We shall discuss some results which classify the smallest degree representations of simple groups. We will then discuss various applications of these ideas including the solution of a conjecture of Serre on semisimplicity of certain representations and a conjecture of Kollar and Larsen on the non-irreducibility of symmetric powers.
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 show the convergence of Kahler Ricci flow on every 2-dimensional orbifold if the underlying orbifold has big $\alpha_{\nu, 1}$
or $\alpha_{\nu, 2}$ (Tian's invariants). We then find some new Kahler Einstein metrics on orbifolds by calculating Tian's invariants.