Algebra Seminar Archive

May 05, 2008

Stable Group Homology and Hecke Operators

Speaker: Avner Ash, Boston College

April 18, 2008

Cells in Hecke algebras

Speaker: Thom Pietraho, Bowdoin

Title: Cells in Hecke algebras

Abstract: Cells in Hecke algebras play a central role in the representation theory of semisimple Lie groups. We will examine the combinatorics which they give rise to in the setting of classical groups and survey what is not yet known.

April 11, 2008

Poonen's definition of the integers inside the rationals

Speaker: Carol Wood, Wesleyan University
Abstract: Sixty years ago, Julia Robinson proved in her PhD thesis that the ring of integers is first-order definable in the rationals. Her proof involved several facts about the representation of integers by quadratic forms. The formula which picks out the integers from among the rationals is complicated, involving several alternations of quantifiers. Recently Bjorn Poonen produced a formula which does the same job, but with only one alternation of quantifiers. I will describe the algebraic ingredients of Poonen's proof in which he uses quaternion algebras to select the integers from among the rationals. This result may bring us closer to, but does not solve, Hilbert's 10th problem over the rationals, which asks whether an algorithm exists for deciding which polynomial equations with rational coefficients have rational solutions.

April 04, 2008

A representation of convex semilinear sets

Speaker: Philip Scowcroft, Wesleyan University
Abstract: If F is an ordered field, a subset of n-space over F is said to be semilinear just in case it is a finite Boolean combination of closed halfspaces, where a closed halfspace is the set of all points obeying a weak linear inequality defined over F. Andradas, Rubio, and Velez have shown that closed (open) convex semilinear sets are finite intersections of closed (open) halfspaces (an open halfspace is defined as before, but with a strict inequality). This talk will discuss a representation of arbitrary convex semilinear sets analogous to the result of Andradas, Rubio, and Velez.

March 28, 2008

Faithful and fully faithful abelian groups

Speaker: Jim Reid, Wesleyan University
Abstract: Let A be an abelian group with R = EndZ(A): In the literature A is called faithful (resp. fully faithful) if, for every finitely generated (resp. arbitrary) nonzero module MR, the tensor functor is faithful on the class of finitely generated (resp. arbitrary) right R-modules. Faithful and fully faithful abelian groups are interesting in their own right and are closely connected with the splitting of exact sequences. We introduce the concept of topological faithfulness and explore the extent to which these three notions are equivalent. (Joint work with W. J. Wickless)

February 22, 2008

Nagao's heuristic

Speaker: Keith Conrad, UCONN
Abstract: We will discuss a formula used by Nagao to find examples of elliptic curves which achieved record (at the time) lower bounds on the rank, and whether such a formula is really valid. This is joint with R. Murty.

February 15, 2008

Strong Approximation of Quadrics and Representations of Quadratic Forms

**PLEASE NOTE CORRECTED DATE OF FEBRUARY 15TH, 2008, AT 2:00 P.M. (NOT SATURDAY, FEBRUARY 16TH) Wai Kiu Chan, Wesleyan University
Abstract: Let V be an indefinite quadratic space over a number field F and U be a nondegenerate subspace of V. Suppose that M is a lattice on V, and that N is a lattice on U which is represented by M locally everywhere. I will describe a necessary and sufficient condition for which there exists a representation of N by M that approximates a given family of local representations. This is applied to determine when the variety of representations of U by V has strong approximation with respect to a finite set of primes of F that contains all the archimedean primes. This is a joint work with Nicu Beli.