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Talks at the University of Western Ontario

February 13, 2004

1. Probability: Fact, Fiction and Quantum.

Abstract. In the Theory of Evolution one separates "the fact of evolution" (that species have evolved) from "the theory of evolution" (natural selection, mutations). Softcore critics accept the fact but attack the theory, often replacing it by things divine (hardcore critics attack even the fact).

In my talk I will formalize in precise terms what I believe is the undisputed "fact" of probability - that stochastic things happen. I will then discuss three theories "explaining" that fact: a tautological theory which explains nothing at all, the classical "Kolmogorov" theory (aka "fiction") and the Quantum Probability theory which seems to be the one really running our universe. I will give a beautiful example that underlines the difference between the classical and the quantum theories and discuss the (proper) inclusion of the former by the latter.

This is a service talk. Everything I will talk about is well known and nothing is original, and I will make every effort to make the talk accessible to anyone not afraid of diagonalizing a matrix.

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2. Khovanov Homology for Knots and Links.

Abstract. Over the last 20 years, knot theorists have been extremely good at borrowing ideas from other fields. We've borrowed from Mathematical Physics and borrowed from Algebra and we have a Beautiful Theory of Knot Invariants that can claim deep heritage on either side. But we haven't been so good at returning. While not entirely impossible, is remains difficult to point at developments in quantum field theory or quantum algebra (our lenders) that owe something to our Beautiful Theory of Knot Invariants.

Came Khovanov in 1999 and changed the picture dramatically by offering Mathematical Physics and Algebra the most valuable prize known to mathematicians - a challenge. For none of them can yet explain whence comes his "Categorification of the Jones Polynomial" - a far reaching generalization of the most celebrated member of our Beautiful Theory of Knot Invariants. The Mathematical Physics and Algebra underlying the Jones polynomial are deep and substantial, and there are all reasons to believe that a successful resolution of Khovanov's challenge will be the same.

In my talk I will quickly describe the Jones polynomial (it's so easy) and then move on to describe Khovanov's homological generalization thereof.

(Khovanov's homology is also a stronger invariant than the Jones polynomial and it is "functorial" in some 4-dimensional sense).

Handout side 1: QRG.pdf.
Handout side 2: NewHandout-1.pdf.