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Combinatorics, Algebra,
Number Theory Seminar
Fall 2005 Archive:
- December 5
- Sung-Yell Song : Perfect e-codes in graphs and Lloyd theorem
Abstract: The classical problem of the existence of
perfect e-error correcting codes can be generalized as follows.
Let G be a finite
graph. Let x belong to the vertex-set V(G ), and let e be a non-negative
integer; define Be(x) to be the set of
vertices of G whose distance
from x is not greater than e. Then a perfect
e-code in G is a subset
C of V(G ) such that
the sets Be(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 distance-transitive graphs. A simple
connected graph G with
distance function ? is said to be distance-transitive 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 e-code 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 L-functions.
Abstract: This talk will be an elementary
introduction to the subject of special values of L-functions.
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 L-functions 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 multi-restricted Stirling numbers and their
q-analog.
Abstract: In this talk, we will give a proof of the
reciprocity law for multi-restricted 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 q-analog of the same question.
October 17
Jerome William Hoffman (Louisiana State University): Modular forms on
noncongruence subgroups and Atkin-Swinnerton-Dyer relations.
Abstract in Portable Document Format
October 10
Christian Roettger: Primitive prime divisors of Mersenne numbers and
other sequences.
Abstract: The n-th Mersenne number
Mn is of course 2n-1. It is
well-known that Mn 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 record-holder primes are Mersenne primes
because there are special primality tests available for these
numbers). A prime p is a primitive divisor of
Mn if it divides Mn, 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 p-adic numbers.
September 26, October 3
Richard Ng: Frobenius-Schur indicators for pivotal monoidal
categories.
Abstract: The notion of Frobenius-Schur
(FS-)indicators of a representation of a finite group has been
well-known for many years. It was not known, until recently, that they
are invariants of the tensor categories of all the finite-dimensional
representations of groups. In the lecture, we will talk about some
recent developments of FS-indicators for Hopf algebras, quasi-Hopf
algebras, and finally pivotal categories. Unlike the group case, the
FS-indicators are, in general, cyclotomic integers. The number
theoretic properties of these indicators have some important
applications to the structure of semisimple quasi-Hopf 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 non-associative 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.
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