Joseph Krasil'shchik

I.S. (Joseph) KRASIL'SHCHIK

Date of birth: February 10, 1948.
E.mail: josephk @ diffiety.ac.ru

1966 - 1971: A student of the Moscow State University, Mechanical & Mathematical Faculty.
1972 - 1980: Researcher at the All-Union Institute for Scientific and Technology Information (USSR Academy of Sciences).
1981 - 1989: Senior researcher at the same Institute.
1989 - 1992: Senior lecturer at the Moscow Institute for Municipal Economy and Civil Engineering.
1992 - 1999: Associate professor at the same Institute.
1999 - 2003: Full professor at the same Institute.
2004 - 2007: Full professor at the Moscow State Technical University of Civil Aviation.
1997 up to now: Full professor at the Independent University of Moscow
2007 up to now: Full professor at the Russian State University for the Humanities
1991 - 1993: Scientific deputy director of the Moscow Sophus Lie Center.
1978: Ph.D. degree in Information Science.
1997: Doctor of Science degree in Mathematics.

Current research: algebraic models and cohomological methods in PDE, nonlocal structures, Poisson-Nijenhuis and Hamiltonian structures in PDE

Publications in Diffeotopy include:

V. Golovko, P. Kersten, I. Krasil'shchik, A. Verbovetsky, On integrability of the Camassa-Holm equation and its invariants, Acta. Appl. Math. 2008 (to appear).
A. Verbovetsky, V. Golovko, I. Krasil'shchik, Variational Poisson-Nijenhuis structures for partial differential equations, Theor. Math. Phys., 154 (2008) no. 2 (to appear).
I.S. Krasil'shchik, Natural coverings and integrable systems, In Symmetries: Theoretical and Methidological Aspects, Astrakhan, 2007, pp. 46-53 (in Russian).
I. Krasil'shchik, Nonlocal geometry of PDEs and integrability, In: G. Gaeta, R. Vitolo, R. Walcher (editors), Symmetry and Pertubation Theory, World Scietific, 2007, pp. 100-108.
A. Verbovetsky, V. Golovko, I. Krasil'shchik, Lie bracket for nonlocal shadows, Scientific Bulletin of MSTUCA, 114 (2007) 4, 9-23 (in Russian).
P.H.M. Kersten, I.S. Krasil'shchik, The Cartan covering and complete integrability of the KdV-mKdV system. In: Constructive Algebra and Systems Theory, B. Hanzon and M. Hazewinkel (Editors) Royal Netherlands Academy of Arts and Sciences, 2006.
P.H.M. Kersten, I.S. Krasil'shchik, and A.M. Verbovetsky, A geometric study of the dispersionless Boussinesq type equation, Acta Appl. Math., 90 (2006) no. 1, pp. 143--178. arXiv:nlin.SI/0511012
P.H.M. Kersten, I.S. Krasil'shchik, and A.M. Verbovetsky, The Monge-Ampere equation: Hamiltonian and symplectic structures, recursions, and hierarchies. Memorandum 1727, Faculty EEMCS University of Twente, 2004, 75 pp.
P.H.M. Kersten, I.S. Krasil'shchik, and A.M. Verbovetsky, (Non)local Hamiltonian and symplectic structures, recursions, and hierarchies: a new approach and applications to the N=1 supersymmetric KdV equation, J. Phys. A: Mathematical and General, 37 (2004) no. 18, 5003-5019.
P.H.M. Kersten, I.S. Krasil'shchik, and A.M. Verbovetsky, On the integrability conditions for some structures related to evolution differential equations, Acta Appl. Math., 83 (2004) no. 1-2, 167-173. arXiv:math.DG/0310451
P.H.M. Kersten, I.S. Krasil'shchik, and A.M. Verbovetsky, Nonlocal Constructions in the Geometry of PDE, Proc. Conf. Symmetry in Nonlinear Mathematical Physics, Kyiv, Ukraine, 2003. Proc. Inst. Mathematics of NAS of Ukraine, Eds A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A.Yehorchenko, Kyiv, Institute of Mathematics, 2004, 50, Part I, 412-423.
I.S. Krasil'shchik, The long exact sequence of a covering: three applications, The Diffiety Inst. Preprint Series, DIPS 6/2003.
P.H.M. Kersten, Krasil'shchik I.S., and A.M. Verbovetsky, A geometric approach to Hamiltonian structures for evolution equations. Proc. Conf. Kolmogorov and Contemporary Mathematics, Moscow, Russia, 2003.
I.S. Krasil'shchik, A.M. Verbovetsky, and P.H.M. Kersten, Nonlocal Hamiltonian, symplectic and recursion structures for N=1 supersymmetric KdV equation. Proc. Conf. Kolmogorov and Contemporary Mathematics, Moscow, Russia, 2003.
P.H.M. Kersten, I.S. Krasil'shchik, and A.M. Verbovetsky, The N=1 supersymmetric KdV equation: (non)local Hamiltonian and symplectic structures, recursions, and hierarchies, 15 pp. arXiv:nlin.SI/0305026
S.A. Igonin, P.H.M. Kersten, and I.S. Krasil'shchik, On symmetries and cohomological invariants of equations possessing flat representations, Memorandum 1639, Faculty of Mathematical Sciences University of Twente, 2002, 24 pp. J. of Diff. Geom. and Its Appl. 19 (2003) no. 3, 319-342. The Diffiety Inst. Preprint Series, DIPS 7/2002. See also: arXiv:math.DG/0301344
P.H.M. Kersten and I.S. Krasil'shchik, From recursion operators to Hamiltonian structures: the factorization method. Memorandum 1624, Faculty of Mathematical Sciences University of Twente, 2002, 16 pp. Lectures delivered at MRI Spring School Frobenius Manifolds in Mathematical Physics.
P.H.M. Kersten, I.S. Krasil'shchik, and A.M. Verbovetsky, Hamiltonian operators and -coverings, 23 pp. J. Geom. and Phys., 50 (2004) 273--302. The Diffiety Inst. Preprint Series, DIPS 6/2002. See also: arXiv:math.DG/0304245
P.H.M. Kersten, I.S. Krasil'shchik, and A.M. Verbovetsky, An extensive study of the N=1 supersymmetric KdV equation, Memorandum 1656, Faculty of Mathematical Sciences University of Twente, 2002, 63 pp.
I.S. Krasil'shchik, Geometry of differential equations: a concise introduction, Acta Appl. Math., 72, (2002) no. 1-2, 1--17.
I.S. Krasil'shchik, A simple method to prove locality of symmetry hierarchies, 4 pp. The Diffiety Inst. Preprint Series, DIPS 9/2002.
I.S. Krasil'shchik, P.H.M. Kersten, Complete integrability of the coupled KdV-mKdV system. In: T. Morimoto, H. Sato, and K. Yamaguchi, eds., Lie Groups, Geometric Structures and Differential Equations - One Hundred Years After Sophus Lie, Advanced Studies in Pure Mathematics, vol.37, Math. Soc. of Japan, 2002, 151-171. arXiv:nlin.SI/0010041
I.S. Krasil'shchik, S. Igonin, On one-parametric families of Baecklund transformations. In: T. Morimoto, H. Sato, and K. Yamaguchi, eds., Lie Groups, Geometric Structures and Differential Equations - One Hundred Years After Sophus Lie, Advanced Studies in Pure Mathematics, vol.37, Math. Soc. of Japan, 2002, 99-114. arXiv:nlin.SI/0010040
I.S. Krasil'shchik, Integrability and supersymmetry. Proc. Conf. Lie Groups, Geometric Structures and Differential Equations, Kyoto--Nara, Japan, December 1999 (RIMS Lecture Notes, v. 1150 (2000). Kyoto, Japan).
I.S. Krasil'shchik,, Deformations and integrable systems. Proc. Conf. Differential Equations and Related Topics dedicated to 100th Anniversary of birthday of I.G. Petrovskii, Moscow, Russia, 2001.
I.S. Krasil'shchik, On one-parametric families of Baecklund transformations, The Diffiety Inst. Preprint Series, DIPS 1/2000.
I.S. Krasil'shchik, P.H.M. Kersten, Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations, - Mathematics and its applications, Volume 507, Kluwer Acad. Publ., Dordrecht/Boston/London, 2000, 400 pp. For more information see here.
I.S. Krasil'shchik, M. Marvan, Coverings and Integrability of the Gauss-Mainardi-Codazzi Equations, Acta Appl. Math. 56, no. 2-3, 1999. PS and PDF-files see on the ESI WWW-server. See also arXiv:solv-int/9812010
I.S. Krasil'shchik, Cohomological approach to Poisson structures on nonlinear evolution equations, Lobachevsky Journal of Mathematics 1999, 3.
I.S. Krasil'shchik. (in collaboration with B. Prinari), Lectures on Linear Differential Operators over Commutative Algebras (The 1st Italian Diffiety School, July, 1998). The Diffiety Inst. Preprint Series, DIPS 1/99.
V.N. Chetverikov, A.B. Bocharov, S.V. Duzhin, N.G. Khor'kova, I.S. Krasil'shchik, A.V. Samokhin, Yu.N. Torkhov, A.M. Verbovetsky, A.M. Vinogradov - Symmetries and Conservation Laws for Differential Equations of Mathematical Physics - I.S. Krasil'shchik, and A.M. Vinogradov, Editors - AMS, 1999, approximately 347 pp., MMONO/182.
I.S. Krasil'shchik Cohomology background in geometry of PDE, in M. Henneaux, I. Krasil'shchik, and A.M. Vinogradov (Eds.) Secondary Calculus and Cohomological Physics, Contemporary Mathematics, vol.219, American Mathematical Society, Providence, RI, 1998, pp. 121-139. See also The Diffiety Inst. Preprint Series, DIPS 1/98.
I.S. Krasil'shchik and A.M. Verbovetsky, Homological methods in equations of mathematical physics, Open Education, Opava, 1998, 150 pp. Diffiety Inst. Preprint Series, DIPS 7/98. Russian PS-file see on the IUM WWW-server. See also arXiv:math.DG/9808130
I.S. Krasil'shchik Algebras with flat connections and symmetries of differential equations, in Lie Groups and Lie Algebras: Their Representations, Generalizations and Applications, Kluwer Acad. Publ., Dordrecht/Boston/London, 1998, pp. 407-424.
I.S. Krasil'shchik, Calculus over commutative algebras: a concise user guide, Acta Appl. Math., Volume 49, Issue 3, December 1997, pp. 235-248. See also The Diffiety Institute Preprints, DIPS 1/96
I.S. Krasil'shchik, Characteristics of linear differential operators over commutative algebras, Acta Appl. Math., Volume 49, Issue 3, December 1997, pp. 257-269. See also The Diffiety Institute Preprints, DIPS 2/96
I.S. Krasil'shchik, Poincare delta-lemma for smooth algebras, Acta Appl. Math., Volume 49, Issue 3, December 1997, pp. 249-255. See also The Diffiety Institute Preprints, DIPS 3/96
Symmetries of Differential Equations in Mathematical Physics and Natural Sciences (a monograph by V.N. Chetverikov, A.B. Bocharov, S.V. Duzhin, N.G. Khor'kova, I.S. Krasil'shchik, A.V. Samokhin, Yu.N. Torkhov, A.M. Verbovetsky, A.M. Vinogradov, edited by A.M. Vinogradov and I.S. Krasil'shchik). Factorial Publ. House, 1997, second edition 2005 (in Russian). For English translation see above.
I.S. Krasil'shchik, A supersymmetry approach to Poisson structures over differential equations, Proc. Conf. Diff. Geom. Appl., Brno, 1996, 14 pp. TeX-file see on the EMIS WWW-server
I.S. Krasil'shchik, Poisson structures on nonlinear evolution equations, Memorandum of the Twente University (1996), no. 1320, 24pp.
I.S. Krasil'shchik, Symmetries and recursion operators for soliton equations, Proc. 1st Non-Orthodox School on Nonlinearity and Geometry, Warsaw 21 - 28 September 1995. Banach Center Publ, Warsaw, 1996.
I.S. Krasil'shchik, P.H.M. Kersten, Graded Froelicher - Nijenhuis brackets and the theory of recursion operators for super differential equations, Memorandum of the Twente University (1993), no.1147, Enschede, 56 pp. Also in: V.V. Lychagin, The Interplay between Differential Geometry and Differential equations, Advances in the Mathematical Sciences, 24. Amer. Math. Soc., Providence RI, 1995.
I.S. Krasil'shchik, Hamiltonian formalism and supersymmetry for nonlinear differential equations, Preprint of the Erwin Schroedinger International Inst. for Mathematical Physics (1995), no. 257, Wien, 61 pp. PS and PDF-files see on the ESI WWW-server.
I.S. Krasil'shchik, Notes on coverings and Baecklund transformations, Preprint of the Erwin Schroedinger International Inst. for Mathematical Physics (1995), no. 260, Wien, 15 pp. PS and PDF-files see on the ESI WWW-server.
I.S. Krasil'shchik, P.H.M. Kersten, Graded differential equations and their deformations: a computational theory for recursion operators, Acta Appl. Math. 41 (1995) 167 - 191. Also in: P.H.M. Kersten and I.S. Krasil'shchik (eds.) Geometric and Algebraic Structures in Differential Equations, Kluwer Acad. Publ., Dordrecht/Boston/London, 1995.
I.S. Krasil'shchik, P.H.M. Kersten, Deformations and recursion operators for evolution equations, Memorandum of the Twente University (1992), no. 1104, Enschede, 47 pp. Also in: Prastaro, A. and Rassias, Th.M. (eds.), Geometry in partial differential equations, World Scientific, Singapore, 1994.
I.S. Krasil'shchik, Lie algebra structures for the symmetries of differential equations possessing recursion operators, Preprint of the Erwin Schroedinger International Inst. for Mathematical Physics (1993), no. 47, Wien, 14 pp. PS and PDF-files see on the ESI WWW-server.
I.S. Krasil'shchik, An algebraic model for characteristics of differential equations Preprint of the Erwin Schroedinger International Inst. for Mathematical Physics (1993), no. 48, Wien, 14 pp. PS and PDF-files see on the ESI WWW-server.
I.S. Krasil'shchik, Deformation algebras of differential equations and recursion operators, Transactions of the Sci. Conf. of MIMECI, Moscow Inst. for Municipal Economy and Civil Engineering, Moscow, 1992 (in Russian).
I.S. Krasil'shchik, Some new cohomological invariants of nonlinear differential equations, Differential Geometry and Its Appl., 2 (1992) no. 4.
I.S. Krasil'shchik, Differential operators of constant growth and Jacobi structures of infinite order, Publ. IRMA 30 (1992), no. XI, Lille.
I.S. Krasil'shchik, Supercanonical algebras and Schouten brackets, Mat. Zametki 49 (1991), no. 1 (in Russian, English translation in: Soviet Mat. Zametki).
I.S. Krasil'shchik, A.M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Baecklund transformations, Acta Appl. Math. 15 (1989) no. 1 - 2. Also in: Vinogradov, A.M. (ed.), Symmetries of partial differential equations. Conservation laws - Applications - Algorithms, Kluwer Acad. Publ., Dordrecht, 1989.
I.S. Krasil'shchik, Schouten brackets and canonical algebras - Global Anal. and Math. Physics, Voronezh State University, 1987 (in Russian, English translation in: Springer Lecture Notes in Math. 1334, Springer-Verlag, 1988).
I.S. Krasil'shchik, Why Baecklund transformation form a group? (1987), Mimeographed notes (in Russian).
I.S. Krasil'shchik, V.V. Lychagin, A.M. Vinogradov, Introduction into the Geometry of Nonlinear Differential Equations, Nauka Publ. House, Moscow, 1986 (in Russian).
I.S. Krasil'shchik, V.V. Lychagin, A.M. Vinogradov, Geometry of Jet Spaces and Nonlinear Differential Equations, Gordon and Breach Sci. Publ., New York, 1986.
I.S. Krasil'shchik, Poincare delta - lemma, Proc. of the Seminar on Algebra and Geometry of Differential Equations, Moscow State University, Moscow, 1986 (in Russian).
I.S. Krasil'shchik, A.M. Vinogradov, On the theory of nonlocal symmetries of nonlinear partial differential equations, Dokl. AN SSSR 275 (1984) no. 5 (in Russian, English translation in: Soviet Math. Dokl.).
I.S. Krasil'shchik, A.M. Vinogradov, Nonlocal symmetries and the theory of coverings, Acta Appl. Math. 2 (1984), no. 1.
I.S. Krasil'shchik, V.V. Lychagin, A.M. Vinogradov , Geometry of Nonlinear Differential Equations, Moscow Institute of Electronic Engineering, Moscow, 1982 (in Russian).
I.S. Krasil'shchik, A.M. Vinogradov, What is the Hamiltonian formalism? Uspekhi Mat. Nauk 30 (1975) no. 1 (in Russian, English translation in: London Math. Soc. Lecture Notes Ser., 60 (1981)).
I.S. Krasil'shchik, A.M. Vinogradov, A method for computation of higher symmetries of nonlinear evolution equations and nonlocal symmetries, Dokl. AN SSSR 253 (1980) no. 6 (in Russian, English translation in: Soviet Math. Dokl.).
I.S. Krasil'shchik, Hamiltonian cohomologies of canonical algebras, Dokl. AN SSSR 251 (1980) no. 6 (in Russian, English translation in: Soviet Math. Dokl.).
I.S. Krasil'shchik, V.V. Lychagin, A.M. Vinogradov, Applications of nonlinear differential equations in civil aviation, Moscow Institute of Civil Aviation Engineers, Moscow, 1977 (in Russian).
I.S. Krasil'shchik, Hamiltonian formalism in canonical rings, The 8th Winter Mathematical School, Voronezh State University, Voronezh, 1974 (in Russian).