Many modern problems in data science aim to efficiently and accurately extract important features and make predictions from high dimensional and large data sets. Naturally occurring structure in the data underpins the success of many contemporary approaches, but large gaps between theory and practice remain. In this talk, I will present recent progress on two different methods for nonparametric regression that can be viewed as the projection of a lifted formulation of the problem with a simple stochastic or convex geometric description, allowing the projection to encapsulate the data structure. In particular, I will first describe how the theory of stationary random tessellations in stochastic geometry can address the computational and theoretical challenges of random decision forests with non-axis-aligned splits. Second, I will present a new approach to convex regression that returns non-polyhedral convex estimators compatible with semidefinite programming. These works open many directions of future work at the intersection of stochastic and convex geometry, machine learning, and optimization.

The Kronecker coefficient $g_{\lambda \mu \nu}$ is the multiplicity of an irreducible $\mathcal{S}_n$-module $M_\nu$ in the tensor product $M_\lambda \otimes M_\mu$. A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. We give such a formula in the case that one of the partitions is a hook shape. Our main tool is Haiman's mixed insertion, which is a generalization of Schensted insertion to colored words. Prior familiarity with combinatorics of words and tableaux will not be assumed.

A major goal in complex dynamics is to understand "dynamical moduli spaces"; that is, conformal conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to $\mathbb C$. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. In this talk, we will begin with an overview of complex dynamics, focusing on the moduli space of quadratic rational maps, which is isomorphic to $\mathbb C^2$. We will explore this space, finding many interesting objects along the way. Note: special tea at 2:45.

Richard Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Loosely speaking, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by the Fourier expansions of a distinguished set of modular functions. This conjecture arose from astonishing coincidences noticed by finite group theorists and arithmetic geometers in the 1970s. Recently, mathematical physicists have revisited moonshine, and they discovered evidence of undiscovered moonshine which some believe will have applications to string theory and 3d quantum gravity. The speaker and his collaborators have been developing the mathematical facets of this theory, and have proved the conjectures which have been formulated. These results include a proof of the Umbral Moonshine Conjecture, and Moonshine for the first sporadic finite simple group which does not occur as a subgroup or subquotient of the Monster. The most recent Moonshine (announced here) yields unexpected applications to the arithmetic elliptic curves thanks to theorems related to the Birch and Swinnerton-Dyer Conjecture and the Main Conjectures of Iwasawa theory for modular forms. This is joint work with John Duncan, Michael Griffin and Michael Mertens.

Vaccines protect their recipients by inducing long-term structural changes in populations of immune cells. Part of that restructuring is exactly analogous to Darwinian Selection. New antibody molecules are created by somatic mutation of existing antibody genes. Subsequently, the immune cell populations that possess these mutated receptors overtake the "wild-type" immune cells due  to the selective advantage they have acquired. Thus the immune system is vastly better prepared to recognize and eliminate the eliciting pathogen the next time around.

New sequencing and biosynthesis technologies, together with mathematical and computational tools, now allow us to investigate this fascinating and important phenomenon more deeply than ever before. I will illustrate this development with examples from the immune response to HIV infection.