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.
-
January 23: Organizational meeting
-
January 30: Tom Head
Title: Higher Natural Numbers
Abstract: The arithmetic of the natural numbers N will be reviewed using only
'tally notation'. Thus N = {|, ||, |||, ... }; the sum of || and ||| is
|||||; and the product of || and ||| is ||||||. The set N of natural
numbers will then be regarded as the set of all words over the one letter
'alphabet' A = {|}. If we think of A as a group of order one, we may regard
the multiplication in N as having been provided by the trivial group
operation in A combined with a distributivity requirement.
These elementary observations suggest how any alphabet A, augmented with
a binary operation, can provide a system of 'Higher Natural Numbers'. Only
the special case in which the alphabet A is provided with the structure of an
Abelian group will be discussed in any detail.
The resulting number systems N(A) provide a pristine new mathematical
landscape for exploration. Questions for investigation are suggested in
various branches of mathematics: Symbolic dynamics, Abstract algebra, Formal
languages, and Number theory analogs. A solid 'classical' connection is
exposed by noting that, with the two element group A = {1,a}, the famous cube
free Thue-Morse sequence of words becomes the cyclic multiplicative
subsemigroup generated by the word 1a. With the three element group A, a
cyclic multiplicative subsemigroup can be specified with each word square
free.
The speaker's special THANKS go to Brett Bernstein for the initiating the
development of this proposed area of research in his 2005 M.S. thesis
Computer Science done here at B.U.
-
February 6 : Benjamin Brewster
Title: Extending partial permutations
Abstract: An abstract for this talk is available as a pdf file
at the following link:
Abstract
-
February 13 : Marcin Mazur
Title: A problem in elementary number theory
Abstract:
-
February 20 : Joseph Petrillo (Alfred University)
Title: An Introduction to Transitive and Persistent Subgroups
Abstract: Given subgroup properties α and β, a subgroup U of a group G may or may not possess one or both of the following properties:
αβ-transitivity: Every α-subgroup of U is a β-subgroup of G.
αβ-persistence: Every β-subgroup of G in U is an α-subgroup of U.
We will present some elementary results and discuss examples of αβ-transitive and αβ-persistent subgroups for various α and β.
-
February 27 : Joanne Redden (SUNY Cortland)
Title: On simple groups as the union of proper subgroups
Abstract: Bernhard Neumann showed that a group is the union of finitely many proper
subgroups if and only if it has a finite noncyclic homomorphic image. J.H.E.
Cohn defined s(G) to be the smallest integer n such that the group G is the
set-theoretic union of n proper subgroups. It is well known that there is no
group with s(G) = 2. The question arises what integers n can occur as s(G)
for a group G. By a result of M.J. Tomkinson, s(G) is always congruent to 1
modulo a prime power in case G is solvable, and there is no group with
s(G) = 7. Cohn showed that s(G) = 10 and 16 for G the alternating and
symmetric group on five letters, respectively. Tomkinson conjectured that
there are no
groups with s(G) = 11, 13 or 15, respectively.
With the help of GAP we determine s(G) for nonsolvable and simple groups
in particular. We found that s(PSL(2,7)) = 15 and that there are no
nonabelian finite simple groups with s(G) = 11 or 13, respectively. Current
evidence supports Tomkinson's conjecture that there are no groups at all
with s(G) = 11 or 13.
This is joint work with Luise-Charlotte Kappe.
-
March 6 : Anders Buch (Rutgers)
Title: Quantum cohomology of Grassmannians
Abstract: The (small) quantum ring of a homogeneous space is a
deformation of
the singular cohomology ring, which uses the (3-point, genus zero)
Gromov-Witten invariants as structure constants. These Gromov-Witten
invariants count the number of curves meeting generic Schubert
varieties, and the associativity of the quantum ring provides an
efficient tool for computing them. The quantum ring of a Grassmann
variety is described by structure theorems of Bertram, which were
originally proved by applying degeneracy locus formulas on
compactified moduli spaces of curves. I have found a simpler method
for proving such structure theorems, by way of defining a "kernel" and
"span" of a curve and applying classical Schubert calculus to study
them. In joint work with A. Kresch and H. Tamvakis, we have proved
that curves can be counted by counting their possible kernel-span
pairs. As a result we prove that the Gromov-Witten invariants on
Grassmannians are special cases of the classical Schubert structure
constants on two-step flag varieties.
-
March 13 : Nor Haniza Sarmin (University Teknologi Malaysia)
Title: Homological invariants of 2-generator non-torsion
groups of class 2
Abstract: Let R be the class of 2-generator non-torsion groups of nilpotency
class 2. Using their classification and non-abelian tensor squares, we
determine certain homological invariants of groups in R, such as the
exterior square, the symmetric square and the Schur multiplier.
This is
joint work with Luise-Charlotte Kappe and Nor Muhainiah Mohd Ali.
-
March 20 : Ilir Snopce
Title: Lie algebras and finite presentability of pro-p-groups.
Abstract: An abstract for this talk is available as a pdf file
at the following link:
Abstract
-
March 27 : Par Kurlberg (Royal Institute of Technology, Sweden)
Title: Lower bounds on the order of some pseuderandom number
generators
Abstract: Given coprime integers b and n, let ord(b,n) be the
multiplicative order of b modulo n. The length of the periods of some
popular pseuderandom number generators (the power generator, the
linear congruential generato, and the Blum-Blum-Shub generator) turns
out to be related to ord(b,n) for apropriately chosen b and n. (Note
that the case n=p, where p is prime is related to Artin's primitive
root conjecture.) We will give lower bounds on ord(b,n) for b fixed
and n ranging over certain subsets of the integers, e.g., the set of
primes, the set of "RSA moduli" (i.e., products of two primes), the
full set of integers, and the images of these sets under the
"Carmichael lambda function". Assuming the generalized Riemann
hypothesis, we can show that the order is essentially maximal for
almost all n in the above mentioned subsets. We can also give weaker
unconditional bounds. The lower bounds in the case of RSA moduli
shows that certain "cycling attacks" on the RSA crypto system are
ineffective.
April 3 : Recess
Title: ``''
Abstract:
-
April 10 : Sarah Rees (Univ. of Newcastle UK)
Title: Group geodesics: regularity, star-freedom and local testability.
Abstract: The geodesic words in a finitely generated group are known to form a
regular set whenever the
group is either word hyperbolic or free abelian. For selected generated
sets, the same is true
for virtually abelian groups, geometrically finite hyperbolic groups, all
Coxeter groups,
Artin groups of finite type and indeed all Garside groups; this list does
not claim to be
exhaustive.
I report on an investigation to look for connections between algebraic
properties of such a
group, combinatorial properties of its presentations,
the structure of its regular set of geodesics, and the complexity of its
word problem. That work is joint work with Gilman, Hermiller and Holt.
Terms such as regularity, star-freedom and local testability will be
defined in the talk; each can be shown to have several different
disguises (set-theoretic, geometric, or algebraic, in terms of an
associated finite semigroup). We shall see in particular that certain
small cancellation conditions on a presentation (which imply word
hyperbolicity) force the set
of geodesic words to be star-free, that
a rather restrictive (but also natural) form of local testability of
geodesics implies that the word problem for that group is context-free,
and hence characterises virtually free groups, that 1-local testability
characterises free
abelian groups, and that in general a group with locally testable
geodesics can have only
finitely many conjugacy classes of torsion elements.
-
April 17 : Darryl Daugherty
Title: Properties of Lie Algebras
Abstract: I will be presenting examples and describing
properties of Lie algebras.
Material covered will include but not limited to ideals, homomorphisms, and
representations of Lie algebras. Time permitting I will also cover some
theorems and lemmas that deal with solvability of Lie algebras. Material
covered will be coming from Hans Samelson's "Notes on Lie Alebras".
-
April 24 : Elizabeth Wilcox
Title: On the Minimum Length of the Chief Series of Odd Order
Finite Solvable Complete Groups
Abstract: This seminar will provide a brief introduction to
complete groups and then discuss the highlights of a 1982 article by Brigitte
Schuhmann, which provided the title of this talk. If there is time, we
will discuss some of my work inspired by Ms. Schuhmann's paper.
-
May 1 : Eric Sponza
Title: Supersolvable Groups
Abstract: A group is supersolvable whenever its chief factors
are all cyclic. In this talk, we'll discuss
properties and theorems involving supersolvable groups
and will finish with a proof of Iwasawa's Theorem,
that a groups is supersolvable iff it is equichained.
-
May 8 : Dandrielle Lewis
Title: Hall Subgroups and Hall Systems of a Finite Solvable Group
Abstract: Hall subgroups and Hall systems of a finite solvable
group are named after the group theorist Philip Hall. A Hall π-subgroup is a
subgroup H of G such that |H| is a π-number and |G:H| is a π'-number,
where π is a set of primes and π' = P \ π
(P being the set of all primes).
Hall's Theorem states: "A finite group G is solvable if and only if
Hall π-subgroups exist in G, for each π contained in P." This theorem
and interesting results about Hall systems of a finite solvable group will be
discussed.
-
May 15 : Erin Decker
Title: Constructing nilpotent groups with prescribed properties
Properties
Abstract: It is well known that the set of commutators in a
group not necessarily
forms a subgroup and that 96 is the order of a minimal example with this
property. Most of these examples which can be found in the literature are
p-groups. Thus the following question arises: For a given prime p, what is
the smallest integer n such that there exists a group of order p^n in which
the set of commutators does not form a subgroup?
L.C. Kappe and R.F. Morse have shown in "On commutators in p-groups" (JGT
8 (2005), 415-429) that n = 6 for any odd prime p and n = 7 for p = 2. The
examples for p = 2 and 3 were found with GAP though explicit
constructions for these groups and verifications are given in the paper. For
p > 3 a separate construction is needed.
The topic of this talk is to present the construction of this group of
order p^6, p > 3, together with verification of the claim that the set of
commutators does not form a subgroup. The group is obtained by two split
extensions starting from an elementary abelian p-group of rank 4. This will
be preceded by a general introduction to group constructions using repeated
split extensions.