GEOMETRIC FLUID DYNAMICS

HENAN UNIVERSITY, Sept -- Oct 2021

Boris Khesin (University of Toronto)


Date/Time:
Beijing Time: Tue and Thu 8:30 -- 10am (Sept 14, 16, 21, 23, 28, 30, Oct 5 and 7)
Toronto Time (EST): Mon and Wed 8:30 -- 10pm (Sept 13, 15, 20, 22, 27, 29, Oct 4 and 6)

Zoom Meeting ID: 567 306 5241

Code: 123456

Poster of the course


Course description:

The course outlines group-theoretic, geometric, and topological approaches to hydrodynamics. We start by describing the Eulerian dynamics of an ideal fluid and the Korteweg--de Vries equation of shallow water from the group-theoretic and Hamiltonian points of view. We move on to cover the geometry of Casimirs for the Euler equation and helicity of vector fields. The Hamiltonian framework will also allow us to recover the motion of point vortices, vortex filaments and membranes. Finally, we will relate the differential geometry of diffeomorphism groups to problems of optimal mass transport.

References:

1) V.Arnold and B.Khesin: Topological Methods in Hydrodynamics, Appl. Math. Series, v. 125, Springer-Verlag 1998/1999, Second edition Springer-Nature 2021.

2) J.Marsden and T.Ratiu: Introduction to Mechanics and Symmetry, Texts in Applied Math., v. 17, Springer-Verlag, 1994/1999.

3) B.Khesin and R.Wendt: The Geometry of Infinite-Dimensional Groups, Ergebnisse der Math. und Grenzgebiete 3.Folge, v. 51, Springer-Verlag, 2009.

Tentative Plan of Lectures:
(slides and recordings will be linked to this page after each lecture)

I. Introducing the Euler equations. Its description as the geodesic flow.
Slides; Recording (or Recording from the Cloud, Access Passcode: mkRRP95# )

II. Equations on the dual Lie algebra, Lie-Poisson structures, Euler--Arnold equations.
Slides; Recording (or Recording from the Cloud, Access Passcode: 0l?ex4W& )

III. The Virasoro algebra and the KdV as an Euler equation.
Slides; Recording (or Recording from the Cloud, Access Passcode: 4FJ7H?#w )

IV. The Hamiltonian framework for hydrodynamics. Conservation laws for the Euler equations.
Slides; Recording (or Recording from the Cloud, Access Passcode: Ze0R7M%6 )

V. Geometry of helicity.
Slides; Recording (or Recording from the Cloud, Access Passcode: FP@5!6yp )

VI. Point vortices and enstrophies in 2D.
Slides; Recording (or Recording from the Cloud, Access Passcode: @!KeZr?9 )

VII. The Marsden--Weinstein symplectic structure on knots and vortex membranes.
Slides; Recording from the Cloud ( Access Passcode: P=F5@vxv )

VIII. Geometry of diffeomorphism groups and optimal mass transport.
Slides; Recording (or Recording from the Cloud, Access Passcode: r9FDLm&L )


A talk for a general math audience:

"The mystery of pentagram maps"


Slides; Recording (or Recording from the Cloud, Access Passcode: 04@Jq2NY )