## Books and papers by Boris Khesin:

**Books**

3.* Geometry of infinite-dimensional groups.* with Robert Wendt,
* Ergebnisse der Mathematik und Grenzgebiete 3.Folge,* ** 51**, Springer-Verlag, 2009, xviii+304pp.; Russian edition: MCCME 2014, 368pp. ( Introduction and table of contents , extended **Lecture notes** )

2. * Lectures on topological fluid mechanics.*
(with M.Berger, L.Kauffman, K.Moffatt, R.Ricca, and De W.Sumners)
* Lecture Notes in Math.,* **1973**, Springer-Verlag, 2009, 221pp.

1.* Topological methods in hydrodynamics.* with Vladimir I. Arnold, * Applied Math. Series * ** 125**, Springer-Verlag, 1998, xv+374pp.; Second printing 1999, xv+376pp.; Russian edition: MCCME 2007, 392pp. ( Introduction and table of contents,
Chapters 1,
2,
3,
4,
5,
6,
Bibliography,
Index)

**Books translated**

1.* Fundamentals of geophysical hydrodynamics.* by Felix V. Dolzhansky, * Encyclopaedia of Math. Sci.: Mathematical Physics * ** 103**, Springer-Verlag, 2013, xiv+272pp.

**Papers**

70. *Vortex sheets and diffeomorphism groupoids.* with Anton Izosimov,
preprint, 46pp. (2017), arXiv:1705.01603

69. *Classification of Casimirs in 2D hydrodynamics.* with Anton Izosimov, to appear in *Moscow Math Journal*, 17pp. (2017), arXiv:1702.01843

68. *Characterization of steady solutions to the 2D Euler equation.* with Anton Izosimov,
*Int. Math. Research Notices (IMRN)*, 34pp. (2017), DOI: https://doi.org/10.1093/imrn/rnw230; arXiv:1511.05623

67. *Coadjoint orbits of symplectic diffeomorphisms of surfaces
and ideal hydrodynamics.* with Anton Izosimov and Mehdi Mousavi,
*Annales de l'Institut Fourier*, vol.66, no.6 (2016), 2385-2433; arXiv:1504.05629

66. *The geometry of dented pentagram maps.* with Fedor Soloviev,
*J. Europ. Math. Soc.*, vol. 18 (2016), 147-179; arXiv:1308.5363;

65. *Non-integrability vs. integrability in pentagram maps.* with Fedor Soloviev, *J. of Geometry and Physics*, vol. 87 (2015), 275-285; (arXiv:1404.6221)

64. *Vortex filaments and the Hasimoto transform.* in *Proceedings of Shape Analysis Workshop in Bad Gastein* (2014), preprint hal-01076953, 19-22;

63. *Comments on "A Mathematical Trivium" by V.Arnold.* with Serge Tabachnikov, in *
ARNOLD: Swimming Against the Tide.*, Amer. Math. Soc. (2014), 57-66;

62. *KAM theory and the 3D Euler equation.* with Sergei Kuksin and Daniel Peralta-Salas, *Advances in Math.*, vol. 267 (2014), 498-522; (arXiv:1401.5516)

61. *The vortex filament equation in any dimension.* *Procedia IUTAM*, vol. 7 (2013), 135-140;

60. *The Euler and Navier-Stokes equations on the hyperbolic plane.* with Gerard Misiolek, *Proc. Nat. Acad. Sci.*, vol. 109, no.45 (2012), 18324-18326;
(arXiv:1205.5322)

59. *The pentagram map in higher dimensions and KdV flows.* with Fedor Soloviev, *Electron. Res. Announc. Math. Sci. (ERA-MS)*, vol.19 (2012), 86-96; (arXiv:1205.3744)

58. *Integrability of higher pentagram maps.* with Fedor Soloviev, *Mathem. Annalen*, vol. 357 (2013), 1005-1047; (arXiv:1204.0756)

57. *On V.I.Arnold and hydrodynamics.
Notices of the AMS,* vol. 59 (2012), 492-495.

56. *Symplectic structures and dynamics on vortex membranes.* *Moscow Math. Journal*, vol.12, no.2 (2012), 413-434; (arXiv:1201.5914)

55. *Curvatures of Sobolev metrics on diffeomorphism groups.* with Jonatan Lenells, Gerard Misiolek, and Stephen C. Preston, *Pure and Appl. Math. Quaterly*, vol.9, no.2 (2013), 291-332; (arXiv:1109.1816)

54. * Geometry of diffeomorphism groups, complete integrability and geometric statistics.* with Jonatan Lenells, Gerard Misiolek, and Stephen C. Preston, *Geom. and Funct. Anal.*, vol.23, no.1 (2013), 334-366; (arXiv:1105.0643)

53. *Dynamics of symplectic fluids and point vortices.* *Geom. and Funct. Anal.*, vol.22, no.5 (2012), 1444-1459; (arXiv:1106.1609)

52. *Discrete spherical means of directional derivatives and Veronese maps.* with Alexander Belyaev and Serge Tabachnikov, *J. of Geometry and Physics*, vol.62 (2012), 124-136; (arXiv:1106.3691)

51. * Contact complete integrability.* with Serge Tabachnikov, *Regular and Chaotic Dynamics*, vol.15, no.4-5 (2010), 504-520; (arXiv:0910.0375)

50. * A nonholonomic Moser theorem and optimal mass transport.* with Paul Lee, *J. of Sympl. Geometry*, vol.7, no.4 (2009), 381-414; (arXiv:0802.1551)

49. * Pseudo-Riemannian geodesics and billiards.* with Sergei Tabachnikov, * Advances in Math.*, vol. 221 (2009), 1364-1396; (math.DG/0608620)

48. * Generalized Hunter-Saxton equation and the geometry of the group of circle
diffeomorphisms.* with Jonatan Lenells and Gerard Misiolek, *Mathem. Annalen*, vol.342 (2008), 617-656; (arXiv:0803.3078)

47. * Groups and topology in the Euler hydrodynamics and KdV.* in *Hamiltonian dynamical systems and applications,* Ed. W.Craig, * NATO Science series B,* ** XVI** Springer-Verlag (2008), 93-102; * The group and Hamiltonian descriptions of hydrodynamical systems. CIME lectures,* 16pp.

46. * Poisson geometry and first integrals of geostrophic equations.* with Paul Lee, *Physica D*, vol.237 (2008), 2072-2077; (arXiv:0802.4439)

45. * Geodesics on an ellipsoid in Minkowski space.* with Daniel Genin and Sergei Tabachnikov, *L'Enseignement Mathématique*, vol. 53 (2007), 307-331; (arXiv:0705.0188)

44. * Shock waves for the Burgers equation and curvatures of diffeomorphism groups.* with Gerard Misiolek, *math.DG/0702196* (2007), *Proc. Steklov Math. Inst.*, vol. 259, 73-81 (pdf)

43. * Pseudodifferential symbols on Riemann surfaces and Krichever-Novikov algebras.* with Dmitry Donin, *Comm. Math. Phys.,* ** 272:2** (2007), 507-527; DOI 10.1007/s00220-007-0234-2.

42. * Topological fluid dynamics. Notices of the AMS,* **52:1** (2005), 9-19; (pdf ).

41. * Polar linkings, intersections, and Weil pairing.* with Alexei Rosly, *Proc. of Royal Soc. London A,*** 461** (2005), 3505-3524 (pdf).

40. * Asymptotic directions, Monge-Ampere equations and the geometry of diffeomorphism groups.* with Gerard Misiolek, * Journal of Math. Fluid Mech.*** 7** (2005), S365-S375; (math.DG/0504556, pdf).

39. * A polar de Rham theorem.* with Alexei Rosly and Richard Thomas, * Topology * ** 43** (2004), 1231-1246; (math.AG/0305081, pdf).

38. * Bihamiltonian structures and quadratic algebras in hydrodynamics and on non-commutative torus.* with Andrey Levin and Mikhail Olshanetsky, *Comm. Math. Phys.*** 250** (2004), 581-612; (nlin.SI/0309017, pdf).

37. * Comments to Arnold's Problems*, in *Arnold's Problems*, Phasis, Moscow (2000), 454pp; Springer (2004), 639pp.

36. * Geometry of fluid motion. Séminaire Équat. Dér. Part., Ecole Polytechnique*** X** (2003), 1--10

35. * Euler equations on homogeneous spaces and Virasoro orbits.* with Gerard Misiolek, * Advances in Math.* ** 176** (2003), 116-144; (pdf).

34. * Geometry of higher helicities. * * Moscow Math. Journal * ** 3:3** (2003), 989-1011.

33.* Polar homology.* with Alexei Rosly, * Canad. J. Math. * ** 55** (2003), 1100-1120; (pdf).

32. * A Poisson--Lie framework for rational reductions of the KP hierarchy.* * Letters in Math. Physics * ** 58** (2001), 101-107; (pdf).

31.* Polar homology and holomorphic bundles.* with Alexei Rosly, * Phil. Trans. Royal Soc. London A * ** 359** (2001), 1413-1427; (pdf).

30. * Topology bounds the energy. An Introduction to the Geometry and Topology of Fluid Flows* (Ed.: R.L. Ricca), Kluwer, Dordrecht, The Netherlands (2001), 229-238.

29.* Symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces.* with Alexei Rosly, *The Arnoldfest* (Eds.: E.Bierstone et al.), Fields Institute Communications ** 24** (1999), 311-323.

28. * Homotopy classification of nondegenerate quasiperiodic curves on
the $2$-sphere.* (with B.Shapiro) * Publ. Inst. Math. (Beograd) (N.S.),* **66(80) **(1999), 127--156.

27. * Extensions and contractions of the Lie algebra of q-pseudodifferential symbols on the circle.* with Volodymyr Lyubashenko and Claude Roger, * J. of Funct. Anal.* **143:1** (1997), 55-97; (pdf).

26. * Informal complexification and Poisson structures on moduli spaces. AMS Transl., Ser. 2,* ** 180** (1997), 147-155.

25. * Four-dimensional realization of two-dimensional current groups. * with Igor Frenkel, *Comm. Math. Phys.* ** 178** (1996), 541-562.

24. * Universal Drinfeld-Sokolov reduction and matrices of complex size. * with Feodor Malikov, *Comm. Math. Phys.* **175** (1996), 113-134.

23. *On cohomology of the Lie algebra of pseudodifferential symbols on a
circle.* with C.Roger, *J. Math. Sci.* **82:6** (1996), 3800--3806.

22. * The Lie-Poisson group of pseudodifferential symbols. * with Ilya Zakharevich, * Comm. Math. Phys.* ** 171** (1995), 475-530.

21. * Affine Gelfand-Dickey brackets and holomorphic vector bundles.* with Pavel Etingof, * Geom. and Funct. Anal.* ** 4** (1994), 399-423.

20. * Steady fluid flows and symplectic geometry.* with Viktor Ginzburg, * J. of Geometry and Physics* **14** (1994), 195-210.

19. * The Lie-Poisson group of pseudodifferential symbols and fractional
KP-KdV-hierarchies.* (with I.Zakharevich) * C.R. Acad. Sci. Paris,* **316 **(1993), Serie I, 621-626.

18. * The logarithm of the derivative operator and higher spin algebras
of $W_{\infty}$-type.* (with I.Bakas and E.Kiritsis) * Comm.
Math. Phys.,* **151 **(1993), 233-243.

17. * Topological methods in hydrodynamics.* (with V.Arnold) * Annual Reviews in Fluid Mechanics,* **24 **(1992), 145-166.

16. * Ergodic interpretation of integral hydrodynamic invariants.* * J. of Geometry and Physics,* **9 **(1992), 101-110.

15. *A hierarchy of centrally extended algebras and the logarithm of the
derivative operator.* * Int. Math. Research Notices
(Duke Math. J.),* **1:1 **(1992), 1-5.

14. *Topology of steady fluid flows.* (with V.Ginzburg) * Topological Aspects of the Dynamics of Fluid and
Plasmas, eds. H.K.Moffatt, G.M.Zaslavsky, P.Comte,
M.Tabor (Kluwer Acad. Publ.),* (1992), 265-272.

13. *Swallowtails and Whitney umbrellas are homeomorphic.* (with B. Shapiro) * J. of Alg. Geometry,* **1 **(1992), 549-560.

12. *Non-degenerate curves on $S^2$ and orbit
classification of the Zamolodchikov algebra.* (with B. Shapiro) * Comm. Math. Phys.,* **145:2 **(1992), 357-362.

11. *Central extension of the algebra of pseudodifferential symbols.*(with O. Kravchenko)
* Funct. Anal. Appl.,* **25:2 ** (1991), 152-154.

10. *Singularities of light hypersurfaces and the structure of
hyperbolicity sets for systems of partial differential
equations.* * Advances in Soviet
Mathematics, AMS, ed. V.I.Arnold,* **1 ** (1990),105-118.

9. *Symplectic leaves of the Gelfand-Dikii brackets and homotopy classes
of non-degenerate curves.*(with V. Ovsienko)
* Funct. Anal. Appl.,* **24:1 ** (1990), 33-40.

8. *Invariants of Hamiltonian KdV-structures.*
* Russ. Math. Surveys,* **45:1 ** (1990), 209-210.

7. *Versal deformations of intersections of invariant submanifolds of
dynamical systems.*
* Russ. Math. Surveys,* **44:3 ** (1989), 181-182.

6. * Invariants of the Euler equations for ideal or barotropic hydrodynamics and superconductivity in D dimensions.* with Yuri Chekanov, * Physica D* **40:1** (1989), 119-131.

5. * Integrals of motion of the Euler equation of multidimensional
hydrodynamics and superconductivity.*(with Yu.Chekanov and V.Ovsienko)
* Diff.Geom., Lie Groups and Mechanics, Zap. Sem. LOMI,* **172 ** (1988), 105-113 (in Russian). English transl.: * J. of Sov. Math.,* ** 59:5** (1992), 1096-1102.

4. * Bifurcations of gradient dynamical systems.*
* Modern Problems in Math., New Achievements, VINITI AN USSR,* **33 ** (1988), 113-155 (in Russian). English transl.: *J. of Sov. Math.* ** 52** (1990), 3279-3305.

3. *
The (super) Korteweg-de Vries equation as an Euler equation.* with Valentin Ovsienko, *Funktsional. Anal. i Prilozhen.* **21:4**
(1987), 81--82; English transl: *Funct. Anal. and Appl.* **21:4** (1987),
329 - 331.

2. *Homogeneous vector fields and Whitney umbrellas.*
* Russ. Math. Surveys,* **42:5 ** (1987), 171-172.

1. * Bifurcations of singular points of gradient vector fields.*
* Funct. Anal. Appl.,* **20:3 ** (1986), 250-252.

**Books/papers edited**

8E. * Vladimir I. Arnold. Collected Works. *
Editors: A.Givental, B.Khesin, M.Sevryuk et al.
* Volume III: Singularity theory,
1972-1979; * Springer (2016), 509pp.

7E. *
ARNOLD: Swimming Against the Tide. *
Editors: B.Khesin and S.Tabachnikov, AMS (2014), 224pp.

6E. * Vladimir I. Arnold. Collected Works. *
Editors: A.Givental, B.Khesin, A.Varchenko et al.
* Volume II: Hydrodynamics, bifurcation theory, and algebraic geometry,
1965-1972; * Springer (2014), 466pp.

4-5E. * Tribute to Vladimir Arnold; Memories of Vladimir Arnold. * (coord. eds. with Serge Tabachnikov),
* Notices of the AMS* ** 59**, March and April (2012), 378-399 and 482-502.

3E. * Vladimir I. Arnold. Collected Works. *
Editors: A.Givental, B.Khesin, J.Marsden, A.Varchenko et al.
* Volume I: Representation of functions, Celestial mechanics,
and KAM theory, 1957-1965; * Springer (2009), 487pp.

2E. * Symplectic and contact topology: Interactions and perspectives.*
Editors: Ya.Eliashberg, B.Khesin, and F.Lalonde, * Fields
Institute Communications*, v.35, AMS, Providence (2003), 201pp.

1E. * The Arnoldfest: Proceedings of a conference in
honour of V.I.~Arnold for his sixtieth birthday,* Toronto (1997),
Editors: E.Bierstone, B.Khesin, A.Khovanskii, and J.Marsden,
* Fields
Institute Communications*, v.24, AMS, Providence (1999), 556pp.