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Contributed Papers / Communications libres
(Org: Walter Burgess and/et Abdellah Sebbar)

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MARTIN ARGERAMI, Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan  S4S 0A2
The Schur-Horn Theorem in II₁ factors [PDF]

The Schur-Horn theorem expresses a relation, in real n-space, of two apparently unrelated notions: majorization and convexity. In finite dimensional von Neumann algebras, it can be expressed in terms of maximal abelian subalgebras (masas) of factors:

Let M be a finite dimensional factor, A Ì M a masa in M, and let P be the unique conditional expectation from M to A. Then, for every a Î A,

P æ è UM(a) ö ø =co{vav*:v Î N(A)},

where U_M is the unitary group of M and N(A) is the normalizer of A.

The author believes (and has evidence, although not a proof) that the Schur-Horn property characterizes masas of finite dimensional factors (that is, if a subalgebra of a finite dimensional factor satisfies the Schur-Horn relation, then it is a masa).

II₁ factors are a natural setting to try the Schur-Horn property, because there always exist a normal conditional expectation onto any masa, and also because the main notion used in the finite dimensional proof is majorization, which can be reasonably extended to the setting of II₁ factor, thanks to the existence of the trace. One cannot expect masas in II₁ factors to be characterized by the Schur-Horn property (it fails clearly for singular masas) but it may, for instance, characterize regular masas.

In our attempts to prove this theorem, we have had to work with the notion of majorization in II₁ factors, and several interesting characterizations appear for unitary orbits of states and automorphisms.

MURRAY BREMNER, Department of Mathematics, University of Saskatchewan, Saskatoon, Saskatchewan  S7N 5E6
Quantization of Lie and Jordan triple systems [PDF]

This talk will show how the decomposition of the group ring of the symmetric group into a direct sum of full matrix subrings can be used to give a complete classification of n-ary operations. Roughly speaking, row equivalence of matrices corresponds to quasi-equivalence of operations. In particular, the Lie and Jordan products represent the two non-trivial quasi-equivalence classes of binary operations. For ternary operations, there are infinitely many quasi-equivalence classes, which divide into eight classes, and four infinite families of classes each with a single parameter. The Lie triple product is contained in one of the infinite classes, and the other operations in the class can be regarded as quantizations of that product. Similar remarks apply to the Jordan triple product. For special values of the parameter, the operation satisfies an identity of degree 5. This identifies some new ternary operations which define varieties of triple systems, similar to Lie and Jordan triple systems, which seem to be an interesting direction for further research.

VAHID DABBAGHIAN, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario  K1S 5B6
An efficient algorithm to construct representations of finite groups [PDF]

Let G be a finite group. It is easy to compute the character of G corresponding to a given complex representation, but much more difficult to compute a representation affording a given character. In part this is due to the fact that a class of equivalent representations contains no natural canonical representation.

Although there is a large literature devoted to computing representations, and methods are known for particular classes of groups, no general method has been proposed which is practical for any but very small groups. For example, the function ``IrreducibleRepresentationsDixon'' which is supplied in the latest version 4.3 of the computer algebra system GAP, is very slow in computing representations for even moderately sized groups and fails to compute a representation in many cases.

We shall describe an algorithm to compute an irreducible matrix representation R which affords a given character c of a given group G. The algorithm uses properties of the structure of G which can be computed efficiently by a program such as GAP, theoretical results from representation theory, theorems from group theory (including the classification of finite simple groups), and linear algebra. The algorithm has been implemented in GAP and appears to work well for a general group G when the character supplied has degree up to about 30.

HANS HEINIG, McMaster University, Hamilton, Ontario  L8S 4K1
Exponential inequalities with weights [PDF]

Characterizations of weight pairs are obtained for which a class of weighted exponential-logarithmic integral inequalities are satisfied. Special cases yield weight characterizations for the geometric mean operator as well as operators of Riemann-Liouville and Laplace type.

TREVOR JONES, University of New Brunswick
Computing some nonstandard Betti numbers [PDF]

In previous work, we established an abstract generalized Gauss-Bonnet theorem for surfaces. We discuss the computation of our nonstandard Betti numbers in the theorem, at least for the special case of Riemann surfaces of constant curvature. By the uniformization theorem we may as well consider hyperbolic space modulo a discrete subgroup of isometries and we discuss the case of hyperbolic space in detail.

VALENTINA KIRITCHENKO, University of Toronto, Toronto, Ontario  M5S 3G3
A Gauss-Bonnet theorem for constructible sheaves on reductive groups [PDF]

We prove an analog of the Gauss-Bonnet formula for constructible sheaves on reductive groups. This formula holds for all constructible sheaves equivariant under the adjoint action and expresses the Euler characteristic of a sheaf in terms of its characteristic cycle. As a corollary from this formula we get that if a perverse sheaf on a reductive group is equivariant under the adjoint action, then its Euler characteristic is nonnegative.

KUNQUAN LAN, Ryerson University
Multiple positive solutions of higher order boundary value problems [PDF]

We consider the existence of one or several nonzero positive solutions for a higher order nonlinear ordinary differential equation with n-sets of separated boundary conditions. The boundary value problems can be changed into a Hammerstein integral equation with a suitable kernel. We shall show that the kernel has upper and lower bounds. This enables us not only to exhibit a new property of positive solutions for the boundary value problems but also to derive new results on these boundary value problems from the well-known results on the existence of one or several positive solutions of Hammerstein integral equations with singularities obtained by the author recently. This avoids utilizing the theory of fixed point index for compact maps defined on cones directly.

CLAUDE LEVESQUE, Département de mathématiques et de statistique, Université Laval, Québec  G1K 7P4
A fundamental system of units for some fields of degree 9 [PDF]

We exhibit a fundamental system of units for some families of composita of two pure cubic number fields of the form Q(w, q) where w³ and q³ are certain positive integers.

RAVIL MOUKHOMETOV, Ottawa
On problem of the reconstruction of metric on the Riemannian manifold [PDF]

On a compact Riemannian manifold M with a boundary D we consider the problem of the reconstruction of the Riemannian metric g if are known the lengths of geodesics with endpoints on the boundary D of M. First results, namely an uniqueness and stability for this non-linear problem in a general formulation, have been got by the auther in 1977. In present time the auther obtains some results when geodesics are reflected from the part L of the boundary D of M and also for the linearized problem that is the integral geometry problem. The auther also gets in connection of this problem some formula: the symplectic volume and from here the Riemannian volume of M is expressed only by the lengths of geodesics with endpoints on the boundary D without L. In this formula the metric g is unknown and the manifold M is not known as the part L of the boundary D is unknown. The considered problems refer to the Riemannian geometry, also to the symplectic geometry (in proofs is used 1-contact form, see: C. Godbillon, Geometrie Differentielle et Mecanique Analytique). The near similar problems are also used in geophysics (structure of Earth) and the obtained results may stimulate the investigations of the new problems.

AMRITANSHU PRASAD, CRM, CP 6128, Succursale centre-ville, Montreal, Quebec  H3C 3J7
Almost unramified automorphic representations for slpit groups over F_q(t) [PDF]

Let G be a split reductive group over a finite field F_q. Let F=F_q(t) and A be the adeles of F. We describe the local constituents at each valuation of F of all the irreducible representations of G(A) that occur discretely in L²(G(F)\G(A)) and have non-zero vectors invariant under the compact open subgroup K of G(A) which is a product Iwahori subgroups at two valuations of F and maximal compact subgroups at all the other valuations.

This is done by showing, firstly, that the space generated by the representations mentioned above is spanned by residues of Eisenstein series associated to an unramified automorphic characters of a maximal split torus of G. This imposes consistency conditions on the local constituents at different valuations of a fixed automorphic representation. An earlier result of the speaker describes the local constituents of the aforementioned representations at the two places where the local factor of K is Iwahori. This, together with the consistency conditions uniquely determines the local constituents at all places of the automorphic representation.

RONALD SKLAR, St. John's University, Jamaica, New York  11439, USA
Mathematical techniques for circumventing query-set-size control in a statistical database [PDF]

This talk will discuss the method of attacking a statistical database by the use of a tracker. Different types of trackers will discussed, and it will be shown that one of the commonly used techniques, namely query-set-size control, for protecting a statistical database can easily be subverted by the use of a tracker. The simple mathematics employed makes this interesting and timely topic one that can be successfully incorporated into a mathematics liberal arts course for non-majors.

VLADLEN TIMORIN, University of Toronto, Tornoto, Ontario  M5S 3G3
Circles and Clifford algebras [PDF]

We study smooth maps from an open subset of R^m to an open subset of Rⁿ that take germs of straight lines to germs of circles. ``Degenerate circles'', i.e. lines or points, are counted as circles. We give a construction of such maps based on representations of Clifford algebras. It provides a complete purely geometric description of Clifford algebras representations in terms of circles.