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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August 30: Organizational meeting
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September 6: (W+L). Kappe
Title: Products of commutators are not commutators: Cassidy's
Example revisited
Abstract: In her 1979 paper, entitled "Products of commutators are not always
commutators:
an example", P.J.Cassidy presents a group in which the set of commutators is
not equal
to the commutator subgroup (Monthly, vol. 86, p. 772). In fact, a much
stronger
statement is true for this group: there is no bound on the number of
commutators in
the product representing an element of the commutator subgroup. Since this
example is
widely known and often quoted, Robert Morse and LCK have included it in
their survey
paper on commutators. At the last moment we found out that Cassidy's proof
is
incorrect, but the statement is correct. In this talk we provide the
context in which
this example is of most interest, give a correct proof, show where Cassidy
went wrong
in hers, and finally explore what other types of examples can be constructed
using
Cassidy's idea.
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September 13: Dikran B. Karagueuzian
Title: The Components of a Variety of Matrices and Free Actions
of Groups on Products of Spheres
Abstract:
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September 20 (joint with the Combinatorics Seminar)
Speaker: Thomas Zaslavsky
Title: Algebra of Tutte Functions
Abstract: The Tutte polynomial, and a kind of generalization that involves
arbitrary parameters, have become famous recently due to connections with
knot invariants and statistical physics. The exact form of the
generalization is not fully understood. I will talk about one aspect of
this, namely, the classification problem for parametrized weak Tutte
functions, which are the natural generalization of the parametrized Tutte
polynomial.
A parametrized weak Tutte function (of matroids) is a function
that satisfies a linearity relation of the form
f(M) = de f(M\e) + ce f(M/e)
for "most" elements of a matroid M on ground set E. M\e and M/e are
matroids on ground set E\e called, respectively, the deletion of e and the
contraction of e. The quantities de and ce (belonging
to some commutative ring) are called the parameters. Consider an
arbitrary class C of matroids such that every deletion and
contraction of a matroid in C is also in C. The problem is to
find all parametrized weak Tutte functions defined on C. The first
step is to set up the universal module for parametrized weak Tutte functions
and relate it to the universal algebra for parametrized strong Tutte
functions (weak functions that satisfy an additional, multiplicative rule).
I will define a matroid and explain the universal Tutte module and
algebra.
This is joint work with Joanna Ellis-Monaghan.
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September 27: Ross Geoghegan
Title: Associativity and Thompson's group.
Abstract: TBA
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October 4: No talk
Title:
Abstract:
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October 11: No talk
Title:
Abstract:
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October 18: Alex Feingold
Title: Introduction to Lie Algebras and Connections with Physics
Abstract: This will be an introduction to the basics of Lie
algebras, including many examples, and showing some connections with physics.
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October 25: Collin Bleak
Title: Solvability in Groups of Piecewise-linear
Homeomorphisms of the Unit interval
Abstract: We investigate subgroups of the group of
PL_o(I) of piecewise-linear, orientation-preserving homeomorphisms
of the unit interval with finitely many breaks in slope, and also
subgroups of Thompson's group F. We find geometric criteria
determining the derived length of any such group, and use these criteria
to classify the solvable and non-solvable subgroups of PL_o(I) and of F.
Let H be a subgroup of PL_o(I) or F. We find that H is solvable if and
only if H is isomorphic to a group in a well described class R of groups.
We also find that H is non-solvable if and only if we can embed a copy of
a specific non-solvable group W into H.
We strengthen the non-solvability classification by finding weak geometric
criteria under which we can embed other groups (all containing W) into
non-solvable subgroups of PL_o(I) or F.
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November 3 (Thursday, usual time and place): Yan Bulgak
Title: C-Groups
Abstract: The rank rank(G) of a group G is the minimal number of
generators of G. We define G to be a C-Group if rank(G) is smaller
than rank(Z(G)). An existence
theorem for a class of such groups will be presented, along with relevant
definitions and results.
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November 8: Tairi Roque
Title: Elements of order p in powerful p-groups
Abstract: We will present a proof that the number
of elements of order at most p in a powerful p-group is equal to
the index of the Frattini subgroup.
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November 15: Gabriela Mendoza
Title: On the n-permutational property of groups
Abstract: A group G has the n-permutational property for some positive integer n,
where n>1, if every product of n elements can be rewritten in at least
one other way.
In the first of these talks, we will give a general introduction to the
original 1983 paper by Curzio, Longobardi, and Maj, entitled "Su di un
problema combinatorio in teoria dei gruppi". Also, we will present an
overview of later research that was inspired by this paper, among them the
extensive contributions of D.J.S. Robinson and R.D. Blyth who showed that
all groups with the 7-permutational property are solvable and provided the
classification of insolvable groups in the class of groups with the
8-permutational property.
The second talk will consist of a presentation of selected theorems and
proofs from the paper by Curzio, Langobardi and Maj.
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November 22: Gabriela Mendoza
Title: On the n-permutational property of groups (continuation)
Abstract: A group G has the n-permutational property for some positive integer n,
where n>1, if every product of n elements can be rewritten in at least
one other way.
In the first of these talks, we will give a general introduction to the
original 1983 paper by Curzio, Longobardi, and Maj, entitled "Su di un
problema combinatorio in teoria dei gruppi". Also, we will present an
overview of later research that was inspired by this paper, among them the
extensive contributions of D.J.S. Robinson and R.D. Blyth who showed that
all groups with the 7-permutational property are solvable and provided the
classification of insolvable groups in the class of groups with the
8-permutational property.
The second talk will consist of a presentation of selected theorems and
proofs from the paper by Curzio, Langobardi and Maj.
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November 29 : Gina Baird
Title: The Amalgamation Property in Varieties of Lattices
Abstract: A variety V has the amalgamation property if for
every A, B, C in
V and embeddings f from A into B and g from A into C, there is a D in V and
embeddings f' from B into D and g' from C into D such that f'f=g'g. We will
discuss this property in varieties of lattices.
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December 6: Andrew Sills (Rutgers)
Title: Polynomial generalizations of Rogers-Ramanujan type
identities
Abstract: The Rogers-Ramanujan identities are a pair of analytic/formal power series
identities, each of which assert the equality of a certain infinite series
with an infinite product. They were first discovered by the
English mathematician L.J. Rogers in 1894. Later, it was
realized that the series and products could be viewed as generating
functions for certain classes of integer partitions, and thus the
Rogers-Ramanujan identities are also combinatorial identities.
Around 1980, Australian physicist Rodney Baxter showed
that the Rogers-Ramanujan identities were intimately linked to the
solution of the hard hexagon model in statistical mechanics. Around
the same time, Rutgers mathematicians Jim Lepowsky and Robert Wilson
gave the first Lie theoretic interpration and proof of the
Rogers-Ramanujan identities. This work eventually led to the discovery
of vertex operator algebras.
In my talk, I plan to give a brief but motivating introduction to
q-series, and then denomonstrate an elementary method by which any
q-series/infinite product identity of
Rogers-Ramanujan type can be
generalized to a polynomial identity, which in turn has important
implications in statistical physics and algorithmic proof theory.