This talk is aimed (mostly) at undergraduate students. 

Abstract: In the 1930’s and 1940’s, Murray and von Neumann developed a theory of operators on Hilbert spaces, which heuristically may be thought of as infinite matrices acting on infinite dimensional vector spaces. Their works include a procedure which starts with an infinite group, a discrete object, and generates a von Neumann algebra, an analytic object which is a continuous analog to the n × n matrices. Much of the active research in this field has been generated by the following question: are structural properties of groups able to classify the resulting algebras? Obtaining a satisfactory resolution to this problem has been surprisingly difficult since standard group invariants are often not invariants of the algebras. We give a brief survey of the evolution of this problem, the surprising broader impacts including the emergence Jones polynomial, and the recent rapid progress in this classification program due to the advent of S. Popas deformation/rigidity theory. We close by describing recent developments in this program which have been made by my collaborators and myself. 

About the speaker: Rolando de Santiago is currently an Assistant Adjunct Professor and a UC Presidential Postdoctoral Fellow at UCLA working under S. Popa. His work is in the classification of type II1 von Neumann algebras, a subfield of functional analysis, and his research interests extend into group theory, topology, fractal geometry, and mathematical physics. He was born and raised up in the South-Eastern part of Los Angeles with 6 of his siblings. He spent 27 years studying at numerous public institutions including Pasadena City College and Cal Poly Pomona. After approximately 8 years of undergraduate work, he finally earned his B.S. in Mathematics. He completed his Masters in Mathematics at Cal Poly Pomona shortly thereafter. His mentors at Cal Poly, J. Rock and R. Wilson, strongly suggested that he pursue his Ph.D. After a significant amount of convincing, he threw all his belongings into a U-Haul, moved to Iowa City, and started grad school at the University of Iowa. He worked under of I. Chifan, the advisor who would help his launch his research career.

You may RSVP here:  https://docs.google.com/forms/d/e/1FAIpQLScYrvrod7lMjOBmMt3Hhz4YSZmqjdOEKmHIsTg70pa4FpQOSA/viewform

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.