Pre-seminar
Title: What is essential dimension?
Abstract: Everyone knows that a general polynomial of degree at least 5 cannot be solved in radicals. However, this is not the end of the story. Attempts to quantify how much a given polynomial can be simplified by a Tschirnhaus transformation naturally leads to the notion of essential dimension. The essential dimension of an algebraic object is the minimal number of independent parameters required to define it. I will try to motivate this notion in the framework of the classical problem of solving (or at least simplifying) polynomials in one variable, where the objects in question are field extensions of finite degree. Surprisingly, even in this classical setting long-standing open questions remain.
Seminar
Title: Hilbert's 13th Problem for algebraic groups.
Abstract: The algebraic form of Hilbert's 13th Problem asks for the resolvent degree rd(n) of the general polynomial f(x) = x^n + a_1 x^{n-1} + ... + a_n of degree n, where a_1, ..., a_n are independent variables. Here rd(n) is the minimal integer d such that every root of f(x) can be obtained in a finite number of steps, starting with C(a_1, ..., a_n) and adjoining an algebraic function in <= d variables at each step. It is known that rd(n) = 1 for every n <= 5. It is not known whether or not rd(n) is bounded as n tends to infinity; it is not even known whether or not rd(n) > 1 for any n. Recently Farb and Wolfson defined the resolvent degree rd_k(G), where G is a finite group and k is a field of characteristic 0. In this setting rd(n) = rd_C(S_n), where S_n is the symmetric group on n letters and C is the field of complex numbers. In this talk I will define rd_k(G) for any field k and any algebraic group G over k. Surprisingly, Hilbert's 13th Problem simplifies when G is connected. My main result is that rd_k(G) <= 5 for an arbitrary connected algebraic group G defined over an arbitrary field k. If time permits, I will also discuss recent joint work with Oakley Edens on Hilbert's 13th Problem (this time, the original version, for the group G = S_n) in prime characteristic.