Past Talks 2024/2025

Seminar Organisers: Marc Truter and David Hubbard

Reading groups:

Term 1 (GIT) - Marc Truter: The term 1 reading group will be on GIT, following Victoria Hoskins' notes. See my webpage for the notes from our talks.

Term 2 (K-Stability) - Marc Truter: The term 2 reading group will be on K-stability, following Harold Blum's notes. See my webpage for the notes from our talks.

Rooms and Times

Term 1: D1.07 12-1pm Thursdays for reading group, D1.07 3-4pm Thursday for talks

Term 2: B1.12 12-1pm Thursdays for reading group, D1.07 3-4pm Thursdays (Week 7: 11-12 A1.01)

Term 3: MB0.08 3-4pm Thursdays

Term 3

Week 1 April 24

Antoine Pinardin - Edinburgh

Linearization problem for finite subgroups of the plane Cremona group (joint work with Arman Sarikyan and Egor Yasinsky)

The plane Cremona group is the group of birational self-maps of the projective plane. Over the field of complex numbers, its subgroups have been extensively studied, and the most complete classification dates back to 2006, with the work of Dolgachev and Iskovskikh. They outline a question yet to be answered, which consists in describing thelinearizable subgroups of the plane Cremona group, those which are conjugated to a subgroup of linear automorphisms of the projective plane. This problem is of particular importance, because it is equivalent to the question of G-equivariant rationality. We give a complete answer over the field of complex numbers.

Week 2 May 1

Gabriel Frey, Shing Tak Lam, Yaoqi Yang - Glasgow

Week 3 May 8

James Jones - Loughborough

(Explicit) Tyurin Degenerations of K3 Surfaces of Degree 4

Moduli spaces of K3 surfaces have long been studied, beginning with the proof of the Torelli theorem for K3s in the 70s. The most well-known compactification of this moduli space remains the Baily-Borel compactification, whose boundary consists of components of dimension 1 and of dimension 0. In this talk we consider the moduli space of K3 surfaces of degree 4 and investigate the 1-dimensional boundary components of its Baily-Borel compactification. Each such boundary component is associated with a rank 17 lattice, and corresponds to a Type II degeneration. We consider Tyurin degenerations, which are the simplest Type II degenerations, and exhibit an explicit correspondence between them and the rank 17 lattices. We then describe 18-dimensional families of these degenerations, each of which is distinguished by a choice of points on an elliptic curve. We conclude by establishing all log canonical models for very general Tyurin degenerations of degree 4. This talk is based on the preprint arxiv:2502.04301.

Week 4 May 15

Charlotte Satchwell - Essex

K-moduli of exceptional del Pezzo hypersurfaces

The minimal model program predicts that Fano varieties are one of the building varieties of all other varieties. The finite number of Fano varieties allows their study however the large number of automorphisms associated to them can cause problems. K-moduli was introduced as a solution to this to give a well behaved moduli space. In this research we use new techniques from non reductive Geometric Invariant Theory to classify the K-moduli of the exceptional del Pezzo hypersurfaces.

Week 5 May 22

Peize Liu - Warwick

Bridgeland stability via quadric fibrations

Bridgeland stability conditions have been successfully constructed on the Kuznetsov component of cubic 3-folds and 4-folds using conic fibration techniques. In this talk, I will introduce the key ideas behind this method and generalise it to give a construction on the cubic 5-folds.

Week 6 May 29

Francesca Rizzo - Paris (IRM PRG)

Double covers of normal varieties and application to quadratic degeneracy loci

The aim of this talk is to present the general construction of double covers of normal varieties and its application to double covers of quadratic degeneracy loci, as developed by Debarre and Kuznetsov in [1].

Double covers of smooth varieties are classically parametrized by 2-torsion line bundles. This correspondence extends to normal schemes, where double covers are parametrized by self-dual rank-one reflexive sheaves. Starting from this framework, the authors of [1] construct double covers of degeneracy loci arising from families of quadratic forms, under suitable regularity assumptions. This approach generalizes the classical case of degeneracy loci of symmetric matrices.

In this talk, we will introduce double covers and explain their relation with reflexive sheaves. We will then show the example of degeneracy loci of symmetric matrices, in which we obtain some double covers that are branched in codimension >1. Finally, we will discuss the general result of double covers of quadratic degeneracy loci associated with families of quadratic forms.

Week 9 Jun 19

Chunkai Xu

Derived category of Grassmannians

In this talk, we will introduce full exceptional collections on Grassmannians constructed by Kapranov in 1990s, and some applications in K-theory of Grassmannians. Based on joint work in progress with Sunny Sood.

Week 10 Jun 26

Sara Veneziale - I-X Centre for AI

Graph Networks for Graph Theory

There have been many new examples of AI guiding discovery in pure mathematics -- from suggesting new conjectures to accelerating computational routines. In this talk we will survey some recent work (Wagner 2021) about using reinforcement learning to find counterexamples in graph theory and we will discuss current work (with Fanglan Feng) about how to combine this approach with Graph Neural Networks. We will finish by talking about possible new research directions.

Term 2

Week 1 Jan 9

Reading group [Peize Liu, GIT and blowups](B1.12 12-1pm Thursday)

Week 3 Jan 23

Reading group [Marc Truter, Introduction to K-stability](B1.12 12-1pm Thursday)

Arun Soor - Oxford (D1.07 3-4pm Thursday)

Six-functor formalisms and derived rigid-analytic geometry

The “yoga” of the six operations is a notion which goes back to Grothendieck. It was originally an evocative notion alluding to the functorial properties satisfied by various categories of coefficients on geometric objects X, for instance D-modules or l-adic sheaves.

In recent years, due to the breakthrough works of Liu--Zheng, Gaitsgory--Rozenblyum, and Mann, our understanding of the six operations has developed from a “yoga" to a rigorously defined notion. I will try to explain this development.

Then I will try to explain how to construct a six-functor formalism for quasi-coherent sheaves in derived rigid-analytic geometry, using the recent work of Ben-Bassat--Kelly--Kremnitzer. Along the way, I'll try to explain why it's quite natural to derive rigid-analytic geometry for this and other reasons.

Week 4 Jan 30

Reading group [Marc Truter, K-stability of polarised varieties](B1.12 12-1pm Thursday)

Week 5 Feb 6

Reading group [Alvaro Gonzalez Hernandez, Odaka's theorems](B1.12 12-1pm Thursday)

Hamdi Dërvodelli - Warwick (D1.07 3-4pm Thursday)

Reducible varieties and tropical geometry

The factoring locus of a polynomial is a list of conditions on its coefficients under which the polynomial factors. The aim of this talk is to explore any potential connections tropical geometry has with this factoring locus. More generally, we want to know if the reducibility of a variety is detected by the tropicalization of its defining ideal. We wonder the same about the smoothness of a variety as well.

Week 6 Feb 13

Amy Li - Texas Austin (D1.07 3-4pm Thursday)

Double covers and nontautological classes on M_{g,n}

The cohomology ring of the moduli space of curves is an object of ongoing study, but is generally too big to work with. In this talk, I will provide an introduction to a special subring, called the tautological ring, which has an explicit generating set but turns out to contain many natural algebraic cycles. A basic question, then, is to ask when the tautological ring is equal to the full cohomology ring. In other words, when do nontautological classes exist?

I will present on recent work which produces an infinite family of nontautological classes on M_{g,n} coming from a particular bielliptic locus. After careful study of the geometry of these curves, we use a Hodge-Kunneth decomposition and the existence of holomorphic forms in certain degrees to prove the result. This is joint work with V Arena, S Canning, E Clader, R Haburcak, SC Mok, and C Tamborini.

There will be pictures!

Week 7 Feb 20

Ruth Wye - Bath [A1.01 11-12 (Rescheduled due to COW)]

Hilbert schemes of points on crepant partial resolutions

The Hilbert scheme of points on smooth surfaces has been well-studied, and is well known. In the case of the minimal resolution of an ADE singularity, there is a nice description of the Hilbert scheme of points as a quiver variety due to Kuznetsov (2007). Much more recently, it was proven that the Hilbert scheme of points on an ADE singularity is also a quiver variety for the same quiver (Craw-Yamagishi, 2023). In this talk, I will present a description of the Hilbert scheme of points of any crepant partial resolution of an ADE singularity as a quiver variety, and explain some of the consequences of this description, based on joint work with Alastair Craw.

Week 10 Mar 13

Joris Köfler - Max Planck (Leipzig) (D1.07 3-4pm Thursday)

Taking the amplituhedron to the limit

The amplituhedron is a semi-algebraic set given as the image of the positive Grassmannian under a linear map subject to a choice of additional parameters. We define the limit amplituhedron as the limit of amplituhedra by sending one of the parameters to infinity. We derive its algebraic boundary in terms of Chow varieties and explain its geometric interpretation from the point of view of Positive Geometry in terms of secants of the rational normal curve. This discussion shows that the limit amplituhedron is a positive geometry with a unique differential form.

Joint work with Rainer Sinn.

Term 1

Week 2 Oct 10

Reading group [Marc Truter, Introduction to Moduli problems and GIT] (2-3pm Thursday C1.06)

Week 3 Oct 17

Reading group [Arnaud Vilpert, Moduli problems] (12-1pm Thursday D1.07)

Week 4 Oct 24

Reading group [Chunkai Xu, Groups and actions] (12-1pm Thursday D1.07)

Joe Malbon - Edinburgh (3-4pm Thursday D1.07)

Classification of Algebraic Varieties

Classification - the identification of similar objects and the distinction of different ones, as well as the construction of spaces that parametrise such equivalence classes - is a guiding meta-principle in algebraic geometry. It was realised around the turn of the last century that isomorphisms are too rigid a notion of equivalence for classification to be achievable, and that the more flexible notion of birational equivalence should be used instead. This point of view was successfully applied to algebraic surfaces, whose birational classification was more or less completed by the 1950s.

For higher-dimensional varieties, it is often said that all varieties are birationally (and conjecturally) constructed from Fano, Calabi-Yau, and canonically polarised varieties, and thus classification may proceed inductively based on dimension. In this talk, I will explain the birational classification of varieties from the viewpoint of the minimal model program, which is an algorithm that constructs varieties in this inductive way. If time permits, I will explain the application of the minimal model program to the construction of moduli spaces, which parametrise equivalence classes of varieties with similar properties, and illustrate some of my own work on the construction of the moduli space of a particular family of K-polystable Fano threefolds.

Week 5 Oct 31

Reading group [Alvaro Gonzalez Hernandez, Affine GIT](12-1pm Thursday D1.07)

Week 6 Nov 7

Reading group [Tommaso Faustini, Projective GIT] (12-1pm Thursday D1.07)

Thamarai Valli - UCL (LSGNT) (3-4pm Thursday D1.07)

Complete intersections in Toric varieties: A Combinatorial Adventure!

Toric varieties are a fascinating class of algebraic varieties with a rich combinatorial structure that simplifies many computations. Seeking to extend this combinatorial advantage to broader classes, we focus on complete intersections within toric varieties, which inherit some of the combinatorial features of toric varieties. This additional structure makes them particularly useful for the classification of Fano varieties, a central problem in algebraic geometry.

In this talk, we’ll begin with a formal introduction to toric varieties as GIT quotients, then explore complete intersections within these varieties and their associated combinatorial properties. We will then examine how these properties aid in classifying Fano varieties. If time permits, I’ll also discuss my recent work on the classification of Fano complete intersections in toric varieties.

Week 7 Nov 14

Reading group [Marc Truter, Stability] (12-1pm Thursday D1.07)

Week 8 Nov 21

Reading group [Peize Liu, Moduli of vectors bundles I] (12-1pm Thursday D1.07)

Michela Barbieri - Imperial (LSGNT) (3-4pm Thursday D1.07)

Derived Categories and Mirror Symmetry Conjecture

Derived categories were originally developed as a homological algebra tool, but in fact are very interesting from a geometric point of view too. One example of this is Homological Mirror Symmetry (HMS), which comes from a mysterious and poorly understood relationship between complex and symplectic geometry, and originates from physics, specifically string theory. Generally, the HMS conjecture says that given some 'complex geometry' X, there is a mirror 'symplectic geometry' Y such that the derived category of coherent sheaves on X, denoted D^b(Coh X), is equivalent to a Fukaya category of Y, denoted Fuk(Y). In this talk I'll talk about toric HMS from the geometric invariant theory (GIT) perspective - this story is nice because it's well-understood and very combinatorial. No prior knowledge of derived categories, symplectic geometry, GIT, etc will be assumed. One of the main goals of this talk will be to explain what derived categories are informally, with the HMS stuff hopefully giving a motivation for studying them.

Week 9 Nov 28

Reading group [Chunkai Xu, Moduli of vectors bundles II] (12-1pm Thursday D1.07)

Week 10 Dec 5

Reading group [Peize Liu, Hypersurfaces and blowups] (12-1pm Thursday D1.07)

Xuanchun Lu - Cambridge (3-4pm Thursday D1.07)

Double Ramification cycles in the punctured setting

The projective line $\mathbb{P}^1$ is the only algebraic curve on which every degree 0 divisor is principal. For any other algebraic curve, it is thus natural to ask when a 'random' divisor on the said curve becomes principal.

Generalising classical works of Abel--Jacobi, a solution to the above problem may be given as a collection of cycles on the moduli spaces $M_{g, n}$ of genus g, n-marked algebraic curves. These cycles are known as the double ramification (DR) cycles. Since $M_{g, n}$ is not compact, these cycles do not fully capture the geometry of our situation. We thus need to seek an extension of DR cycles to some compactification of $M_{g, n}$.

In this talk, I will explain some of the challenges involved in constructing such an extension, and how they can be resolved using ideas and techniques from logarithmic geometry. I will also highlight some downstream enumerative applications of DR cycles. Time permitting, I will attempt to shed some light on the appearance of the word 'punctured' in the title.