*On the occasion of his 60th birthday*.

Joseph Semenovich Krasil'shchik was born on February 10, 1948 in Moscow (USSR).

He attended one of the best Moscow high schools with specialization in Physics and Mathematics. The school (#52) had rich ties with Moscow State University and he entered its Department of Mechanics and Mathematics in 1966 after graduating the school with honors. Among the department faculty at that time there was a number of excellent mathematicians: Kolmogorov, Manin, Arnold, Novikov, Kurosh, Gelfand to name but a few. In the university Krasil'shchik specialized in geometry and topology under the direct of A.M. Vinogradov. He graduated from the university in 1971.

Being an active member of the famous Vinogradov’s seminar, he followed his mentor when the latter switched his scientific studies to applications of algebraic topology and differential geometry to differential equations. And it became the principal area of research for Krasil'shchik ever since. He made a considerable contribution in the field.

From the very first papers till now a subject of constant interest of Krasil'shchik is the algebraic aspects of differential calculus. His works in this field include Hamiltonian formalism in (super)commutative algebras; algebraic study of differential equations; generalization of the δ-Poincaré lemma; algebraic theory of Frölicher-Nijenhuis, Richardson-Nijenhuis, and Schouten brackets.

Joseph Krasil'shchik and his teacher and collaborator Alexander Vinogradov are the principal founders of the nonlocal theory in geometry of differential equations. The central notion of this theory—a covering—is a key to decipher the structure of Bäcklund transformations, zero-curvature representations, Miura transformation, recursion operators, nonlocal symmetries, conservation laws, and Hamiltonian structures, Estabrook-Wahlquist algebras, etc.

In 1972–1989 Krasil'shchik worked as a Researcher at the All-Union Institute for Scientific and Technology Information (USSR Academy of Sciences). There he obtained his Ph.D. degree in Information Science. Yet the institute loose discipline allowed finding time for studies in abstract mathematics. Krasil'shchik publications of that period dealt with hamiltonian cohomologies of canonical algebras and nonlocal symmetries. These articles laid a foundation for his further research in the field.

From 1989 to 2003 Krasil'shchik worked at the Moscow Institute for Municipal Economy, first as senior lecturer and later as professor. After he obtained the Doctor of Science degree in Physics and Mathematics at the Moscow State University, 1997 he also took a position of the full professor at the Independent University of Moscow and still holds it.

It was a period of a great scientific activity. The publications by Krasil'shchik treats such topics as cohomology background in geometry of PDE, deformations and integrable systems, Bäcklund transformations, the connection between integrability and supersymmetry, etc. He was invited to short-time visiting positions at University of Tromso, Norway; University of Twente, the Netherlands; Erwin Schrodinger Inst. for Math. Physics, Austria; University of Seville, Spain; the Lille University, France; Bar-Ilan University, Israel; Universities of Florence, Salerno, and Lecce, Italy, etc. He took part (as a plenary lecturer) in numerous conferences around the world; coauthored five monographs in a collaboration with A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Khor'kova, P.H.M Kersten, V.V. Lychagin, A.V. Samokhin, Yu.N.Torkhov, A.M. Verbovetsky, A.M. Vinogradov.

In 1990th Krasil'shchik began a long, fruitful collaboration with Paul Kersten from Twente University in Enschede, Holland. Together, they developed the theory of deformations of differential equations into a tool for computing major invariants of differential equations, such as recursion operators and Hamiltonian operators. Krasil'shchik’s C-cohomology, Kersten’s REDUCE package, and nonlocal theory all together boiled down to a beautiful geometric theory of recursion operators of PDEs, which was summarized in the monograph “Symmetries and recursion operators for classical and supersymmetric differential equations” (2000).

In 2004–2007 Krasil'shchik had a position of a full professor at the Moscow State Technical University of Civil Aviation. Since 2007 he is a full professor at the Russian State University for the Humanities. In 2007 he also had a short-time visiting positions at University of Angers, France. In this period Krasil'shchik’s research concentrates mainly on Hamiltonian and symplectic structures, recursions, and hierarchies. The specter of application includes KdV-mKdV system, dispersionless Boussinesq type equation, Monge-Ampère equation, supersymmetric KdV equation, Camassa-Holm equation, etc. Overall, the list of his papers include more than 70 papers.

In recent years, Krasil'shchik undertook a project to extend these results to (non)local Hamiltonian and symplectic operators. He carries out this project in collaboration, by now, with V. Golovko, S. Igonin, P. Kersten, A. Verbovetsky, and R. Vitolo.

Krasil'shchik has a prominent place in the mathematical community. For many years he heads a famous “Krasil'shchik’s” research seminar at the Independent University of Moscow. It produces ideas, aspirants, international cooperation; and it spreads the elevated spirit of a pure mathematical research at the times of collapsing mathematical education. Krasil'shchik is a member of Moscow Mathematical Society and American Mathematical Society. He is an editor to eight volumes of collected works.

Outside mathematics he is a passionate angler. His other hobbies are extreme water tourism in wilderness of Siberia, Polar Urals, etc., and also classical music and jazz. His family life is a source of lasting happiness.

We, friends and colleagues of Joseph Semenovich Krasil'shchik wish him good health, happiness and new trophies: in fishing and mathematics.

V. Goldberg, S. Igonin, B. Kruglikov, A. Kushner, V. Lychagin, V. Rubtsov, A. Samokhin, A. Verbovetsky, R. Vitolo.

### Mathematical publications of Joseph Krasil’shchik

**Books**

A. M. Vinogradov, I. S. Krasil'shchik, and V. V. Lychagin, Primenenie nelinejnykh differentsial'nykh uravnenij v grazhdanskoj aviatsii, MIIGA, Moscow, 1977 (Russian).

A. M. Vinogradov, I. S. Krasil'shchik, and V. V. Lychagin, Geometriya nelineinykh differentsialnykh uravnenii, MIEM, Moscow, 1982 (Russian).

A. M. Vinogradov, I. S. Krasil'shchik, and V. V. Lychagin, Vvedenie v geometriyu nelineinykh differentsialnykh uravnenii, Nauka, Moscow, 1986 (Russian).

I. S. Krasil'shchik, V. V. Lychagin, and A. M. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, Gordon and Breach, 1986.

A. V. Bocharov, A. M. Verbovetsky, A. M. Vinogradov (editor), S. V. Duzhin, I. S. Krasil'shchik (editor), A. V. Samokhin, Yu. N. Torkhov, N. G. Khor'kova, and V. N. Chetverikov, Simmetrii i zakony sokhraneniya uravnenij matematicheskoj fiziki, Factorial, Moscow, 1997 (Russian), Second edition 2005.

I. S. Krasil'shchik and A. M. Verbovetsky, Homological methods in equations of mathematical physics, Open Education & Sciences, Opava, 1998, arXiv:math/9808130.

A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor'kova, I. S. Krasil'shchik (editor), A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, and A. M. Vinogradov (editor), Symmetries and conservation laws for differential equations of mathematical physics, AMS, 1999.

I. S. Krasil'shchik and A. M. Verbovetsky, Gomologicheskie metody v geometrii differentsial'nykh uravnenij, IUM, Moscow, 1999, Lecture notes, (Russian).

I. S. Krasil'shchik and P. H. M. Kersten, Symmetries and recursion operators for classical and supersymmetric differential equations, Kluwer, 2000.

I. S. Krasil'shchik, P. H. M. Kersten, M. Marvan, and Verbovetsky A. M., Gomologicheskie metody geometrii differentsial'nykh uravnenij, MCCME, Moscow, 2008 (Russian), to appear.

**Book Translations**

Zh. Pommare, Sistemy uravnenii s chastnymi proizvodnymi i psevdogruppy Li, Mir, Moscow, 1983 (Russian), Translated from the English by A. V. Bocharov, M. M. Vinogradov and I. S. Krasil'shchik, Translation edited and with a preface and an appendix by A. M. Vinogradov.

A. M. Vinogradov, Cohomological analysis of partial differential equations and secondary calculus, Translations of Mathematical Monographs, vol. 204, AMS, 2001, Translated from the Russian manuscript by Joseph Krasil'shchik.

Jet Nestruev, Smooth manifolds and observables, Graduate Texts in Mathematics, vol. 220, Springer, 2003, Joint work of A. M. Astashov, A. B. Bocharov, S. V. Duzhin, A. B. Sossinsky, A. M. Vinogradov, and M. M. Vinogradov. Translated from the 2000 Russian edition by A. B. Sossinsky, I. S. Krasil'shchik, and S. V. Duzhin.

**Books and Collections Editing**

P. H. M. Kersten and I. S. Krasil'shchik (eds.), Geometric and algebraic structures in differential equations, Kluwer, 1995, Papers from the Workshop on Algebra and Geometry of Differential Equations held in Enschede, 1993, Reprint of Acta Appl. Math. 41 (1995), no. 1-3.

I. S. Krasil'shchik and A. M. Vinogradov (eds.), Algebraic aspects of differential calculus, Kluwer, 1997, Acta Appl. Math. 49 (1997), no. 3.

Marc Henneaux, Joseph Krasil'shchik, and Alexandre Vinogradov (eds.), Secondary calculus and cohomological physics, Contemporary Mathematics, vol. 219, AMS, 1998.

B. P. Komrakov, I. S. Krasil'shchik, G. L. Litvinov, and A. B. Sossinsky (eds.), Lie groups and Lie algebras. Their representations, generalisations and applications, Mathematics and its Applications, vol. 433, Kluwer, 1998.

I. S. Krasil'shchik and A. M. Vinogradov (eds.), Geometrical aspects of nonlinear differential equations, Kluwer, 1999, Acta Appl. Math. 56 (1999), no. 2-3.

Joseph Krasil'shchik (ed.), Symmetries of differential equations and related topics, Kluwer, 2002, Acta Appl. Math. 72 (2002), no. 1-2.

Joseph Krasil'shchik (ed.), Geometry of PDE in action: zero-curvature representations, recursion operators, and control systems, Kluwer, 2004, Acta Appl. Math. 83 (2004), no. 1-2.

I. S. Krasil'shchik and B. S. Kruglikov (eds.), Algebra and geometry of PDEs, Springer, 2008, Acta Appl. Math. 101 (2008), no. 1-3.

**Papers and other publications**

**1974**

I. S. Krasil'shchik, Gamil'tonov formalizm v kanonicheskikh kol'tsakh, Voronezh Winter Math. School, Voronezh State Univ., 1974, pp. 23–24 (Russian).

**1975**

A. M. Vinogradov and I. S. Krasil'shchik, What is Hamiltonian formalism?, Russ Math. Surv. 30 (1975), no. 1, 177–202, Russian original: Uspehi Mat. Nauk 30 (1975), no. 1, 173–198; also in Integrable systems: selected papers, London Math. Soc. Lect. Note Ser., 60, 1981, 241–266.

**1980**

I. S. Krasil'shchik, Hamiltonian cohomology of canonical algebras, Sov. Math. Dokl. 21 (1980), 625–629, Russian original: Dokl. Akad. Nauk SSSR 251 (1980), 1306–1309.

A. M. Vinogradov and I. S. Krasil'shchik, A method of calculating higher symmetries of nonlinear evolutionary equations, and nonlocal symmetries, Sov. Math. Dokl. 22 (1980), 235–239, Russian original: Dokl. Akad. Nauk SSSR 253 (1980), 1289–1293.

**1984**

A. M. Vinogradov and I. S. Krasil'shchik, On the theory of nonlocal symmetries of nonlinear partial differential equations, Sov. Math. Dokl. 29 (1984), 337–341, Russian original: Dokl. Akad. Nauk SSSR 275 (1984), 1044–1049.

I. S. Krasil'shchik and A. M. Vinogradov, Nonlocal symmetries and the theory of coverings: an addendum to Vinogradov’s “Local symmetries and conservation laws” [Acta Appl. Math. 2 (1984), 21–78], Acta Appl. Math. 2 (1984), 79–96.

**1986**

I. S. Krasil'shchik, Pochemu preobrazovaniya Beklunda obrazuyut gruppu?, Proceedings of the seminar on the algebra and geometry of differential equations (Moscow), MSU, 1986 (Russian).

**1987**

I. S. Krasil'shchik, Del'ta-lemma Puankare, Mimeographed notes, 1987.

**1988**

I. S. Krasil'shchik, Schouten bracket and canonical algebras, Global analysis—studies and applications, III, Lecture Notes in Math., vol. 1334, Springer, 1988, pp. 79–110, Russian original: Global'nyj analiz i matematicheskaya fizika, Nov. Global'nom anal., Voronezh, 1987, 73–94.

**1989**

I. S. Krasil'shchik and A. M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math. 15 (1989), 161–209.

**1991**

I. S. Krasil'shchik, Supercanonical algebras and Schouten brackets, Math. Notes 49 (1991), no. 1, 50–54, Russian original: Mat. Zametki 49 (1991), no. 1, 70–76.

**1992**

I. S. Krasil'shchik, Algebry deformatsij differentsial'nykh uravnenij i operatory rekursii, Proceedings XXX MIKKHiS Scientific Conference (Moscow), MIKKHiS, 1992, pp. 18–21 (Russian).

I. S. Krasil'shchik, Differential operators of constant growth and Jacobi structures of infinite order, Publ. IRMA Lille 30 (1992), no. XI, 1–21.

I. S. Krasil'shchik, Some new cohomological invariants for nonlinear differential equations, Differential Geom. Appl. 2 (1992), 307–350.

**1993**

I. S. Krasil'shchik, Some new cohomological invariants of nonlinear differential equations. I, Russ. Math. 37 (1993), no. 1, 25–35, Russian original: Izv. Vyssh. Uchebn. Zaved. Mat. 1993, no. 1, 27–37.

I. S. Krasil'shchik, Some new cohomological invariants of nonlinear differential equations. II, Russ. Math. 37 (1993), no. 2, 52–65, Russian original: Izv. Vyssh. Uchebn. Zaved. Mat. 1993, no. 2, 54–68.

I. S. Krasil'shchik, Lie algebra structures for the symmetries of differential equations possessing recursion operators, ESI Preprint 47 (1993).

I. S. Krasil'shchik, An algebraic model for characteristics of differential equations, ESI Preprint 48 (1993).

**1994**

I. S. Krasil'shchik and P. H. M. Kersten, Deformations and recursion operators for evolution equations, Geometry in partial differential equations, World Sci., 1994, pp. 114–154, Also: Memorandum of the Twente University, 1992, no. 1104.

**1995**

I. S. Krasil'shchik, Hamiltonian formalism and supersymmetry for nonlinear differential equations, ESI Preprint 257 (1995).

I. S. Krasil'shchik, Notes on coverings and Bäcklund transformations, ESI Preprint 260 (1995).

I. S. Krasil'shchik and P. H. M. Kersten, Graded differential equations and their deformations: a computational theory for recursion operators, Acta Appl. Math. 41 (1995), 167–191.

P. H. M. Kersten and I. S. Krasil'shchik, Graded Frölicher-Nijenhuis brackets and the theory of recursion operators for super differential equations, The interplay between differential geometry and differential equations, Amer. Math. Soc. Transl. Ser. 2, vol. 167, AMS, 1995, pp. 91–129, Also: Memorandum of the Twente University, 1993, no. 1147.

**1996**

I. S. Krasil'shchik, A supersymmetry approach to Poisson structures over differential equations, Differential geometry and applications, Masaryk Univ., Brno, 1996, pp. 381–391.

I. S. Krasil'shchik, Poisson structures on nonlinear evolution equations, Memorandum of the Twente University 1320 (1996).

**1997**

I. S. Krasil'shchik, Calculus over commutative algebras: a concise user guide, Acta Appl. Math. 49 (1997), 235–248.

I. S. Krasil'shchik, Poincaré δ-lemma for smooth algebras, Acta Appl. Math. 49 (1997), 249–255.

I. S. Krasil'shchik, Characteristics of linear differential operators over commutative algebras, Acta Appl. Math. 49 (1997), 257–269.

I. S. Krasil'shchik, Algebraicheskie metody v teorii integriruemykh sistem, MSU, Moscow, 1997, Dr. Sci. Thesis (Russian).

**1998**

I. S. Krasil'shchik, Symmetries and recursion operators for soliton equations, Nonlinearity and geometry, PWN, Warsaw, 1998, pp. 141–156.

Joseph Krasil'shchik, Cohomology background in geometry of PDE, Secondary calculus and cohomological physics, Contemp. Math., vol. 219, AMS, 1998, pp. 121–139.

I. S. Krasil'shchik, Algebras with flat connections and symmetries of differential equations, Lie groups and Lie algebras. Their representations, generalisations and applications, Math. Appl., vol. 433, Kluwer, 1998, pp. 407–424.

**1999**

Joseph Krasil'shchik in collaboration with Barbara Prinari, Lectures on linear differential operators over commutative algebras. (The 1st Italian diffiety school, July, 1998), Diffiety Inst. Preprint Series 1 (1999), DIPS 1/99.

I. S. Krasil'shchik, Cohomological approach to Poisson structures on nonlinear evolution equations, Lobachevskii J. Math. 3 (1999), 127–145 (electronic).

Joseph Krasil'shchik and Michal Marvan, Coverings and integrability of the Gauss-Mainardi-Codazzi equations, Acta Appl. Math. 56 (1999), 217–230, arXiv:solv-int/9812010.

**2000**

Joseph Krasil'shchik, Integrability and supersymmetry, RIMS Kokyuroku 1150 (2000), 147–152.

Joseph Krasil'shchik, On one-parametric families of Bäcklund transformations, Diffiety Inst. Preprint Series 1 (2000), DIPS 1/2000.

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, MSU, Moscow, 2001.

**2002**

Sergei Igonin and Joseph Krasil'shchik, On one-parametric families of Bäcklund transformations, Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie (Kyoto/Nara, 1999), Adv. Stud. Pure Math., vol. 37, Math. Soc. Japan, Tokyo, 2002, pp. 99–114, arXiv:nlin/0010040.

Paul Kersten and Joseph Krasil'shchik, Complete integrability of the coupled KdV-mKdV system, Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie, Adv. Stud. Pure Math., vol. 37, Math. Soc. Japan, 2002, pp. 151–171, arXiv:nlin/0010041.

Joseph Krasil'shchik, Geometry of differential equations: a concise introduction, Acta Appl. Math. 72 (2002), 1–17.

Iosif Krasil'shchik, A simple method to prove locality of symmetry hierarchies, Diffiety Inst. Preprint Series 9 (2002), DIPS 09/2002.

P. H. M. Kersten and I. S. Krasil'shchik, From recursion operators to Hamiltonian structures. The factorization method, Memorandum of the Twente University 1624 (2002), Lectures delivered at the MRI Spring School “Frobenius Manifolds in Mathematical Physics”.

P. H. M. Kersten, I. S. Krasil'shchik, and A. M. Verbovetsky, An extensive study of the N=1 supersymmetric KdV equation, Memorandum of the Twente University 1656 (2002).

**2003**

S. Igonin, P. H. M. Kersten, and I. Krasil'shchik, On symmetries and cohomological invariants of equations possessing flat representations, Differential Geom. Appl. 19 (2003), 319–342, arXiv:math/0301344.

P. H. M. Kersten, I. S. Krasil'shchik, and A. M. Verbovetsky, A geometric approach to Hamiltonian structures for evolution equations, Proc. Int. Conf. Kolmogorov and Contemporary Mathematics (Moscow), 2003, pp. 815–816.

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. Int. Conf. Kolmogorov and Contemporary Mathematics, 2003, pp. 817–818.

I Krasil'shchik, The long exact sequence of a covering: three applications, Diffiety Inst. Preprint Series 6 (2003), DIPS 6/2003.

**2004**

P. Kersten, I. Krasil'shchik, and A. Verbovetsky, Hamiltonian operators and l*-coverings, J. Geom. Phys. 50 (2004), 273–302, arXiv:math/0304245.

P. Kersten, I. Krasil'shchik, and A. 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 37 (2004), 5003–5019, arXiv:nlin/0305026.

P. Kersten, I. Krasil'shchik, and A. Verbovetsky, On the integrability conditions for some structures related to evolution differential equations, Acta Appl. Math. 83 (2004), 167–173, arXiv:math/0310451.

Paul Kersten, Iosif Krasil'shchik, and Alexander Verbovetsky, Nonlocal constructions in the geometry of PDE, Symmetry in nonlinear mathematical physics. Part 1, Inst. Math. NAS Ukr., Kiev, 2004, pp. 412–423.

P. H. M. Kersten, I. Krasil'shchik, and A. Verbovetsky, The Monge-Ampère equation: Hamiltonian and symplectic structures, recursions, and hierarchies, Memorandum of the Twente University 1727 (2004).

**2005**

I. S. Krasil'shchik, Prostoj metod dokazatel'stva lokal'nosti simmetrij evolyutsionnykh uravnenij, Scientific Bulletin of MSTUCA 91 (2005), 12–19 (Russian).

**2006**

Paul Kersten, Iosif Krasil'shchik, and Alexander Verbovetsky, A geometric study of the dispersionless Boussinesq type equation, Acta Appl. Math. 90 (2006), 143–178, arXiv:nlin/0511012.

P. H. M. Kersten and I. S. Krasil'shchik, The Cartan covering and complete integrability of the KdV-mKdV system, Constructive Algebra and Systems Theory (B. Hanzon and Hazewinkel M., eds.), Royal Netherlands Academy of Arts and Sciences, 2006, pp. 251–265.

**2007**

A. M. Verbovetsky, V. A. Golovko, and I. S. Krasil'shchik, Skobka Li dlya nelokal'nykh tenej, Scientific Bulletin of MSTUCA 114 (2007), 9–23 (Russian).

J. Krasil'shchik, Nonlocal geometry of PDEs and integrability, Symmetry and perturbation theory (G. Gaeta, R. Vitolo, and Walcher S., eds.), World Sci., 2007, pp. 100–108.

I. S. Krasil'shchik, Estestvennye nakrytiya i integriruemye sistemy, Symmetries: theoretical and methodological aspects, Astrakhan Univ., Astrakhan, 2007, pp. 46–53 (Russian).

**2008**

V. A. Golovko, I. S. Krasil'shchik, and A. M. Verbovetsky, Variational Poisson-Nijenhuis structures for partial differential equations, Theor. Math. Phys. 154 (2008), 227–239, Russian original: Teor. Mat. Fiz. 154 (2008), 268–282.

V. Golovko, P. Kersten, I. Krasil'shchik, and A. Verbovetsky, On integrability of the Camassa-Holm equation and its invariants, Acta Appl. Math. 101 (2008).