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.

We construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches, without performing an intervening surgery. In the restrictive context of rotational symmetry, the construction gives evidence in favor of Perelman's hope for a "canonically defined Ricci flow through singularities". This is joint work with Sigurd Angenent and Cristina Caputo.