相关论文: Symplectic and Lie algebraic techniques in Geometr…

We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.

In this paper we apply the Symplectic Projectors Method to gauge theories with second class constraints in various space-time dimensions. The conclusion is that this method is equivalent to the standard quantization methods. Although it…

This is the preliminary manuscript of a book on symplectic field theory based on a lecture course for PhD students given in 2015-16. It covers the essentials of the analytical theory of punctured pseudoholomorphic curves, taking the…

For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the…

A new method of algebraic nature is proposed for the study of the asymptotic properties of special polynomials. The technique we foresee is based on the use of umbral operators, allowing a unified treatment of a large body of polynomial…

In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on…

A presymplectic structure on odd dimensional manifold is given by a closed 2-form which is nondegenerate, i.e., of maximal rank. We investigate geometry of presymplectic manifolds. Some basic theorems analogous to those in symplectic and…

The purpose of this note is to make some connection between the sub-Riemannian geometry on Carnot-Caratheodory groups and symplectic geometry. We shall concentrate here on the Heisenberg group, although it is transparent that almost…

We compare the performances of symplectic and non-symplectic integrators for the computation of normal geodesics and conjugate points in sub-Riemannian geometry at the example of the Martinet case. For this case study we consider first the…

An algebraic technique adapted to the problems of the fundamental theoretical physics is presented. The exposition is an elaboration and an extension of the methods proposed in previous works by the aut

We construct Grassmann spaces associated with the incidence geometry of regular and tangential subspaces of a symplectic copolar space, show that the underlying metric projective space can be recovered in terms of the corresponding…

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised…

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to…

This paper develops new aspects of the interplay between shifted symplectic geometry and classical Poisson geometry, focusing on lagrangian morphisms into 2-shifted symplectic groups. We establish a Lie-type correspondence between such…

These are lecture notes from my talks at the "Current Developments in Mathematics" conference (Harvard, 2006). They cover a variety of topics involving symplectic cohomology. In particular, a discussion of (algorithmic) classification…

The aim of this paper is to present remarkable classes of Lie-admissible algebras containing in particular the associative algebras, the Vinberg algebras and pre-Lie algebras. We determine the associated quadratic operads and their dual…

We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.

New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the…

A study is made of real Lie algebras admitting compatible complex and product structures, including numerous 4-dimensional examples. If g is a Lie algebra with such a structure then its complexification has a hypercomplex structure. It is…

The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the…