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, “On two-dimensional polynomially integrable billiards on surfaces of constant curvature”, Doklady Mathematics, 98:1 (2018), 382–385
4.
A. A. Glutsyuk, Yu. G. Kudryashov, “On quadrilateral orbits in planar billiards”, Doklady Mathematics, 83:3 (2011), 371–373
5.
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. 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
9.
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
10.
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
11.
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
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, “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
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
26.
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. 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
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
36.
Alexey A. Glutsyuk, “On Hamiltonian Projective Billiards on Boundaries of Products of Convex Bodies”, Proc. Steklov Inst. Math., 327 (2024), 37–54
37.
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, “On convergence of generalized continued fractions and Ramanujan conjecture”, C. R. Math. Acad. Sci. Paris, 341:7 (2005), 427–432 , arXiv: math/0412445
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
41.
A. A. Glutsyuk, C. Sabot, “Stokes matrices of hypergeometric integrals”, Annales de l'Institut Fourier, 60:1 (2010), 291–317 , arXiv: 0712.1707
42.
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
43.
A. A. Glutsyuk, “Simultaneous metric uniformization of foliations by Riemann surfaces”, Commentarii Mathematici Helvetici, 79:4 (2004), 704–752
44.
A. A. Glutsyuk, “Confluence of singular points and the nonlinear Stokes Phenomena”, Transactions of Moscow Mathematical Society, 62 (2001), 49–95
45.
A. A. Glutsyuk, Yu. S. Ilyashenko, “Uniqueness theorem for completely non-degenerate B-groups”, Izv. RAN. Ser. matem., 90:1 (2026) (to appear) ; (to appear)
46.
A. A. Glutsyuk, Yu. L. Sachkov, “Sub-Riemannian geodesics on the 3-dimensional Heisenberg nilmanifold”, Russian Math. Surveys, 80:1 (2025), 144–146
47.
A. A. Glutsyuk, V. S. Matveev, “Esli billiard Minkovskogo yavlyaetsya proektivnym, to on est standartnyi billiard”, Matem. sb., 216:5 (2025), 64–82;
48.
Alexey A. Glutsyuk, “Extended Model of Josephson Junction, Linear Systems with Polynomial Solutions, Determinantal Surfaces, and Painlevé III Equations”, Proc. Steklov Inst. Math., 326 (2024), 90–132
49.
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
50.
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
51.
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
52.
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
53.
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
54.
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
55.
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
56.
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
57.
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
58.
T. Bakiev, A. I. Bufetov, V. A. Vassiliev, S. M. Voronin, A. A. Glutsyuk, A. S. Gorodetski, A. V. Dukov, V. Yu. Kaloshin, A. V. Klimenko, V. V. Kozlov, S. B. Kuksin, S. K. Lando, V. S. Oganesyan, G. I. Olshanski, S. Yu. Pilyugin, O. V. Pochinka, Ya. G. Sinai, A. S. Skripchenko, A. L. Skubachevskii, I. A. Taimanov, V. A. Timorin, V. M. Tikhomirov, D. V. Treschev, D. A. Filimonov, K. M. Khanin, H. Hedenmalm, A. G. Khovanskii, M. A. Tsfasman, A. I. Shafarevich, I. S. Shilin, S. Yu. Yakovenko, “On the 80th birthday of Yulij Sergeevich Ilyashenko”, Russian Math. Surveys, 80:2 (2025), 345–357
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
