The soliton began its mathematical life being chased by a horseman along the
Union Canal at Herminston, close to Heriot-Watt University, Edinburgh,
and is now pervasive in the sciences and applied mathematics. Still,
it is somewhat rare that a non-linear PDE admit soliton solutions; and still
rarer that they have the algebraic and geometric properties progressively
brought to light after startling discoveries in the 1990s.
This talk will review three prototypes of such equations, from areas of
Hamiltonian Mechanics, Statistical Mechanics, and Random-Matrix Theory, with
emphasis on open problems. Soliton solutions typically organize into
manifolds that have classical theoretical interest, ``moduli spaces''.
Most recently, the Cosmological Constant has been written in terms of
the bosonic string measure over such a moduli space, using the same
type of classical special functions that give the algebro-geometric
solitons.