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Multiplicities of Maxwell sets of Pham singularities
Functional Analysis and Its Applications. 2016. Vol. 50. No. 3. P. 225–227.
A two-sided estimate of local multiplicities of Maxwell sets of isolated singularities of smooth functions is proved. This estimate is sharp for semi-homogeneous functions.
Glutsyuk A., Inventiones Mathematicae 2026 Vol. 245 P. 977–1058
A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture, studied by many mathematicians, states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then it is an ellipse. In the paper we study ...
Added: September 8, 2026
Glutsyuk A., Regular and Chaotic Dynamics 2026 Vol. 31 No. 1 P. 132–149
A caustic of a billiard is a curve whose tangent lines are reflected to its own tangent lines. A billiard is called Birkhoff caustic-integrable, if there exists a topological annulus adjacent to its boundary from inside that is foliated by closed caustics. The famous Birkhoff Conjecture, studied by many mathematicians, states that the only Birkhoff ...
Added: January 27, 2026
Vassiliev V., , in: Handbook of Geometry and Topology of Singularities VII.: Springer, 2025.
Classification of real function singularity and topological properties of the spaces of their singular and non-singular perturbations are described ...
Added: October 31, 2025
Vassiliev V., Journal of Singularities 2024 Vol. 27 P. 47–67
Spaces of Morse perturbation of real simple and parabolic function singularities are studied, in particular the classification of isotopy classes for simple singularities is completed ...
Added: October 31, 2025
Glutsyuk A., Ergodic Theory and Dynamical Systems 2024 Vol. 44 No. 5 P. 1418–1467
Reflection in a strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reflected by the billiard to lines tangent to C. The famous Birkhoff conjecture states that the only strictly convex billiards with a foliation by ...
Added: December 29, 2023
Campillo A., Delgado F., Sabir M. Gusein-Zade, Bulletin of the London Mathematical Society 2024 Vol. 56 No. 1 P. 449–459
There exist several equivalent equations for the Poincaré series of a collection of valuations on the ring of germs of functions on a complex analytic variety. We give definitions of the Poincaé series of a collection of valuations in the real setting (i.e., on the ring of germs of functions on a real analytic variety), ...
Added: December 5, 2023
Glutsyuk A., / Series arXiv "math". 2021.
A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then the billiard is an ellipse. It was studied by many ...
Added: February 3, 2022
Glutsyuk A., / Series "Working papers by Cornell University". 2021. No. 2104.01362.
Reflection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reflected by the billiard to lines tangent to C. The famous Birkhoff conjecture states that the only strictly convex billiards with a foliation by closed ...
Added: November 4, 2021
Glutsyuk A., Izmestiev I., Tabachnikov S., Israel Journal of Mathematics 2021 Vol. 241 P. 693–719
By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they ...
Added: October 19, 2020
Glutsyuk A., Pacific Journal of Mathematics 2020 Vol. 305 No. 2 P. 577–595
We consider two nested billiards in ℝd, d≥3, with C2-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due ...
Added: October 12, 2019
Vassiliev V., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 825–836
The local multiplicities of the caustics, the Maxwell sets, and the complex Stokes' sets in the spaces of versal deformations of Pham singularities (that is, of germs of holomorphic functions C^n --> C^1 which are expressed by the sums of degrees of the coordinate functions) are calculated ...
Added: December 27, 2017
Vassiliev V.A, Bulletin of the London Mathematical Society 2012 Vol. 44 No. 4 P. 637–641
The Arnold theorem (generalizing a consideration by Jacobi) states that on a generic Riemannian surface, which is sufficiently close to a sphere, the kth caustic of a generic point has at least four semi-cubical vertices. We prove this fact by the methods of the Morse theory; in particular, we replace the previous analytical condition of ...
Added: February 4, 2013