I will discuss a new "addition" operation which given bounded open sets
 in
  R^d produces a domain whose volume is the sum of their volumes.  This
  operation arises as the scaling limit of a lattice model in which
 particles
  starting in the intersection of the two domains perform simple random
 walks
  in Z^d until reaching unoccupied sites outside the union.  It also has
  equivalent descriptions in terms of a Hele-Shaw fluid flow problem and
 a
  simple problem in electrostatics.
 
  In the first part of the talk I'll motivate how we were led to define
 the
  smash sum, and describe what we know about its properties so far.  In
 the
  second part, I'll open up the discussion to some topics we are
 currently
  exploring, including: inverse problems, an axiomatic characterization
 of the
  smash sum, and conformal invariance.  Joint work with Yuval Peres.