Professor Jarod Alper will speak to graduate students at the Current Topics Seminar.
Title: A tale of two polynomials
Abstract: We will investigate various notions of complexity of a given polynomial. While these notions are interesting from a purely mathematical perspective, they are also of deep interest to computer scientists and engineers with vast applications in the real world. We will mostly focus on two polynomials—-the determinant and the permanent (which is just like the determinant but without all of the minus terms). Measuring the difference between the complexity of these two polynomials will yield an algebraic analogue of the famous P vs NP problem.
While I will try to say something about my own work, the focus will be on discussing some really cool open problems in the field. We will introduce Valiant’s conjecture (the algebraic analogue of the P vs NP problem), Comon’s conjecture and Strassen’s conjecture. Can sophisticated techniques in algebraic geometry, commutative algebra, homological algebra and representation theory shed any light on these problems? Come to find out!