Combinatorics, Algebra,
Number Theory Seminar
Spring 2010 Archive:

Abstract: By comparing the  n-level correlation function attached to a product of  L-functions with the correlation function attached to a single  L-function we give necessary conditions for factorization of a primitive automorphic  L-function over a number field into a product of  L-functions over cyclic algebraic number fields.

Abstract: In recent years quantum walks have emerged as one of the most useful tools to design quantum algorithms. For this reason they are currently the subject of intensive study. In this talk, I will introduce the concept of quantum walk on a graph as a generalization of the classical concept of random walk. I will describe the main issues in the study of the dynamics of quantum walks and how the classification of these systems is related to some old problems in algebra.

Abstract: We define relation algebras and discuss some of their connections with logic. We then show how to construct representable relation algebras from the (weaker) class of semiassociative relation algebras. This construction allows new proofs of old results: e.g., set theory may be coded into the logic of three variables (Tarski), the equational theory of semiassociative relation algebras is undecidable (Maddux), and free semiassociative relation algebras are not atomic (Nemeti). The methods and results are directed toward the still unsolved problem of coding set theory into the equational theory of diagonal-free cylindric algebras.

Abstract: A representation is an embedding of a group into a general linear group over some appropriate field. Representations can be useful for calculations, and are interesting in their own right. Representation theory is well developed for groups, and provides valuable insight into groups and modules through the use of characters and other tools. Since matrix multiplication is inherently associative, representations in the traditional sense have no use in nonassociative structures. This talk will introduce some general concepts of representation theory, and explore some possible nonassociative analogues.

Abstract: In this talk, I will briefly introduce the history of Hopf algebras and the problem of classification for some finite dimensional Hopf algebras. I will also talk about Hopf algebras of dimension  4p  when  p  is an odd prime and 2 < p < 12, joint work with Richard Ng.

Abstract: If  Q'  and  Q  are quasigroups of the same order, an ordered triple  ( f₁ , f₂ , f₃ )  of bijections from  Q'  to  Q  is an isotopy if

xf₁ . yf₂ = (x.y) f₃

for all  x  and  y  in  Q'. An autotopy of  Q  is an isotopy from  Q  to  Q. In his 2009 Monash University Ph.D. thesis, Douglas Stones observed that the components of an autotopy of a quasigroup of prime order  p  up to 23 either all have the same cycle type, or else comprise two  p-cycles and the identity permutation. He asked whether this pattern always held. In this talk, an affirmative answer to Stones' question is given for group isotopes of prime order.

Abstract: Full idempotent reducts of abelian groups, or equivalently  Z-modules  ( A , + , Z ), are given by integral affine spaces, reducts  ( A , P )  of  ( A , + , Z ), where  xyzP = x - y + z  is the ternary Mal'cev operation. An abstract characterization of such affine spaces provided a basis for a number of well-known theorems characterizing algebras equivalent to subreducts of modules. In this talk we will describe full idempotent reducts of commutative monoids. We will show that they may be described as certain ternary or certain  n-ary reducts of commutative monoids belonging to some special varieties of commutative monoids. We will provide their abstract characterization, and discuss their meaning for the problem of embedding modes (idempotent and entropic algebras) as (sub-)reducts into semimodules over commutative semirings.

Abstract: In 1988, L. Carini, I. R. Hentzel and G. M. Piacentini-Cattaneo proved that there are four different families of commutative algebras satisfying an identity of degree four, over a field of characteristic not 2 or 3. In this talk, we will show that for three of these families, there exists a trace form. Moreover, we analyze what happens when this form is non-degenerate. Related to the other family, we give an example to show that there exists a nonzero trace form, but it is an open problem to find a general pattern for trace forms of these algebras.

This is joint work with Manuel Arenas and Carlos Rojas-Bruna