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Plenary Speakers / Conférenciers principaux

MARTIN GOLUBITSKY, University of Houston
Oscillations in coupled systems and animal gaits

Collins and Stewart noted that many quadruped gaits can be described by spatio-temporal symmetries. For example, when a horse paces it moves both left legs in unison and then both right legs and so on. The motion is described by two symmetries: Interchange front and back legs, and swap left and right legs with a half-period phase shift.

Central pattern generators (CPGs) in the neural system are thought to send time periodic signals to the `legs'. We model CPGs by coupled systems of differential equations with symmetries based on leg permuation. We describe how periodic solutions with prescribed spatio-temporal symmetry can be formed in symmetric systems; construct a CPG architecture that naturally produces quadrupedal gait rhythms; and make several testable predictions about gaits.

This is joint work with Luciano Buono, Jim Collins, and Ian Stewart.

JOHN OCKENDON, Oxford University
Simulating mathematics in industry

This talk will illustrate how a few basic mathematical ideas can help industry understand some of its problems in a helpful and quantitative way. What transcends all successful mathematics in industry is the construction of a consistent, accessible theoretical framework within which a class of processes can be placed. Examples of such frameworks will include several stages of glass manufacture, the electrical heating of metals and wave propagation around stealthy objects. This range of topics will also help me show how to set about stimulating mathematics in industry and, also how to find ``stimulating mathematics in industry''.

ARTURO PIANZOLA, University of Alberta, Edmonton, Alberta  T6G 2G1
Local triviality and infinite dimensional Lie algebras

Using the beautiful classification of complex finite dimensional simple Lie algebras for inspiration, my intention is to explain how some fundamental questions about an interesting family of infinite dimensional Lie algebras (which includes the affine Kac-Moody case), can be answered by using principal fibrations.

I will use many examples, stay away from technicalities, and try to convey to a general audience what some of the central ideas at hand are.

DAVID PIMM, Faculty of Education, University of Alberta, Edmonton, AB  T6G 2G5
Interactions between language and mathematics: fluency, understanding and time

In this talk, I will take a look at some significant ways in which language interacts with mathematics (despite Dutch philosopher of mathematics L. E. J. Brouwer's claim that `Mathematics is a languageless activity of the human mind, having its origin in a move in time'), illustrated with some examples from mathematics classrooms. In particular, I will cautiously revisit the still-smouldering debate about mathematical fluency vs mathematical understanding (and I use the term `vs' advisedly) and offer some reasons connected with the nature of symbols themselves for why these need to be twin goals of any mathematics education. Lastly, I will try out some new (i.e. unrefined) thoughts about connections between the language of mathematics, the nature of mathematical knowledge and time.

RICHARD SCHOEN, Stanford University
Constructing calibrated submanifolds

Calibrated submanifolds include complex submanifolds of Kähler manifolds as well as special lagrangian submanifolds. This latter class of submanifolds is of substantial interest in string theory. This talk will give a general introduction to the subject with particular emphasis on existence questions special lagrangian and minimal lagrangian submanifolds. We will describe deformation results, gluing methods, heat equation methods, and variational methods for constructing them.

DAN VOICULESCU, University of California, Berkeley, California  94720-3840, USA
Free entropy

Shannon's information-theoretic entropy has an analogue in a highly non-commutative probabilistic context, the free entropy of free probability theory. There are important connections to random matrix theory and applications to the solution of a number of problems on von Neumann algebras.

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