The Algebra Seminar

Spring 2008

The seminar meets Tuesdays in room LN 2205 at 2:50 p.m. There will be refreshments in the Anderson Reading Room at 4:00.

Organizers: Alex Feingold and Marcin Mazur

January 29 : Organizational meeting

January 31 (Thursday!), 2.50 pm in LN 2205 : Elizabeth Wilcox
Title: Thompson's Normal p-Complement Theorem
Abstract: In 1964 John G. Thompson published an article in the Journal of Algebra giving a sufficient condition for a group to have a normal p-complement (NPC) for an odd prime p, a result which has become known as Thompson's NPC Theorem. In this first part of my candidacy exam I will define key concepts like NPC and the Thompson subgroup, and then we will work through the most of the details of the proof of Thompson's NPC Theorem.

February 5: Tom Head
Title: Computing with Light: Permutations Light Up!
Abstract: An algorithmic problem (that subsumes the classical problem of finding all Hamiltonian cycles in a directed graph) is: Given a finite set A and binary relation R contained in A X A, find all the subsets of R that are permutations of A. We will solve an instance of this problem by 'taking at most r+2a photos', with r & a the numbers of elements in R and A respectively. As our lab for optical computing, a xerox machine suffices. Each 'photo' becomes a xerox onto a transparency. This computation will be carried out 'before your eyes' by displaying in a time sequence the transparencies as they arise during the computation. It will be clear that the procedure used applies to all instances of the given algorithmic problem.
Through computing with light we have previously given linear time procedures for: (1) Satisfiability of sets of boolean clauses, (2) Finding the maximum cliques in a graph, and (3) Finding the minimum dominating sets in a graph. In all cases the new procedures have been obtained by adapting the solutions we developed previously in wet labs through collaborations with S. Gal & M. Yamamura in Binghamton and with H. Spaink in Leiden.

February 7 (Thursday!), 10.05 am in LN 2205: Elizabeth Wilcox
Title: The Frobenius Kernel of a Frobenius Group is Nilpotent
Abstract: Frobenius groups are an interesting type of group, the structure of which is well known. In this second part of my candidacy exam I will define Frobenius group, and subsequently Frobenius kernel, and then establish some of the structural properties of these groups. Using Thompson's NPC Theorem we will then prove that the Frobenius kernel of a Frobenius group is always nilpotent, a long standing hypothesis that Thompson proved in the 1960's.

February 12 : Simon J. Joyce
Title: Reversibility in well know classes of groups.
Abstract: Given a group G and g an element of G we say g is reversible (in G) if g is conjugate to its own inverse.
This talk is based on a research project I did one Summer. I will present some basic results about reversible group elements and determine set of reversible elements for abelian , dihedral, symmetric and alternating groups. I'll also talk a bit about reversibility for groups of small finite order.

February 19: Viji Thomas
Title: The non-abelian tensor product of finite groups is finite: a homology free proof.
Abstract: Abtsract

February 26: Allen Mann (Colgate University)
Title: Independence-friendly cylindric set algebras
Abstract: Independence-friendly logic is a conservative extension of first- order logic that has the same expressive power as existential second- order logic. We algebraize independence-friendly logic in the same way that Boolean algebra is the algebra of propositional logic and cylindric algebra is the algebra of first-order logic.
We will show that every independence-friendly cylindric set algebra over a structure is an expansion of a Kleene algebra and that the equational theory of the class of such Kleene algebras is finitely- axiomatizable.

March 4 (joint with Combinatirics), 1.15-2.15 in LN 2205 : Catherine Stenson (Juniata College)
Title: Line-Shelling Zonotopes of Polytopes
Abstract: A line shelling of a polytope P is a permutation of its facets which corresponds to a particularly nice way of constructing P. The set of all shellings of P by lines through the origin is the vertex set of yet another polytope, the line-shelling zonotope of P. I'll describe the construction of this polytope and discuss some of its properties.

March 4 : Ben Brewster
Title: Pronormality and Local Pronormality
Abstract: TBA

March 11 : Quincy Loney
Title: Alternative Algebras
Abstract: In 1898 Hurwitz showed that every normed division algebra over the reals is isomorphic to either a real, complex, quaternion, or octonion algebra. In this talk we will discuss some of the properties of these alternative composition algebras. We will also use the Cayley-Dickson construction to show the existence of such algebras.

March 18 : Alex Feingold
Title: Matrix group actions on hyperbolic spaces
Abstract: I will describe models of hyperbolic spaces as 2x2 Hermitian matrices over K, where K is one of the real normed division algebras, R, C, H or O, and show how certain matrix groups of 2x2 matrices over K act naturally. I will discuss how certain subgroups of these matrix groups, with entries from a discrete subring of K, are Coxeter groups, which include the Weyl groups of certain hyperbolic Kac-Moody Lie algebras. This is joint work with Hermann Nicolai and Axel Kleinschmidt.

March 25 : Spring recess
Title:
Abstract:

April 1 : Matthew Short
Title: On Diagram Groups
Abstract: In this talk we will give basic definitions, examples, some motivations and a few results on diagram groups.

April 8

Title

Abstract

April 15 : Darryl Daugherty
Title: Lie Algebra Modules
Abstract: In this talk I will first provide some background on Lie Algebras, and then provide for the construction of specific Lie Algebra modules, and how to compute dimensions of their weight spaces.

April 22 : Elizabeth Dittrick
Title: Cooper's Theorem
Abstract: In this talk, I will prove Cooper's Theorem which states that if f is an automorphism of a group G such that f(g) is a power of g for each g in G (we call such automorphisms power automorphisms) then f(g)g^{-1} is in the center of G for any g in G. Cooper's Theorem has some strong consequences that are beyond the scope of this talk. For example, using Cooper's Theorem we can show power automorphisms centralize G/Z(G) and the commutator subgroup of G' of G, as well as, a result of Schenkman that the norm of a group G is contained in the second center.

April 28 (Monday!), 3.30 pm in LN 2402 : Bogdan Petrenko (SUNY Brockport)
Title: When are two pairs of 2x2 matrices conjugate?
Abstract: The goal of the talk is to give a criterion of when 2 pairs of 2x2 matrices over a field are conjugate by the same matrix. This criterion is valid in an open and dense subset of pairs of 2x2 matrices, but does not work for all such pairs. I don't know how to extend this criterion to all such pairs (and to pairs of matrices of other sizes).

April 29 : Mark Kleiner, Syracuse University
Title: Integral quadratic forms in representation theory of algebras
Abstract: For the variable X = (X_1,...,X_n), a quadratic form q(X) = \sum_{i=1}^n X_i^2 + \sum_{j>i} q_{ij} X_i X_j is said to be integral if q_{ij} is an integer for all j>i. A vector z = (z_1,...,z_n) in Z^n is said to be positive, written z>0, if z is nonzero and z_i is non-negative for all i. The vector z is a root if q(z) = 1. The form q is called positive definite if q(z) > 0 for all nonzero z, and q is called weakly positive if q(z) > 0 for all z > 0. We present some facts about positive roots of a positive definite or weakly positive integral quadratic form, discuss the relationship with the classical root systems that come from Lie theory, and explain how integral quadratic forms ``control" representations of quivers or of partially ordered sets.

May 1 (Thursday), 11:40 AM - 1:05 PM in LN 2402 : Erin Decker (MA-thesis defense)
Title: On the Construction of Groups with Prescribed Properties
Abstract: In the literature, many examples of groups with prescribed properties are merely presented in terms of their generators and defining relations without considering the calculations required to find them. In this talk we will present two types of construction. The first is by iterated split extensions and will be used to construct finite p-groups with prescibed properties on their commutators and autocommutators. The second one uses methods from linear algebra and finite fields and is used to construct finite nonabelian solvable groups with all proper homomorphic images cyclic. These groups play a role in finite coverings of groups by proper subgroups.

May 6 : Nor Muhainiah Mohd Ali (University Teknologi Malaysia)
Title: Using GAP to compute homological invariants of 2-generator non-torsion groups of nilpotency class two.
Abstract: Let R be the class of 2-generator non-torsion groups of nilpotency class 2. Using their classification and non-abelian tensor squares, we determine certain homological invariants of groups in R, such as the exterior square, the symmetric square and the Schur multiplier. With the help of GAP, we first compute the invariants for some representative groups, then extrapolate from there to obtain invariants in the general case. This is joint work with Luise-Charlotte Kappe and Nor Haniza Sarmin.

PREVIOUS SEMESTERS

Departmental home page.