There are 3 main goals, which will likely occupy more than a single semester:

1. Understanding the connection between probability distributions, A-discriminants, and the expected behavior of roots of systems of polynomial equations.

2. Understanding the algorithmic implications of (1), with an eye toward Smale's 17th Problem. (The latter problem deals with solving systems of polynomial equations in polynomial time, on average, in a sense that is useful for numerical analysis.)

3. Developing the background necessary to understand recent work of Shiffman and Zelditch on random holomorphic sections of line bundles over Kahler manifolds. (Their theory includes the theory of random matrices as a very special case.)

We will also explore relevant number-theoretic topics along the way, such as the distribution of primes in arithmetic progression. The latter is actually intimately connected with polynomial system solving over the complex numbers. Needless to say, we will also define basic algorithmic notions (such as complexity classes) along the way.

The facilitator (Rojas) will begin with some introductory talks, giving many examples along the way. THIS SEMINAR IS INTENDED FOR GRADUATE STUDENTS, so questions (and contributed talks!) are most welcome.

NOTE: We will meet in Milner, which has outside doors that close around 5:00 pm. So please try to come early, or call Rojas' cell number (979) 220-4273, if you have trouble entering the building.