The rank of an elliptic curve is a measure of the number of solutions of the equation that defines the curve. In recent years there has been spectacular progress in the theory of elliptic curves, but the rank remains very mysterious. Even basic questions such as how to compute the rank, or whether the rank can be arbitrarily large, are not settled.
In this lecture we will introduce elliptic curves and discuss what is known, as well as what is conjectured but not known, about ranks.
In 1967 Furstenberg discovered a very surprising phenomenon:
while both $T: x \to 2 x \bmod 1$ and $s: x \to 3 x \bmod 1$ on $\R / \Z$ have many closed invariant sets, closed sets which are invariant under both $T$ and $S$ are very rare (indeed, are either finite sets of rationals or $\R / \Z$). Furstenberg also conjectured that a similar result holds for invariant measures. This conjecture is of course still open.
As has been shown by several authors, including Katok-Spatzier and Margulis this phenomenon is not an isolated curiosity but rather a deep property of many natural $\Z ^ d$ and $\R ^ d$ actions ($d > 1$) with many applications.
Recently, there has been substantial progress in the study of measures invariant under such actions. While we are at present still far from full resolution of this intriguing question, the partial results we currently
have are already powerful enough to prove results in other fields. In particular, these techniques enable proving a special but important case of Rudnick and Sarnak's Quantum Unique Ergodicity Conjecture, as well as a partial result towards Littlewood's Conjecture on simultaneous diophantine approximations (the later is in a joint paper with M. Einsiedler and A. Katok).
Let X be a smooth complex variety and Y be a closed subvariety of X. We discuss different methods describing the complexity of the singularities of the pair (X,Y) from its resoultion of singularities, analysis and the geometry of the spaces of jets and arcs. We'll also describe appliactions to singularities of theata divisors and commutative algebra.
