St Andrews Geometry seminar

For the first semester of 2025-2026, the seminar will run on Fridays, 13:00 to 14:00. The talks are in Lecture Theatre A unless specified otherwise.

September 5: Amanda Hirschi (Sorbonne)

On stabilisations of symplectic 4-manifolds

Two simply-connected smooth 4-manifolds are homeomorphic if and only if their stabilisations, i.e. products with S^2, are diffeomorphic. The Donaldson 4-6 question asks whether this fact can be lifted to the symplectic category. It asks whether the underlying smooth manifolds of two (simply-connected) symplectic manifolds are diffeomorphic if and only if their products with S^2, equipped with the standard area form, are symplectic deformation equivalent. I will describe one example of a smooth 4-manifold admitting two symplectic forms that remain deformation inequivalent after taking the product with S^2, giving counterexamples to one implication of the conjecture. On the other hand, I will explain why two symplectic manifolds, whose stabilisations are deformation equivalent, have the same Gromov-Witten invariants. This is joint work with Luya Wang. 

September 19: Violeta Lopez (St Andrews)

Brill-Noether general tropical curves

Brill–Noether theory provides a way to study smooth algebraic curves and their moduli space. In 2012, Cools, Draisma, Payne and Robeva “tropicalized” this theory by showing that the tropical curve known as the chain of loops is “Brill–Noether general”. In this talk I will review some concepts and results related to tropical Brill–Noether theory and I will present a new tropical curve that is Brill–Noether general. This is my current research work.

September 26: Alison La Porta (St Andrews)

The connection between Maximum Likelihood Thresholds and Rigidity Theory

The MLT of a graph

Any graph G=([n],E) has an associated Gaussian graphical model, i.e. a statistical model of n-variate normal distributions such that two random variables x and y are conditionally independent given the other random variables whenever the vertices representing x and y in G do not share an edge.

Given d independent samples X1,...,Xd, we then ask if the Maximum Likelihood Estimator (MLE) of G exists. An important piece of information lies in the Maximum Likelihood Threshold (MLT) of G, i.e. the minimum d such that the MLE of G exits almost surely.

Rigidity theory

A (bar-joint) framework is a geometric structure composed of stiff bars, linked together by freely rotational joints. A framework is flexible if it can be deformed continuously into a non-congruent framework without changing its bar-lengths, and it is rigid otherwise. There are variations to the definition of rigidity, such as continuous, infinitesimal and global rigidity, amongst others.

Independently of the various notions, the main goal in rigidity theory is to determine when a framework is rigid or flexible. This is a relatively new area of research, which uses a wide range of mathematical tools, and find applications in a variety of applied sciences.

The MLT of a graph and geometric rigidity

In particular, research from recent years connects rigidity theory to the MLT problem. In this talk, we explore how this connection can be exploited. We will see recent results which use techniques from rigidity theory to find new and significantly improved bounds for the MLT of a graph.

October 17: Two talks - Diane Maclagan (Warwick) and Alexia Corradini (Cambridge)

Diane Maclagan: Tropical vector bundles

I will describe a new definition, joint with Bivas Khan, for a tropical toric vector bundle on a tropical toric variety. This builds on the tropicalizations of toric vector bundles, and can be used to define tropicalizations of vector bundles on subvarieties of toric varieties.  I will discuss when these bundles do and do not behave as in the classical setting. 

Alexia Corradini: The Lagrangian Ceresa cycle

I will introduce an equivalence relation on Lagrangians in a symplectic manifold called algebraic Lagrangian cobordism. Its purpose is to mirror algebraic equivalence of cycles, in the same way Sheridan—Smith observed that cylindrical Lagrangian cobordisms are “mirror” to rational equivalences of cycles. I then show that this equivalence relation captures non-trivial symplectic geometry through a Lagrangian version of the Ceresa cycle story. The Ceresa cycle of a curve is a 1-cycle in its Jacobian; it provided one of the first examples of a cycle proven to be homologically, but not algebraically trivial. The Lagrangian construction builds upon on Zharkov’s tropical version of this story.

October 31: Rachael Boyd (Glasgow)

Diffeomorphisms of reducible 3-manifolds

I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. We prove that when M is orientable and has non-empty boundary, B Diff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough. The theory we develop to prove this theorem has other applications, and I’ll provide an overview of these.

November 7: Ruadhai Dervan (Warwick)

Metric wall-crossing

The quotient theory of symplectic geometry goes by the name of symplectic reduction, and involves a moment map along with a symplectic form. A basic question is to understand the behaviour of symplectic reduction as one varies the moment map or symplectic form, for a fixed action of a Lie group on a smooth manifold, with Duistermaat-Heckman and Guillemin-Sternberg having proven foundational results in this direction.

I will explain new results that apply when the symplectic manifold is further a compact complex manifold, such as a smooth projective variety. In this setting, I will give a fairly complete metric/symplectic description of how the symplectic reductions vary, giving a metric upgrade to (and a non-projective extension of) the algebro-geometric story of variation of geometric invariant theory. I will use these results to state some general conjectures about the metric geometry of moduli spaces in wall-crossing problems. This is work in progress.

November 14: Aporva Varshney (UCL)

Stringy Kahler moduli of flops using GIT 

According to predictions coming from physicists and mirror symmetry, the fundamental group of the "stringy Kahler moduli space" of a variety acts on the derived category by autoequivalences. However, it is unclear how to compute or even define this space in full generality. I will discuss two approaches given in the literature, one using Bridgeland stability manifolds and the other based in GIT and window subcategories. We will see that these two approaches agree for single curve threefold flops of lengths one and two. This is based on work in arxiv:2508.05285. 

November 21: Elliot Gathercole (Lancaster)

Superrigidity in complements of singular divisors 

A complex Fano projective variety M (for example, projective space) is canonically a monotone symplectic manifold. Given an effective anticanonical divisor D, we obtain an isotropic cell complex L inside the complement X=M-D, called the skeleton, and an increasing family of compact neighbourhoods of L which exhaust X.

In a symplectic manifold, we can ask if a subset is rigid: that is, can it be displaced from itself by a Hamiltonian isotopy? In the above setting, we can ask a quantitative refinement of this question: what is the smallest neighbourhood of L in our family which is rigid?

For certain kinds of singular (i.e. possibly not SNC) divisors D, I will describe how we can answer this question in terms of spectral invariants, give some examples of interesting rigid isotropic cell complexes obtained in this manner, and, if time allows, describe a possible connection of this question to birational geometry

November 28: Two talks - Darragh Glynn (Warwick) and Giovanni Ambrosioni (ETH Zurich)

Darragh Glynn: Boundary stratifications of Hurwitz spaces 

A Hurwitz space is a moduli space parametrising regular maps of algebraic curves with prescribed branching data. Building on work of Harris-Mumford, Abramovich-Corti-Vistoli describe a smooth compactification whose boundary parametrises degenerate maps. The boundary decomposes into connected subsets, or strata, corresponding to combinatorial types of degenerations. I will present recent results giving an explicit, implementable description of the strata. This description leads to a natural definition of a tropical Hurwitz space and has applications in complex dynamics. 

Giovanni Ambrosioni: Categorical metric approximability and applications to Lagrangian topology

In this talk I will introduce a new notion of approximability for metric spaces that can be seen as a categorification of a concept introduced by Turing for metric groups in 1938 and as a generalization of total-boundedness. Categorical metric approximability relies on the theory of triangulated persistence categories introduced by Biran-Cornea-Zhang and allows for the definition of refinements of classical measurements of complexity of objects of triangulated categories such as categorical entropy.  I will discuss approximablity of spaces of Lagrangian submanifolds and present some examples. If time permits, I will discuss applications to rigidity and complexity of Lagrangians, as well as potential relations to open problems in Lagrangian topology. This talk is based on joint work with Paul Biran and Octav Cornea.