Scientific

Many scientific questions are considered solved to the best possible degree when we have a method for computing a solution. This is especially true in mathematics and those areas of science in which phenomena can be described mathematically: one only has to think of the methods of symbolic algebra in order to solve equations, or laws of physics which allow one to calculate unknown quantities from known measurements. The crowning achievement of mathematics would thus be a systematic way to compute the solution to any mathematical problem. The hope that this was possible was perhaps first articulated by the 18th century mathematician-philosopher G. W. Leibniz. Advances in the foundations of mathematics in the early 20th century made it possible in the 1920s to first formulate the question of whether there is such a systematic way to find a solution to every mathematical problem. This became known as the decision problem, and it was considered a major open problem in the 1920s and 1930s. Alan Turing solved it in his first, groundbreaking paper "On computable numbers" (1936). In order to show that there cannot be a systematic computational procedure that solves every mathematical question, Turing had to provide a convincing analysis of what a computational procedure is. His abstract, mathematical model of computability is that of a Turing Machine. He showed that no Turing machine, and hence no computational procedure at all, could solve the Entscheidungsproblem.

Turing 2012 - Calgary

This talk is part of a series celebrating the Alan Turing Centenary in Calgary. The following mathtube videos are also part of this series

1. Alan Turing and the Decision Problem, Richard Zach.
2. Turing's Real Machine, Michael R. Williams.
3. Alan Turing and Enigma, John R. Ferris.

After reviewing ordinary finite-dimensional Morse theory, I will explain how Morse generalized Morse theory to loop spaces, and how Floer generalized it to gauge theory on a three-manifold. Then I will describe an analog of Floer cohomology with the gauge group taken to be a complex Lie group (rather than a compact group as assumed by Floer), and how this is expected to be related to the Jones polynomial of knots and Khovanov homology.

I will discuss techniques to get upper and lower bounds for moments of zeta and L-functions. The lower bounds are unconditional and the upper bounds in general rely on the Riemann Hypothesis. In several cases of low moments, one can obtain asymptotics, and I may discuss a couple of such recent cases.

This lecture is part of a series of 3

1. Lecture 1: distribution-values-zeta-and-l-functions-1-3
2. Lecture 2: Moments of zeta and L-functions on the Critical Line, I
3. Lecture 3: Moments of zeta and L-functions on the critical line, II

I will discuss the distribution of values of zeta and L-functions when restricted to the right of the critical line. Here the values are well understood by probabilistic models involving “random Euler products”. This fails on the critical line, and the L-values here have a different flavor here with Selberg’s theorem on log normality being a representative result.

This lecture is part of a series of 3

1. Lecture 1: distribution-values-zeta-and-l-functions-1-3
2. Lecture 2: Moments of zeta and L-functions on the Critical Line, I
3. Lecture 3: Moments of zeta and L-functions on the critical line, II