Titles and Abstracts

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LMS Regional Meeting (Monday 4 April)

Keynote speakers

Frances Kirwan, Moduli spaces, moduli stacks and quotients in algebraic geometry
When studying classification problems in algebraic geometry, we are faced with the issue that geometric objects usually vary in families depending on continuous parameters. The ideal solution to a classification problem is to find a moduli space, whose points correspond bijectively to equivalence classes of the objects to be classified, and whose geometric structure reflects the way the objects can vary in families. This is typically too much to hope for, unless we either restrict ourselves to very well-behaved objects, or extend our notion of a space and work with moduli stacks. The construction and study of moduli spaces and stacks can often be reduced to the study of algebraic group actions on algebraic varieties, and quotients of such actions; this will be the main focus of the talk.

Diane Maclagan, Solving equations with polynomial coefficients (with applications to toric and tropical geometry)
Despite the fact that both algebraically closed fields and rational functions in several variables can be introduced in a first undergraduate algebra class, the algebraic closure of K(t_1,...,t_n) is surprisingly poorly understood. In this talk I will discuss the state of the art, including a new algebraically closed field containing K(t_1,...,t_n) introduced recently in joint work with Gandini, Hering, Mohammadi, Rajchgot, Wheeler and Yu. Our motivation, as I will explain, was to generalise a toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. One application of this is a Bertini theorem for tropical varieties.

Richard Thomas, Counting sheaves on Calabi-Yau 4-folds, I

I will give an introduction to virtual cycles on moduli spaces. These are the foundations on which any theory in modern enumerative algebraic geometry is built. At the end I will explain what goes wrong for moduli spaces of sheaves on Calabi-Yau 4-folds, and give an overview (with no details) of two ways to fix it. This talk will serve as an introduction to Jeongseok Oh’s more detailed talk tomorrow.

20-minute morning talks (at 11.20 am)

Yannik Schuler (Sheffield), Correspondences in enumerative geometry
I would like to talk about a correspondence between logarithmic, local, and open Gromov--Witten invariants. Even though these three types of invariants are all curve counts they all have a very different flavour. However, it turns out that starting from a pair (X,D) where X is a smooth complex projective surface and D an anti-canonical divisor satisfying certain positivity conditions one can construct three different geometries for which the above mentioned GW counts are related by explicit formulae. This talk can be seen as a warm-up and advertisement for the talk of Andrea Brini on Wednesday were we will learn that the here mentioned correspondence is actually a piece of a much larger web of dualities.

Alberto Cobos Rabano (Sheffield), Genus-0 quasimaps to toric varieties
We study enumerative invariants of toric varieties. In the genus-0 case, we are interested in morphisms from P^1 to X. The moduli space of stable maps produces a compactification by allowing the source curve to be not just P^1, but a tree of P^1's. On the other hand, we could take a different compactification, such as the moduli space of stable quasimaps. There, points in the boundary may also have a tree of P^1's as source but also the morphism to X may be replaced by a rational functions with certain base-points. The goal of the talk is to introduce these moduli spaces and explain how they are related geometrically, with possible applications to enumerative invariants.

Shengxuan Liu (Warwick), Stronger Bogomolov-Gieseker inequality and Bridgeland stability conditions

In this talk, I will first introduce the definition of Bridgeland stability conditions, which generalize the stability conditions on curves to higher dimensional varieties. Then I will talk about stronger Bogomolov-Gieseker inequality and its connection with Bridgeland stability conditions, especially in the case of X_{2,4} in P^5.

Samuel Johnston (Cambridge), Gross-Siebert and Keel-Yue mirror families
The past few years have seen much progress in the construction of mirror families associated with (log) Calabi-Yau varieties. We will briefly review two of these constructions, one due to Gross and Siebert using log Gromov-Witten invariants, and the other due to Keel and Yue in a slightly more restrictive setting using naive non-archimedean curve counts. I will sketch a proof demonstrating that when both mirror families can be constructed, they agree in most situations. The proof for this fact mostly boils down to showing certain log Gromov-Witten invariants are enumerative, and time permitting, I will show how to apply this fact to derive solutions to certain concrete enumerative problems.

MMM in Midlands Workshop (Tuesday 5 April--Thursday 8 April)

Christian Böhning, Skew matrices of linear forms, matrix factorisations and intermediate Jacobians of cubic threefolds

I will report on some joint work with Hans-Christian von Bothmer (Hamburg).

Results due to Druel and Beauville show that the blowup of the intermediate Jacobian of a smooth cubic threefold X in the Fano surface of lines can be identified with a moduli space of semistable sheaves of Chern classes c_1=0, c_2=2, c_3=0 on X. We identify this space with a space of matrix factorizations. This has the advantage that this description naturally generalizes to singular and even reducible cubic threefolds. In this way, given a degeneration of X to a reducible cubic threefold X_0, we obtain an associated degeneration of the above moduli spaces of semistable sheaves.

Andrea Brini, Curve counting on surfaces and topological strings
Enumerative geometry is a venerable subfield of Mathematics, with roots dating back to Greek Antiquity and a present inextricably linked with developments in other domains. Since the early 90s, in particular, the interaction with String Theory has sent shockwaves through the subject, giving both unexpected new perspectives and a remarkably powerful, physics-motivated toolkit to tackle several traditionally hard questions in the field.
I will survey some recent developments in this vein for the case of enumerative invariants associated to a pair (X, D), with X a complex algebraic surface and D a singular anticanonical divisor in it. I will describe a surprising web of correspondences linking together several a priori distant classes of enumerative invariants associated to (X, D), including the log Gromov-Witten invariants of the pair, the Gromov-Witten invariants of an associated higher dimensional Calabi-Yau variety, the open Gromov-Witten invariants of certain special Lagrangians in toric Calabi–Yau threefolds, the Donaldson–Thomas theory of a class of symmetric quivers, and certain open and closed Gopakumar-Vafa-type invariants. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants.

Matthew Buican, Comments on Galois Actions in TQFT
2+1 dimensional TQFTs are naturally organised into orbits under the action of certain Galois groups. This basic fact has important physical implications when searching for observables that might classify topological phases of matter. At the same time, the Galois structure of TQFT is tied to better understanding certain number theory questions underlying TQFT. I will discuss and re-examine some of these mathematical and physical implications of the TQFT Galois action in light of a series of results on Galois action in TQFT.

Mathew Bullimore, Generalised Symmetries and Moduli Stacks

I will review aspects of generalised global symmetries and their 't Hooft anomalies in supersymmetric quantum field theories and explain how this information is encoded geometrically in moduli stacks, which lift the traditional course moduli spaces of supersymmetric vacua to incorporate the presence of a non-trivial TQFT at a point on the moduli space. Examples will include Higgs and Coulomb branch moduli stacks in 3d N = 4 gauge theories.

Tudor Dimofte, Aspects of 3d N=4 TQFT
I will discuss aspects of the (putative) TQFT's that arise from topological twists of 3d N=4 gauge theories, including 1) their (necessarily) non-semisimple and derived structure; 2) their deformations by background connections associated to flavor symmetry; and 3) the presence and role of boundary vertex algebras. I will also briefly discuss an interesting family of examples from recent work with T. Creutzig, N. Garner, and N. Geer, related to quantum groups at roots of unity.

Lotte Hollands, Partition functions, BPS states and abelianization

Recently, there have been various exciting developments in the interplay between BPS structures, topological string partition functions and exact WKB analysis. In this talk I will report on this from the perspective of four-dimensional N=2 field theory and its lift to five dimensions. I will introduce the four-dimensional Nekrasov partition function and rewrite this partition function in the Nekrasov-Shatashvili limit as an integral of a ratio of Wronskians of solutions to an associated differential equation, with the AD2 theory and the pure SU(2) theory as two main examples. This motivates the introduction of a generalized Nekrasov partition function which encodes the four-dimensional BPS spectrum and may be defined entirely in terms of exact WKB analysis. In five dimensions this partition function captures the non-perturbative quantum geometry of the topological string. This talk is based on 2109.14699, 2203.08249 and work in progress.

Cristina Manolache, Reduced Gromov--Witten invariants
In a series of papers Zinger, Li--Zinger and Vakil--Zinger constructed reduced genus one Gromov--Witten invariants and related them to usual Gromov--Witten invariants. I will discuss a way to generalise this construction.

Jeongseok Oh, Counting sheaves on Calabi-Yau 4-folds, II
Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry.

We construct an algebraic virtual cycle, which reproduces the invariants of Borisov-Joyce. A key step is a localisation of Edidin-Graham’s square root Euler class for SO(2n) bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We also give a K-theoretic refinement by defining K-theoretic square root Euler classes and their localised versions.

This is joint work with Richard Thomas.

Alice Rizzardo, Many examples of non-Fourier-Mukai functors

Functors between the derived categories of two smooth projective varieties are a fundamental object of study. Almost all known such functors are so-called Fourier-Mukai: roughly speaking, they are well behaved with respect to the geometry of the varieties. They admit a lift to a functor between the enhancements of the two derived categories.
The first example of a non-Fourier-Mukai functor was given by myself and Van den Bergh in 2015. I will show that this is not a pathological example by providing a way to construct a non-Fourier-Mukai functor from the derived category of any smooth projective variety of dimension greater or equal to 3 admitting a tilting bundle. This is joint work with Theo Raedschelders and Michel Van den Bergh.

Sakura Schafer-Nameki, Canonical Singularities and 5d Superconformal Field Theories
I will report on recent progress on 5d Superconformal Field Theories (SCFTs) and their realization in M-theory on canonical three-fold singularities, including their moduli spaces (Higgs and Coulomb branches) as well as the imprint of generalized symmetries in the geometry.

Nick Sheridan, Quantum cohomology as a deformation of symplectic cohomology
Let X be a compact symplectic manifold, and D a normal crossings symplectic divisor in X. We give a criterion under which the quantum cohomology of X is the cohomology of a natural deformation of the symplectic cochain complex of X \ D. The criterion can be thought of in terms of the Kodaira dimension of X (which should be non-positive), and the log Kodaira dimension of X \ D (which should be non-negative). We will discuss applications to mirror symmetry. This is joint work with Strom Borman and Umut Varolgunes.

Dimitri Wyss, BPS-invariants from non-archimedean integrals
Let $M(\beta,\chi)$ be the moduli space of one-dimensional semi-stable sheaves on a del Pezzo surface $S$, supported on an ample curve class $\beta$ and with Euler-characteristic $\chi$. Working over a non-archimedean local field $F$, we define a natural measure on the $F$-points of $M(\beta,\chi)$ and prove that the integral of a certain gerbe on $M(\beta,\chi)$ with respect to this measure is independent of $\chi$. A recent result of Maulik-Shen then implies that these integrals compute the so-called BPS-invariants of $M(\beta,\chi)$, which appear in the context of Donaldson-Thomas theory. This is joint work with Francesca Carocci and Giulio Orecchia.