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## Log terminal singularities, platonic tuples and iteration of Cox rings

Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension coming with a torus action of complexity one. In this setting, the previously finite groups become solvable torus extensions. As explicit examples, we investigate compound du Val threefold singularities. We give a complete classification and exhibit all the possible chains of iterated Cox rings.

Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We give a slightly improved version of Local Structure Theorems obtained by Knop and Timashev, which describe the action of some parabolic subgroup of $G$ on an open subset of $X$. We also extend various results of Vinberg and Timashev on the set of horospheres in $X$. We construct a family of nongeneric horospheres in $X$ and a variety $\Hor$ parameterizing this family, such that there is a rational $G$-equivariant symplectic covering of cotangent vector bundles $T^*\Hor \dashrightarrow T^*X$. As an application we recover the description of the image of the moment map of $T^*X$ obtained by Knop. In our proofs we use only geometric methods which do not involve differential operators.

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.

Cox rings are significant global invariants of algebraic varieties, naturally generalizing homogeneous coordinate rings of projective spaces. This book provides a largely self-contained introduction to Cox rings, with a particular focus on concrete aspects of the theory. Besides the rigorous presentation of the basic concepts, other central topics include the case of finitely generated Cox rings and its relation to toric geometry; various classes of varieties with group actions; the surface case; and applications in arithmetic problems, in particular Manin's conjecture. The introductory chapters require only basic knowledge in algebraic geometry. The more advanced chapters also touch on algebraic groups, surface theory, and arithmetic geometry. Each chapter ends with exercises and problems. These comprise mini-tutorials and examples complementing the text, guided exercises for topics not discussed in the text, and, finally, several open problems of varying difficulty.

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.

Fascinating and surprising developments are taking place in the classification of algebraic varieties. Work of Hacon and McKernan and many others is causing a wave of breakthroughs in the Minimal Model Program: we now know that for a smooth projective variety the canonical ring is finitely generated. These new results and methods are reshaping the field. Inspired by this exciting progress, the editors organized a meeting at Schiermonnikoog and invited leading experts to write papers about the recent developments. The result is the present volume, a lively testimony of the sudden advances that originate from these new ideas. This volume will be of interest to a wide range of pure mathematicians, but will appeal especially to algebraic and analytic geometers.

Preface

The workshop “Algebraic Varieties and Automorphism Groups” was held at the Research Institute of Mathematical Sciences (RIMS), Kyoto University during July 7-11, 2014. There were over eighty participants including twenty people from overseas Canada, France, Germany, India, Korea, Poland, Russia, Singapore, Switzerland, and USA.

Recently, there have been remarkable developments in algebraic geometry and related fields, especially, in the area of (birational) automorphism groups and algebraic group actions.

The purpose of this workshop was to provide the experts and young researchers with the opportunities to interact in the fields of affine and complete algebraic geometry, group actions and commutative algebra related to the topics listed below as well as to publicize the new results. We are confident that these purposes were achieved by the endeavors of the participants.

The main topics of the workshop were the following:

Algebraic varieties containing An-cylinders; Algebraic varieties with fibrations; Algebraic group actions and orbit stratifications on algebraic varieties; Automorphism groups and birational automorphism groups of algebraic varieties.There were 19 talks on the above and related topics by experts from the viewpoints of (affine) algebraic geometry, transformation groups, and commutative algebra. Inspired by the talks, there were active discussions and communication among participants during and after the workshop.

The present volume is the proceedings of the workshop and contains 15 articles on the workshop topics. We hope that this volume will contribute to the progress in the theories of algebraic varieties and their automorphism groups.

The workshop was financially supported by the RIMS and Grant- in-Aid for Scientific Research (B) 24340006, JSPS. We wish to thank all those who supported us in organizing the workshop and preparing this volume.

June, 2016

Kayo Masuda, Takashi Kishimoto, Hideo Kojima, Masayoshi Miyanishi, Mikhail Zaidenberg

All papers in this volume have been refereed and are in final form. No version of any of them will be submitted for publication elsewhere.

Exploring Bass’ Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass’ Triangulability Problem in the affirmative. To this end we prove a theorem on invariant subfields of 1-extensions. We also obtain a general construction of all rationally triangulable subgroups of the Cremona groups and, as an application, classify rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy.

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.

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.