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## Extended $r$-spin theory and the mirror symmetry for the $A_{r-1}$-singularity

By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau-Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of $r$-spin curves. In this paper we show that the parameters of the miniversal deformation, considered as functions of the flat coordinates, also have a simple geometric interpretation using the extended $r$-spin theory, first considered by T. J. Jarvis, T. Kimura and A. Vaintrob, and studied in a recent paper of E. Clader, R. J. Tessler and the author. We prove a similar result for the singularity $D_4$ and present conjectures for the singularities $E_6$ and $E_8$.

Proposed a model of financial bubbles and crises based upon the methodology of complex systems analysis. The irrationality of financial investors, as it was well known, had been empirically explained by «the greater fool theory». This process, in modern terms, was represented as the autocatalytic process leading to a system's singularity. It was shown how the procedures (slice and dice) of a CDO synthesis generated the excess growth of the securitized assets value. The latter being coupled with the high le-verage might produce the total collapse of a financial system. On a macrolevel the behaviour the of a system was modeled by a differential equation depending on three parameters. Such an outcome was explained on the system's microlevel as a process of financial percolation which was modeled, quite surprisingly, by the same equation of a Bernoulli type. Invariant constants of percolation were used to estimate different parameters of a model. The model application to the study of 2007-2010 credit crunch has given rise to the impressively coherent results in terms of probabilities and the return time periods of critical events that took place on the global financial markets.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 R. 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 of G is bijective; this answers Grothendieck's question. 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; this answers the other Grothendieck's question.

This book introduces a 'Big History' perspective to understand the acceleration of social, technological and economic trends towards a near-term singularity, marking a radical turning point in the evolution of our planet. It traces the emergence of accelerating innovation rates through global history and highlights major historical transformations throughout the evolution of life, humans, and civilization. The authors pursue an interdisciplinary approach, also drawing on concepts from physics and evolutionary biology, to offer potential models of the underlying mechanisms driving this acceleration, along with potential clues on how it might progress. The contributions gathered here are divided into five parts, the first of which studies historical mega-trends in relation to a variety of aspects including technology, population, energy, and information. The second part is dedicated to a variety of models that can help understand the potential mechanisms, and support extrapolation. In turn, the third part explores various potential future scenarios, along with the paths and decisions that are required. The fourth part presents philosophical perspectives on the potential deeper meaning and implications of the trend towards singularity, while the fifth and last part discusses the implications of the Search for Extraterrestrial Intelligence (SETI). Given its scope, the book will appeal to scholars from various disciplines interested in historical trends, technological change and evolutionary processes.

Proposed a model of financial bubbles and crises based upon the methodology of complex systems analysis. It was shown how the procedures (slice and dice) of a CDO synthesis generated the excess growth of the securitized assets value. The latter being coupled with the high leverage might produce the total collapse of a financial system. On a macrolevel of a system its behaviour was modeled by a differential equation depending on three parameters. The irrationality of financial investors, as it was well known, had been empirically explained by «the greater fool theory». This process, in modern terms, was represented as the autocatalytic process leading to a system's singularity. Such an outcome was explained on the system's microlevel as a process of financial percolation which was modeled, quite surprisingly, by the same equation of a Bernoulli type. Invariant constants of percolation were used to estimate different parameters of a model. The model application to the study of 2007-2010 credit crunch has given rise to the impressively coherent results in terms of probabilities and the return time periods of critical events that took place on the global financial markets.

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.

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas.

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.