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 24: Organizational meeting
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January 31: Collin Bleak
Title: Embedded Wreath Products in a Group of Homeomorphisms
Abstract: We characterize the top group of any standard restricted wreath product (of
non-trivial groups) embedded in the group PLo(I) of piecewise-linear,
orientation preserving homeomorphisms of the unit interval. We show that the
top group must be isomorphic with the integers. This result answers a
question of Mark Sapir.
The talk features a nice interplay between the geometry of elements of PLo(I)
and some purely algebraic operations (such as computing commutators).
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February 7 : Prof. Dikran Karagueuzian
Title: Ideal Class Groups and Wall Obstructions
Abstract: This is an expository talk. I will explain the algebra necessary to
produce a nonzero Wall Finiteness Obstruction. No topology background is required.
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February 14 : Prof. Alex Feingold
Title: Representations of Heisenberg Lie algebras
Abstract: I will give an expository talk about the Fock space
representations of Heisenberg Lie algebras, and discuss possible generalizations
where the integral modes are replaced by certain algebraic integers.
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February 21 : No speaker this week
Title:
Abstract:
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February 28: Joseph Smith
Title: Centre and norm
Abstract: The norm is the intersection of all the normalizer subgroups of in
a given group G and is denoted N(G). I will prove a theorem of Beidleman, J. C.; Heineken,
H.; and Newell, M. which states that in a p-group G either N(G)/Z(G) or [G,N(G)] is cyclic.
If there is time, I will discuss some applications of this theorem within capable p-groups.
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March 7 : Gabriela Mendoza
Title: Characterizing 3-Rewriteability in Groups
Abstract: Let G be a group and n an integer greater than 1. Then G is said to be
n-rewriteable if for each n-element subset {x_1,x_2,...,x_n} of G there
exists a permutation sigma in S_n such that
x_1...x_n=x_(sigma(1))...x_(sigma(n)).
In this talk we will give the following characterization of 3-rewriteable
groups.
Theorem. For a group G the following conditions are equivalent:
1) G is 3-rewriteable.
2) The order of G' is one or two.
3) For all x in G, the centralizer of x in G has index 1 or 2 in G.
4) For all x in G, the order of the conjugacy class containing x is one or
two.
5) The probability that two elements in G commute is greater than 1/2.
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March 14 : Spring Break
Title:
Abstract:
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March 21 : Xiaochun Rong (Rutgers)
Title: Finite subgroups of compact Lie groups and equivariant embedding
Abstract: We will report a recent work with Yusheng Wang (Peking
University), in which we prove that any finite group of a connected
compact Lie group of dimension n contains an abelian subgroup
of index bounded above by a constant depending only on n. Our
method is from the comparison geometry. We will also discuss
some applications.
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March 28 : Professor Zoran Sunik, Texas A&M University, Joint talk with the
Geometry-Topology Seminar
Title: Hanoi Towers, Schreier Graphs, Iterated Monodromy Groups and Julia sets
Abstract: We model the well known Hanoi Towers Problem on k pegs by a self-similar group H(k)
acting on a k-regular rooted tree. The Schreier graph of the action of the group H(k) on level n in the tree
models the n-disk version of the problem. As n goes to infinity the obtained limiting graph is the Schreier
graph of the action of H(k) on the boundary of the k-ary tree.
For the original version of Hanoi Towers on 3 pegs the obtained group H(3) is an automaton group branching
regularly over its commutator. The corresponding finite Schreier graphs are 3-regular graphs that approximate
the Sierpinski gasket. The group H(3) can be described as the iterated monodromy group of a post-critically
finite map f on the Riemann sphere. The Julia set of the map f is homeomorphic to the limiting infinite Schreier
graph.
The action of H(3) on the levels of the tree provides permutational representations that can be used to
determine the spectrum of the Markov operator on the associated Schreier graphs. The spectrum can be described
as the closure of the backward orbit of a quadratic polynomial p. It consists of a countable set of isolated
points that accumulate to a Cantor set, which is the Julia set of the polynomial p.
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April 4: Radhika Ganapathy
Title: On the power structure of powerful p-groups
Abstract: A p-group is said to have regular power structure
if:
1) Gp^k is the set of pk
th powers in G.
2) \Omegak(G) : = < g in G | gp^k
= 1 > has exponent at most pk .
3) | Gp^k | = [G : \Omegak(G)
] .
Powerful p-groups are already known to satisfy 1). In this talk we will
prove that when p is odd, powerful p-groups satisfy 2) and 3).
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April 11 : No Talk This Week
Title:
Abstract:
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April 18 : Darryl Daugherty
Title: Free groups from fields
Abstract: In the nineteenth century, field theory brilliantly resolved a number of questions that
had taxed mathematicians for centuries; for example, ``The circle cannot be squared" by straight edge and compass,
and solving polynomial equations by radicals is not always possible. These successes have continued to be held up
as superb examples of the power of mathematical thought, and are demonstrated at an undergraduate level. The
purpose of this article is to provide another such natural example which leads to a concrete realisation of the
free group on 2 generators.
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Thursday, April 20 : JOINT WITH THE COMBINATORICS SEMINAR
Time: 1:15 - 2:15 , Room: LN-2201
Speaker: Stephanie van Willigenburg (Univ. of British Columbia)
Title: ``Coincidences Amongst Skew Schur Functions: A Pictorial Approach"
Abstract: In the intersection of algebra, combinatorics, algebraic geometry
and more, are functions called skew Schur functions. These functions are invaluable in
translating problems from one area to another in order to make them more accessible.
For example, going from algebra to combinatorics, "When are two skew Schur or Weyl
modules equivalent over C?" becomes "When are two skew Schur functions equal?",
which becomes "When are two pictures of boxes the same?"
In this talk we'll attack the former by studying the latter. More precisely, we'll
introduce skew Schur functions pictorially before determining conditions under which
these pictures are "the same", and then see where else in combinatorics these conditions
arise.
No prior knowledge of any of the above is required.
This is joint work with Vic Reiner and Kris Shaw.
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April 25 : Prof. Fernando Guzman
Time: 3:00 PM (Note the starting time is slightly later than usual)
Title: "Four-variable associative laws for group commutators"
Abstract: In 1941 Levi proved that a group is nilpotent of class 2 iff
the commutator operation is associative. Recently, Geoghegan and
Guzman showed that a group is solvable iff the commutator operation
eventually satisfies all instances of the generalized associative law.
In this talk we discuss solvability and nilpotency of groups whose
commutator operation satisfies one of the four-variable instances of
the associative law.
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May 2 : Olga Salazar-Diaz
Time: 10:15 AM - Noon (Note the morning starting time)
Title: "Thompson's group V from a dynamical viewpoint"
Abstract: Thompson's group V can be thought of as the group of
automorphisms of a certain algebra or as a subgroup of the group of self-homeomorphisms
of the Cantor set. Thus, the dynamics of an element of V can be studied.
We analyze the dynamics in detail and use the analysis to give another solution of
the conjugacy problem in V.
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May 2 : David Biddle (Cornell University)
Time: 2:50 PM (The usual starting time)
Title: ``Computing Homological Invariants of Nonabelian Tensor Products"
Abstract: The classifying
space functor B: Groups ---> CW-complexes is
adjoint to the functor \pil:
CW-complexes ---> Groups in the sense that
the group of homotopy classes of pointed spaces [X,BG]* is
isomorphic to
Hom(\pil(X),G). We
give this a fancy name and say that `groups' are a model for
`homotopy 1-types.' The functor B allows us to define a cohomology
theory for
groups by setting H*(G) = H*(BG). After Cech
introduced the higher homotopy
groups, it was immediately pointed out by Hopf and others that the
functors
\pi<k, k > 1,
gave
rise to abelian groups. In the past 25 years, there have been a
plethora of papers building on Whitehead's 1949 notion of a `crossed
module'
P ---> M. There are now constructions of classifying functors B(P
---> M) in a
suitable category with `new' functors Hkcr. The
crossed module which we are
interested in is the nonabelian tensor square of a group G, G \otimes G
---> G.
This talk will provide an introduction to the notion of `homotopy
2-types' and how one
can begin a nonabelian homology theory. In particular, we use machinery
of
Graham Ellis and others to determine H*(G \otimes G) and $H*(G
\otimes G ---> G).
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May 9 : Three 20-minute talks, dress rehearsals of talks to be given
at the Zassenhaus Conference on May 19-21 in Ohio:
Speaker: Prof. Luise-Charlotte Kappe
Title: "Products of commutators are not always 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). However, a typographical error in Cassidy's paper impacts the
verification of the claim. In this talk we give a corrected proof of Cassidy's claim in a slightly more
general setting and take another look at her example in context with the function \lambda(G) for a group G.
For a group G the function \lambda(G) denotes the smallest integer n such that every element of the
commutator subgroup of G is a product of n commutators. The statement that the set of commutators of a group
does not form a subgroup is equivalent to \lambda(G) > 1. For every prime p, Guralnick has constructed a
p-group P of order pn+n2 with \lambda(P) = n. For Cassidy's group C we have \lambda(C) is infinite.
However, for given n and any prime p, the group C has a suitable homomorphic image G = G(n,p) such that
\lambda(G) = n and G has order pf(n), where f(n) is a linear function of n.
Speaker: Gabriela Mendoza
Title: ``For a given prime p, what is the smallest simple nonabelian group
whose order is divisible by p?"
Abstract: In this talk we will show that for any prime p > 3, PSL(2,p) with
with order p(p2 - 1)/2 is the smallest nonabelian simple group whose order is divisible by
p. For p = 2 and p = 3 the group in question is A5, the alternating group on 5 letters.
Speaker: Joseph Smith
Title: ``Groups whose normalizers form a lattice"
Abstract: We will examine groups whose norm is a normalizer in the group. I
will first show these groups are nilpotent. Then I will outline the proof
that every subgroup that contains the norm is also a normalizer in the group.