Dynamical systems,
real and complex billiards,
integrable billiards,
periodic orbits,
ordinary differential equations in real and complex time,
holomorphic foliations,
Stokes phenomena,
model of Josephson junction in superconductivity.
Dynamical systems, real and complex billiards, analytic theory of ordinary differential equations in real and complex time, foliations, complex geometry and dynamics, transformation groups.
Main publications:
A.A.Glutsyuk, “On polynomially integrable Birkhoff billiards on surfaces of constant curvature”, Klassifikatsiya polinomialno integriruemykh ploskikh bilyardov: reshenie polinomialnoi versii gipotezy Birkhgofa, sformulirovannoi S.V.Bolotinym, J. Eur. Math. Society, 23:3 (2021), 994–1049
A.A.Glutsyuk, “On 4-reflective complex analytic planar billiards”, Vvedenie kompleksnykh bilyardov na dvumernoi kompleksnoi ploskosti. Klassifikatsiya kompleksnykh bilyardov s otkrytym mnozhestvom 4-periodicheskikh orbit. Primenenie k veschestvennym bilyardam, vklyuchaya reshenie dvumernoi gipotezy Tabachnikova o kommutiruyuschikh bilyardakh, J. Geom Analysis, 27:1 (2017), 183–238
A. Glutsyuk, “Instability of nondiscrete free subgroups in Lie groups”, Dokazano, chto vsyakuyu konechno-porozhdennuyu nediskretnuyu svobodnuyu podgruppu v proizvolnoi gruppe Li mozhno prevratit v nesvobodnuyu podgruppu skol ugodno malym vozmuscheniem obrazuyuschikh, Transformations groups, 16:2 (2011), 413–479
Yu. Bibilo; A. Glutsyuk, “On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation”, Preprint, 72 str, podan v pechat. Issledovano semeistvo differentsialnykh uravnenii na dvumernom tore, modeliruyuschee silno shuntirovannyi perekhod Dzhozefsona v sverkhprovodimosti. Resheny dve gipotezy o peremychkakh zon fazovogo zakhvata (o raspolozhenii na odnoi pryamoi i o polozhitelnosti) s pomoschyu novykh razrabotannykh metodakh, ispolzuyuschikh teoriyu kompleksnykh lineinykh uravnenii (yavlenie Stoksa i izomonodromnye deformatsii, opisyvaemye uravneniyami Penleve 3) i teoriyu bystro-medlennykh sistem, Preprint series arxiv.org (Working papers of Cornell University), 2020, https://arxiv.org/abs/2011.07839
A. A. Glutsyuk, Yu. L. Sachkov, Uspekhi Mat. Nauk (to appear)
2.
A. A. Glutsyuk, V. S. Matveev, “If a Minkowski billiard is projective, it is the standard billiard”, Matem. sb. (to appear)
2024
3.
A. A. Glutsyuk, “On extended model of Josephson junction, linear systems with polynomial solutions, determinantal surfaces and Painlevé 3 equations”, Topology, Geometry, Combinatorics, and Mathematical Physics, Collected papers. Dedicated to Victor Matveevich Buchstaber on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 326, Steklov Math. Inst., Moscow, 2024
4.
A. A. Glutsyuk, “On Hamiltonian projective billiards on boundaries of products of convex bodies”, Mathematical Aspects of Mechanics, Collected papers. Dedicated to Dmitry Valerevich Treschev on the occasion of his 60th birthday and to Sergey Vladimirovich Bolotin on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 327, Steklov Math. Inst., Moscow, 2024 (to appear)
2023
5.
Alexey Glutsyuk, “On germs of constriction curves in model of overdamped Josephson junction, dynamical isomonodromic foliation and Painlevé 3 equation”, Mosc. Math. J., 23:4 (2023), 479–513;
A. A. Glutsyuk, I. V. Izmestiev, S. L. Tabachnikov, “Four equivalent properties of integrable billiards”, Israel Journal of Mathematics, 241 (2021), 693–719 , arXiv: 1909.09028
Yu. P. Bibilo, A. A. Glutsyuk, “On families of constrictions in the model of an overdamped Josephson junction”, Russian Math. Surveys, 76:2 (2021), 360–362
2020
9.
A.A.Glutsyuk, “On commuting billiards in higher-dimensional spaces of constant curvature”, Pacific J. Math., 305:2 (2020), 577–595 , arXiv: 1807.10567
A.A.Glutsyuk, I. V. Netay, “On spectral curves and complexified boundaries of phase-lock areas in a model of Josephson junction”, J. Dyn. Control Systems, 26 (2020), 785–820 , arXiv: https://arxiv.org/abs/1908.08491;
A. V. Borisov, I. A. Dynnikov, A. A. Glutsyuk, A. E. Mironov, I. A. Taimanov, A. Yu. Vesnin, “The conference “Dynamics in Siberia”, Novosibirsk, February 24–29, 2020”, Sib. elektron. matem. izv., 17 (2020), 68–96
2019
12.
A. A. Glutsyuk, “On constrictions of phase-lock areas in model of overdamped Josephson effect and transition matrix of the double-confluent Heun equation”, J. Dyn. Control Systems, 25:3 (2019), 323–349 , arXiv: 1805.02624;
V. M. Buchstaber, A. A. Glutsyuk, “Total positivity, Grassmannian and modified Bessel functions”, Functional Analysis and Geometry: Selim Grigorievich Krein Centennial, Contemp. Math., 733, AMS, 2019, 97–107; ;
E. Bakaev, A. Glutsyuk, S. Dorichenko, A. Zaslavskii, P. Kozhevnikov, L. Mednikov, V. Novikov, V. Retinskii, A. Ryabichev, E. Rjabov, A. Semenov, M. Skopenkov, I. Frolov, A. Shapovalov, Kvant, 2019, no. 12, 34–35
15.
A. Bufetov, A. Glutsyuk, N. Goncharuk, S. Gusein-Zade, Yu. Kudryashov, S. Lando, A. Sosinskii, V. Timorin, M. Tsfasman, “Yuliyu Sergeevichu Ilyashenko — 75”, Mosc. Math. J., 19:2 (2019), 185–188
16.
I. A. Dynnikov, A. A. Glutsyuk, A. E. Mironov, I. A. Taimanov, A. Yu. Vesnin, “The Conference “Dynamics in Siberia”, Novosibirsk, February 25 – March 2, 2019”, Sib. elektron. matem. izv., 16 (2019), 16–39
I. A. Dynnikov, A. A. Glutsyuk, G. N. Kulipanov, A. E. Mironov, I. A. Taimanov, A. Yu. Vesnin, “The Conference «Dynamics in Siberia» dedicated to the 90th anniversary of B. V. Chirikov, Novosibirsk, February 26–March 4, 2018”, Sib. elektron. matem. izv., 15 (2018), 10–38
2017
21.
A. A. Glutsyuk, L. G. Rybnikov, “On families of differential equations on two-torus with all phase-lock areas”, Nonlinearity, 30 (2017), 61–72 , arXiv: 1505.06975
V. M. Buchstaber, A. A. Glutsyuk, “On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect”, Proc. Steklov Inst. Math., 297 (2017), 50–89;
24.
I. A. Dynnikov, A. A. Glutsyuk, A. E. Mironov, I. A. Taimanov, A. Yu. Vesnin, “The conference “Dynamics in Siberia”, Novosibirsk, February 26–March 4, 2017”, Sib. elektron. matem. izv., 14 (2017), 7–30
2016
25.
V. M. Buchstaber, A A Glutsyuk, “On determinants of modified Bessel functions and entire solutions of double confluent Heun equations”, Nonlinearity, 29 (2016), 3857–3870 , arXiv: 1509.01725;
A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov, I. V. Shchurov, “On the Adjacency Quantization in an Equation Modeling the Josephson Effect”, Funct. Anal. Appl., 48:4 (2014), 272–285
A. I. Bufetov, A. A. Glutsyuk, S. M. Gusein-Zade, A. S. Gorodetski, V. Yu. Kaloshin, A. G. Khovanskii, V. A. Kleptsyn, S. Yu. Nemirovskii, A. B. Sossinsky, M. A. Tsfasman, V. A. Vassiliev, S. Yu. Yakovenko, “Yulij Sergeevich Ilyashenko”, Mosc. Math. J., 14:2 (2014), 173–179
2012
30.
A. A. Glutsyuk, Yu. G. Kudryashov, “No planar billiard possesses an open set of quadrilateral trajectories”, Journal of Modern Dynamics, 6:3 (2012), 287–326
A. A. Glutsyuk, “Unique ergodicity of horospheric foliations revisited”, Journal of fixed point theory and its applications, 8, The Steve Smale Festschrift:1 (2010), 113–149
2007
36.
A. A. Glutsyuk, Yu. S. Ilyashenko, “Restricted version of the infinitesimal Hilbert 16th problem”, Mosc. Math. J., 7:2 (2007), 281–325;
A. A. Glutsyuk, “On convergence of generalized continued fractions and Ramanujan conjecture”, C. R. Math. Acad. Sci. Paris, 341:7 (2005), 427–432 , arXiv: math/0412445
D. V. Anosov, V. I. Arnol'd, A. A. Glutsyuk, A. S. Gorodetski, V. Y. Kaloshin, A. B. Katok, A. G. Khovanskii, S. K. Lando, A. B. Sossinski, M. A. Tsfasman, S. Yu. Yakovenko, “Yulij S. Ilyashenko”, Mosc. Math. J., 5:1 (2005), 3–4
2004
43.
A. A. Glutsyuk, “Simultaneous metric uniformization of foliations by Riemann surfaces”, Commentarii Mathematici Helvetici, 79:4 (2004), 704–752
44.
A. A. Glutsyuk, “Resonant confluence of singular points and Stokes phenomena”, Journal of Dynamical and Control Systems, 10:2 (2004), 253–302
A. A. Glutsyuk, “Confluence of singular points and the nonlinear Stokes Phenomena”, Transactions of Moscow Mathematical Society, 62 (2001), 49–95
2000
48.
A. A. Glutsyuk, “On the codimension of the set of one-dimensional polynomial foliations on $\mathbb{C}^ n$ and $\mathbb{CP}^n$ that do not satisfy a sufficient condition for the hyperbolicity of leaves”, St. Petersburg Math. J., 11:4 (2000), 567–587
A. A. Glutsyuk, “Limit sets at infinity for liftings of non-self-intersecting curves on the torus to the plane”, Math. Notes, 64:5 (1998), 579–589
1996
51.
A. A. Glutsyuk, “Hyperbolicity of the leaves of a generic one-dimensional holomorphic foliation on a nonsingular projective algebraic variety”, Proc. Steklov Inst. Math., 213 (1996), 83–103
1995
52.
A. A. Glutsyuk, “Asymptotic Stability of Linearizations of a Planar Vector Field with a Singular Point Implies Global Stability”, Funct. Anal. Appl., 29:4 (1995), 238–247
1994
53.
A. A. Glutsyuk, “The Hyperbolicity of Phase Curves of a Generic Polynomial Vector Field in $\mathbb{C}^n$”, Funct. Anal. Appl., 28:2 (1994), 77–84
54.
A. A. Glutsyuk, “A complete solution of the Jacobian problem for vector fields on the plane”, Russian Math. Surveys, 49:3 (1994), 185–186
1993
55.
A. A. Glutsyuk, “An analogue of Cayley's theorem for cyclic symmetric connected graphs with one cycle connected with generalized Lyashko–Looijenga covers”, Russian Math. Surveys, 48:2 (1993), 182–183
56.
A. A. Glutsyuk, “An analogue of Cayley's theorem for cyclically symmetric connected graphs with a single cycle that are associated with generalized Lyashko–Looijenga covers”, Russian Math. Surveys, 48:4 (1993), 247–248
On rationally integrable projective billiards A. A. Glutsyuk Conference «Hyperbolic Dynamics and Structural Stability» Dedicated to the 85th Anniversary of D. V. Anosov November 10, 2021 17:00
О квантовании перемычек в уравнениях Джозефсона A. A. Glutsyuk Seminar of the Department of Geometry and Topology "Geometry, Topology and Mathematical Physics", Steklov Mathematical Institute of RAS (Novikov Seminar) March 21, 2012 18:30
2024
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