• A
• A
• A
• ABC
• ABC
• ABC
• А
• А
• А
• А
• А
Regular version of the site
Of all publications in the section: 7
Sort:
by name
by year
Article
Glutsyuk A. Israel Journal of Mathematics. 2021.

Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary infinitely-smooth hypersurface in Euclidean space that is either a global strictly  convex closed hypersurface, or a germ of hypersurface. We deal with the pseudogroup generated by compositional ratios of reflections from it and of reflections from its small deformations. In the case of  global convex hypersurface, we show that the latter pseudogroup is  dense  in the pseudogroup of Hamiltonian diffeomorphisms between subdomains of the phase cylinder: the space of oriented lines intersecting the hypersurface transversally. We prove an analogous local result for a  germ of hypersurface. The derivatives of the  above compositional differences in the deformation parameter are Hamiltonian vector fields calculated by Ron Perline. To prove the main results, we  find the Lie algebra generated by them and prove its $C^{\infty}$-density in the Lie algebra of Hamiltonian vector fields. We also prove analogues of the above results for hypersurfaces in Riemannian manifolds.

Article
Khoroshkin S. M., Огиевецкий О. В. Israel Journal of Mathematics. 2017. Vol. 221. P. 705-729.

We describe the diagonal reduction algebra D(gl(n)) of the Lie algebra gln in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl(n)).

Article
Gavrilovich M., Hasson A. Israel Journal of Mathematics. 2015. Vol. 209.

We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of niteness, countability and in nite equi-cardinality. We argue that from the homotopy theoretic point of view our construction is essentially automatic following basic existing methods, and so is (almost all) the veri cation that the construction works.

We use the posetal model category to introduce homotopy-theoretic intu- itions to set theory. Our main observation is that the homotopy invariant version of cardinality is the covering number of Shelah's PCF theory, and that other combinatorial objects, such as Shelah's revised power function - the cardinal function featuring in Shelah's revised GCH theorem | can be obtained using similar tools. We include a small \dictionary" for set theory in QtNaamen, hoping it will help in nding more meaningful homotopy-theoretic intuitions in set theory.

Article
Alexey Talambutsa, Bucher M. Israel Journal of Mathematics. 2016. Vol. 212. No. 2. P. 521-546.
Article
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 form a Liouville net. Let us say that a geodesically convex curve on a Riemannian surface has the Poritsky property if it can be parametrized in such a way that all of its string diffeomorphisms are shifts with respect to this parameter. In 1950, Poritsky has shown that the only closed plane curves with this property are ellipses. In the present article we show that a curve on a Riemannian surface has the Poritsky property if and only if it is a coordinate curve of a Liouville net. We also recall Blaschke's derivation of the Liouville property from the Ivory property and his proof of Weihnacht's theorem: the only Liouville nets in the plane are nets of confocal conics and their degenerations. This suggests the following generalization of Birkhoff's conjecture: If an interior neighborhood of a closed geodesically convex curve on a Riemannian surface is foliated by billiard caustics, then the metric in the neighborhood is Liouville, and the curve is one of the coordinate lines.