Height functions in Diophantine Geometry
Time and place
1 PM on Tuesday, March 2nd, 2010; NAC 6113
Prof. Sonal Jain (New York University)
Abstract
The search for integer or rational number solutions to polynomial equations has intrigued mathematicians for at least two and half thousand years. Such equations are called Diophantine equations, named after the Greek mathematician Diophantus who wrote about such equations. The simplest examples of Diophantine equations are linear equations such as aX + bY = c where a, b and c are integers. Perhaps the most famous is the Fermat equation X^n + Y^n= Z^n; here n is a positive integer. There is an algorithm for finding integer solutions to the former. As for the latter, a proof that the n larger than 2 case has no non-trivial solutions requires some of the most sophisticated mathematics yet developed.
So where does geometry come in, and what is Diophantine Geometry? The idea is to view the solutions to a Diophantine equation as defining a surface or related geometric object. In this talk, I will give an introduction to height functions in Diophantine geometry. A height function tells us about the arithmetic complexity of a rational solution to the given equation, and it also tells us about the complexity of the geometry of the corresponding surface near the solution point. Height functions lie at the heart of several important theorems in Diophantine geometry. In this talk will discuss some classical applications of height functions as well as some recent results and open problems.
