Page 691 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 691
GENERAL TOPICS
networks, improvement of traffic management, optimisation of the transport route system are
becoming especially important. The paper is dedicated to the method of constructing the short-
est movement trajectory on a 2D surface, for which the formulas can be obtained according to
GPS navigation or using the level lines information on the surface. The movement trajectory
from point A to point B considers the angle of the greatest upturn (downturn) where the parts of
the trajectory unfit for movement are eliminated. The mass, friction coefficient, tractive power
of the car engine of the moving vehicle on the surface at each point of the trajectory are known.
A numerical experiment was performed for the method.
Transformations of Hamiltonian systems connected with the fifth Painlevé
equation
Adam Ligeza, a.ligeza@mimuw.edu.pl
University of Warsaw, Poland
The talk will be about the Painlevé equations, especially about the fifth one PV . I am going
to present three different Hamiltonians and Hamiltonian systems connected with PV (KNY
Hamiltonian, Okamoto’s Hamiltonian and Rational Hamiltonian) and present a method how to
match them by using algebraic geometry tools. I will show how that can be done by matching
surface roots on the level of the Picard lattice. Moreover I will check whether our matching is
cannonical.
This is a joint work with Galina Filipuk, Anton Dzhamay and Alexander Stokes.
References
[1] https://dlmf.nist.gov/32
[2] K. Kajiwara, M. Noumi, and Y. Yamada, “Geometric aspects of Painlevé equations”, J.
Phys. A 50 (2017), no. 7, 073001, 164.
[3] K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, “From Gauss to Painlevé. A mod-
ern theory of special functions”, Aspects of Mathematics, E16, Friedr. Vieweg & Sohn,
Braunschweig, 1991.
[4] M. Noumi, “Painlevé Equations through Symmetry”, Translations of Mathematical Mono-
graphs, Vol. 233, American Mathematical Society, Providence, RI, 2004.
[5] K. Okamoto, “Sur les feuilletages associés aux équations du second ordre á points cri-
tiques fixes de P. Painlevé.(French)[On foliations associated with second - order Painlevé
equations with fixed critical points]”, Japan. J. Math.(N.S.) 5 (1979), no. 1, 1-79.
[6] K. Okamoto, “Studies on the Painlevé equations. III. Second and fourth Painlevé Equa-
tions PII and PIV .”, Math. Ann. 275 (2), pp. 221–255.
[7] H. Sakai, “Rational surfaces associated with affine root systems and geometry of the
Painlevé equations (1999)”, Comm. Math. Phys. 220 (2001), no. 1, 165–229.
[8] I. R. Shafarevich, “Basic Algebraic Geometry 1. Varieties in projective space”, Third ed.,
Springer, Heidelberg, 2013.
689
networks, improvement of traffic management, optimisation of the transport route system are
becoming especially important. The paper is dedicated to the method of constructing the short-
est movement trajectory on a 2D surface, for which the formulas can be obtained according to
GPS navigation or using the level lines information on the surface. The movement trajectory
from point A to point B considers the angle of the greatest upturn (downturn) where the parts of
the trajectory unfit for movement are eliminated. The mass, friction coefficient, tractive power
of the car engine of the moving vehicle on the surface at each point of the trajectory are known.
A numerical experiment was performed for the method.
Transformations of Hamiltonian systems connected with the fifth Painlevé
equation
Adam Ligeza, a.ligeza@mimuw.edu.pl
University of Warsaw, Poland
The talk will be about the Painlevé equations, especially about the fifth one PV . I am going
to present three different Hamiltonians and Hamiltonian systems connected with PV (KNY
Hamiltonian, Okamoto’s Hamiltonian and Rational Hamiltonian) and present a method how to
match them by using algebraic geometry tools. I will show how that can be done by matching
surface roots on the level of the Picard lattice. Moreover I will check whether our matching is
cannonical.
This is a joint work with Galina Filipuk, Anton Dzhamay and Alexander Stokes.
References
[1] https://dlmf.nist.gov/32
[2] K. Kajiwara, M. Noumi, and Y. Yamada, “Geometric aspects of Painlevé equations”, J.
Phys. A 50 (2017), no. 7, 073001, 164.
[3] K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, “From Gauss to Painlevé. A mod-
ern theory of special functions”, Aspects of Mathematics, E16, Friedr. Vieweg & Sohn,
Braunschweig, 1991.
[4] M. Noumi, “Painlevé Equations through Symmetry”, Translations of Mathematical Mono-
graphs, Vol. 233, American Mathematical Society, Providence, RI, 2004.
[5] K. Okamoto, “Sur les feuilletages associés aux équations du second ordre á points cri-
tiques fixes de P. Painlevé.(French)[On foliations associated with second - order Painlevé
equations with fixed critical points]”, Japan. J. Math.(N.S.) 5 (1979), no. 1, 1-79.
[6] K. Okamoto, “Studies on the Painlevé equations. III. Second and fourth Painlevé Equa-
tions PII and PIV .”, Math. Ann. 275 (2), pp. 221–255.
[7] H. Sakai, “Rational surfaces associated with affine root systems and geometry of the
Painlevé equations (1999)”, Comm. Math. Phys. 220 (2001), no. 1, 165–229.
[8] I. R. Shafarevich, “Basic Algebraic Geometry 1. Varieties in projective space”, Third ed.,
Springer, Heidelberg, 2013.
689
