arXiv subjects

Boris Khesin

Publications and source records attributed to Boris Khesin.

At least 19 recordsLinked to original sources

arXiv2026

Madelung hydrodynamics and Poisson geometry of wave functions

We describe the Poisson geometry of the Madelung transform between quantum mechanics and hydrodynamics for generic wave functions. We prove that for arbitrary oriented manifolds this transform, being regarded as a momentum map, naturally defines prequantum bundles for coadjoint orbits of semidirect extensions of diffeomorphism groups. Furthermore, we show that the Madelung framework provides a natural infinite-dimensional version of the convexity results for Hamiltonian torus actions, thus giving a partial answer to Atiyah's question. In particular, for wave functions without zeros our results provide a K\"ahler map between the infinite-dimensional Fubini--Study and Fisher--Rao geometries, thus extending previous results to non-simply-connected manifolds. Furthermore, for wave functions with noncritical zeros, the Madelung transform is shown to be a symplectomorphism to the coadjoint orbits with Morse--Bott densities. The latter, in turn, furnishes a novel momentum map point of view on the Wallstrom quantization condition for the hydrodynamical form of quantum mechanics. We also comment on the relation between the Madelung setting and the Marsden--Weinstein symplectic structures on knots and membranes.

math.DG↗

arXiv2026

Virasoro extensions for diffeomorphisms with breaks

We study homeomorphisms of the circle that are smooth diffeomorphisms away from a finite set of $n$ points. These "broken diffeomorphisms" do not form a Lie group, but instead naturally assemble into a Lie groupoid. We construct an explicit nontrivial $n$-dimensional central extension of this groupoid, which restricts to the classical Virasoro group when confined to smooth diffeomorphisms. We further describe the associated "broken Virasoro" algebroid, defined as a nontrivial $n$-dimensional central extension of the Lie algebroid of vector fields on the circle that are smooth except at $n$ points. This construction generalizes the Virasoro algebra. As a byproduct, we analyze a related setting on an interval: we construct a nontrivial central extension of the Lie algebra of vector fields vanishing at the endpoints, together with the corresponding central extension of the group of diffeomorphisms fixing the endpoints. We also describe the associated Lie algebroid and groupoid obtained by allowing the endpoints to vary.

math.DG↗

arXiv2026

Infinite-dimensional nonholonomic and vakonomic systems

In this paper, we present a collection of infinite-dimensional systems with nonholonomic constraints. In finite dimensions the two essentially different types of dynamics, nonholonomic or vakonomic ones, are known to be obtained by taking certain limits of holonomic systems with Rayleigh dissipation, as in [Koz83]. We visualize this phenomenon for the classical example of a skate on an inclined plane. The infinite-dimensional examples of nonholonomic and vakonomic systems revisited in the paper include subriemannian and Euler-Poincare-Suslov systems on Lie groups, the Heisenberg chain, the general Camassa-Holm equation, infinite-dimensional geometry of a nonholonomic Moser theorem, subriemannian approximations of an ideal hydrodynamics, parity-breaking nonholonomic fluids, and potential solutions to Burgers-type equations arising in optimal mass transport. Finally, we return to a higher-dimensional analogue of the skate, the kinematics of a car with $n$ trailers, as well as its limit as $n\to \infty$. We show that its infinite-dimensional version is a snake-like motion of the Chaplygin sleigh with a string, and it is subordinated to an infinite-dimensional Goursat distribution.

math.DG↗

arXiv2026

Open problems in billiards and quantitative symplectic geometry

This document collects contributions to the Open Problem List in Billiards and Quantitative Symplectic Geometry, compiled following discussions during the workshop ``Billiards and quantitative symplectic geometry'' that took place at the University of Heidelberg on July 14--18, 2025.

math.SG↗

arXiv2026

On Arnold's and Pushkin's puzzles

We discuss and draw the reader's attention to several passages in Vladimir Arnold's note on the epigraph to the novel in verse "Evgenii Onegin" by A$.$S$.$Pushkin, as well as to puzzles hidden in the novel by the poet himself.

math.HO↗

arXiv2025

Universal vector and matrix optimal transport

In this paper we propose a gauge-theoretic approach to the problems of optimal mass transport for vector and matrix densities. This resolves both the issues of positivity and action transitivity constraints. Bures-type metrics on the corresponding semi-direct product groups of diffeomorphisms and gauge transformations are related to Wasserstein-type metrics on vector half-densities and matrix densities via Riemannian submersions. We also describe their relation to Poisson geometry and demonstrate how the momentum map allows one to prove the Riemannian submersion properties. The obtained geodesic equations turn out to be vector versions of the Burgers equations.

math.DG↗

arXiv2025

Pensive billiards, point vortices, and the silver ratio

We define a new class of plane billiards - the `pensive billiard' - in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting, while preserving the billiard rule of equality of the angles of incidence and reflection. This generalizes so called `puck billiards' proposed by M.Bialy, as well as a `vortex billiard', i.e. the motion of a point vortex dipole in 2D hydrodynamics on domains with boundary. We prove the variational origin and invariance of a symplectic structure for pensive billiards, as well as study their properties including conditions for a twist map, the existence of periodic orbits, etc. We also demonstrate the appearance of both the golden and silver ratios in the corresponding hydrodynamical vortex setting. Finally, we introduce and describe basic properties of pensive outer billiards.

math.DS↗

arXiv2025

Morse-Bott Volume Forms

A Morse-Bott volume form on a manifold is a top-degree form which vanishes along a non-degenerate critical submanifold. We prove that two such forms are diffeomorphic (by a diffeomorphism fixed on the submanifold) provided that their relative cohomology classes with respect to the submanifold coincide. For a zero submanifold of codimension at least 2, this means that two Morse-Bott volume forms with the same zero set are diffeomorphic if and only if they have equal total volumes. We show how "Moser's trick" for establishing equivalence of non-degenerate volume forms can be adapted to this setting.

math.DG↗

arXiv2025

Relative Helicity and Tiling Twist

We consider domino tilings of 3D cubiculated regions. The tilings have two invariants, flux and twist, often integer-valued, which are given in purely combinatorial terms. These invariants allow one to classify the tilings with respect to certain elementary moves, flips and trits. In this paper we present a construction associating a divergence-free vector field $ξ_t$ to any domino tiling $t$, such that the flux of the tiling $t$ can be interpreted as the (relative) rotation class of the field $ξ_t$, while the twist of $t$ is proved to be the relative helicity of the field $ξ_t$.

math.CO↗

arXiv2025

Curvature of Measure-Preserving Diffeomorphism Groups of Non-Orientable Surfaces

We study curvatures of the groups of measure-preserving diffeomorphisms of non-orientable compact surfaces. For the cases of the Klein bottle and the real projective plane we compute curvatures, their asymptotics and the normalized Ricci curvatures in many directions. Extending the approach of V. Arnold, and A. Lukatskii we provide estimates of weather unpredictability for natural models of trade wind currents on the Klein bottle and the projective plane.

math.DG↗

arXiv2024

Information geometry of diffeomorphism groups

The study of diffeomorphism groups and their applications to problems in analysis and geometry has a long history. In geometric hydrodynamics, pioneered by V.~Arnold in the 1960s, one considers an ideal fluid flow as the geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms of the fluid domain with respect to the metric defined by the kinetic energy. Similar considerations on the space of densities lead to a geometric description of optimal mass transport and the Kantorovich-Wasserstein metric. Likewise, information geometry associated with the Fisher-Rao metric and the Hellinger distance has an equally beautiful infinite-dimensional geometric description and can be regarded as a higher-order Sobolev analogue of optimal transportation. In this work we review various metrics on diffeomorphism groups relevant to this approach and introduce appropriate topology, smooth structures and dynamics on the corresponding infinite-dimensional manifolds. Our main goal is to demonstrate how, alongside topological hydrodynamics, Hamiltonian dynamics and optimal mass transport, information geometry with its elaborate toolbox has become yet another exciting field for applications of geometric analysis on diffeomorphism groups.

math.DG↗

arXiv2024

The Euler non-mixing made easy

The non-transitivity without extra constraints in the Euler equation in any dimension is almost evident and can be derived, e.g., from Morse theory.

math.DS↗

arXiv2024

Coadjoint orbits of area-preserving diffeomorphisms of non-orientable surfaces

We give a classification of generic coadjoint orbits for the group of area-preserving diffeomorphisms of a closed non-orientable surface. This completes V. Arnold's program of studying invariants of incompressible fluids in 2D. As an auxiliary problem, we also classify simple Morse pseudo-functions on non-orientable surfaces up to area-preserving diffeomorphisms.

math.SG↗

arXiv2024

Simple unbalanced optimal transport

We introduce and study a simple model capturing the main features of unbalanced optimal transport. It is based on equipping the conical extension of the group of all diffeomorphisms with a natural metric, which allows a Riemannian submersion to the space of volume forms of arbitrary total mass. We describe its finite-dimensional version and present a concise comparison study of the geometry, Hamiltonian features, and geodesics for this and other extensions. One of the corollaries of this approach is that along any geodesic the total mass evolves with constant acceleration, as an object's height in a constant buoyancy field.

math.DG↗

arXiv2024

Singular vortex pairs follow magnetic geodesics

We consider pairs of point vortices having circulations $Γ_1$ and $Γ_2$ and confined to a two-dimensional surface $S$. In the limit of zero initial separation $\varepsilon$, we prove that they follow a magnetic geodesic in unison, if properly renormalized. Specifically, the ``singular vortex pair" moves as a single charged particle on the surface with a charge of order $1/\varepsilon^2$ in a magnetic field $B$ which is everywhere normal to the surface and of strength $|B|=Γ_1 +Γ_2$. In the case $Γ_1=-Γ_2$, this gives another proof of Kimura's conjecture (Kimura 1999) that singular dipoles follow geodesics.

physics.flu-dyn↗

arXiv2023

Geometry of generalized fluid flows

The Euler equation of an ideal (i.e. inviscid incompressible) fluid can be regarded, following V.Arnold, as the geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. In this paper we describe the common origin and symmetry of generalized flows, multiphase fluids (homogenized vortex sheets), and conventional vortex sheets: they all correspond to geodesics on certain groupoids of multiphase diffeomorphisms. Furthermore, we prove that all these problems are Hamiltonian with respect to a Poisson structure on a dual Lie algebroid, generalizing the Hamiltonian property of the Euler equation on a Lie algebra dual.

math.DG↗

arXiv2023

The Toda flow as a porous medium equation

We describe the geometry of the incompressible porous medium (IPM) equation: we prove that it is a gradient dynamical system on the group of area-preserving diffeomorphisms and has a special double-bracket form. Furthermore, we show its similarities and differences with the dispersionless Toda system. The Toda flow describes an integrable interaction of several particles on a line with an exponential potential between neighbours, while its continuous version is an integrable PDE, whose physical meaning was obscure. Here we show that this continuous Toda flow can be naturally regarded as a special IPM equation, while the key double-bracket property of Toda is shared by all equations of the IPM type, thus manifesting their gradient and non-autonomous Hamiltonian origin. Finally, we comment on Toda and IPM modifications of the QR diagonalization algorithm, as well as describe double-bracket flows in an invariant setting of general Lie groups with arbitrary inertia operators.

math.DS↗

arXiv2023

Geometric Hydrodynamics in Open Problems

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and achievements in this area. The topics discussed include variational settings for different types of fluids, models for invariant metrics, the Cauchy and boundary value problems, partial analyticity of solutions to the Euler equations, their steady and singular vorticity solutions, differential and Hamiltonian geometry of diffeomorphism groups, long-time behaviour of fluids, as well as mechanical models of direct and inverse cascades.

math.DG↗