Blyth 2004

Department of Mathematics
University of Toronto

Eleventh Annual R.A. Blyth Lectures in Mathematics


Professor Yu. Manin

Max-Planck-Institut für Mathematik, Bonn and Northwestern University

Lecture 1
Wednesday, October 27, 2004, 4:10 p.m.
Sidney Smith Hall, 100 St. George St. Room 2102

Quantum Computing Project

Quantum computing is a vast research project. Starting with Alan Turing's vision of computation as time evolution of a classical physical system, quantum computing enhances the theoretical model by allowing parts of the system to be quantum. For example, if the central processor could be initialized, and then induced to evolve in a superposition of various classical states, the resulting quantum parallel computation could provide considerably better performance level than the classical one. Several basic ideas and algorithms of quantum information processing will be explained in this non-technical talk.

Lecture 2
Thursday, October 28, 2004, 4:10 p.m.
Sidney Smith Hall, 100 St. George Street
Room 5017A

Fractional Dimensions in Geometry and Algebra

Dimension is one of the central notions of geometry and its various chapters. Most common geometric spaces have integer dimension. The idea that one can define sets of non-integer dimension, and the first definition and examples of such sets, go back to Hausdorff and Besicovich. Although Hausdorff dimension and its variations arguably furnish the best known examples of fractional dimension, there are more and more instances in noncommutative geometry, homological algebra and theoretical physics, where fractional dimensions appear in various guises and with totally different definitions. Some of these instances will be discussed in the lecture whose main motivation is the search of common ground for different constructions.

Lecture 3
Friday, October 28, 2004, 3:10 p.m.
Sidney Smith Hall, 100 St. George Street
Room 5017A

Real Multiplication and Quantum Tori: A Program in Number Theory and Noncommutative Geometry

Classical theory of Complex Multiplication shows that all abelian extensions of a complex quadratic field K are generated by the values of appropriate modular functions at the points of finite order of elliptic curves whose endomorphism rings are orders in K. For real quadratic fields, a similar description is not known. However, the relevant case of Stark conjectures strongly suggests that such a description must exist. In this lecture I will discuss a proposal to use two - dimensional quantum tori corresponding to real quadratic irrationalities as a replacement of elliptic curves with complex multiplication. I will review some basic constructions of theory of quantum tori, in particular, recent results of A, Polishchuk, from the perspective of this Real Multiplication research project.

The Blyth Lecture Reception will follow the First Lecture on Wednesday, October 27, 2004, at Massey College, 4 Devonshire Place All are invited to attend.