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Regular version of the site
Of all publications in the section: 5
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Article
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 the 'sufficient closeness to the sphere' by a geometric one, which probably is considerably less restrictive.

Added: Feb 4, 2013
Article
Gusein-Zade S. Bulletin of the London Mathematical Society. 2018. Vol. 50. No. 2. P. 261-273.

Homological index of a holomorphic 1‐form on a complex‐analytic variety with an isolated singular point is an analogue of the usual index of a 1‐form on a non‐singular manifold. One can say that it corresponds to the top Chern number of a manifold. We offer a definition of homological indices for collections of 1‐forms on a (purely dimensional) complex‐analytic variety with an isolated singular point corresponding to other Chern numbers. We also define new invariants of germs of complex‐analytic varieties with isolated singular points related to ‘vanishing Chern numbers’ at them.

Added: Oct 27, 2020
Article
Gorsky E., Гусейн-Заде С. М. Bulletin of the London Mathematical Society. 2018. Vol. 50. No. 2. P. 261-273.

Homological index of a holomorphic 1-form on a complex analytic variety with an isolated singular point is an analogue of the usual index of a 1-form on a non-singular manifold. One can say that it corresponds to the top Chern number of a manifold. We offer a definition of homological indices for collections of 1-forms on a (purely dimensional) complex analytic variety with an isolated singular point corresponding to other Chern numbers. We also define new invariants of germs of complex analytic varieties with isolated singular points related to "vanishing Chern numbers" at them.

Added: Feb 4, 2018
Article
V.A. Vassiliev. Bulletin of the London Mathematical Society. 2015. Vol. 47. No. 2. P. 290-300.

We prove that there are no bounded domains with smooth boundaries in even-dimensional Euclidean spaces, such that the volumes cut off from them by affine hyperplanes depend algebraically on these hyperplanes. For convex ovals in R2, this is Lemma XXVIII from Newton's ‘Philosophiae naturalis principia mathematica’.

Added: Jan 19, 2016
Article
Buryak A., Rossi P. Bulletin of the London Mathematical Society. 2021.

In this paper we compute the intersection number of two double ramification (DR) cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic DR integrals are the main ingredients for the computation of the DR hierarchy associated to the infinite‐dimensional partial cohomological field theory given by exp(μ^2 Θ) , where μ is a parameter and Θ is Hain's theta class, appearing in Hain's formula for the DR cycle on the moduli space of curves of compact type. This infinite rank DR hierarchy can be seen as a rank 1 integrable system in two space and one time dimensions. We prove that it coincides with a natural analogue of the Korteweg‐de‐Vries (KdV) hierarchy on a noncommutative Moyal torus.

Added: Feb 1, 2021