University of Northern Iowa Department of Mathematics

Title: Does not compute: Hilbert's 10th problem and Diophantine Equations

Speaker: Todd Eisworth

ABSTRACT

In his famous 1900 address at the International Congress of Mathematicians, David Hilbert asked (as problem 10 in his famous list) if there could exist an algorithm that could, when given a polynomial with integer coefficients, determine whether or not there exist integer solutions for the equation obtained by setting the polynomial equal to zero. importance of this question rests on the fact that many of the most notorious questions inmathematics (like the twin prime conjecture or Fermat's Last Theorem) can be translated into this context -- such an algorithm could in theory tell us the answer to all of them! Hilbert's question was finally answered in the negative in 1970 by a twenty year old Russian student Yuri Matiyasevich. The solution is a wonderful blend of number theory, mathematical logic, and the theory of computation. We will take a look at how the solution came about and give some interesting consequences.

The only pre-requisite required for this lecture is an interest in mathematics or computer science. The topics touched on in this lecture are very much related to current mathematical research going on here at UNI.