In recent years, many mathematicians have contributed to a combinatorial theory for the polynomial ring ${\mathbb C}[x_1, x_2, \ldots]$ similar to symmetric function theory. Beginning with Schubert polynomials and later key polynomials, numerous bases have been introduced whose monomials have combinatorial interpretations. In the theory of harmonic polynomials, there is a natural inner product for the polynomial ring with monomials as an orthogonal basis. Duality with respect to this inner product is characterized by a Cauchy type identity. We show how to interpret this duality combinatorially. As a byproduct, we recover Postnikov and Stanley’s dual Schubert polynomials and introduce a novel family of dual key polynomials whose further properties remain uninvestigated.  

Computing the derivative of long-time-averaged observables with respect to system parameters is a central problem for many numerical applications. Conventionally, there are three straight-forward formulas for this derivative: the pathwise perturbation formula (including the backpropagation method used by the machine learning community), the divergence formula, and the kernel differentiation formula. We shall explain why none works for the general case, which is typically chaotic (also known as the gradient explosion phenomenon), high-dimensional, and small-noise.

We present the fast response formula, which is a 'Monte-Carlo' type formula for the parameter-derivative of hyperbolic chaos. It is the average of some function of u-many vectors over an orbit, where u is the unstable dimension, and those vectors can be computed recursively. The fast response overcomes all three difficulties under hyperbolicity assumptions. Then we discuss how to further incorporate the kernel differentiation trick to overcome non-hyperbolicity.

The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking bundle-theoretic questions. However, in general, many non-equivalent bundles represent the same K-theory class. Bridging the gap between K-theory and actual bundles is challenging even for the simplest CW complexes.

For example, given random r and n, the number of rank r bundles on complex projective r-space that are trivial in K-theory is unknown. In this talk, we will compute the p-primary portion of the number of rank r bundles on $\mathbb CP^n$ in infinitely many cases. We will give lower bounds for this number in more cases.

Building on work of Hu, we use Weiss-theoretic techniques in tandem with a little chromatic homotopy theory to translate bundle enumeration to a computation of the higher real K-theory of particular simple spectra. The result will involve actual numbers!  This is joint work with Hood Chatham and Yang Hu.

Nonparametric models are of great interest in various scientific and engineering disciplines. Classical kernel methods, while numerically robust and statistically sound in low-dimensional settings, become inadequate in higher-dimensional settings due to the curse of dimensionality.

In this talk, we will introduce a new framework called Variance-Reduced Sketching (VRS), specifically designed to estimate density functions and nonparametric regression functions in higher dimensions with a reduced curse of dimensionality. Our framework conceptualizes multivariable functions as infinite-size matrices, facilitating a new matrix-based bias-variance tradeoff in various nonparametric contexts.

We will demonstrate the robust numerical performance of VRS through a series of simulated experiments and real-world data applications. Notably, VRS shows remarkable improvement over existing neural network estimators and classical kernel methods in numerous density estimation and nonparametric regression models. Additionally, we will discuss theoretical guarantees for VRS to support its ability to deliver nonparametric estimation with a reduced curse of dimensionality.

We proved closed/dense dichotomy of $\operatorname{PGL}_2(\mathbb{Q}_p)$-orbit closures in the renormalized frame bundle of a $p$-adic homogeneous space of infinite volume. Our result is a generalization of Ratner’s theorem and the result of McMullen, Mohammadi, and Oh in 2017 into non-Archimedean local fields.

Let $\mathbb{K}$ be an unramified quadratic extension of $\mathbb{Q}_p$. Our homogeneous space is a quotient space of $\operatorname{\mathbb{K}}$ by a certain class of Schottky subgroups. Using the main tools of McMullen, Mohammadi, and Oh, we introduced the necessary properties of Schottky subgroups and used the Bruhat-Tits tree $\operatorname{PGL}_2$. In this talk, we introduce the highly-branched Schottky subgroups and steps for the proof of the main theorem.

This is a joint work with Seonhee Lim.

One day, mathematicians started thinking really hard about moving piles of dirt around, and the Wasserstein distance was born. It measures the difference between two probability distributions, in a way that is different (and sometimes better) than entropy and the L^p metrics.  In this talk, I will introduce the field known as optimal transport, and talk about some applications, mainly to machine learning.