geometry of nonlinear differential equations; application of homological methods to partial differential equations.
A theory of nonlocal symmetries and conservation laws for nonlinear differential equations was constructed (jointly with A. M. Vinogradov). A cohomological theory of recursion operators for symmetries of partial differential equations was constructed. New recursion operators for supersymmetric integrable systems were found (jointly with P. H. M. Kersten).
Graduted from Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University (MSU) in 1971 (Chair of Higher Geometry and Topology). Ph. D. thesis was defended in 1978. D. Sci. thesis was defended in 1997. A list of my works contains more than 60 titles. Since 1991 had let a research seminar on geometry of differential equations (at MSU until 1997 and at the Independent University of Moscow afterwards).
Krasil'shchik I. S., Kersten P. H. M., Symmetries and recursion operators for classical and supersymmetric differential equations, Kluwer Acad. Publ., Dordrecht, 2000
Krasil'shchik I. S., Vinogradov A. M. (eds.), Symmetries and conservation laws for differential equations of mathematical physics, Amer. Math. Soc., Providence, RI, 1999
Krasil'shchik I. S., “Cohomology background in geometry of PDE”, Secondary calculus and cohomological physics (Moscow, 1997), Contemp. Math., 219, 1998, 121–139
Krasil'shchik I. S., “Some new cohomological invariants for nonlinear differential equations”, Differential Geom. Appl., 2:4 (1992), 307–350
Krasil'shchik I. S., Lychagin V. V., Vinogradov A. M., Geometry of jet spaces and nonlinear partial differential equations, Gordon and Breach Science Publishers, New York, 1986
H. Baran, I. S. Krasil'shchik, O. I. Morozov, P. Vojčák, “Nonlocal symmetries of integrable linearly degenerate equations: A comparative study”, TMF, 196:2 (2018), 169–192; Theoret. and Math. Phys., 196:2 (2018), 1089–1110
H. Baran, I. S. Krasil'shchik, O. I. Morozov, P. Vojčák, “Coverings over Lax integrable equations and their nonlocal symmetries”, TMF, 188:3 (2016), 361–385; Theoret. and Math. Phys., 188:3 (2016), 1273–1295
A. M. Verbovetsky, R. Vitolo, P. Kersten, I. S. Krasil'shchik, “Integrable structures for a generalized Monge–Ampère equation”, TMF, 171:2 (2012), 208–224; Theoret. and Math. Phys., 171:2 (2012), 600–615
V. A. Golovko, I. S. Krasil'shchik, A. M. Verbovetsky, “Variational Poisson–Nijenhuis structures for partial differential
equations”, TMF, 154:2 (2008), 268–282; Theoret. and Math. Phys., 154:2 (2008), 227–239
I. S. Krasil'shchik, “Cohomological approach to poisson structures on nonlinear evolution equations”, Lobachevskii J. Math., 3 (1999), 127–145
I. S. Krasil'shchik, “Some new cohomological invariants of nonlinear differential equations. II”, Izv. Vyssh. Uchebn. Zaved. Mat., 1993, 2, 54–68; Russian Math. (Iz. VUZ), 37:2 (1993), 52–65
I. S. Krasil'shchik, “Some new cohomological invariants of nonlinear differential equations. I”, Izv. Vyssh. Uchebn. Zaved. Mat., 1993, 1, 27–37; Russian Math. (Iz. VUZ), 37:1 (1993), 25–35
I. S. Krasil'shchik, “Supercanonical algebras and Schouten brackets”, Mat. Zametki, 49:1 (1991), 70–76; Math. Notes, 49:1 (1991), 50–54
A. M. Vinogradov, I. S. Krasil'shchik, “On the theory of nonlocal symmetries of nonlinear partial
differential equations”, Dokl. Akad. Nauk SSSR, 275:5 (1984), 1044–1049
A. M. Vinogradov, I. S. Krasil'shchik, “A method of computing higher symmetries of nonlinear evolution equations, and nonlocal symmetries”, Dokl. Akad. Nauk SSSR, 253:6 (1980), 1289–1293
I. S. Krasil'shchik, “Hamiltonian cohomology of canonical algebras”, Dokl. Akad. Nauk SSSR, 251:6 (1980), 1306–1309
A. M. Vinogradov, I. S. Krasil'shchik, “What is the hamiltonian formalism?”, Uspekhi Mat. Nauk, 30:1(181) (1975), 173–198; Russian Math. Surveys, 30:1 (1975), 177–202
A. M. Astashov, I. V. Astashova, A. V. Bocharov, V. M. Buchstaber, V. A. Vassiliev, A. Verbovetsky, A. M. Vershik, A. P. Veselov, M. M. Vinogradov, L. Vitagliano, R. F. Vitolo, Th. Th. Voronov, V. G. Kac, Y. Kosmann-Schwarzbach, I. S. Krasil'shchik, I. M. Krichever, A. P. Krishchenko, S. K. Lando, V. V. Lychagin, M. Marvan, V. P. Maslov, A. S. Mishchenko, S. P. Novikov, V. N. Rubtsov, A. V. Samokhin, A. B. Sossinsky, J. Stasheff, D. B. Fuchs, A. Ya. Khelemsky, N. G. Khor'kova, V. N. Chetverikov, A. S. Schwarz, “Alexandre Mikhailovich Vinogradov (obituary)”, Uspekhi Mat. Nauk, 75:2(452) (2020), 185–190; Russian Math. Surveys, 75:2 (2020), 369–375