The rank of an elliptic curve comes in three flavors: arithmetic (which measures the set of rational points), analytic, and cohomological. Conjecturally all three coincide, but in many cases one knows more about one than another. In this talk we will introduce elliptic curves and their ranks, and discuss various conjectures and theorems relating these ranks.

Informally speaking, fractals are sets of non-integer dimension. The
standard Cantor set is a simplest example of a fractal set. An important
characteristic of a fractal set is its fractal (or Hausdorff) dimension. We
will give two examples of recent results (oscillatory motions in the three
body problem and spectrum of a discrete Schrodinger operator with Fibonacci
potential) where fractals appear in a natural way, and their Hausdorff
dimension can be estimated.