Abstracts for the Moshe Flato Lecture Series 2008

Starting from the first paper by Mendel and followed by the Hardy-Weinberg principle there were many mathematical treatments of algebraic/stochastic mechanisms of heredity. I will present some of these ideas in the context of contemporary mathematics and then talk on the mathematical potential of the method of Sturtivant used by him in the first gene mapping.

I will describe recent geometric constructions in the classical subject of algebraic integrable systems (like the Euler top), related with supersymmetric gauge theories and topological string theory in physics, and such diverse subjects as hyperkähler geometry and Bridgeland stability in triangulated categories in mathematics.

In this talk I will review the basic ideas behind the heuristic replica approach to spin glasses and the recent rigorous results that have been obtained using probabilistic method. Although these theorems confirm the results of the replica approach they use a quite different language and the justification of the replica approach still remains open. In this framework I will propose a new conjecture that could justify the replica approach.

In the study of large time behavior of interacting particle systems large deviations play an important role. While studying the evolution towards equilibrium of large systems over large scales in space and time the probability measures involved are often far away from global equilibrium. Methods that involve entropy and large deviations are useful in this context. We will explore this with the help of some examples.

In 1975 when Fred Sanger was developing dideoxy termination sequencing, he found Roger Staden who developed the first computer program to assemble longer DNA sequences from the reads. The reads were randomly located and oriented along the target DNA. Until recently all DNA sequence assembly programs were further elaborations of his original technique. They often consist of three major steps: compare all pairs of reads, find an approximate arrangement of the significant overlaps, and multiple alignment for this arrangement. Staden used a greedy assembly version of this method. In 1995 an elegant and entirely new approach was proposed in which each read is broken down into shorter overlapping words, and then a certain graph is constructed so that Eulerian paths in this graph correspond to the target DNA sequence. In this talk I will show how this graph is constructed and give some examples of its operation. Today for new-generation sequencing, this Eulerian method is the method of choice.