Back in the 1970s and 1980s, a bunch of people proved a host of results about finite-dimensional irreducible representations of simple Lie algebras over the complex numbers. They discovered a startling dichotomy: in various ways, small representations behave similarly to each other and large representations behave similarly to each other. Moreover, the difference between “small”…
If you love semisimple Lie algebras, like I do, then you should take a look at Meinolf Geck’s paper On the construction of semisimple Lie algebras and Chevalley groups that just appeared in Proceedings of the AMS (journal version, preprint version). He gives an elementary and canonical construction of a semisimple complex Lie algebra from its…
Writing math papers is more like technical writing than writing essays, but general advice on technical writing only gets you so far. For more advanced advice, Igor Pak offers his opinionated thoughts. I especially recommend his tips on how to create a productive Final Remarks section. (Real beginners at writing math papers make a standard…
My recent article Essential dimension of algebraic groups, including bad characteristic with Bob Guralnick was named an “Editor’s Choice“. As a consequence, it will be available free open access on the journal website for the next year. Here’s the abstract: We give upper bounds on the essential dimension of (quasi-) simple algebraic groups over an…
My survey E8, the most exceptional group will appear in an upcoming issue of Bulletin of the American Mathematical Society. It talks about the history and recent results on E8, focusing on the algebraic perspective of Lie algebras, Lie groups, and algebraic groups. You can read the article here or get an inkling of some the…
The publisher is making a new edition of Optimization in microeconomics! This is a textbook for a senior-level capstone course in mathematical economics. It was developed for Math/Econ 425 at Emory University. The text is a polishing of the extensive course notes that economist Chris Curran developed over the many years that he taught that…
Updated June 10, 2016 I have two new papers out, both joint with Bob Guralnick: Spinors and essential dimension: We show that certain representations of the spin group Spinn are generically free for n > 14 and calculate the generic stabilizer for n ≤ 14, for all base fields. This shows that the formula for the…
New preprint out now: Tits p-indexes of algebraic groups, joint with Charles De Clercq. We give a `localized at a prime p‘ version of the results in Jacques Tits‘s classic 1966 paper Classification of semisimple algebraic groups. In that paper, he introduced the index of a semisimple group and classified all the possible values it…
My paper Some people have all the luck, giving the details about the mathematical analysis behind Lawrence Mower’s 2014 expose of the Florida Lottery, has appeared. (See also this description from MAA FOCUS.) Our analysis has since been applied to identify criminals in many other states. This paper was written with Richard Arratia (USC), Lawrence…
My paper “Simple groups stabilizing polynomials“, written with Bob Guralnick, has appeared in Forum of Mathematics Pi. In it, we show that, if a simple algebraic group G stabilizes a polynomial function f on a vector space V, then with a very short list of exceptions G is actually the identity component of the stabilizer of f. We give various concrete applications of…