# Manjul Bhargava

Princeton University
Visit:
Tuesday, May 19, 2015 to Thursday, May 21, 2015
Lectures:

Values of integer polynomials

Polynomials are one of the most basic objects in mathematics - in particular, in number theory - yet the values they take remain largely mysterious!  In number theory, one is generally interested in polynomials having integer or rational number coefficients, and the values they take at integral or rational arguments.  In these three lectures, we examine three classical problems in number theory that involve understanding the values taken by integer polynomials, and give an overview of the long history of these problems and what is now known in each case.

May 19th  3pm

Lecture 1: The representation of integers by quadratic forms

The famous "Four Squares Theorem" of Lagrange asserts that any positive integer can be expressed as the sum of four square numbers. That is, the quadratic form $a^2 + b^2 + c^2 + d^2$ represents all (positive) integers. When does a general quadratic form represent all integers?  When does it represent all odd integers?  When does it represent all primes?  We show how all these questions turn out to have very simple and surprising answers.  In particular, we describe joint work with Jonathan Hanke that led to a proof of Conway's "290-Conjecture".

May 20th  3pm

Lecture 2: How likely is it for an integer polynomial to take a square value?
It is well known that the density of integers that are squarefree is $6/\pi^2$, giving one of the more intriguing occurrences of $\pi$ where one might not a priori expect it!  A natural next problem that has played an important role in number theory is that of understanding the density of squarefree values taken by an integer polynomial.  We survey a number of recent results on this problem for various types of polynomials - some of which use the ABC Conjecture and some of which do not.