### Book chapter

## Слайд-многочлены и комплексы подслов

We define simplicial complexes for slide polynomials and show that they are always homeomorphic to balls or spheres.

### In book

We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope.

In [K], a convex-geometric algorithm was introduced for building new analogs of Gelfand–Zetlin polytopes for arbitrary reductive groups. Conjecturally, these polytopes coincide with the Newton–Okounkov polytopes of flag varieties for a geometric valuation. I outline an algorithm (geometric mitosis) for finding collec- tion of faces in these polytopes that represent a given Schubert cycle. For GL_n and Gelfand–Zetlin polytopes, this algorithm reduces to a geometric version of Knutson–Miller mitosis introduced in [KST].

A new approach is described to the Schubert calculus on complete flag varieties, using the volume polynomial associated with Gelfand- Zetlin polytopes. This approach makes it possible to compute the intersection products of Schubert cycles by intersecting faces of a polytope. Bibliography: 23 titles.

Toric geometry exhibited a profound relation between algebra and topology on one side and combinatorics and convex geometry on the other side. In the last decades, the interplay between algebraic and convex geometry has been explored and used successfully in a much more general setting: first, for varieties with an algebraic group action (such as spherical varieties) and recently for all algebraic varieties (construction of Newton-Okounkov bodies). The main goal of the conference is to survey recent developments in these directions. Main topics of the conference are: Theory of Newton polytopes and Newton-Okounkov bodies; Toric geometry, geometry of spherical varieties, Schubert calculus, geometry of moduli spaces; Tropical geometry and convex geometry; Real algebraic geometry and fewnomial theory; Polynomial vector fields and the Hilbert 16th problem.

We construct generalized Newton polytopes for Schubert subvarieties in the variety of complete flags in C^n . Every such “polytope” is a union of faces of a Gelfand–Zetlin polytope (the latter is a well-known Newton–Okounkov body for the flag variety). These unions of faces are responsible for Demazure characters of Schubert varieties and were originally used for Schubert calculus.

I describe a convex geometric procedure for building generalized Newton polytopes of Schubert varieties. One of the goals is to extend to arbitrary reductive groups our joint work with Evgeny Smirnov and Vladlen Timorin on Schubert calculus (in type A) in terms of Gelfand-Zetlin polytopes.

The material of the present paper is grounded on the holist algebraic method (Q-analysis) proposed by English mathematician and physicist R.H.Atkin. At its core, the approach is aimed at both analysis of systems structures (in the form of simplicial complexes K, which is formed by a set of properly adjoined objects called simplexes) and calculation of numeric estimates of structural complexity of systems based on the results of such analysis.

Turning complexity estimate of system’s structure into a real number creates additional difficulties in the comparison of two different complexes because there is no real verbal scale, which would have been accustomed to human beings and would allow a group of experts to express opinions and draw easily conclusions about degree of complexity of K at each particular dimensional level of its analysis. Therefore, the present paper deals with consideration of the approach that is more focused on human perception of characteristics obtained, mental comprehension and formation (comparison) of personal constructs in psychological space (or, P-space) – modified structural complexity estimate is based right on notions of distance and similarity within psychological space.