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The Algebra Seminar
Spring 2010
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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: Marcin Mazur
To receive announcements of seminar talks by email, please join the seminar's
mailing list.
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January 26: Organizational meeting
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February 2 : Viji Thomas (Binghamton University)
Title: The weak dimensions of Gaussian rings.
Abstract: We will show that a Gaussian ring is a domain iff it
is a Prufer domain. We provide necessary and sufficient conditions for a
Gaussian ring R to be semihereditary. We will also discuss Gaussian Coherent
rings and their weak and finitistic dimensions.
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February 9 : Tom Head (Binghamton University)
Title: How to Slip from Abelian Groups into Commutative Semigroups
Abstract: We are all familiar with the elements of Abelian
Groups. I'm guessing that few have taken the time to peruse the theory of
commutative semigroups. You may have thought that what structure you would
find there would be too frivolous to merit your close attention. Perhaps this
is true, but I find real delight in surveying the landscape of this theory.
My approach is to assume that any commutative semigroup really should have been
an Abelian group and to try to see it as some sort of group or diagram of
groups - perhaps with some extra fluff that can be excused. My purpose is to
conduct a sightseeing tour of this landscape.
This talk arises from my attention being drawn back a half century ago when I
studied this sort of thing using the work of Takayuki Tamura who died this last
summer at the age of 90. He began working in this area in the 1940's and never
stopped.
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February 16 : Zoran Sunic (Texas A&M University)
Title: Finite self-similar p-groups with abelian first level stabilizers
Abstract: We determine all finite p-groups that admit a faithful, self-similar
action on the p-ary rooted tree such that the first level stabilizer is
abelian group. A group is in this class if and only if it is a split
extension of an elementary abelian p-group by a cyclic group of order p.
The proof is based on use of virtual endomorphisms. In this context the
result says that if G is a finite p-group with abelian subgroup H of
index p, then there exists a virtual endomorphism of G with trivial core
and domain H if and only if G is a split extension of H and H is
elementary abelian p-group.
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February 23: Alex Feingold (Binghamton University)
Title: Hyperbolic Weyl Groups and the Four Normed Division
Algebras
Abstract: Hyperbolic Weyl groups are the Weyl group symmetries of
the hyperbolic Kac-Moody Lie algebras. Those infinite dimensional algebras have
appeared in some theoretical physics papers on supergravity, a theory combining
general relativity with supersymmetry. They have also been conjectured to play
a role in string theory, especially the algebra known as E10, and in the theory of extremal black holes.
This talk will give an introduction to recently published joint work with
Hermann Nicolai and Axel Kleinschmidt on the Weyl groups of hyperbolic
Kac-Moody algebras of ranks 3, 4, 6 and 10. These are intimately linked to the
four normed division algebras, K, real numbers, complex numbers, quaternions,
and octonions, respectively. A crucial role is played by integral lattices of
the division algebras and associated discrete matrix groups. Our findings can
be summarized by saying that the even subgroups, W+,
of the Kac-Moody Weyl
groups, W, are isomorphic to generalized modular groups over K for the simply
laced algebras, and to certain finite extensions thereof for the non-simply
laced algebras.
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March 2: Dmytro Savchuk (Binghamton University)
Title: Tableaux representation of iterated wreath products of
cyclic groups and Suschansky p-groups of intermediate growth
Abstract: One of the pioneering examples of infinite periodic
groups was constructed by Suschansky using the language of tableaux describing
the iterated wreath products of cyclic groups of prime order p.
This language
was intruduced by Kaloujnine in the end of 1940's to study Sylow subgroups of
Sym(pn). We will discuss this tool and its applications,
and give the original
definition of Suschansky p-groups. It was realized slightly later
that these
groups also act on p-ary tree and can be generated by finite
automata. This
approach yields another proof of periodicity, and also allows to prove that
these groups have intermediate growth. This talk is based on a joint work
with Ievgen Bondarenko.
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March 9 : Elizabeth Wilcox (Binghamton University)
Title: When the Base of a Finite Wreath Product isn't
Characteristic
Abstract: We'll discuss the consequences of knowing that the
base of a
finite permutational wreath product is not characteristic. Much has been
discovered in this topic in the case of standard wreath products and
wreath products $G \wr H$ where $H$ acts transitively, but we'll discuss
finite wreath products where the action of $H$ is unlimited. The talk
will develop an understanding of centralizers in a wreath product from a
new viewpoint and examine the number of conjugates with which a given
element may commute. This will ultimately lead to the conclusion that if
the base of a finite permutational wreath product $G \wr H$ is not
characteristic then $G$ must equal $A \rtimes \langle x \rangle$ where $A$
is an odd order abelian group on which $x$, an element of order 2, acts by
inversion.
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March 16: Joe Kirtland (Marist College)
Title: Finite Groups with Many Permutable Subgroups
Abstract: A group G is semi-permutable if every subgroup not
contained in the Frattini subgroup of G is permutable in G. Note that a
subgroup H of a group G is permutable in G if HK = KH for all subgroups K of G.
This talk will examine the properties of semi-permutable groups and present
conditions for when semi-permutability implies that all subgroups in a
group are permutable.
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March 23 : Amanda Taylor (Binghamton University)
Title: The Burnside Problem for Periodic Groups
Abstract: The Burnside problem says if G is a k-generated group of exponent
n, is G finite? The answer in general is no. In this talk, we will discuss
some history of the problem in its various reincarnations and a proof by
Gupta and Sidki showing the existence of an infinite finitely generated
p-group for each odd prime p. If time permits, we may discuss some other
interesting properties of these groups.
This talk should be very accessible, even for first year students,
assuming they have had or are taking group theory.
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March 30 : Spring Break (at Binghamton University)
Title:
Abstract:
April 6 : No meeting (but see below).
Title:
Abstract:
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April 8 (CROSS LISTING WITH THE COLLOQUIUM; SPECIAL DAY THURSDAY and TIME 4:30 p.m.) : Richard M. Foote (University of Vermont)
Title: Strongly Closed Subgroups of Finite Groups: The Local--Global Principle in Action
Abstract:
Let G be a finite group with an abelian Sylow p-subgroup A, for some prime p.
A classical result of W. Burnside shows that if A factors as A_1 \times A_2
under the action of the normalizer, N_G(A), of A in G, and if N_G(A) acts
trivially by conjugation on A_1, then G possesses a normal subgroup G_2 that
has A_2 for a Sylow p-subgroup. A basic (fusion) ingredient in Burnside's proof
augurs the notion of {\it strongly closed subgroups} and the wealth of results
on recognizing the existence of normal subgroups of a finite group from the
conjugacy patterns in its p-subgroups. The talk will survey the history and
latest results in this vein, including as a special case a complete
generalization of Burnside's Theorem that provides necessary and sufficient
p-local conditions for G to factor as a direct product.
Ramifications to other areas such as algebraic topology, fusion systems, and number theory will be mentioned.
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April 13: Chris Mauriello (Binghamton University)
Title: Root Systems and Lie Algebras
Abstract: This talk introduces root systems through a few key examples that correspond
to particular finite dimensional Lie algebras. Specifically we look at the
root system and the root space decomposition of the Lie algebras:
sl(2,\C), sl(3,\C), and so(8,\C).
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April 20 : Quincy Loney (Binghamton University)
Title: Chevalley's Spinor Construction of the Octonions
Abstract: Many who are familiar with the octonions may have
seen their construction via the Cayley-Dickson process. In this talk we will
present Chevalley's "spinor" construction of this non-associative normed
composition algebra. This method uses a Clifford algebra to obtain three
8-dimensional irreducible representations of the special orthogonal Lie
algebra, so(8); the natural and two spinor representations. Their direct sum,
C, is given a commutative, non-associative algebra structure. The action
of the symmetric group, S_3, on that "Chevalley algebra", C, and on
so(8), is called "triality", and allows us to construct the octonions.
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April 27: No meeting (there will be two next week though).
Title:
Abstract:
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May 4 : Alexander Ushakov (Stevens Institute of Technology )
Title: Strong law of large numbers for graph(group)-valued random elements.
Abstract: We introduce a notion of the mean-set (expectation) of
a graph- (group-) valued random element ξ and prove a generalization of the
strong law of large numbers. Furthermore, we prove an analogue of the
Chebyshev's inequality and Chernoff type inequalities for ξ . Finally, we
discuss applications of our techniques in group-based cryptography.
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May 4 (SPECIAL TIME 4:15 p.m.) : Drazen Adamovic (University of Zagreb,
Croatia)
Title: On the representation theory of certain vertex algebras.
Abstract: In the theory of vertex algebras and in conformal field
theory it is important to construct and classify vertex algebras whose
representation-categories have finitely many simple objects. This leads to the
notions of rational and C2-cofinite vertex algebras. In this
talk I will review basic concepts and examples related to these vertex algebras.
I will also give an overview of recent results (joint work with A. Milas) on
the representation theory of certain C2-cofinite,
non-rational vertex algebras.
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