differential equation on complex manifolds; monodromy and isomonodromy deformation equations; KZ equation associated with different Coxeter groups and their monodromy; knots and links with additional symmetries and their finite order invarints; quantum integrable models of the Calogero–Moser–Sutherland type; bispectral problem for KZ operators and their connections Isomonodromic deformations.
For linear pfaffian systems of the Fuchs type on complex manifolds, the Frobenius conditions of the integrability were expressed across iterated Chen integrals and commutator relations in fundamental group of complement to the singularity divisor of the system. The multi-dimensional generalization of the Lappo–Danilevskii theory for fuchsian systems on complex projective spaces was formulated. For any hyperplane configuration in a complex projective space and an analytic deformation of the unit representation of the fundamental group of the complement to the hyperplane configuration the multi-dimensional Riemann–Hilbert problem is solved. The universal KZ operators (uKZ) and universal Calogero–Mozer–Sutherland(uCMS) hamiltonians associated to a root system are defined. On Bete and Dunkl manifolds are defined. The connections between uKZ and uCMS on Bete–Dunkl manifolds were studied. The $B_n$-Coxeter analog of the Drinfeld theory of the braided quasi-bialgebras and the $B_n$-analog of the main theorem in this theory (Drinfeld-Kohno theorem) was formulated and proved. On base of the $B_n$ quasi-bialgebra theory the universal Kontsevich–Vassiliev invarint of knots and links with additional order two symmetry was constructed. With point of view of the KZ equations the characterization of a simplest solutions of Schlesinger equations was given.
Graduated from Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University (MSU) in 1972 (department higher geometry and topology). Ph.D. thesis was defended in 1991. A list of my works contains more than 30 titles. Since 1990 I and V. A. Golubeva and E. E. Petrov have lrd the research seminar in Kolomna State Pedagogical Institute.
Golubeva V. A., Leksin V. P. On two types of representations of the braid groups associated with the Knizhnik–Zamolodchikov equations // Journal Dynamical and Control Systems, 1999, 5, no. 4, 565–596.
Lexin V. P. Monodromy for the KZ equations of the $B_n$ type and accompanying algebraic structures // Functional Differential Equations, 2001, 8, no. 3–4, 330–344.
Dmitrii Viktorovich Anosov (obituary) S. M. Aseev, V. M. Buchstaber, R. I. Grigorchuk, V. Z. Grines, B. M. Gurevich, A. A. Davydov, A. Yu. Zhirov, E. V. Zhuzhoma, M. I. Zelikin, A. B. Katok, A. V. Klimenko, V. V. Kozlov, V. P. Leksin, M. I. Monastyrskii, A. I. Neishtadt, S. P. Novikov, E. A. Sataev, Ya. G. Sinai, A. M. Stepin Uspekhi Mat. Nauk, 70:2(422) (2015), 181–191
Reminiscences about Andrei Andreevich Bolibrukh A. V. Chernavskii, V. P. Leksin, M. Butuzov, I. V. Vyugin, R. R. Gontsov, V. A. Poberezhnyi, Yu. S. Ilyashenko, A. G. Sergeev, S. P. Konovalov Uspekhi Mat. Nauk, 59:6(360) (2004), 207–215
Andrei Andreevich Bolibrukh (obituary) D. V. Anosov, V. I. Arnol'd, V. M. Buchstaber, V. A. Golubeva, A. A. Gonchar, A. B. Zhizhchenko, Yu. S. Ilyashenko, V. V. Kozlov, S. P. Konovalov, L. D. Kudryavtsev, V. P. Leksin, O. B. Lupanov, A. A. Mal'tsev, E. F. Mishchenko, S. P. Novikov, Yu. S. Osipov, M. M. Postnikov, V. A. Sadovnichii, A. G. Sergeev, L. D. Faddeev, A. V. Chernavskii Uspekhi Mat. Nauk, 58:6(354) (2003), 139–142