Algebra Seminar Archive

November 30, 2009

Computer Science Seminar: Brendan Dolan-Gavitt

Title: TBA Speaker: Brendan Dolan-Gavitt '06, Georgia Tech

Abstract: TBA

April 17, 2009

Ranks of elliptic curves in towers of function field extensions

Speaker: Lisa Berger
Abstract: Given an elliptic curve E over a global field K, it is a theorem of Mordell and Weil that the K-rational points on E form a finitely generated abelian group. For K=\mathbb(Q) it is conjectured that there exist elliptic curves with Mordell-Weil groups E(K) of arbitrarily large rank. We'll discuss some current work on ranks of elliptic curves and higher dimensional abelian varieties in towers of function field extensions.

April 10, 2009

Field descent method and its applications

Speaker: Ka Hin Leung, National University of Singapore
Abstract In the study of some combinatorial objects with a suitable group G of symmetries, it often involves certain equations in cyclotomic integers. These equations arise from applying complex representations to group ring equations characterizing the combinatoric objects in question. The three almost exclusively used methods in this direction are multiplier theorems, the self-conjugacy approach and the field descent method. Field descent method deals with equation of the form |X|2 = n where n is an integer and X is in Z[fm]. The main objective is to find a computable integer F(m,n), a divisor of m such that X f m j is in Z[fF(m,n)].

In this talk, we will highlight the main idea of the method and demonstrate how it can be applied in the study of difference sets, circulant Hadamard matrix and Barker conjectures.

April 03, 2009

Schur-sigma groups and unramified p-extensions

Speaker: Michael Bush, Smith College
Abstract The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways which will be explained in the first part of the talk along with an overview of some earlier results on an explicit family of finite groups of this type. In the second part I will report on work in progress to catalog all such finite groups of small order when p = 3. In the process, some new infinite families have been discovered.

March 27, 2009

Eisenstein series on loop groups

Speaker: Kyu-Hwan Lee, UCONN
Abstract: After reviewing affine Lie algebras and loop groups, we will define Eisenstein series on loop groups and calculate their constant terms. It will be shown that the constant terms satisfy functional equations given by affine Weyl group symmetries.

February 20, 2009

Pure generation by pure-projective modules

Speaker: Philipp Rothmaler, CUNY
Abstract: The behavior of, for instance, flat modules (resp. torsion-free abelian groups) is determined by that of the finitely generated projectives (resp. free abelian groups of finite rank)--whose direct limits they are--within the context of finitely presented modules (resp. finitely generated abelian groups). As an example, direct products of flat modules are flat if and only if the ring (a cyclic projective module!) is coherent, a property that can be expressed in terms of finitely presented modules. (The ring of integers is coherent, and products of torsion-free abelian groups are torsion-free.) It will be shown where this leads when the context is broadened to arbitrary pure-projective modules (in case of the integers, to arbitrary direct sums of finitely generated abelian groups). I will try and keep the talk accessible to a general algebraic audience.

February 06, 2009

Deformations of Galois Representations

Speaker: Michael Broshi, UMASS
Abstract: In this talk we will introduce the basic notions of the deformation theory of Galois representations. This study became popularized after its use in the proof of Fermat's Last Theorem. We will discuss some applications and, time permitting, some recent developments.

This will be an introductory talk and no familiarity with the subject is assumed.

December 05, 2008

Class number indivisibility for function fields

Speaker: Siman Wong, UMass Amherst
Abstract: We will review facts about class numbers of quadratic function fields, and we will discuss new results by way of quadratic forms over function fields.

*Note the unusual time.

October 31, 2008

Injectivity and surjectivity of maps between curves

Speaker: Mike Zieve, IAS
Abstract: Let f be a morphism of curves over a finite field k. If k is sufficiently large compared to the degree of f and genera of the curves, then injectivity of f on k-rational points is equivalent to surjectivity. Surprisingly, the bounds on the size of k for these two implications seem to have wildly different orders of magnitude. I will explain this and give examples and properties of the bijective maps -- for instance, if f is bijective on k-rational points for a sufficiently large field k, then f is bijective on r-rational points for infinitely many finite extensions r of k. I will also discuss recent progress towards classifying all maps with the latter property in case the curves have genus zero, based on connections with curves having large automorphism group and instances of a positive characteristic analogue of Riemann's existence theorem.

October 24, 2008

Arithmetic Topology

Speaker: Cam McLeman
Abstract: One might measure the strength of a mathematical analogy by seeing how few independent identifications are needed before two seemingly disparate theories begin to look like one and the same. In this talk, well explain a blossoming field of mathematics called arithmetic topology, stemming from the remarkably strong analogy obtained only by identifying primes in number fields with knots embedded in 3-manifolds. In addition to being a rather quirky and amusing analogy, the analogy has furthered both fields of study by adopting the others techniques and by taking cues from the others results. This talk will motivate the original analogy between knots and primes and give some examples of current research being fueled by this analogy.

October 17, 2008

Representations of positive ternary quadratic forms

Speaker: Byeong-Kweon Oh, Seijong University
Abstract: A positive definite quadratic form f is said to be regular if it globally represents all integers that are represented by the genus of f. In 1997, Jagy, Kaplansky and Schiemann provided a list of 913 candidates of primitive positive definite regular ternary quadratic forms, and all but 22 of them are verified to be regular. In this talk we show that 8 of these 22 candidates are, in fact, regular. We also show some finiteness result on ternary forms that represent every eligible integer in some arithmetic progression.

October 10, 2008

Arithmetic of Fundamental Groups

Speaker: Chris Rasmussen, Wesleyan University
Abstract: In this talk, we introduce the arithmetic fundamental group attached to an algebraic curve, and describe the Galois representations attached to its automorphism group. These representations are quite useful in answering questions about the arithmetic of algebraic varieties, and we will try to make this explicit with lots of examples and open problems.

May 05, 2008

Stable Group Homology and Hecke Operators

Speaker: Avner Ash, Boston College

April 18, 2008

Thom Pietraho: 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

Jim Reid: 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

Keith Conrad: 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

Wai Kiu Chan: 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.