**Abstracts of Talks**

**Professor Brian Alspach**

Department of Mathematics & Statistics, University of Regina

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Cayley graphs are formed from groups as a means of providing vertex-transitive graphs. A fundamental problem for Cayley graphs is recognition and determining when two Cayley graphs are isomorphic. I'll discuss old and new results on this topic.

**Professor Mik Bickis**

Mathematical Sciences Group, University of Saskatchewan

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Classical central limit theorems are concerned with the weak convergence of sums of independent random variables. However, there are other random processes which will produce normal distributions as a weak limit. One such situation is the case of randomised linear functions, defined by T=x'Gy where x and y are fixed vectors and G is a random element from a group of linear transformations. If G is sampled from a uniform distribution on the orthogonal group, then T will have a Beta distribution which approaches a Normal distribution as the dimension goes to infinity. Of greater interest are cases where G is sampled from a discrete group that merely permutes coordinates. When sampling from the symmetric group of all coordinate permutations, asymptotic normality has been established provided the corresponding sequence of vectors satisfies certain regularity conditions. The situation with smaller permutation groups is less clear, with the approach to normality appearing to depend on properties of both the group and the vectors being permuted. These issues will be examined using Monte Carlo methods.

**Professor Murray Bremner**

Department of Mathematics & Statistics, University of Saskatchewan

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An irreducible representation of a simple Lie algebra can be a direct summand of its own tensor square. In this case the representation admits a nonassociative algebra structure which is invariant in the sense that the Lie algebra acts as derivations.

I will first review the representation theory of the simple Lie algebra sl(2). A version of the Clebsch-Gordan theorem, giving the decomposition of the tensor product of two sl(2)-modules and explicit highest weight vectors for the direct summands, will be presented. From this we see that the irreducible representation with highest weight n (and dimension n+1) occurs as a summand of its symmetric square when n = 1 (mod 4) and as a summand of its exterior square when n = 3 (mod 4). For n = 4m+3 we obtain a series of anticommutative sl(2)-invariant algebras beginning with sl(2) itself (n = 2) and the simple non-Lie Malcev algebra (n = 6). The structure constants are given for the 11-dimensional algebra arising in the next case n = 10; this algebra is the main focus of the talk. (I am especially interested in the antisymmetric case since this generalizes Lie and Malcev algebras.)

I will describe a computer search for an identity satisfied by this 11-dimensional algebra. (This is joint work with Irvin Hentzel of Iowa State University.) The smallest identity it satisfies, which does not follow from anticommutativity, has degree 7. We have shown by computation that there are no identities in degree < 7. We have classified the identities of degree 7 in the weak sense that we know, from the point of view of the representation theory of the symmetric group S7, where the identities occur and how many there are. However we still do not have simple explicit forms of these identities.

These ideas can also be applied to other simple Lie algebras. The software system LiE can be used to determine all cases for fundamental representations of simple Lie algebras of rank up to 8 in which the representation occurs as a summand of its exterior square. This provides a large number of new anticommutative algebras, with simple Lie algebras in their derivation algebras, which deserve further study.

**Professor David Cowan**

College of Arts & Science, University of Saskatchewan

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If A is a finite set, a subset of the free monoid A* is called a formal language over A. The class of rational languages over A is the least class containing the finite languages and closed under the boolean operations, concatenation, and the star operation, which, for a given language L, is just the free submonoid L* of A* generated by L. In the early seventies, a hierarchy of families of star-free languages was introduced by Brzozowski as a means by which such languages might be studied and classified (a star-free language is a rational language that can be obtained from the finite languages without resorting to the star operation). A fundamental question in this approach is one of decidability: given a star-free language L, is it decidable where in the hierarchy L first appears. This question for Brzozowski's and other related hierarchies will be discussed.

**Professor Douglas R. Farenick**

Department of Mathematics & Statistics, University of Regina

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A Fukamiya-Kaplansky-type lemma is established:

**Professor Jonathon Funk**

Department of Mathematics & Statistics, University of Regina

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It is known that a topological space has a best approximation by a locally connected space: if X is a topological space, then there is a locally connected space Y and a continuous map f:Y-->X with the property that any continuous map from a locally connected space L-->X factors uniquely through f. We call Y the locally connected coreflection of X. Any Grothendieck topos also has a locally connected coreflection, but this topos is not always easy to describe. A Jonnson-Tarski algebra is a set A equipped with an isomorphism A -> A x A (x = cartesion product). The category of JT-algebras is a topos that is not locally connected. I will describe its locally connected coreflection in terms of the dual Kennison algebras: a Kennison algebra is a set B equipped with an isomorphism B + B -> B (+ = disjoint union = coproduct).

Jonsson-Tarksi algebras and their duals the Kennison algebras are related to inverse semigroups, ordered groupoids, left cancellative categories and etendue (Grothendieck's term). They are also related to what Lawvere has called a topos distribution. Indeed, the category of Kennison algebras is canonically equivalent to the distribution dual of the Jonsson-Tarski topos.

**Professor Andrei Volodin**

Department of Mathematics & Statistics, University of Regina

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For a sequence of random variables $\{X_n, n\ge 1\}$, the convergence rates (that is Hsu - Robbins complete convergence and an iterated logarithm type results) are obtained for bootstrapped means. No assumptions are made concerning either the marginal or joint distributions of the random variables $\{X_n, n\ge 1\}$. As special cases, new results for pairwise i.i.d. sequence and stationary ergodic sequences.

*Last update: January 31, 2003
--------- created and maintained by Franz-Viktor Kuhlmann*