A Bachelor's Degree in Mathematics offers sharp intellectual depth and the breadth to apply technical knowledge to a variety of disciplines. The forma mentis of mathematicians makes them attractive to a number of industries, from Wall Street to engineering firms, and K-12 education. In view of this, it is not hard to understand why many technically gifted UCI students choose to major in mathematics. But how many of our students think about a career in mathematical research? When do they start even considering the possibility of pursuing graduate studies in mathematics? How do they learn what it takes to craft a successful application for a PhD degree in our top universities? Naturally, individual one-on-one interactions with our faculty and graduate students certainly take place and their role is invaluable, but in this talk I would like to explore an alternative and synergistic mechanism to address these questions in a more 'formalized' manner.

I will present a number of ideas, with the overarching goal of creating a platform to provide information, support, enthusiasm and critical encouragement to all our undergraduates that want to know what graduate school is about.

Fourier coefficients of automorphic forms are the building blocks for automorphic L-functions. While these coefficients are often quite mysterious, there is one family of automorphic forms whose Fourier coefficients do have an explicit and rather uniform description -- Eisenstein series. In fact, Langlands' initial study of Eisenstein series' coefficients in the 1960's led him to make conjectures about equalities of L-functions which inform much of modern number theory. I'll discuss two new explicit descriptions for Fourier coefficients of Eisenstein series which hold in great generality and hint at undiscovered connections among automorphic forms, representation theory, and physics. One description makes use of Kashiwara crystal graphs and the other uses the 6-vertex model in statistical mechanics. Both objects possess beautiful combinatorial structure that deserves to be more widely known, though we do not assume familiarity with either and all concepts mentioned above will be defined over the course of the talk.

This talk presents a strategy for computational wave propagation that consists in decomposing the solution wavefield onto a largely incomplete set of eigenfunctions of the weighted Laplacian, with eigenvalues chosen randomly. The recovery method is the ell-1 minimization of compressed sensing. For the mathematician, we establish three possibly new estimates for the wave equation that guarantee accuracy of the numerical method in one spatial dimension. For the engineer, the compressive strategy offers a unique combination of parallelism and memory savings that should be of particular relevance to applications in reflection seismology. Joint work with Gabriel Peyre.

We explain how to construct knot and tangle invariants (such as the Jones polynomial or Khovanov homology) by studying holomorphic vector bundles on certain compact, complex manifolds. Topologically these complex manifolds are just products of the same projective space P1. Conjecturially, if one used Grassmannians Gr(k,n) instead of projective spaces this would give a series of new knot invariants.