# Scientific

## Alan Turing and the Decision Problem

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

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

## Time and chance happeneth to them all: Mutation, selection and recombination

Many multi-cellular organisms exhibit remarkably similar patterns of aging and mortality. Because this phenomenon appears to arise from the complex interaction of many genes, it has been a challenge to explain it quantitatively as a response to natural selection. I survey attempts by me and my collaborators to build a framework for understanding how mutation, selection and recombination acting on many genes combine to shape the distribution of genotypes in a large population. A genotype drawn at random from the population at a given time is described in our model by a Poisson random measure on the space of loci, and hence its distribution is characterized by the associated intensity measure. The intensity measures evolve according to a continuous-time, measure-valued dynamical system. I present general results on the existence and uniqueness of this dynamical system, how it arises as a limit of discrete generation systems, and the nature of its equilibria.

## Gauge Theory and Khovanov Homology

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.

## Ranks of elliptic curves

We show how to use conjectures for moments of L-functions to get insight into the frequency of rank 2 elliptic curves within a family of quadratic twists.

- Read more about Ranks of elliptic curves
- 2485 reads

## Moments of zeta and L-functions on the critical Line II (3 of 3)

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

## Moments of zeta and L-functions on the critical Line I (2 of 3)

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

## Distribution of Values of zeta and L-functions (1 of 3)

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

## PIMS/UBC Distinguished Colloquium: Dusa McDuff

## Inaugural Hugh C Morris Lecture - George Papanicolaou

PIMS was pleased to present the inaugural lecture in the new Hugh C Morris series with speaker George Papanicolaou from Stanford University.

## Special values of Artin L-series (3 of 3)

Dirichlet’s class number formula has a nice conjectural generalization in the form of Stark’s conjectures. These conjectures pertain to the value of Artin L-series at s = 1. However, the special values at other integer points also are interesting and in this context, there is a famous conjecture of Zagier. We will give a brief outline of this and display some recent results.

This lecture is part of a series of 3.