We will construct a noncommutative polynomial, P, in 2n variables such that every 2n-tuple of nxn matrices vanish when plugged into P. The Cayley-Hamilton theorem will be the key ingredient.
 

In this talk we will discuss a new approach to the problem of emergence in hydrodynamic systems of collective behavior. The problem seeks to establish convergence to a flocking state in a system with self-organization governed by strictly local laws of communication. The typical results in this direction insist on propagation of flock connectivity which translates into a quantitative non-vacuum condition on macroscopic level. With the introduction of small noise one can relax such a condition considerably, and even allow for vacuum, in the context of the corresponding Fokker-Planck-Alignment equations. The flocking behavior becomes the problem of establishing hypocoercivity and relaxation of solutions to the global Maxwellian. We will describe a model which does precisely that in the non-perturbative settings.

Nash equilibrium problems (NEPs) are games for several players . A Nash Equilibrium (NE) is a tuple of strategies such that each player's benefits cannot be improved when the other players' strategies are fixed. For NEPs given by polynomial functions, we formulate efficient polynomial optimization problems for computing NEs. The Moment-SOS relaxations are used to solve them. Under genericity assumptions, the method can find a Nash equilibrium if there is one; it can also find all NEs if there are finitely many ones. The method can also detect nonexistence if there is no NE.

This is a joint work with Dr. Xindong Tang.
 

The moduli stack of oriented formal groups embodies, in the world of spectral algebraic geometry, the fundamental chromatic connection between the stable homotopy category and formal groups. As such, it validates the folklore picture of Morava, Hopkins, et al. Somewhat surprisingly, it is also closely related to a more recent development: the "cofiber of $\tau$ philosophy" of Gheorghe-Isaksen-Wang-Xu.

In this talk, we will introduce the moduli stack of oriented formal groups, and explain how the algebro-geometric structure of its connective cover reflects and gives rise to the $\tau$-deformation structure of cellular motivic spectra over $\mathbb{C}$.

The optimal transport problem is known to suffer the curse of dimensionality. A recently proposed approach to mitigate the curse of dimensionality is to project the sampled data from the high dimensional probability distribution onto a lower-dimensional subspace, and then compute the optimal transport between the projected data. However, this approach requires to solve a max-min problem over the Stiefel manifold, which is very challenging in practice. In this talk, we propose a Riemannian block coordinate descent (RBCD) method to solve this problem. We analyze the complexity of arithmetic operations for RBCD to obtain an $\epsilon$-stationary point, and show that it significantly improves the corresponding complexity of existing methods. Numerical results on both synthetic and real datasets demonstrate that our method is more efficient than existing methods, especially when the number of sampled data is very large. We will also discuss how the same idea can be used to solve the projection robust Wasserstein barycenter problem.

Motivated by questions about a p-adic Fourier transform, we study invariant norms on the p-adic Schrödinger representations of Heisenberg groups. These Heisenberg groups are p-adic, and the Schrodinger representations are explicit irreducible smooth representations that play an important role in their representation theory.

Classically, the field of coefficients is taken to be the complex numbers and, among other things, one studies the unitary completions of the representations (which are well understood). By taking the field of coefficients to be an extension of the p-adic numbers, we can consider completions that better capture the p-adic topology, but at the cost of losing the Haar measure and the $L^2$-norm. Nevertheless, we establish a rigidity property for a family of norms (parametrized by a Grassmannian) that are invariant under the action of the Heisenberg group.

The irreducibility of some Banach representations follows as a result. The proof uses "q-arithmetics".

Let $\mu$ be a measure on a finite group $G$. We define the spectral gap of $\mu$ to be the operator norm of the map that sends $\phi \in L^2(G)^{\circ}$ to $\mu * \phi$. We say that $\mu$ is symmetric if $\mu(x) = \mu(x^{-1})$. Now fix $G = SL_2(\mathbb{Z}/n\mathbb{Z}) \times SL_2(\mathbb{Z}/n\mathbb{Z})$, with $n \in \mathbb{N}$ being arbitrary. Suppose that $\mu$ is a measure on $G$ such that it's pushforwards to the left and right component have spectral gaps lesser than $\lambda_0 < 1$ and $\mu$ takes a minimum of $\alpha_0$ on it's support. Further suppose that the support of $\mu$ generates $G$. Then we show that there are constants $L, \beta > 0$ depending only on $\lambda_0,\alpha_0$ such that $\mu^{(*)L\log |G|}(\Gamma) \leq \frac{1}{|G|^{\beta}}$, where $\Gamma$ is the graph of any automorphism of $SL_2(\mathbb{Z}/n\mathbb{Z})$. We will discuss this result and its implications. This talk is based on joint work with Professor Alireza Salehi-Golsefidy.

We consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler 20. The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. In these cases, we show that the tangent flow is unique and the singularity is of Type I. This talk is based on a joint work with Zilu Ma and Yongjia Zhang.

We will discuss some of the known power savings for the number of $G$-extensions of a number field with discriminant bounded above by $X$. We will put a focus on the existence of secondary terms in the asymptotic growth rate, and in particular will discuss a proof of the existence of some secondary terms when $G$ is abelian.

Let M be a smooth submanifold of $\mathbb{R}^n$ equipped with the Euclidean(chordal) metric. This talk will consider the smallest dimension, m, for which there exists a bi-Lipschitz function f : M → $\mathbb{R}^m$ with biLipschitz constants close to one. We will begin by presenting a bound for the embedding dimension m from below in terms of the bi-Lipschitz constants of f and the reach, volume, diameter, and dimension of M. We will then discuss how this lower bound can be applied to show that prior upper bounds by Eftekhari and Wakin on the minimal low-distortion embedding dimension of such manifolds using random matrices achieve near-optimal dependence on dimension, reach, and volume (even when compared against nonlinear competitors). Next, we will discuss a new class of linear maps for embedding arbitrary (infinite) subsets of $\mathbb{R}^n$ with sufficiently small Gaussian width which can both (i) achieve near-optimal embedding dimensions of submanifolds, and (ii) be multiplied by vectors in faster than FFT-time. When applied to d-dimensional submanifolds of $\mathbb{R}^n$ we will see that these new constructions improve on prior fast embedding matrices in terms of both runtime and embedding dimension when d is sufficiently small.

This is joint work with Benjamin Schmidt (MSU) and Arman Tavakoli (MSU).