Of all the algebraic curves around, the rational ones have the most rational points. (And yes, the terminology is mostly accidental!) So maybe if an algebraic variety of higher dimension has lots of rational points, then it ought to have lots of rational curves too? The answer is "probably yes", but is complicated enough that I can spin it out to a whole seminar's worth of talking. Vojta's Conjectures, the Batyrev-Manin Conjectures, and Diophantine approximation are but three of the threads that I will weave into the narrative.
Algebraic topology was invented by Poincare in 1895 to study the behavior of algebraic functions. In his seminal ICM address 5 years later, Hilbert posed a fundamental challenge to the field: find a topological obstruction to reducing the solution of the general degree 7 polynomial to an expression in functions of two or fewer variables. In this talk, I'll review some of the beautiful history of algebraic topology and algebraic functions, discuss Hilbert's problem, (including why it is still open, and why this matters), and outline ongoing work in applying the topology of braids and algebraic functions to this problem. This is joint work with Benson Farb.
Abstract: Basic questions in analytic number theory concern the density of one set in another (e.g. square-free integers in all integers). Motivated by Weil's number field/function field dictionary, we introduce a topological analogue measuring the “homological density” of one space in another. In arithmetic, Euler products can be used to show that many seemingly different densities coincide in the limit. By combining methods from manifold topology and algebraic combinatorics, we discover analogous coincidences for limiting homological densities arising from spaces of 0-cycles (e.g. configuration spaces of points) on smooth manifolds and complex varieties. We do not yet understand why these topological coincidences occur. This is joint work with Benson Farb and Melanie Wood.