The water wave equations describe the motion of an incompressible,
inviscid fluid under the influence of gravity and bounded above by a
free surface. This problem has a long history, with important
contributions from Lagrange, Cauchy, Poisson, and Levi-Civita.

In this talk, we will provide the first construction of exact rotational
solitary water waves of large amplitude. Starting from a uniform shear
flow with a flat free surface, we use a degree-theoretic continuation
argument to construct a global connected set of symmetric solitary waves
of elevation, whose profiles decrease monotonically on either side of a
central crest.