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## Schubert calculus for algebraic cobordism

Journal fuer die reine und angewandte Mathematik. 2011. No. 656. P. 59-85.

Kiritchenko V., Hornbostel J.

We establish a Schubert calculus for Bott-Samelson resolutions in the algebraic cobordism ring of a complete flag variety G/B.

Kiritchenko V., Smirnov E., Timorin V., Успехи математических наук 2012 Т. 67 № 4 С. 89-128

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. ...

Added: September 19, 2012

Valentina Kiritchenko, Mathematical Research Letters 2016 Vol. 23 No. 4 P. 1069-1096

We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand{Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipe dreams introduced by Knutson and Miller. For Sp_4 and a Newton{Okounkov polytope of the symplectic flag variety, ...

Added: February 25, 2016

Kiritchenko V., Padalko M., Schubert calculus on Newton-Okounkov polytopes / Cornell University. Series arXiv "math". 2018.

A Newton-Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytope so that the intersection product of cycles corresponds to the set-theoretic intersection of faces (whenever the latter are transverse). We explain the ...

Added: October 15, 2019

Kiritchenko V., Timorin V., Smirnov E., Oberwolfach Reports 2011 Vol. 8 No. 3 P. 2341-2344

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 ...

Added: November 17, 2012

Kiritchenko Valentina, Krishna A., Transformation Groups 2013 Vol. 18 No. 2 P. 391-413

We obtain an explicit presentation for the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation for the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational equivariant cobordism rings of wonderful symmetric varieties of minimal rank. ...

Added: February 18, 2013

Kiritchenko V., Geometric mitosis / Cornell University. Series math "arxiv.org". 2014.

We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand--Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipe dreams introduced by Knutson and Miller. For Sp_4 and a Newton--Okounkov polytope of the symplectic flag variety, ...

Added: September 17, 2014

Mathematical Society of Japan, 2016

This volume contains the proceedings of the 5th MSJ Seasonal Institute on Schubert Calculus, held at Osaka City University, from September 17–27, 2012. It is recommended for all researchers and graduate students who are interested in Schubert calculus and its many connections and applications to related areas of mathematics, such as geometric representation theory, combinatorial ...

Added: October 19, 2020

Sechin P., International Mathematics Research Notices 2017 P. 1-24

In this article, we calculate the ring of unstable (possibly nonadditive) operations from algebraic Morava K-theory K(n)^∗ to Chow groups with ℤ_(p) -coefficients. More precisely, we prove that it is a formal power series ring on generators c_i:K(n)^∗→CH^i⊗ℤ_(p) , which satisfy a Cartan-type formula. ...

Added: March 25, 2017

M. : Higher School of Economics Publishing House, 2012

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 ...

Added: November 17, 2012

Valentina Kiritchenko, , in : Oberwolfach Reports. Vol. 11. Issue 2.: Zürich : European Mathematical Society Publishing house, 2014. P. 1484-1487.

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 ...

Added: June 23, 2014

Sechin P., On the Structure of Algebraic Cobordism / Cornell University. Series arXiv "math". 2018.

In this paper we investigate the structure of algebraic cobordism of Levine-Morel as a module over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with respect to the topological filtration Ω∗_(r)(X) are unions of finitely presented 𝕃-modules of very specific structure. Namely, these ...

Added: December 6, 2018

Kiritchenko V., , in : Oberwolfach Reports. Issue 9.: Oberwolfach : European Mathematical Society Publishing house, 2012. P. 5-7.

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. ...

Added: November 17, 2012

Evgeny Smirnov, Anna Tutubalina, European Journal of Combinatorics 2023 Vol. 107 Article 103613

Schubert polynomials for the classical groups were defined by S.Billey and M.Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety over a classical group. We provide a combinatorial description for these polynomials, as well as their double versions, by introducing analogues of pipe dreams, or RC-graphs, for Weyl groups ...

Added: September 14, 2022

Smirnov E., В кн. : Тезисы докладов седьмой школы-конференции "Алгебры Ли, алгебраические группы и теория инвариантов". : Самара : Инсома-пресс, 2018. С. 43-44.

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

Added: October 7, 2019

Kiritchenko V., Smirnov E., Timorin V., Russian Mathematical Surveys 2012 Vol. 67 No. 4 P. 685-719

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. ...

Added: February 4, 2013

Kiritchenko V., Квант 2014 № 1 С. 2-6

The article popularizes Schubert's method (Schubert calculus) for solving enumerative geometry problems. In particular, this method is applied to the classical problem on the number of lines that intersect 4 given lines in 3-space. The article in intended for high school students. ...

Added: May 16, 2014

Kiritchenko V., Padalko M., , in : Interactions with Lattice Polytopes Magdeburg, Germany, September 2017. : Springer, 2022. P. 233-249.

A Newton–Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytope so that the intersection product of cycles corresponds to the set-theoretic intersection of faces (whenever the latter are transverse). We explain the ...

Added: January 31, 2023

191574970, Functional Analysis and Its Applications 2006 Vol. 40 No. 2 P. 81-90

It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...

Added: September 23, 2016

Losev A. S., Slizovskiy S., JETP Letters 2010 Vol. 91 P. 620-624

Added: February 27, 2013

Ilyashenko Y., Яковенко С. Ю., М. : МЦНМО, 2013

Предлагаемая книга—первый том двухтомной монографии, посвящённой аналитической теории дифференциальных уравнений.
В первой части этого тома излагается формальная и аналитическая теория нормальных форм и теорема о разрешении особенностей для векторных полей на плоскости.
Вторая часть посвящена алгебраически разрешимым локальным задачам теории аналитических дифференциальных уравнений , квадратичным векторным полям и проблеме локальной классификации ростков векторных полей в комплексной области ...

Added: February 5, 2014

Kalyagin V.A., Koldanov A.P., Koldanov P.A. et al., Physica A: Statistical Mechanics and its Applications 2014 Vol. 413 No. 1 P. 59-70

A general approach to measure statistical uncertainty of different filtration techniques for market network analysis is proposed. Two measures of statistical uncertainty are introduced and discussed. One is based on conditional risk for multiple decision statistical procedures and another one is based on average fraction of errors. It is shown that for some important cases ...

Added: July 19, 2014

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020