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I will discuss some recent work on localized "solitary" waves propagating along the surface of an incompressible fluid. First, I will outline a surprisingly simple proof, joint with Vladimir Kozlov and Evgeniy Lokharu, that such waves travel faster than a certain critical speed. This was a longstanding open problem, and implies that the waves are symmetric and monotone about a central crest. Next, I will outline an existence theory, joint with Susanna Haziot, for large waves with constant vorticity. One of the main novelties is that we treat the problem as a elliptic system for two scalar functions, one describing a conformal mapping of the fluid domain and another describing the motion inside the fluid.

Please note the time change to accommodate our speaker's time zone.