The Algebra Seminar
Fall 2009
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: Marcin Mazur
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September 1 : Organizational meeting
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September 8: Viji Thomas
Title: Automorphisms of direct products of finite groups
Abstract: If H and K are finite groups with no common direct
factor and if G=H x K, then the structure and order of AutG can be simply
expressed in terms of AutH, AutK and the homomorphism groups Hom(H,Z(K)) and
Hom(K,Z(H)).
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September 15: Collin Formanek
Title: Growth of Algebras and Gelfand-Kirillov Dimension
Abstract: A presentation of the basic definitions and properties of the growth of a
finitely generated k-algebra, and the Gelfand-Kirillov Dimension of an arbitrary
k-algebra.
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September 22: Bogdan Petrenko (SUNY Brockport)
Title: The structure of maximal subalgebras of matrix algebras
over finite fields.
Abstract: We will describe the structure of maximal subalgebras
of the matrix algebra over finite field. Then we'll describe the structure of
all possible intersections of such subalgebras for the cases of 2-by-2
and 3-by-3 matrices. The understanding of these intersections yields the
formulas for the smallest number of generators for finite direct sums of
integer matrix rings and for the probability that several randomly
chosen elements generate such a ring. (This is join work with Rostyslav
Kravchenko and Marcin Mazur.)
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September 29: Joanne Redden (Elmira College)
Title: On the covering number of small alternating groups.
Abstract: According to Bernhard Neumann, every group with a
noncyclic finite homomorphic
image is the union of finitely many proper subgroups. The minimal number of
subgroups needed to cover a group G is called the covering number of G.
Tomkinson showed that for a solvable group the covering number is of the
form prime power plus one and he suggested the investigation of the covering
number for families of finite simple groups. So far, a few results are known,
among them some for alternating groups. Exact covering numbers are known for
alternating groups of rank 5 or rank n, where n is even and congruent to
2 mod 4,
and estimates have been given for others. Using GAP and a little help from graph
theory we have been able to determine the covering number for alternating
groups of rank 7 and 8 and find a better estimate for rank 9.
This is joint work with Luise-Charlotte Kappe.
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October 6: Joseph Evan (Kings College)
Title: Characterizing subgroups of a direct product satisfying
the strong Frattini argument
Abstract: Several authors have recently been working on a project of characterizing
subgroup properties in direct products of groups. The latest contribution to this
research is the characterization of subgroups satisfying the strong Frattini
argument. In this talk, I will give some general background on subgroups satisfying
the strong Frattini argument, present the aforementioned characterization, and
elaborate on some of its key elements. The work presented in this talk is joint
with Ben Brewster and P. Hauck and S. Reifferscheid (Tubingen).
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October 13: Benjamin Brewster
Title: Subgroups of a direct product satisfying the Frattini Argumen
Abstract: This talk has overlap with last week's talk;even a prequel.
Information will be presented about subgroups satifying the Frattini
Argument and emphasis will converge toward direct products.We will see
that the "critical section" of such a subgroup is nilpotent but is not
necessarily abelian,as in the case with the Strong Frattini Argument.
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October 20: Luise-Charlotte Kappe
Title: Finite coverings: a journey through groups, loops, rings and semigroups.
Abstract: A group is said to be covered by a collection of subsets if each element
of the group belongs to at least one subset in the collection: the
collection of subsets is called a covering of the group.
On the bottom of page 105 of Derek Robinson's "Finiteness Conditions and
Generalized Soluble Groups I", there are two theorems which served as my
roadmap for exploring finite coverings of groups, loops, rings and
semigroups. The first one, an unpublished result by Reinhold Baer, is stated
as follows.
Baer's Theorem: A group is central-by-finite if and only if it has a finite
covering by abelian subgroups.
The second one, due to Bernhard Neumann, is stated as follows.
Neumann's Lemma: Let G be a group having a covering by finitely many cosets
by not necessarily distinct subgroups. If we omit any cosets of subgroups of
infinite index, the remaining cosets will still cover the group.
In my talk I will report on my journeys through groups, loops, rings and
semigroups, on what I discovered there about finite coverings together with
several fellow travelers and on some discoveries which might still lie
ahead.
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October 27: Martin Kassabov (Cornell University)
Title: Subspace arrangements and property T
Abstract: I will mainly talk about (my viewpoint at) a method for proving property T
started by Dymara and Januszkiewicz. Their original motivation came
from groups acting on dimensional buildings, but the refined idea does not
use anything more than angles between subspaces in an finite dimensional
Euclidian space.
Parts of the talk are based on a work of M. Ershov and A. Jaikin.
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November 3: (CROSS LISTING WITH NUMBER THEORY SEMINAR; SPECIAL TIME 4:15 p.m.): Benjamin Lundell, Cornell University.
Title: New Parts of Hecke Rings.
Abstract: In 1970s, Barry Mazur studied certain completed Hecke algebras and related their ring-theoretic properties to deep arithmetic results. In this talk, we will discuss recent progress towards answering a modified version of a question of Mazur's about the rank of such a Hecke algebra and some of the arithmetic corollaries.
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November 10: Martha Kilpack
Title: Pseudocomplementation and Completeness of Closure
Operators on a Poset (First of two talks for Admission to Candidacy Exam).
Abstract We will take a look at the set of closure operators
(uco(P)) on a poset
(P). In particular, we will focus on properties of a poset, P, that yield
uco(P) to be a complete lattice. This talk is based on a paper by
Francesco Ranzato, Universita di Padova.
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November 12 (Thursday) 1:15 PM, LN-2206 : Martha Kilpack
Title: Pseudocomplementation and Completeness of Closure
Operators on a Poset (Second of two talks for Admission to Candidacy Exam).
Abstract: We will take a look at the set of closure operators (uco(P)) on a poset
(P). In particular, we will focus on properties of a poset, P, that yield
uco(P) to be a pseudocomplemented lattice. This talk is based on a paper
by Francesco Ranzato, Universita di Padova.
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November 17: (CROSS LISTING WITH COMBINATORICS SEMINAR; SPECIAL TIME 1:15 p.m.): Justin Lambright, Lehigh University.
Title: A combinatorial interpretation for computations in the quantum polynomial ring.
Abstract: A Hopf algebra called the quantum coordinate ring of SL(n,C) is often studied in terms of a related noncommutative ring called the quantum polynomial ring in n2 variables. Various bases of these rings and their representation-theoretic applications lead to the study of transition matrices whose entries are commutative polynomials having nonnegative integer coefficients. Examples of such polynomials include Brenti's modified R-polynomials. I generalize Brenti's work to give combinatorial interpretations for coefficients in a larger class of transition matrices. As an application, I simplify somewhat the previous formulation of the dual canonical basis of the quantum polynomial ring.
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November 24: (CROSS LISTING WITH COMBINATORICS SEMINAR; SPECIAL TIME 1:15 p.m.): Matthias Beck (San Francisco State University)
Title: Symmetrically Constrained Compositions of an Integer
Abstract:
Abstract
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December 1: Adam Perry
Title: A Theorem of Szigmondy (First of two talks for Admission to Candidacy Exam)
Abstract: We will see a result about the existence of certain prime divisors
of a^n - 1 which don't divide a^i - 1 for i=1,...,n-1 and positive integer
a. This theorem from elementary number theory has applications in
group theory, but that will be part of the second talk.
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December 4: (SPECIAL DAY, TIME, PLACE: FRIDAY 2:30 in SL 306): Ilir Snopce, Binghamton University.
Title: Lie Methods in Pro-p Groups (Ph.D. thesis defense).
Abstract: I will talk about two different problems on pro-p
groups that I have discussed in my thesis. The first one is related to normal
zeta functions of pro-p groups, and the second one is a problem raised by
Iwasawa. I use Lie ring techniques to solve these problems.
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December 8: Rostyslav Kravchenko (Texas A&M University)
Title: The number of generators of algebras of matices over rings of integers of a number field.
Abstract: We give another proof of Pleasant's result on the
generators of rings of integers of a number field and generalize it to the case of algebras which are sums of matrices over rings of integers.
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