A. Sergyeyev, On
local time-dependent symmetries of integrable evolution equations,
Proc.
3^{rd }Int. Conference "Symmetry in Nonlinear Mathematical Physics"
(Kyiv, July 13-18, 1999), published in Proceedings of Institute
of Mathematics of NAS of Ukraine, 2000, 30, Part 1, p.196-203.
A. Sergyeyev, Nonlocal symmetries and formal symmetries for evolution equations,
Group
and Analytic Methods in Mathematical Physics, Proceedings of Institute
of Mathematics of NAS of Ukraine, 2001, 35, p.227-236.
A. Sergyeyev, On time-dependent symmetries and formal symmetries of evolution
equations, in Symmetry and Perturbation Theory, A. Degasperis and
G. Gaeta eds., Singapore, World Scientific, 1999, p.303-308.
A.G. Sergyeyev, On the class of equations for the massless particles that
are invariant under Poincaré parasupergroup, in Proc. 2nd All-Ukrainian
Conference of young scientists (Kyiv, 16-18 May 1995), Kyiv, 1995,
p.41-46 (in Ukrainian).
C. Other Works
A. Sergyeyev, Time dependence and (non)commutativity of symmetries of evolution
equations, in "Noncommutative Structures in Mathematics in Physics"
(Proc.
NATO ARW), S. Duplij and J. Wess eds., Kluwer, 2001, p.379-390.
Preprint
version (fulltext in arxiv.org a.k.a.
LANL
preprint archive)
A.G. Sergyeyev, Higher Symmetries and Parasupersymmetries of Evolution
Equations, Ph.D. thesis, Kyiv, Institute of Mathematics of NAS of Ukraine,
2000, 130 p. (in Ukrainian).
A.G. Sergyeyev, On the structure of generalized symmetries for nonlinear
evolution equations, Visnyk Kyivs'kogo Universytetu (Bulletin of the
University of Kiev), Series: Physical and mathematical Sciences, 1999,
no.3, p.61-70 (in Ukrainian).
A.G. Sergheyev, On time-dependent symmetries of evolution equations, Symmetry
and Analytic Methods in Mathematical Physics, Proceedings of Institute
of Mathematics of NAS of Ukraine, 1998, 19, p. 216-220.
O.F. Kalaida, A.G. Sergyeyev, O.V. Galkin, New variant of matrix algorithms
of numerical integration for regular, weakly singular, and singular integrals,
Visnyk
Kyivs'kogo Universytetu (Bulletin of the University of Kiev), Series: Physical
and Mathematical Sciences, 1993, no.3, p.18-22 (in Ukrainian).
Fellowships and Grants
Team Grant MSM4781305904 of the Ministry of Education, Youth and Sports
of Czech Republic
The personal development project No. 254/b for the year 2004 of the Ministry
of Education, Youth and Sports of Czech Republic
Internal Grant IGS 1/2004 of Silesian University in Opava
Team Grant No. 201/04/0538 of the Czech Grant Agency (Principal Investigator:
Michal Marvan)
Postdoctoral Fellowship at Graduiertenkolleg "Geometrie und Nichtlineare
Analysis", Institut für
Mathematik, Humboldt-Universität zu Berlin (April 1 - December
31, 2001)
Team Grant No. 201/00/0724 of the Czech Grant Agency (Principal Investigator:
Demeter Krupka)
Team Grants MSM:J10/98:192400002 and VS 96003 "Global Analysis" of the
Ministry of Education, Youth and Sports of Czech Republic
ISSEP (Soros Students Program) Personal Grant GSU052172 (1995)
Personal Studentship from Kyiv National Taras Shevchenko University (1994--1996)
Main Scientific Results
Proof of maximal superintegrability for a large class of Benenti systems
(natural Hamiltonian dynamical systems admitting a special conformal Killing
tensor) -- joint work with M. Blaszak (see [A.3])
For a large class of (1+1)-dimensional evolution systems, new easily verifiable
sufficient conditions ensuring the locality of symmetries generated via
the repeated application of recursion operator (see [A.2,
A.6,
B.1]) or repeated commutation with master symmetry (see
[A.9, B.1]) were obtained. In particular,
using these results we proved the Maltsev--Novikov
conjecture on weak nonlocality of higher recursion operators, Hamiltonian
and symplectic structures of integrable systems in (1+1) dimensions (see
[A.2]).
A new description of Hamiltonian operators compatible with a given nondegenerate
Hamiltonian operator, applicable for both finite- and infinite-dimensional
case (see [A.5])
The complete classification of (1+1)-dimensional evolution systems admitting
a generalized conditional symmetry (GCS) of prescribed form, including
(unlike the known results of Svirshchevskii, Kamran,
Milson, and Olver and Doyle)
the case of explicitly time-dependent GCS (see [A.4,
A.10]).
All local generalized symmetries of the Bakirov system, including those
explicitly dependent on independent variables, were found. In particular,
it was shown that this system has only one non-Lie-point local generalized
symmetry. This result, generalizing earlier
work of Beukers, Sanders, and Wang, completes the refutation of the
popular conjecture that the existence of one non-Lie point generalized
symmetry for a system of PDEs implies the existence of infinitely many
such symmetries (see [A.12]).
An explicit formula for the leading term of commutator of two generalized
symmetries of sufficiently high order was obtained (see [A.11]).
This formula is valid
for a large class of (1+1)-dimensional evolution systems with constraints.
It possesses a number of useful applications and provides the complete
solution of A.M. Vinogradov's
problem of 'evaluation from the top' for the symmetry algebras of these
systems.
The first example of parasupersymmetric relativistic quantum-mechanical
system with non-oscillator-like interaction was found. The complete set
of eigenvalues and eigenvectors belonging to the discrete energy spectrum
for this model was found (see [C.5]).
Talks Given at the Following Conferences
10^{th} International Conference on Differential Geometry and its
Applications (August 27-31, 2007, Olomouc, Czech Republic)
Seventh International Conference "Symmetry in Nonlinear Mathematical Physics"
(June 24-30, 2007, Kyiv, Ukraine)
16^{th} International Colloquium "Quantum Groups and Integrable
Systems" (June 14-16, 2007, Prague, Czech Republic)
The 27^{th} Winter School "Geometry and Physics" (January 13-20,
2007, Srni, Czech Republic)
Eduard Čech Centre Meeting (November 10-11, 2006, Telč, Czech Republic)
15^{th} International Colloquium "Quantum Groups and Integrable
Systems" (June 15-17, 2006, Prague, Czech Republic)
Sixth International Conference "Symmetry in Nonlinear Mathematical Physics"
(June 20-26, 2005, Kyiv, Ukraine)
14^{th} International Colloquium "Quantum Groups and Integrable
Systems" (June 16-18, 2005, Prague, Czech Republic)
The 25^{th} Winter School "Geometry and Physics" (January 15-22,
2005, Srni, Czech Republic)
10^{th} International Conference on Modern Group Analysis (MOGRAN
X) (October 24-31, 2004, Larnaca, Cyprus)
9^{th} International Conference on Differential Geometry and its
Applications (August 30-September 3, 2004, Prague, Czech Republic)
11^{th} International Conference "Symmetry Methods in Physics"
(June 21-24, 2004, Prague, Czech Republic)
13^{th} International Colloquium "Quantum Groups and Integrable Systems" (June
17-19, 2004, Prague, Czech Republic)
Fifth International Conference "Symmetry in Nonlinear Mathematical Physics"
(June 23-29, 2003, Kyiv, Ukraine)
Fifth Negev Workshop on Applied Mathematics (June 17-21, 2003, Sede Boqer,
Israel)
The 23^{rd} Winter School "Geometry and Physics" (January 18-25,
2003, Srni, Czech Republic)
11^{th} International Colloquium "Quantum Groups and Integrable Systems" (June
20-22, 2002, Prague, Czech Republic)
The 22^{nd} Winter School "Geometry and Physics" (January 12-19,
2002, Srni, Czech Republic)
8^{th} International Conference on Differential Geometry and its
Applications (August 27-31, 2001, Opava, Czech Republic)
Banach Center Workshop "Multi-Hamiltonian Structures: Geometric and Algebraic
Aspects" (August 9-17, 2001, Bedlewo, Poland)
Fourth International Conference "Symmetry in Nonlinear Mathematical Physics"
(July 9-15, 2001, Kyiv, Ukraine)
The 21^{st} Winter School "Geometry and Physics" (January 13-20,
2001, Srní, Czech Republic)
NATO Advanced Research Workshop "Noncommutative Structures in Mathematics
and Physics" (September 24-28, 2000, Kyiv, Ukraine)
Third International Conference "Symmetry in Nonlinear Mathematical Physics"
(July 13-18, 1999, Kyiv, Ukraine)
XXXI Symposium on Mathematical Physics (May 18-21, 1999, Torun, Poland)
Workshop Recent Trends in Mathematics (Schlosshotel Klink, Germany,
November 23-25, 2001)
Advanced Course Variational Sequences and Bicomplexes (Opava, Czech
Republic, August 20-24, 2001)
The 5^{th} CIME Session Direct and Inverse Methods in Solving
Nonlinear Evolution Equations (Grand Hotel San Michele, Cetraro, Italy,
September 6-11, 1999)
Short Scientific Visits
Institute of Physics, Adam Mickiewicz University
(Poznan, Poland, May 20-June 9, 2007, host: Prof. M. Blaszak)
Centre de Recherches Mathématiques, Université de Montréal,
(Montréal, Québec, Canada, November 15-December 11, 2004,
host: Pavel Winternitz)
Universität Paderborn (Paderborn, Germany, December 5-6, 2001, host:
Benno
Fuchssteiner)
Vrije Universiteit van Amsterdam (Amsterdam, The Netherlands, February
18 - March 4, 2001, host: Jan A.
Sanders)
Mathematical Institute of Silesian University in Opava (Opava, Czech Republic,
January 10-21, 2000, host: Michal
Marvan)