply certain constraints. (The original application is to genus 0 curves in P^4 when we impose the constraint that they lie in the quintic 3-fold.)

When it doesn’t work there are fixes (like the p-fields of Guffin-Sharpe-Witten/Chang-Li) for special cases associated with curve-counting. I will describe joint work with Jeongseok Oh developing a general theory.

Funding

This event is taking place as part of the UK Algebraic Geometry Network. The network is funded by a Network Support for the Mathematical Sciences grant from the Isaac Newton Institute, via EPSRC grant EP/V521929/1.

Some participant travel costs will also be covered by the COW. 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.