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The Algebra Seminar
Fall 2013
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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
To receive announcements of seminar talks by email, please join the seminar's
mailing list.
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August 27 : Organizational Meeting
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September 3: No Meeting
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Abstract:
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September 10: Karl Lorensen (Pennsylvania State University at Altoona)
Title: The cohomology of solvable groups of finite rank
Abstract: Assume that $G$ is a virtually torsion-free solvable group of finite rank
and $A$ a $\mathbb ZG$-module whose underlying abelian group is torsion-free and has finite rank.
In this talk, I will discuss a theorem, proved jointly with Peter Kropholler, that stipulates a
condition on $A$ that ensures that $H^n(G,A)$ and $H_n(G,A)$ are torsion abelian groups with
finite exponent for all $n\geq 0$.
In addition, I intend to explain how the special case of this result for $n=2$ can be applied to
discern the presence of near supplements and complements in solvable groups of finite rank.
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September 17: Peter Abramenko (University of Virginia)
Title: Presentations of Kac-Moody groups over fields
Abstract: The Kac-Moody groups mentioned in the title should be thought of as generalizations of
Chevalley groups. It is well-known that Chevalley groups over fields admit BN-pairs and act on associated spherical
buildings. Kac-Moody groups over fields admit "twin BN-pairs", and thus act on "twin buildings". I shall try to
describe how these objects arise, and how some standard examples look like. Twin buildings were introduced by Tits
and Ronan in order to make geometric ideas available for the study of Kac-Moody groups (in Tits's sense). As one
application of these geometric ideas (derived by Bernhard Muehlherr and the speaker) one obtains efficient
presentations of "2-spherical" (i.e. all entries of the associated Coxeter matrix are finite) Kac-Moody groups
over fields. These efficient presentations then have some applications of their own as demonstrated by work
of Pierre-Emmanuel Caprace and other recent work by Ralf Koehl.
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September 24: No meeting
Title:
Abstract:
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October 1: Alex Feingold
Title: A Constructive Exploration of Hyperbolic Kac-Moody Groups Through Explicit Examples
Abstract: I will describe a current research project with Walter Freyn which studies the structure
of Cartan subalgebras in hyperbolic Kac-Moody groups, and which may be related to the associated building.
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October 8: Eran Crockett
Title: A Characterization of Congruence Lattices
Abstract: It is well-known in universal algebra that the set of congruences on an
algebra forms an algebraic lattice. In 1963, George Grätzer and E.T.
Schmidt proved the converse: every algebraic lattice is isomorphic to the
congruence lattice of some algebra. After introducing the necessary
background information, I will sketch Grätzer and Schmidt's proof.
Time permitting, I will discuss an open problem related to this theorem.
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October 15: No meeting
Title:
Abstract:
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October 22: David Biddle
Title: Homogeneous Groups of Order p^3 and their
Automorphism Groups.
Abstract: We will show via direct computation that the finite
Heisenberg group T_p of order p^3 and exponent p is indeed homogeneous and we
will investigate its automorphism group, noting that the number of ordered
pairs of generators of T_p is exactly the size of Aut(T_p). In particular
we will directly see that the size of its automorphism group is much
larger the non-homogeneous groups of order p^3 (in fact, the orders of the
automorphism groups of the non-homogeneous groups of order p^3 are proper
factors).
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October 29: Chen Tang
Title: Maximal subgroups of direct products
Abstract: I will talk about the maximal subgroups of direct
products from the paper of Jacques Thevenaz. First, I will show how to describe
the subgroups of direct products and find out a way to count the number of
maximal subgroups of finite group. After these basic information, I will show
some applications of maximal subgroups.
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November 5 : Laura Gray
Title: Groups in Which the Commutator Operation Satisfies Certain
Algebraic Conditions
Abstract: We will discuss groups in which the associative law of
the commutator operation is satisfied through the paper of F.W Levi. We will
investigate equivalent conditions and properties of these groups. In doing this we
will examine groups satisfying a weaker condition pertaining to associativity and
the relevant properties of these groups.
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November 12 : Rachel Skipper
Title: An Introduction to Profinite Groups.
Abstract: Let C be a category of finite groups. In this talk, I will introduce pro-C
groups, which are a certain type of topological group that are constructed
in a particular way using groups from the category C, focusing mainly on
when C is the category of finite groups or finite p-groups. I will be
discussing some of the basic properties of these groups and how a number
of important properties of finite groups can be extended to pro-C groups
when these groups are infinite. This talk will be both accessible and
expository.
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November 19 : Amelia Stonesifer
Title: Isomorphic Subgroups of the Rationals
Abstract: Many of the standard techniques for determining whether two subgroups are isomorphic
are not useful in torsion free abelian groups such as the rationals. Isomorphisms between finitely generated
subgroups of the rationals are easily obtained, but in order to generalize we will discuss the solvability
of equations, the height of elements, and the type of a subgroup. Finally, through a proof originally
introduced by Reinhold Baer and later simplified by Friedrich Kluempen and Denise Reboli, we will see that
subgroups of the rationals are isomorphic when the subgroups have the same type.
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November 26 : Katie Sember
Title: Free groups in quaternion algebras
Abstract: Let R be the ring of algebraic integers of an
imaginary quadratic number field. We will show how to construct explicit
units $u,v$ in the quaternion ring (-1,-1/R) such that u,v generate a
non-abelian free group. The talk is based on the recent paper
"Free Groups in Quaternion Aglebras" by S. O. Juriaans and A. C Souza Filho
(Journal of Algebra 379 (2013)).
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December 3: Diego Penta (First Part of Admission to Candidacy Exam)
Title: Decomposing the rank 3 hyperbolic KM Lie algebra F
with respect to the rank 2 `Fibonacci' subalgebra H(3)
Abstract: The rank 3 hyperbolic Kac-Moody Lie algebra F whose Weyl group is the hyperbolic
triangle group T(2,3,/infty) (isomorphic to PGL(2,Z)) was studied by Feingold and Frenkel in 1983 using
its decomposition with respect to an affine KM subalgebra of type A1(1). Further
work of Feingold and Nicolai showed that F also contains every rank 2 hyperbolic KM algebra whose Cartan
matrix is symmetric. The smallest such example is the `Fibonacci' rank 2 hyperbolic Fib whose off-diagonal
Cartan entries are -3. The decomposition of F with respect to Fib can be seen as a Z-grading, but the
Fib-module structure of each graded piece contains infinitely many irreducible components. We will discuss
the parallels and differences between these two decompositions of F, and give some preliminary results on
the Fib-module structure of the 0-graded piece.
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December 5, 1:15 PM, LN-2201 : Diego Penta (Second Part of Admission to Candidacy Exam)
Title: Decomposing the rank 3 hyperbolic KM Lie algebra F
with respect to the rank 2 `Fibonacci' subalgebra H(3)
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December 10: Slobodan Tanusevski
Title: Generalizations of Thompson's group F
Abstract: For a given group G, I will describe a group F(G) that combines
G with Thompson's group F. I will also discuss some properties of these groups.
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December 17, 2.50 PM, LN G208: Adam Allan
Title: Structure of Modular Centralizer Algebras
Abstract: Given a finite group G with subgroup H and k a field of
characteristic dividing |G|, we may consider the centralizer algebra kG^H
consisting of elements of kG fixed under the conjugation action of H. These
algebras serve as potentially important stepping stones between Z(kG) and kG.
In this talk we shall consider some of the fundamental questions regarding the
structure of such algebras, with particular emphasis on (i) the case where G
is a p-group and (ii) the question of when kG^H is self-injective.
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