The Algebra Seminar
FALL 2004
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.
Organizer: Marcin Mazur
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September 7: Organizational meeting
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September 14: Alex Feingold
Title: Monstrous Moonshine: The First Twentyfive Years
Abstract: The classification of finite simple groups includes 26 sporadic
groups, the largest of which is called the Monster. It was connected to
the theory of modular functions by an observation of J. McKay, and
subsequently Conway and Norton published the paper ``Monstrous
Moonshine" containing many more remarkable numerical connections.
They conjectured the existence of a Z-graded module which would
explain some of the connections. Frenkel, Lepowsky and Meurman
constructed that ``Moonshine module", using vertex operators and
linking the Monster to modular functions through Conformal Field
Theory. This talk will survey the main points of this remarkable story.
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September 21 : Alex Feingold
Title: Monstrous Moonshine: The First Twentyfive Years (part 2).
Abstract:
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September 28 : Marcin Mazur
Title: Connected transversals to nilpotent groups
Abstract:
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October 5 : Antun Milas (SUNY at Albany)
Title: Superconformal characters and modular forms
Abstract: It is known that every rational vertex operator
(with certain finiteness condition) yields a SL(2,Z)--module
on the vector space spanned by graded traces (or characters).
To such a module we will associate a canonical automorphic form
and study its properties by using purely representation theoretical
methods.
It is interesting that a similar picture
persists in the setting of N=1 vertex operator superalgebras (even though
the modular invariance is broken). In particular, we will show how
the representation theory of N=1 superconformal superalgebras
can be used to obtain new proofs of some classical modular identities such
as the famous Jacobi's Four Square Theorem (1829) and a Carlitz's
modular, modulus 8 identity (1953). We will also present some
generalizations.
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October 12 : Joshua Palmatier
Title: l-groups versus l-monoids
Abstract: Lattice-ordered monoids (l-monoids) are a generalization of
lattice-ordered groups (l-groups). In this talk, we will
define these structures and then begin a comparison between the
two. How can l-monoids differ from l-groups? What are some of
the properties that l-groups must satisfy that are not restrictions
on l-monoids? And finally, what properties must a monoid satisfy in order
for it to be the positive cone of an l-group? We will investigate these
questions using multiple examples of both l-groups and l-monoids.
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October 19 : Michael Barry (Allegheny College)
Title: On a conjecture of Alvis
Abstract: In 1990 Dean Alvis conjectured that, for each integer
n >=15, there is an ordinary irreducible character of the alternating group
A_n with the property that its degree is divisible by every prime less
than or equal to n. We describe a proof of this conjecture and its
relation to some general questions about the character theory of finite groups.
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October 26 : Joseph Smith
Title: Fusion and the Transfer
Abstract: We will prove Alperin's main result on fusion. Then we will
define the transfer and use it to prove a useful theorem about the focal
subgroup.
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November 2 : Joseph Smith
Title: Theorems of Burnside, Frobenius, and Grun.
Abstract: We will use Alprin's theorem and the focal subgroup theorem
from the first talk to prove theorms due to Burnside, Frobenius and Grun.
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November 9 (joint with Geometry/Topology) : Alejandro Adem (IAS Princeton and Univ. of British Columbia)
Title: Representation Varieties and K-theory
Abstract:
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November 16 (joint with Combinatorics) : William Schmitt (George Washington University)
Title: The Free Product of Matroids
Abstract: We introduce a noncommutative binary operation on matroids, called the
free product, and discuss some of its properties. In particular, the free
product is characterized by a certain universal property, is associative,
and respects matroid duality. We characterize matroids that are
irreducible with respect to free product and show that, up to isomorphism,
every matroid factors uniquely as a free product of such matroids. We use
these results to prove an inequality involving the number of nonisomorphic
matroids on n elements which was conjectured by Welsh, and to prove the
freeness of the algebra of matroids whose product is dual to the
restriction-contraction coproduct. This is joint work with Henry Crapo.
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November 23 : Gabriela Mendoza
Title: On n-Levi groups
Abstract: A group G is called n-abelian if for an integer n and all x and y in G we
have (x^n)(y^n)=(xy)^n. Likewise, a group is called n-Levi or n-Bell if for
an integer n and all x, y in G we have [x,y^n]=[x,y]^n or [x^n,y]=[x,y^n],
respectively. We have that n-abelian implies n-Levi, and in turn n-Levi
implies n-Bell. The containments are proper in each case. Abelian groups
are n-abelian for all integers n, and 2-Engel groups, i.e. groups
satisfying the law [x,y,y]=1, are n-Levi for all integers n.
For every group G we define three sets of integers as follows: E(G), L(G)
and B(G) are the sets of integers for which G is n-abelian, n-Levi or
n-Bell, respectively. It turns out that these sets are semigroups.
The main topic of the talk is a characterization of the sets E(G), L(G)
and B(G).
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November 30 : Michael Ward
Title: Intersections of Sylow-subgroups
Abstract: It has long been of interest to know the minimum number of Sylow
p-subgroups whose intersection is O_p(G), the largest normal p-subgroup
of G. In the past 10 years, V. I. Zenkov and, independently, B.
Brewster and P. Hauck have also investigated the minimum number of Sylow
p-subgroups needed to obtain the intersection of a fixed Sylow subgroup
with any collection of Sylow subgroups. (Zenkov actually deals with the
more general situation of a nilpotent Hall pi-subgroup.) In this talk,
we will survey some of the existing theory, give some examples, present
(most) of the proof of Zenkov's result, and perhaps speculate on
extensions of Zenkov. Zenkov's proof is another "KGB-proof" and it is
elementary save for one appeal to a consequence of the classification of
finite simple groups, which is of independent interest.
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December 7 : Robert Morse (University of Evansville)
Title: Computing the nonabelian tensor square of polycyclic groups
Abstract: In this talk we will develop a method for computing the
nonabelian tensor square for any polycyclic group. This new method is
amenable to computer implementation. As an application of this method,
we compute the nonabelian tensor square for the free nilpotent of class
3 groups of rank n.
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