Variables Separated Equations and Finite Simple Groups:
Mike Fried, UC Irvine

My talk will start with a variables separated (algebraic) equation:
(*) f(x)-g(y)=0, f and g polynomials.

I will discuss the roots of a paper "Relating two genus 0 problems of John Thompson," from Thompson's 70th birthday volume. Davenport's problem asked what polynomial pairs ( f, g ) over Q have their ranges on Z/p (the integers mod p, p a prime) the same for almost all p.

Davenport's precise conjecture was only 'almost true.' From this came the Genus 0 Problem: Showing that rational functions have severely limited monodromy groups. This was the work of many group theorists, foremost of Bob Guralnick, starting with his collaboration with Thompson.

By walking through Davenport's problem with hindsight, we get lessons on two general tools about algebraic equations. We celebrate that by attending to two questions:
  1. What allows us to produce equations from branch cycles?
  2. What (finite) groups arise in 'nature'?
    3.
Each phrase addresses an aspect of problems based on equations. That is, we seem to need algebraic equations. Yet why, and how much do we lose in using more easily manipulated surregates for them? The full story is in UMStory.html.