**For a General Audience: **

The foundations of real analysis, from Newton and Leibnitz to Weierstrass
and Cantor

Beginning with the development of calculus in the 1680s, mathematical
analysis treaded on new territory with very insecure footings. In
the following century, study of the vibrating string (Fourier series)
exposed many problems in the way mathematicians thought of functions,
limits and even real numbers. It took about 200 years to realize
that a rigorous, non-geometric, approach to the study of functions
was necessary, and to figure out how to resolve the difficulties.
This resolution led to our modern view of mathematics.

**For a Mathematical Audience:**

*Operator theory meets algebraic geometry*

I will discuss how to study commuting sets of operators on Hilbert
space which satisfy polynomial relations. Under a natural norm constraint,
there is a universal operator algebra that models this. In an effort
to classify these algebras up to isomorphism, one must deal with
the variety associated to the polynomial relations. Classification
up to completely isometric isomorphism is very nice. But the algebraic
isomorphism problem raises many difficulties. Ideas from operator
theory and algebraic geometry are combined with a function theoretic
representation of our algebras as multipliers on a Hilbert space
of functions on the variety

Kenneth R. Davidson is a Professor of Pure Mathematics at the University
of Waterloo. He did his undergraduate work at Waterloo and received
his Ph.D. from the University of California, Berkeley in 1976. From
1976 to 1978, he was a Moore instructor at M.I.T. He joined the faculty
at Waterloo in 1978. In 1985, he won the Israel Halperin Prize in
operator algebras. He was appointed a Fellow of the Royal Society
of Canada in 1992, and a Fellow of the Fields Institute in 2006. From
2001 to 2004, he served as Director of the Fields Institute. In 2007
he became University Professor at the University of Waterloo. Professor
Davidson's research interests include operator algebras and functional
analysis, and their applications to other areas of mathematics. He
has published over 120 scientific papers and 4 monographs, and has
supervised 8 Ph.D. students and 20 postdoctoral fellows.

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