I will talk about the use of peers to enhance learning in three different contexts.  The first context is a flipped integral calculus course. Students are expected to prepare for class ahead of time by watching video(s) and taking online quizzes.  The instructor accesses the quiz data before class and uses student responses to tailor the classroom instruction. In-class time focuses on extending student understanding with a variety of active learning techniques, including peer-to-peer instruction. I will report the data we have collected about the impact of this experience on  both student attitudes and learning.

The second context is a summer online bridge program for incoming students. We utilize undergraduate coach/mentors to meet online virtually with a team of 4-5 incoming students throughout the summer to help close some of their mathematical gaps.  I will describe the design of this program, how it enhances Yale's desire to recruit and retain a diverse student body, and the impact it has on student attitudes and learning. I will also highlight data that describes the impact of peer coaches on both learning and the motivation to learn.

The third context is a systematic supervised reading/research program for ~1200 math majors at UC Irvine.  I will provide some suggestions for how this program might be structured to leverage advanced undergraduates and graduate students to help motivated math majors.

Undergraduates are curious about research in mathematics: what kinds of questions do mathematicians ask, what does research entail, how do you begin to solve a new problem. In this talk, we will discuss integrating undergraduate research projects inside the classroom and how to expose students to new mathematical questions in both upper and lower division courses. We will then talk more generally about setting students up for success in the classroom.

Convex-geometric methods, involving random projection operators and coverings, have been successfully used in the study of the largest and smallest singular values, delocalization of eigenvectors, and, among further applications, in establishing the limiting spectral distribution for certain random matrix models. Conversely, random linear operators play a very important role in high-dimensional convex geometry, as a tool in constructing special classes of convex sets and studying sections and projections of convex bodies. In this talk, I will discuss some recent results (by my collaborators and myself) on the borderline between convex geometry and the theory of random matrices, focusing on invertibility of square non-Hermitian random matrices (with applications to the study of the limiting spectral distribution), edges of the spectrum of sample covariance matrices, as well as some applications of random operators to questions in high-dimensional convex geometry.

Representation stability is an exciting new area that combines ideas from commutative algebra and representation theory. The meta-idea is to combine a sequence of objects together using some newly defined algebraic structure, and then to translate abstract properties about this structure to concrete properties about the original object of study. Finite generation is a particularly important property, which translates to the existence of bounds on algebraic invariants, or some predictable behavior. I'll discuss some examples coming from combinatorial representation theory (Kronecker coefficients) and topology (configuration spaces).

Markov duality in spin chains and exclusion processes has found a wide variety of applications throughout probability theory. We review the duality of the asymmetric simple exclusion process (ASEP) and its underlying algebraic symmetry. We then explain how the algebraic structure leads to a wide generalization of models with duality, such as higher spin exclusion processes, zero range processes, stochastic vertex models, and their multi-species analogues.