A fundamental problem in algebraic topology is to compute and
understand the (stable) homotopy groups of a spheres. This remains
mysterious and largely unsolved, despite major advances in the field.
We will give an exposition of some classical theorems, computational
methods and an illuminating example relating to the stable homotopy
groups of spheres. We will also discuss some directions the field is
taking, including how surprising and invaluable applications of other
areas of mathematics such as number theory and algebraic geometry have
aided in its progress. This talk will be accessible to first year
graduate students with a course in topology; basic exposure to
(co)homology would be advantageous, but is not necessary.