Imperial College London

 Is it possible to define gauge theories on singular spaces? The answer to this question is emphatically yes​, and the prime example of such spaces are two-dimensional orbifolds known as spindles​. First, I will introduce spindles from a symplectic geometry perspective. Then I will discuss the notion of orbi-bundles, which allows one to consistently describe regular gauge fields/spinors on orbifolds.

Shimura varieties are highly symmetric algebraic varieties that play an important role in the Langlands program. In the first part of the talk, I will try to give you a sense of what they are like, with a focus on their different kinds of symmetries. In the second part of the talk, I will introduce Igusa stacks, a powerful new tool in the study of Shimura varieties. To illustrate their role, I will discuss how Igusa stacks can shed light on the many structures that exist on the intersection cohomology of Shimura varieties. This is joint work in progress with Linus Hamann and Mingjia Zhang.

Over 30 years has passed since the original proof of Fermat's Last Theorem by Wiles and Taylor—Wiles. There are now several proofs known to humanity, and I'm currently teaching one of them to a computer. This made me try to find out what the most ergonomic route was nowadays, and I found it by asking Richard Taylor what it was. In the talk I will summarise how to prove Fermat's Last Theorem in 2026, highlighting the differences between the modern method and the original route discovered by Wiles (we do use p=3, but in a different way). I won't talk much at all about Lean and essentially none of the work I will present is my own; this will just be a standard number theory seminar, and probably everything in it will already be known to the experts, but hopefully younger people will learn something.

We consider targeted intervention problems in dynamic network and graphon games. First, we study a general dynamic network game in which players interact over a graph and seek to maximize their heterogeneous, concave goal functionals. We establish the existence and uniqueness of a Nash equilibrium in both the finite-player network game and the corresponding infinite-player graphon game, and prove its convergence as the number of players tends to infinity. We then introduce a central planner who implements a dynamic targeted intervention. Given a fixed budget, the central planner maximizes the average welfare at equilibrium by perturbing the players' heterogeneous goal functionals. Using a novel fixed-point argument, we prove the existence and uniqueness of an optimal intervention in the graphon setting, and show that it achieves near-optimal performance in large finite networks. Finally, we study the special case of linear-quadratic goal functionals and derive semi-explicit solutions for the optimal intervention.

 

This is a joint work with Sturmius Tuschmann.  

I will speak about my work with Helge Ruddat on how to construct explicitly log structures and morphisms. I will also discuss some motivation. I will try to stay informal and assume no prior knowledge of log structures.

Let $X_0 $ be a rational surface with a cyclic quotient singularity $(1,a)/r$.  Kawamata constructed a remarkable vector bundle  $F_0$  on $X_0$ such that the finite-dimensional algebra End$(F_0)$ "absorbs'' the singularity of $X_0$ in a categorical sense. If we deform over an irreducible component of the versal deformation space of $X_0$ (as described by Kollár and Shepherd-Barron), the vector bundle $F_0$ also deforms to a vector bundle $F$. These results were established using abstract methods of birational geometry, making the explicit computation of the family of algebras challenging. We will utilise homological mirror symmetry to compute End$(F)$ explicitly in a certain bulk-deformed Fukaya category. In the case of a $Q$-Gorenstein smoothing, this algebra End$(F)$ is a matrix order over $k[t]$ and "absorbs" the singularity of the corresponding terminal 3-fold singularity. This is based on joint work with Jenia Tevelev.

I will discuss the Brill-Noether theory of a general elliptic 𝐾3 surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general 𝑘-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.

I will discuss the Brill-Noether theory of a general elliptic $K3$ surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general $k$-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.

I will review recent progress on extending the Minimal Model Program to algebraically integrable foliations, focusing on applications such as the canonical bundle formula and recent results toward the boundedness of Fano foliations.