We consider the mixed-strategy zero-sum game such that each player’s objective function is quadratic in its own variables. By considering each player’s value function and duality, the bi-quadratic games are reformulated as linear programs over the cone of copositive (COP) and completely positive (CP) matrices. We apply moment and SOS relaxations for the conic constraints of CP and COP matrices, respectively, and obtain a hierarchy of semidefinite relaxations. Under certain conditions, the finite convergence for this hierarchy is guaranteed, and the tightness can be checked via flat truncation. We present numerical experiments to show the effectiveness of our approach.

We study stochastic interacting particle methods with and without field coupling for high dimensional concentration and singularity formation phenomena. In case of entropy production of reverse-time diffusion processes, the method computes concentrated invariant measures mesh-free up to dimension 16 at a linear complexity rate based on solving a principal eigenvalue problem of non-self-adjoint advection-diffusion operators. In case of fully parabolic chemotaxis nonlinear dynamics in 3D, the method captures critical mass for finite time singularity formation and blowup time at low costs through a smoother field without relying on self-similarity.

The talk will cover two new developments in our work on efficient high-order methods for conservation laws: (1) A simple stabilization technique for spectral element methods, which uses continuous solution spaces and is provably convergent for linear problems at arbitrary orders of accuracy. The main motivation for the new scheme is its lower cost, which comes from having fewer degrees of freedom, no Riemann solvers, and a line-based sparsity pattern. However, it also has other attractive properties such as an improved CFL condition and allowing for other solvers including static condensation. (2) A deep reinforcement learning approach for generation of meshes with optimal connectivities. Starting from a Delaunay mesh, we formulate the mesh optimization process as a "game" where the moves are standard topological element operations, and the goal is to maximize the number of regular nodes. The agent is trained in a self-play framework using the proximal policy optimization (PPO) algorithm running on GPUs. Our approach works for 2D triangular and quadrilateral meshes with minimal modification, and it routinely produces close-to-perfect meshes.

Let $G\stackrel{\alpha}{\curvearrowright}(M,\tau)$ be a trace-preserving action of a finite group $G$ on a tracial von Neumann algebra. Suppose that $A \subset M$ is a finitely generated unital $*$-subalgebra which is globally invariant under $\alpha$. We give a formula relating the von Neumann dimension of the space of derivations on $A$ valued on its coarse bimodule to the von Neumann dimension of the space of derivations on $A \rtimes^{\text{alg}}_\alpha G$ valued on its coarse bimodule, which is reminiscent of Schreier's formula for finite index subgroups of free groups. This formula induces a formula for $\dim \text{Der}_c(A,\tau)$ (defined by Shlyakhtenko) and under the assumption that $G$ is abelian we obtain the formula for $\Delta$ (defined by Connes and Shlyakhtenko). These quantities and the free entropy dimension quantities agree on a large class of examples, and so by combining these results with known inequalities, one can expand the family of examples for which the quantities agree.

Inspired by the classical random graph process introduced by Erdos and Renyi in 1960, we discuss two analogous processes for random representable matroids, one introduced by Kelly and Oxley in 1982 and the other one introduced by Cooper, Frieze and Pegden in 2019. In the talk we address the evolution of the rank, circuits, connectivity, and the critical number (corresponding to the logarithm of the chromatic number of graphs) of the first random matroid, and then we focus on the minors in both matroid models. 

This talk discusses the challenging problem of finding global optimal solutions for two-stage stochastic programs with continuous decision variables and nonconvex recourse functions. We introduce a two-phase approach, which does not only generate global lower bounds for the nonconvex stochastic program but also simplifies the computation of the expected value of the recourse function by using moments of random vectors. This makes our overall algorithm particularly suitable for the case where the random vector follows a continuous distribution or when dealing with many scenarios. Numerical experiments are given to demonstrate the effectiveness of our proposed approach.

We will consider the behavior of the Kolmogorov-Sinai entropies of amenable group actions under a Shannon orbit equivalence. Although dynamical entropy is in general not invariant under orbit equivalences, recent works have shown that various notions of restricted orbit equivalences will preserve entropy. We focus on the case where the orbit equivalence is Shannon, and both groups are finitely generated amenable. In this talk, we will present a proof for our main result.

As the shortest path metric on a weighted tree can be embedded isometrically into a finite $\ell_1$ space, a Lipschitz embedding of a given graph into $\ell_1$ can be obtained by constructing a low distortion embedding into a tree. Conversely, while there are various topological properties of graphs that guarantee controlled distortion Lipschitz embeddings into $\ell_1$ ($k$-outerplanar, series-parallel, low Euler characteristic), it is still often the case that such a graph embeds quite poorly into a tree.

By introducing the notion of a stochastic embedding into a family of trees one can find more general concrete embeddings into $\ell_1$ then those limited by a single tree. In fact, it is known that every graph with n vertices embeds stochastically into trees with distortion O(log(n)). Nevertheless, this upper bound is sharp for graphs such as expanders, grids and, by a recent joint work with Schlumprecht, a large class of "fractal-like" series-parallel graphs called slash powers. 

In this talk we introduce an equivalent characterization of stochastic distortion called expected distortion and after proving a mild extension of a result of Gupta regarding poor tree embeddings of a cycle, inductively lower bound the expected distortion of generalized Laakso graphs found in most nontrivial slash power families.

We will discuss questions surrounding automorphic L-functions, particularly Deligne’s conjecture about critical values of motivic L-functions. In particular, we study the adjoint L-function of U(2,1).

Hundley showed that a certain integral, involving an Eisenstein series on the exceptional group G_2, computes this L-function at unramified places. We discuss the computation of this integral at the archimedean place for quaternionic modular forms, and how this relates to Deligne's
conjecture.

Given an automorphism of a variety $X$, there is an induced ''natural'' automorphism on $X^{[n]}$, the Hilbert scheme of $n$ points of $X$. While unnatural automorphisms of $X^{[n]}$ are known to exist for certain varieties $X$ and integers $n$, all previously known examples could be shown to be unnatural because they do not preserve multiplicities. Belmans, Oberdieck, and Rennemo thus asked if an automorphism of a Hilbert scheme of points of a surface is natural if and only if it preserves the diagonal of non-reduced subschemes.

We give an answer in the negative for all $n\ge 2$ by constructing explicit counterexamples on certain abelian surfaces $X$. These surfaces are not generic, and hence we prove a partial converse statement that all automorphisms of the Hilbert scheme of two points on a very general abelian surface are natural for certain polarization types (including the principally polarized case).

When trying to solve partial differential equations, a common practice is to enlarge the space of possible solutions to the class of non-differentiable functions, where it is easier to find “weak” solutions (i.e. potentially very irregular). As we are usually interested in “strong” solutions (i.e very regular), one is then confronted with the following problem: how do we upgrade the regularity? A fundamental tool in these situations is a monotonicity formula, an object that allows to study the infinitesimal behavior of solutions of PDEs by reducing it to a classification problem. More concretely, a monotonicity formula is an identity implying that a certain quantity related to the problem at hand is monotone, or conserved. I will try to convey the gist of this idea that has found applications in many areas at the intersection of geometry and analysis, e.g. harmonic maps, minimal surfaces, free boundary problems, Yang-Mills connections to name just a few. I will try to maintain the level of analysis needed at a minimum, you only need to remember that the first derivative of a smooth function at an interior minimum is zero. I will explain the rest. 

Mean field games are played by populations of competing agents who derive their update rules by comparing their own state variable with that of the mean field. After a brief introduction to several areas where they have been used recently, we will focus on models of competing tumor cell populations based on the replicator dynamics mean field evolutionary game with prisoner’s dilemma payoff matrix. We use optimal and adaptive control theory on both deterministic and stochastic versions of these models to design multi-drug chemotherapy schedules that suppress the competitive release of resistant cell populations (to avoid chemo-resistance) by maximizing the Shannon diversity of the competing subpopulations. The models can be extended to networks where spatial connectivity can influence optimal chemotherapy scheduling.