THE SECOND SASKATCHEWAN MATHEMATICS MINI-MEETING
Abstracts of Talks
Professor Brian Alspach
Department of Mathematics & Statistics, University of Regina
will talk on
Isomorphisms of Cayley graphs
Abstract:
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
will talk on
Asymptotic distributions induced by random permutations
Abstract:
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
will talk on
Invariant Algebra Structures on Irreducible Representations of
Simple Lie Algebras
Abstract:
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
will talk on
Concatenation Hierarchies of Rational Languages
Abstract:
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
will talk on
Positivity and Symmetry in Algebras with Involution
Abstract:
A Fukamiya-Kaplansky-type lemma is established:
aaa has nonnegative
spectrum for every a in a complex algebraic algebra with positive
involution a. The proof of this lemma is
based on results of independent interest that characterise, by algebraic
data alone, finite-dimensional real and complex C*-algebras
among all involutive algebras.
Professor Jonathon Funk
Department of Mathematics & Statistics, University of Regina
will talk on
The locally connected coreflection of
the Jonsson-Tarski topos
Abstract:
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
will talk on
The complete convergence rates of the bootstrap mean
Abstract:
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