Quantum cohomology and characteristic numbers

  The text-book example of quantum cohomology applied to enumerative geometry
  is the celebrated formula of Kontsevich (1994) which solves the classical
  counting problem: ``How many rational plane curves of degree $d$ pass
  through $3d-1$ general points?''  The formula is a recursion whose only
  initial input is the fact that there is a unique straight line through two
  points, and the recursion itself is the expression of the associativity of
  the quantum product (the WDVV equation), a new ring structure on the
  cohomology space.  I'll spend a considerable time explaining this
  now-classical material, because the main result I want to present is a
  direct generalisation.  Namely, the WDVV equation can be generalised (a
  certain deformation constructed by coupling with 2D gravity) to account
  also for tangency conditions, solving the century-old characteristic number
  problem for rational plane curves, i.e. the numbers of degree-$d$ rational
  curves through $a$ points and tangent to $b$ lines, $a+b=3d-1$.  In the
  last two minutes of the talk, I will discuss other enumerative problems,
  and generalisation to higher genus and arbitrary target spaces...