L3

TBA

In this talk I will give a gentle introduction to some aspects of the theory of hypergeometric functions as a natural language for addressing various integrals appearing in quantum field theory (QFT). In particular I will focus on the so-called intersection pairings as well as the differential equations satisfied by the integrals, and I will show how these aspects of the mathematical theory can find a natural interpretation in concrete QFT applications. I will mostly focus on Feynman integrals as paradigmatic example, where the language will shed new light on our most powerful method for computing Feynman integrals as well as their non-local symmetries. I will then give an outlook how these methods could allow us to also learn about integrals appearing in other places in field and string theory, such as Coulomb branch amplitudes, celestial holography and AdS (supergravity and string) amplitudes.

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.  

Using novel arguments as well as techniques developed over the last  twenty years to study mean field games, in this paper (i) we investigate the optimal control of the Dyson equation, which is the mean field equation for the so-called Dyson Brownian motion, that is, the stochastic particle system satisfied by the eigenvalues of large random matrices, (ii) we establish the well-posedness of the resulting infinite dimensional Hamilton-Jacobi equation, 
(iii) we provide a complete and direct proof for the large deviations for the spectrum of large random matrices, and (iv) we study the asymptotic behavior of the transition probabilities of the Dyson Brownian motion.  Joint work with Charles Bertucci and Pierre-Louis Lions.

This illustrated historical talk covers the period from around 1890, when graph theory was still mainly a collection of isolated results, to the 1990s, when it had become part of mainstream mathematics. Among many other topics, it includes material on graph and map colouring, factorisation, trees, graph structure, and graph algorithms.