Georgia Institute of Technology College of Sciences

The late Goro Shimura proposed a question regarding certain invariant differential operators on a Hermitian symmetric space. This was answered by Sahi and Zhang by showing that the Harish-Chandra images of these namesake operators are specializations of Okounkov's BC-symmetric interpolation polynomials. We prove, in the super setting, that the Harish-Chandra images of super Shimura operators are specializations of certain BC-supersymmetric interpolation polynomials due to Sergeev and Veselov. Similar questions include the Capelli eigenvalue problems which are generalized to the quantum and/or super settings. This talk is based a joint work with Siddhartha Sahi.

In the 1990s, Arkadi Nemirovski asked the following question:

How hard is it to estimate a solution to unknown homogeneous linear difference equation with constant coefficients of order S, observed in the Gaussian noise on [0,N]?

The class of all such solutions, or "signals," is parametric---described by 2S complex parameters---but extremely rich: it includes the weighted sums of S exponentials, polynomials of degree S, harmonic oscillations with S arbitrary frequencies, and their algebraic combinations. Geometrically, this class is the union of all S-dimensional shift-invariant subspaces of the space of two-sided sequences, and of interest is the minimax risk on it with respect to the mean-squared error on [0,N]. I will present a recent result that shows this minimax risk to be O( S log(N) log(S)^2 ), improving over the state of the art by a polynomial in S factor, and coming within an O( log(S)^2 ) factor from the lower bound. It relies upon an approximation-theoretic construction related to minimal-norm interpolation over shift-invariant subspaces, in the spirit of the Landau-Kolmogorov problem, that I shall present in some detail. Namely, we will see that any shift-invariant subspace admits a bounded-support reproducing kernel whose spectrum has nearly the smallest possible Lp-energies for all p ≥ 1 at once.

We study nonlinear energy transfer and the existence of stationary measures in a class of degenerately forced SDEs on R^d with a quadratic, conservative nonlinearity B(x, x) constrained to possess various properties common to finite-dimensional fluid models and a linear damping term −Ax that acts only on a proper subset of phase space in the sense that dim(kerA) ≫ 1. Existence of a stationary measure is straightforward if kerA = {0}, but when the kernel of A is nontrivial a stationary measure can exist only if the nonlinearity transfers enough energy from the undamped modes to the damped modes. We develop a set of sufficient dynamical conditions on B that guarantees the existence of a stationary measure and prove that they hold “generically” within our constraint class of nonlinearities provided that dim(kerA) < 2d/3 and the stochastic forcing acts directly on at least two degrees of freedom. We also show that the restriction dim(kerA) < 2d/3 can be removed if one allows the nonlinearity to change by a small amount at discrete times. In particular, for a Markov chain obtained by evolving our SDE on approximately unit random time intervals and slightly perturbing the nonlinearity within our constraint class at each timestep, we prove that there exists a stationary measure whenever just a single mode is damped.

We say a class of graphs $\mathcal{F}$ is degree-bounded if there exists a function $f$ such that $\delta(G)\le f(\tau(G))$ for every $G\in\mathcal{F}$, where $\delta(G)$ denotes the minimum degree of $G$ and $\tau(G)$ is the biclique number of $G$, that is, the largest integer $t$ such that $G$ contains $K_{t,t}$ as a subgraph. Such a function $f$ is called a degree-bounding function for $\mathcal{F}$.

In joint work with Ant\'onio Gir\~ao, Zach Hunter, Rose McCarty and Alex Scott, we proved that every hereditary degree-bounded class $\mathcal{F}$ has a degree-bounding function that is singly exponential in the biclique number $\tau$. A more recent result by Ant\'onio Gir\~ao and Zach Hunter improved this bound to a polynomial function in $\tau$. In this talk, I will discuss examples and the recent results on degree-boundedness.