## Abstracts

**Ian F. Blake,** Dept. of Electrical and Computer
Engineering, University of Toronto

*Directions in Cryptography *

The notion of a one-way function and their use in public key cryptography,
introduced in the mid-seventies, led the field of security in a variety
of directions. An area of particular interest is the effort to determine
the difficulty of certain mathematical tasks, such as integer factorization
and the modular discrete logarithm function. Additionally, the use of
such mathematical functions in the formulation and implementation of
protocols which achieve achieve useful communication functions, has
been an area of intense recent interest. This talk will address recent
progress on certain aspects of these two areas.

**Henri Darmon,** Dept. of Mathematics and
Statistics, McGill University

*Elliptic curves and Kronecker's solution to Pell's equation *

A celebrated 1865 paper of Kronecker describes how to solve Pell's equation
in terms of values of the Dedekind eta-function. I will describe his
approach, assuming no prior number theoretic background, and indicate
how it can be modified to construct rational points on elliptic curves
in terms of the modular symbols of Birch and Manin; the latter topic
is part of a work in progress with Massimo Bertolini.

**Ian Munro**, University of Waterloo

*Succinct Data Structures*

Although computer memories, at all levels of the hierarchy, have grown
dramatically over the past few years, increased problem size continues
to outstrip this growth. Recently developed data compression techniques
attack one major aspect of the problem, but here we focus on structural
information: combinatorial objects such as trees, other classes of graphs,
permutations and the like. The interest is in representations that are
not only terse, but also permit the basic operations one would expect
on the underlying data type to be performed quickly without decoding
large portions of the data. We call such data structures succinct. The
archtypical example is the binary tree, whose usual representation requires
4n lg n bits, if we are to navigate up and down the tree and report
subtree size. The information theoretic minimum, however, is only about
2n bits. We present a representation requiring essentially this mimimal
space while supporting, in constant time, the natural operations used
in traversing a tree. The general approach is then applied to several
other structures to obtain optimal (or near optimal) space bounds while.

**Keith J. Worsley, **Dept. of Mathematics
and Statistics, McGill University

*The geometry of random images in astrophysics and brain mapping *

The geometry in the title is not the geometry of lines and angles
but the geometry of topology, shape and knots. For example, galaxies
are not distributed randomly in the universe, but they tend to form
clusters, or sometimes strings, or even sheets of high galaxy density.
How can this be handled statistically? The Euler characteristic (EC)
of the set of high density regions has been used to measure the topology
of such shapes; it counts the number of connected components of the
set, minus the number of `holes', plus the number of `hollows'. Despite
its complex definition, the exact expectation of the EC can be found
for some simple models, so that observed EC can be compared with expected
EC to check the model. A similar problem arises in functional magnetic
resonance imaging (fMRI), where the EC is used to detect local increases
in brain activity due to an external stimulus. The famous Nash Embedding
Theorem helps us to extend these ideas to manifolds, so that we can
detect changes in brain shape via structure masking, surface extraction,
and 3D deformation fields. Finally, we look at some curious random fields
whose excursion sets are strings, and we show using the Siefert representation
that these strings can be knotted.

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