Title:        Singularities in 2D flows: The Tale of Two Branch Points

Abstract:
   We consider the classical problem of 2D fluid flow with a free boundary. Recent
works strongly suggest that square--root type branch points appear naturally in
2D hydrodynamics. We illustrate how the fluid domain can be complemented by a
``virtual'' fluid, and the equations of motion are transplanted to a branch cut (a vortex sheet)
in the conformal domain. A numerical and theoretical study of the motion of
complex singularities in multiple Riemann sheets is suggested.
Unlike preceding works for dynamics of singularities: the short branch cut approximation,
and the study of viability of meromorphic solutions in fluid dynamics, the present approach
neither simplifies the equations of fluid flow, nor uses local Laurent expansions. Instead
the new approach is based on analytic functions and allows construction of global solutions
in 2D hydrodynamics.
A natural extension of the approach considers fluid flows described by many pairs
of square--root branch points.

Short bio:

Sergey Dyachenko obtained his PhD in applied mathematics at the University
of New Mexico in 2014 under the guidance of Professor Pavel Lushnikov and
Alexander Korotkevich. Before Dr. Dyachenko joined the faculty of SUNY at
Buffalo as tenure-track Assistant Professor in 2020, he was J.L. Doob Research
Assistant Professor at the Department of Mathematics at the University of
Illinois in Urbana-Champaign (UIUC) in 2015-2019. Concurrently with the
appointment at UIUC Dr. Dyachenko has been a Postdoctoral Fellow at the
Institute for Computational and Experimental Research in Mathematics (ICERM)
at Brown University in 2016-2017 where he worked together with Profs. Walter
Strauss and Vera Mikyoung Hur. Sergey discovered new limiting waves in 2D
free surface flows over a shear current, and his work contributed to the
discovery of exact solutions to free surface wave problem with constant
vorticity. When Dr. Dyachenko was Acting Assistant Professor at the
Applied Mathematics department at the University of Washington in 2019-2020 he
found a family of invariants in 2D water waves problem that are attached to
complex singularities describing potential fluid flow. His interests include
topics in water waves, scientific computing and singularity formation in
nonlinear systems.