L3

We present an adjoint method for optimization of the spatially inhomogeneous Boltzmann equation for rarefied gas dynamics. The adjoint method is derived using a "discretize then optimize" approach. Discretization (in time and velocity) is via the Direct Simulation Monte Carlo (DSMC) method, and adjoint equations are derived from an augmented Lagrangian.  The boundary conditions that are included in this analysis include spectral reflection, thermal reflection, and inflow boundary conditions. For thermal reflection, a "score function" is included as a statistical regularization. This is joint work with Yunan Yang (Cornell). This special seminar is jointly held with the Keble Complexity Research Cluster.

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.

What is the common theory of all finite fields equipped with an additive and/or multiplicative character? Hrushovski answered this question in the additive case working in (a mild version of) continuous logic. Motivated by natural number-theoretic examples we generalise his results to the case allowing for both (non-trivial) additive character and (sufficiently generic) multiplicative character. Apart from answering the above question we obtain a quantifier elimination result and a generalisation of the definability of the Chatzidakis-Macintyre-van den Dries counting measure to this context. The proof relies on classical results on bounds of character sums following from the work of Weil.

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.  

Maximum likelihood estimation is a ubiquitous task in statistics and its applications. The task is: given some observations of a random variable, find the distribution(s) in your statistical model which best explains these observations. A modern perspective on this classical problem is to study the "likelihood geometry" of a statistical model. By focusing on models which have a polynomial parametrization, i.e., lie on an algebraic variety, this perspective brings in tools, algorithms and invariants from algebraic geometry and combinatorics.

In this talk, I will explain some of the key ideas in likelihood geometry and discuss its recent application to the study of likelihood asymptotics, i.e., understanding likelihood estimation for very large or very small observation counts. Agostini et al. showed that these asymptotics can be modeled and understood using tools from tropical geometry, and they used this to completely describe the asymptotics for linear models. In our work, we use the same approach to treat the class of log-linear models (also known as Gibbs distributions or maximum entropy models) and give a complete and combinatorial description of the likelihood asymptotics under some conditions.

This talk is based on joint work with Emma Boniface (UC Berkeley) and Serkan Hoşten (San Francisco SU), available at: https://epubs.siam.org/doi/full/10.1137/24M1656839