We say a finite group G is realized as a Galois group over a field K if there is a finite Galois extension L/K such that G is isomorphic to Gal(L/K). We will develop a method of checking whether a given group is realized over a given field. In particular, we will show that A_n is realized over Q.
A Method to Realize Groups as Galois groups: Rigidity in An, Part 2
Oct. 22, 2013 1pm (MATH 350)
Number Theory
Ryan Rosenbaum
X
Tate included in his thesis an analog of Poisson summation on the adeles which he referred to as "Riemann-Roch" for number fields. Our goal is to explain Tate's justification for using this name. We will briefly review necessary properties of local and global fields along with the Fourier transform. We will then introduce the notion of the divisor group for a function field and show that Tate's "Riemann-Roch" implies a more familiar statement of theorem in this context.
The Robinson-Schensted-Knuth (RSK) correspondence is a bijection between finitely supported countably infinite matrices and pairs of semi-standard Young tableaux with many applications to combinatorics and symmetric functions. Berenstein and Kirillov obtained explicit tropical formulas for the RSK bijection, and also considered geometric lifts of these formulas. Recently, O'Connell and other showed that under the geometric RSK (gRSK) correspondence the Schur functions are replaced by analytic objects called Whittaker functions which are ubiquitous in analytic number theory and physics. They also used gRSK to "lift" the Cauchy-Littlewood identity, which gave an alternative proof of a formula conjectured by Bump and proven by Stade.
We will discuss RSK, its connection to the classical Cauchy-Littlewood identity, and the formulas of Berenstein and Kirillov. We will then explain how to obtain the analog of the Cauchy-Littlewood identity from gRSK. Time permitting, we will also discuss how the classical Cauchy-Littlewood shows up in number theory.
The geometric Robinson-Schensted-Knuth correspondence and Whittaker functions II
A Method to Realize Groups as Galois groups: Rigidity in An, Part 2