Imperial College London

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 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.

Differentiable programming is the underpinning technology for the AI revolution. It allows neural networks to be programmed in very high level user code while still achieving very high performance for both the evaluation of the network and, crucially, its derivatives. The Firedrake project applies exactly the same concepts to the simulation of physical phenomena modelled with partial differential equations (PDEs). By exploiting the high level mathematical abstraction offered by the finite element method, users are able to write mathematical operators for the problem they wish to solve in Python. The high performance parallel implementations of these operators are then automatically generated, and composed with the PETSc solver framework to solve the resulting PDE. However, because the symbolic differential operators are available as code, it is possible to reason symbolically about them before the numerical evaluation. In particular, the operators can be differentiated with respect to their inputs, and the resulting derivative operators composed in forward or reverse order. This creates a differentiable programming paradigm congruent with (and compatible with) machine learning frameworks such as Pytorch and JAX. 

In this presentation, David Ham will present Firedrake in the context of differentiable programming, and show how this enables productivity, capability and performance to be combined in a unique way. I will also touch on the mechanism that enables Firedrake to be coupled with Pytorch and JAX.

Please note this talk will take place at Rutherford Appleton Laboratory, Harwell Campus, Didcot. 

A way to study rational points on a variety is by looking at their image in the p-adic points. Some natural questions that arise are the following: is there any obstruction to weak approximation on the variety? Which primes might be involved in it? I will explain how primes of good reduction can play a role in the Brauer-Manin obstruction to weak approximation, with particular emphasis on the case of K3 surfaces.