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.
Biography
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.
Main publications:
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.
V. P. Leksin, “Linear Pfaffian Systems and Classical Solutions of Triangular Schlesinger Equations”, Trudy Mat. Inst. Steklova, 308 (2020), 210–221; Proc. Steklov Inst. Math., 308 (2020), 196–207
V. P. Leksin, “Kohno systems on Manin–Schechtman configuration spaces”, CMFD, 46 (2012), 120–128; Journal of Mathematical Sciences, 201:6 (2014), 796–804
4.
V. P. Leksin, “Multidimensional Jordan–Pochhammer systems and their applications”, Trudy Mat. Inst. Steklova, 278 (2012), 138–147; Proc. Steklov Inst. Math., 278 (2012), 130–138
D. V. Anosov, V. P. Leksin, “Andrei Andreevich Bolibrukh's works on the analytic theory of differential equations”, Uspekhi Mat. Nauk, 66:1(397) (2011), 3–36; Russian Math. Surveys, 66:1 (2011), 1–33
2009
6.
V. P. Leksin, “Topology of Complements of Hyperplane Arrangements and Isomonodromic Deformations of Fuchsian Systems”, Trudy Mat. Inst. Steklova, 267 (2009), 198–203; Proc. Steklov Inst. Math., 267 (2009), 190–194
2007
7.
V. P. Leksin, “Monodromy of Fuchsian Systems on Complex Linear Spaces”, Trudy Mat. Inst. Steklova, 256 (2007), 267–277; Proc. Steklov Inst. Math., 256 (2007), 253–262
V. P. Leksin, “The Lappo–Danilevskii method and trivial intersections of radicals in lower central series terms for certain fundamental groups”, Mat. Zametki, 79:4 (2006), 577–580; Math. Notes, 79:4 (2006), 533–536
S. V. Alenov, V. P. Leksin, “On Diagram Formulas for Knot Invariants”, Trudy Mat. Inst. Steklova, 252 (2006), 10–17; Proc. Steklov Inst. Math., 252 (2006), 4–11
V. P. Leksin, “A. A. Bolibrukh's works on multidimensional regular and Fuchsian systems”, Uspekhi Mat. Nauk, 59:6(360) (2004), 151–160; Russian Math. Surveys, 59:6 (2004), 1155–1164
V. A. Golubeva, V. P. Leksin, “Algebraic Characterization of the Monodromy of Generalized Knizhnik–Zamolodchikov Equations of $B_n$ Type”, Trudy Mat. Inst. Steklova, 238 (2002), 124–143; Proc. Steklov Inst. Math., 238 (2002), 115–133
V. P. Leksin, “Doubling Operations and Monodromy of Generalized Knizhnik–Zamolodchikov Equations”, Trudy Mat. Inst. Steklova, 236 (2002), 218–225; Proc. Steklov Inst. Math., 236 (2002), 206–213
V. A. Golubeva, V. P. Leksin, “Quadratic Relations in the Cohomology of Generalized Pure Braids and Dunkl Identities”, Funktsional. Anal. i Prilozhen., 30:2 (1996), 73–76; Funct. Anal. Appl., 30:2 (1996), 131–134
V. A. Golubeva, V. P. Leksin, “On reconstruction of Veselov–Felder formula in the theory of Calogero–Sutherland operators”, TMF, 106:1 (1996), 62–75; Theoret. and Math. Phys., 106:1 (1996), 50–60
M. P. Zamakhovskii, V. P. Leksin, “Vector version of the Matsuo–Cherednik equation for the root system $G_ 2$”, Mat. Zametki, 58:3 (1995), 456–460; Math. Notes, 58:3 (1995), 992–996
V. A. Golubeva, V. P. Leksin, “Complete integrability of generalized Knizhnik–Zamolodchikov equations of trigonometric type”, Uspekhi Mat. Nauk, 50:3(303) (1995), 151–152; Russian Math. Surveys, 50:3 (1995), 615–617
1993
17.
V. A. Golubeva, V. P. Leksin, “Generalized Knizhnik–Zamolodchikov equations and the factorization of their solutions”, Mat. Zametki, 54:5 (1993), 25–34; Math. Notes, 54:5 (1993), 1100–1105
1991
18.
V. A. Golubeva, V. P. Leksin, “Exponential representation of solutions of Pfaffian systems (the one-parameter case)”, Differ. Uravn., 27:9 (1991), 1526–1538; Differ. Equ., 27:9 (1991), 1078–1088
19.
V. P. Leksin, “Riemann–Hilbert problem for analytic families of representations”, Mat. Zametki, 50:2 (1991), 89–97; Math. Notes, 50:2 (1991), 823–838
V. P. Leksin, “Meromorphic Pfaffian systems on complex projective spaces”, Mat. Sb. (N.S.), 129(171):2 (1986), 201–217; Math. USSR-Sb., 57:1 (1987), 211–227
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, “Dmitrii Viktorovich Anosov (obituary)”, Uspekhi Mat. Nauk, 70:2(422) (2015), 181–191; Russian Math. Surveys, 70:2 (2015), 369–381
2004
22.
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, “Reminiscences about Andrei Andreevich Bolibrukh”, Uspekhi Mat. Nauk, 59:6(360) (2004), 207–215; Russian Math. Surveys, 59:6 (2004), 1213–1224
23.
D. V. Anosov, V. P. Leksin, “Andrei Andreevich Bolibrukh in life and science (30 January 1950 – 11 November 2003)”, Uspekhi Mat. Nauk, 59:6(360) (2004), 3–22; Russian Math. Surveys, 59:6 (2004), 1009–1028
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, “Andrei Andreevich Bolibrukh (obituary)”, Uspekhi Mat. Nauk, 58:6(354) (2003), 139–142; Russian Math. Surveys, 58:6 (2003), 1185–1189
Schlesinger systems with a basis of hypergeometric solutions V. P. Leksin International Conference “Differential Equations and Optimal Control” dedicated to the centenary of the birth of Academician Evgenii Frolovich Mishchenko June 8, 2022 12:00
Заседание посвящено памяти Дмитрия Викторовича Аносова A. M. Stepin, L. M. Lerman, V. Z. Grines, S. Yu. Pilyugin, M. I. Zelikin, V. P. Leksin, A. V. Klimenko, Yu. S. Ilyashenko Meetings of the Moscow Mathematical Society April 7, 2015
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