### Working paper

## Log terminal singularities, platonic tuples and iteration of 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.

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.

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.

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.

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.

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.