### Working paper

## Degenerations, transitions and quantum cohomology

Let *F*_{λ} be a generalized flag variety of a simple Lie group *G* embedded into the projectivization of an irreducible *G*-module *V*_{λ}. We define a flat degeneration *F*_{λ}^{a}, which is a *G*_{a}^{M} variety. Moreover, there exists a larger group *G*^{a} acting on *F*_{λ}^{a}, which is a degeneration of the group *G*. The group *G*^{a} contains *G*_{a}^{M} as a normal subgroup. If *G* is of type *A*, then the degenerate flag varieties can be embedded into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of *F*_{λ}^{a} is generated by the set of degenerate Plüker relations. We prove that the coordinate ring of *F*_{λ}^{a} is isomorphic to a direct sum of dual PBW-graded *g*-modules. We also prove that there exist bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogues of semistandard tableaux.

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of *GL _{n}*. We construct the action of the Yangian of

*sl*in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of

_{n}*sl*[

_{n}*s*

^{±1},

*t*]) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analogue of the Gelfand-Tsetlin basis. The affine analogue of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space

*M*of torsion free sheaves on the plane, of rank

_{n,d}*n*and second Chern class

*d*, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center

*Z*of the Yangian of

*gl*naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on

_{n}*M*is the image of a noncommutative power sum in

_{n,d}*Z*.

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of *GL _{n}.* We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand-Tsetlin subalgebra of

*U*(

*gl*), and formulate a conjectural answer for the small quantum cohomology rings in terms of certain commutative shift of argument subalgebras of

_{n}*U*(

*gl*).

_{n}We prove the non-commutative Hodge-to-de Rham Degeneration Conjecture of Kontsevich and Soibelman.

The purpose of this paper is to assess the size of public sector within the Russian banking industry. We identify and classify at least 78 state-influenced banks. We distinguish between banks that are majority-owned by federal executive authorities or Central Bank of Russia, by sub-federal (regional and municipal) authorities, by state-owned enterprises and banks, and by "state corporations". We estimate their combined market share to have reached 56% of total assets by July 1, 2009. Banks indirectly owned by public capital are the fastest-growing group. Concentration is increasing within the public sector of the industry, with the top five state-controlled banking groups in possession of over 49% of assets. We observe a crowding out and erosion of domestic private capital, whose market share is shrinking from year to year. Several of the largest state-owned banks now constitute a de facto intermediate tier at the core of the banking system. We argue that the direction of ownership change in Russian banking is different from that in CEE countries.

Few economic events have caused such controversy as the privatization process in Russia. Some see it as the foundation of political and economic freedom. For others it was economics gone wrong, and ended in "Russians stealing money from their own country". As Russia reasserts itself, and its new brand of capitalism, it is ever more important that policy makers and scholars understand the roots of the economic structure and governance of that country; what was decided, who made the decisions and why, what actually transpired, and what implications this has for the future of Russia.

This work, written by two senior advisors to the Russian government, has unique access to documentation, tracking the decision making process in the Russian Mass Privatization process. By close reference to events, and supplemented by interviews with many of the key participants, it shows that the policies adopted were often influenced and shaped by different forces than those cited by current popular accounts. The book challenges the interpretation of Russian privatization by some of the West’s most eminent economists. It underlines that economists of all schools, who bring assumptions from the West to the analysis of Russia, may reach false or misleading conclusions. It is an essential guide for anyone interested in Russian economic reform, and anyone who seeks to understand this enigmatic country, and its actions today.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

The purpose of this paper is to carefully assess the size of public sector within the Russian banking industry. We identify and classify at least 78 state-influenced banks. For the state-owned banks, we distinguish between those that are majority-owned by federal executive authorities or Central Bank of Russia, by sub-federal (regional and municipal) authorities, by state-owned enterprises and banks, and by "state corporations". We estimate their combined market share to have reached 56% of total assets by July 1, 2009. Banks indirectly owned by public capital are the fastest-growing group. Concentration is increasing within the public sector of the industry, with the top five state-controlled banking groups in possession of over 49% of assets. We observe a crowding out and erosion of domestic private capital, whose market share is shrinking from year to year. Several of the largest state-owned banks now constitute a de facto intermediate tier at the core of the banking system. We argue that the direction of ownership change in Russian banking is different from that in CEE countries.

In this study, for the first time in Russian practice from a large amount of empirical data on state contracts for procurement of goods, works and services, made a major budgetary organization during 2008-2010., Examines factors affecting the decline in trading, delays in supplies, as well as problems in the performance of obligations under the contracts. The analysis showed that a reduction prices at the auctions directly dependent on the number of applications accepted for review by the competitive commissions. Falling prices are more frequent in the procurement of goods and experimental trust (compared to the benefits of the inspection), as well as a state contract for works. However, the prices are much less likely to have been lowered in auctions (compared to purchasing through quotations and tenders). Delays in supplies occurred in 27% of patients and were more frequent in the procurement of experimental benefits, and were characterized for major purchases and state contracts executed during the I-III quarters of the year. More serious problems in the performance of obligations, full fraught with supply disruptions, have characterized the state contracts, culminating in the IV quarter. The overall risk supply disruptions were reported only 5% of purchases at competitive procedures, but on the contracts accounted for nearly half of all purchases of the budget organization in 2008-2010. Based on the analysis in the formulation of recommendations to improve the system of public procurement.

The chapter aims to explore the special model of flagship universities that emerged in the former socialist Soviet system and analyse their contemporary transition. A planned state economy requires the development of a sophisticated hierarchical typology of higher education institutions with flagship universities positioned at the top of the hierarchy. The Soviet higher education model was different from those found in other parts of the world because of its officially assigned special leadership roles. Soviet flagship universities provided support for other universities in national or regional contexts. This support included the training of teaching staff, curriculum development, and quality control. Flagship universities had exclusive opportunities to conduct research. The first part of the chapter highlights the differences between these flagship universities and the rest of the higher education system.

After the collapse of Soviet Union the hierarchical model of Russian higher education has changed under the influence of market forces, private education, and increasing competition between universities. These changes have affected flagship universities. The second part of the paper examines the reasons why some former flagship universities lost their special role, why some flagship universities managed to keep or strengthen their role, and, why several new leadership universities have emerged. The chapter describes the transformation, changes of internal features and attributes of former leading universities on the path to the contemporary models of flagship universities. The end of the paper discusses the basic factors that allow leading universities in Russia to become contemporary flagship universities.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.