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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