Special public lecture:

## How to solve equations in rational numbers

### Professor Frank Calegari

Frank Calegari grew up in Melbourne, and was one of Australia's representatives in the International Mathematical Olympiad in 1992 and 1993. After graduating from the University of Melbourne he took his PhD at the University of California, Berkeley. He was recently at Harvard, as holder of a prestigious American Institute of Mathematics 5-year Fellowship. He is now at Northwestern University, but is visiting the University of Sydney from April 30th to June 4th.

When: 24 May 2011 at 5:30–6:30pm.

Where: Old Geology Lecture Theatre, Edgeworth David Building, University of Sydney

#### Abstract:

Many problems in number theory, so called diophantine equations, are concerned with finding integer solutions to polynomial equations. Are there positive integers $$x$$, $$y$$, and $$z$$ such that $$x^n+y^n = z^n$$ for $$n \ge 3$$ (Fermat's Last Theorem)? Which primes can be written as the sum of two squares? Which primes can be written as the sum of two rational cubes? We present various approaches to deciding whether a diophantine equation has a solution, and discuss whether it is reasonable to expect that these methods always work.