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.
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January 20 : Organizational meeting
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January 27 : Gaywalee Yamskulna
Title: Vertex operator algebras and their representations
Abstract: Vertex operator algebras were introduced to mathematics in the
work of Borcherds, Frenkel, Lepowsky and Meurman. In Physics, they appear
in the string theory literature, in the work of Belavin, Polyakov and
Zamolodchikov, as the chiral symmetry algebras of two-dimensional
conformal field theory. Since their introduction, Vertex operator algebras
have proven to be a truly universal algebraic structure, having found
applications in many fields including finite group theory, Lie algebras,
topology, and modular forms.
In this talk, I will discuss the representation theory of vertex operator
algebras. In particular, I will discuss fixed-point theory and its
connection to quasi-quantum groups. I will also describe
the connection between the representation theory of vertex operator
algebras and the study of Poisson algebras and Lie algebroids.
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February 3 : Ross Geoghegan
Title: Introduction to Thompson's Groups
Abstract: The title refers to some groups discovered forty years ago by
the logician Richard Thompson. He showed that they are infinite and
finitely presented, and some of them are simple groups; until then there
was no known example of an infinite finitely presented simple group.
Thompson's work was published but not well publicized or disseminated
During the 1970's several topologists rediscovered these groups
independently, knowing nothing of Thompson's work. In the 80's the groups
were proved to have homological, homotopical and group theoretic properties
which had not previously been seen. Since then there has been a steady
flow of interesting results and unanswered questions concerning these
groups, often motivated by their appearance in non-algebraic contexts.
This will be an introductory talk, more like a colloquium than a seminar,
aiming to convey the flavor of the subject.
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February 10 : Benjamin Brewster
Title: The Permutable Core of a Subgroup
Abstract: A subgroup H of a group G is called permutable
provided for
every subgroup A of G, AH = HA. Every normal subgroup is permutable but not
every permutable subgroup is normal.
However, the product of two permutable subgroups is again permutable. Thus
each subgroup has a unique maximal subgroup of it which is permutable in G.
The seminar will consider questions about when this subgroup,the permutable
core, is normal, ie. coincides with the usual core, and related issues.
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February 17 : Luise-Charlotte Kappe
Title: Neumann's Lemma in Groups and Loops
Abstract: Neumann's Lemma can be stated as follows:
"If a group G is equal to the
set-theoretic union of finitely
many cosets of subgroups, then
any of those cosets belonging to
subgroups of infinite index can
be dropped, and the remaining
ones still cover G."
A careful analysis of the proof of Neumann's Lemma
leads to conditions under which it holds in loops.
There exist loops which are not groups satisfying
these assumptions.
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February 24 : Wolfgang Kappe
Title: Some Exercises in Group Theory
Abstract: It is a very easy exercise to show that the
center of a group
is equal to the intersection of the maximal abelian subgroups.
So is it true that the intersection of the maximal class-2-subgroups
is the second center? This and related exercises will be discussed.
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March 2 : Nicholas Koban
Title: An Introduction to the Bieri-Neumann-Strebel Sigma Invariant
Abstract: The purpose of this talk is to describe the
geometric invariant Sigma^1 of
a finitely generated group G. To do so, I will discuss such ideas as the
character sphere of G and ``cutting the Cayley graph of G in half".
I will
give a number of examples of Sigma^1 for particular groups such as
the free abelian group on two generators, the free group on two generators,
and G=< a,b,c: ac = ca, bc =cb>.
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March 9 : Joseph Evan
Title: Results about Subgroups of Direct Products that Satisfy
the Frattini Argument
Abstract: A subgroup U of a group G satisfies the Frattini Argument if for all
normal subgroups N of G, the group G is equal to the product of N with the
normalizer of its intersection with U. In this talk, we will consider
subgroups of direct products of finite solvable groups that satisfy the
Frattini Argument. In particular, for a subgroup U that satisfies the Frattini
argument in GH, where GH is a direct product of finite solvable groups, we will
examine the structure of the projection of U into one of its direct factors, G
or H, modulo its intersection that direct factor.
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March 16 : Fernando Guzman
Title: Lattice Concepts in Groups and Rings
Abstract: Many arguments in Group Theory and Ring/Module Theory are,
on close inspection, Lattice theoretic arguments in disguise, dealing
with the lattice of subgroups, lattice of normal subgroups, lattice of
ideals, lattice of submodules. In this talk, we'll present some basic
Lattice Theory setup, and illustrate the phenomenon with the concept
of "radical".
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March 23 : Joshua Palmatier
Title: A Visual Approach to Finite, Totally-Ordered M-Zeroids
Abstract: Finite, totally-ordered m-zeroids of size n can be visualized
using multiple paths in a n by n grid, where the paths must satisfy
certain specific properties associated to the algebraic properties of the
m-zeroid. In this talk, I will introduce the concept of an m-zeroid in
general, then simplify down to the finite, totally-ordered m-zeroids.
Then I will demonstrate how to represent these finite, totally-ordered
m-zeroids as paths in a grid. I will also discuss the converse: what
properties a set of paths set up in a grid must satisfy in order for the
grid to represent an m-zeroid.
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March 30 : Dikran Karagueuzian
Title: Topological Methods for the Cohomology of Galois Groups
Abstract: TBA
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April 6 : No seminar
Title: TBA
Abstract: TBA
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April 13 : Bronlyn Wassink
Title: Purity in Compactly Generated Modular Lattices
Abstract: The concept of purity has an important role in the study of
abelian groups. After briefly discussing some properties and
examples of pure subgroups, I will present a paper by
T. Head which
generalizes the definition and theorems of purity in abeilan groups into
lattice theoretical terms.
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April 20 : Gina Baird
Title: The Variety of Boolean Semirings
Abstract: The variety of Boolean semirings is the variety generated by
the 2 two-element semirings. We will find a complete set of laws for this
variety and show that the category of Boolean semirings is equivalent to
the category of partially complemented distributive lattices and dual to
the category of partially Stone spaces.
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April 27 : Gabriela Mendoza
Title: On the Power Structure of Finite Groups
Abstract: It is well known that the squares of elements in a group
do not form a subgroup and that the alternating group on four
letters is minimal with this property. For given n, what is
the group of minimal order such that the n-th powers of
elements do not form a subroup? For odd n, it can be shown
that the dihedral group of order 2p is minimal with this
property, where p is the smallest prime dividing n.
If n is even, the situation is more complex. The order
of the group of minimal order with this property depends on
the odd prime factors of n and the exact 2-power dividing n.
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May 4 : Denise Yull
Title: A non-nilpotent group of minimal order with a
fixed-point-free automorphism
Abstract: A group with a fixed-point-free automorphism of prime order is nilpotent.
This is not necessarily the case if the fixed-point-free automorphism does
not have prime-power order. In this talk we construct a non-nilpotent group
of order 48 which has a fixed-point-free automorphism of order 6. This group
is minimal with this property.