
Combinatorics, Algebra, Number Theory Seminar Fall 2005 Archive:
 December 5
 SungYell Song : Perfect ecodes in graphs and Lloyd theorem
Abstract: The classical problem of the existence of
perfect eerror correcting codes can be generalized as follows.
Let G be a finite
graph. Let x belong to the vertexset V(G ), and let e be a nonnegative
integer; define B_{e}(x) to be the set of
vertices of G whose distance
from x is not greater than e. Then a perfect
ecode in G is a subset
C of V(G ) such that
the sets B_{e}(c), as c runs through
C, form a partition of V(G
). N. Biggs shows that the proper setting for the perfect code
question is the class of distancetransitive graphs. A simple
connected graph G with
distance function ? is said to be distancetransitive if whenever
u,v,x,y are vertices of G
satisfying ?(u,v)=?(x,y), then
there is an automorphism g of G such that
g(u)=x and g(v)=y. The
Hamming graph H(n,q) is one of such graphs. Its vertex
set is the set F ^{n }of words of length n over
an alphabet F with q symbols, and two words in F
^{n }are adjacent iff x and y differ in
exactly one coordinate place. Lloyd's theorem states that a necessary
condition for the existence of a perfect ecode in F ^{n
}is that the zeros of a certain polynomial are integers. The
original proof given by Lloyd in 1957 was rather complicated, and
q was required to be a prime power, but over the years several
simplifications and generalizations have been made. In this talk we
will survey some of the literatures in this context.
 November 28
 Theodore Rice: Greedy Algebras
 November 21
 Thankgiving: no seminar.
 November 14
 Key One Chung: Graph Products and Graphic Coalgebras
Abstract: The First part of the talk will be a
survey of graph products. We will discuss several different graph
products and their properties from algebraic and combinatorial views.
Secondly, we will see how the product over the category of graphs is
defined by using the coalgebras. We will also discuss the completeness
of the category of graphs.
 November 7
 Anantharam Raghuram (University of Iowa): Special
Values of Lfunctions.
Abstract: This talk will be an elementary
introduction to the subject of special values of Lfunctions.
The first half should be accessible to a graduate student. The latter
part of the talk will be about Deligne's conjectures on the special
values of symmetric power Lfunctions associated to a
holomorphic modular form.
 October 31
 Ling Long: Modern approaches to counting problems.
Abstract: We will discuss some classical counting
problems such as theory of partition and Diophantine equations, in
particular, Fermat's last theorem. We will see how some basic counting
techniques have involved and integrated with other areas of
mathematics such as combinatorics, geometry, analysis, and topology.
We will try to get some ideas how Fermat's last theorem has been
proved. The whole talk is elementary.
 October 24
 Ling Long: On multirestricted Stirling numbers and their
qanalog.
Abstract: In this talk, we will give a proof of the
reciprocity law for multirestricted Stirling numbers, which implies
the reciprocity law for the ordinary Stirling numbers. This is a joint
result with Choi, Ng, and Smith.
Furthermore, we will discuss
the qanalog of the same question.
 October 17
 Jerome William Hoffman (Louisiana State University): Modular forms on
noncongruence subgroups and AtkinSwinnertonDyer relations.
Abstract in Portable Document Format
 October 10
 Christian Roettger: Primitive prime divisors of Mersenne numbers and
other sequences.
Abstract: The nth Mersenne number
M_{n} is of course 2^{n}1. It is
wellknown that M_{n} can only be prime for prime index
n, but it is wide open whether there are infinitely many
Mersenne primes (as far as I know, 38 Mersenne primes are known, and
virtually all of today's recordholder primes are Mersenne primes
because there are special primality tests available for these
numbers). A prime p is a primitive divisor of
M_{n} if it divides M_{n}, but no
earlier term of the sequence. We have a result about a weighted
asymptotic average of all primitive prime divisors of Mersenne numbers
for n up to a bound T. The proof uses very little number
theory, but a powerful Tauberian theorem for Dirichlet series. There
are intriguing prospects of generalizing the approach to many other
linear recurrence sequences. We will examine consequences and
connections to Wieferich primes and padic numbers.
 September 26, October 3
 Richard Ng: FrobeniusSchur indicators for pivotal monoidal
categories.
Abstract: The notion of FrobeniusSchur
(FS)indicators of a representation of a finite group has been
wellknown for many years. It was not known, until recently, that they
are invariants of the tensor categories of all the finitedimensional
representations of groups. In the lecture, we will talk about some
recent developments of FSindicators for Hopf algebras, quasiHopf
algebras, and finally pivotal categories. Unlike the group case, the
FSindicators are, in general, cyclotomic integers. The number
theoretic properties of these indicators have some important
applications to the structure of semisimple quasiHopf algebras and
spherical fusion categories. This seminar is intended to be accessible
to graduate students.
 September 19
 Anna Romanowska (Warsaw University of Technology): On
embedding.problems.
Abstract: One of the most efficient ways of
describing the structure of an algebra is to embed it into another,
usually one with a better known and richer structure. Prototypical
examples are given by the embedding of integral domains into fields
and commutative cancellative semigroups into commutative groups. Among
many other instances of such techniques, let us mention also the
embedding of cancellative entropic groupoids into quasigroups. Such
methods appear to be quite successful in investigating the structure
of modes, idempotent and entropic algebras. We will give a survey of
results concerning embedding modes (as subreducts) into vector spaces,
modules over commutative rings, and more generally, semimodules over
commutative semirings. We will also discuss the problem of the
existence of such embeddings.
 September 12
 Jonathan Smith: Duality for central quasigroups.
Abstract: Central quasigroups (e.g. real numbers
under subtraction) are the nonassociative analogues of abelian
groups. Duality for central quasigroups is based on the classical
Pontryagin duality for abelian groups. The dual of a central
quasigroup puts a quasigroup structure on the set of characters of its
abelian group isotope. While the combinatorial character theory of
quasigroups, based on association schemes, was previously limited to
finite quasigroups, the duality theory yields characters for some
locally compact central quasigroups.
Archive
of other semesters
Back
to the Mathematics Institute
Back to
Main Street

