This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well.

The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces.

The book does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves – such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points – are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion.

Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework.

I show that Hurwitz numbers may be generated by certain correlation functions which appear in quantum chaos.

The assembly process is extremely complex for aircraft and its management requires to address numerous optimization problems related to the assignment of tasks to workstations, staffing problem for each workstation and finally the assignment of tasks to operators at each workstation. This paper treats the latter problem dealing with the assignment of tasks to operators under ergonomic constraints. The problem of optimal tasks scheduling in aircraft assembly line is modelled as Resource-Constrained Project Scheduling Problem (RCPSP). The objective of this research is to assign tasks to operators and to find an optimal schedule of task processing under economic and ergonomic constraints. Two different models to solve this problem are presented and evaluated on an industrial case study.

This volume, dedicated to the memory of the great American mathematician Bertram Kostant (May 24, 1928 – February 2, 2017), is a collection of 19 invited papers by leading mathematicians working in Lie theory, representation theory, algebra, geometry, and mathematical physics. Kostant’s fundamental work in all of these areas has provided deep new insights and connections, and has created new fields of research. This volume features the only published articles of important recent results of the contributors with full details of their proofs. Key topics include: Poisson structures and potentials (A. Alekseev, A. Berenstein, B. Hoffman) Vertex algebras (T. Arakawa, K. Kawasetsu) Modular irreducible representations of semisimple Lie algebras (R. Bezrukavnikov, I. Losev) Asymptotic Hecke algebras (A. Braverman, D. Kazhdan) Tensor categories and quantum groups (A. Davydov, P. Etingof, D. Nikshych) Nil- Hecke algebras and Whittaker D-modules (V. Ginzburg) Toeplitz operators (V. Guillemin, A. Uribe, Z. Wang) Kashiwara crystals (A. Joseph) Characters of highest weight modules (V. Kac, M. Wakimoto) Alcove polytopes (T. Lam, A. Postnikov) Representation theory of quantized Gieseker varieties (I. Losev) Generalized Bruhat cells and integrable systems (J.-H. Liu, Y. Mi) Almost characters (G. Lusztig) Verlinde formulas (E. Meinrenken) Dirac operator and equivariant index (P.-É. Paradan, M. Vergne) Modality of representations and geometry of-groups (V. L. Popov) Distributions on homogeneous spaces (N. Ressayre) Reduction of orthogonal representations (J.- P. Serre).

This book constitutes extended, revised and selected papers from the 7th International Conference on Optimization Problems and Their Applications, OPTA 2018, held in Omsk, Russia in July 2018. The 27 papers presented in this volume were carefully reviewed and selected from a total of 73 submissions. The papers are listed in thematic sections, namely location problems, scheduling and routing problems, optimization problems in data analysis, mathematical programming, game theory and economical applications, applied optimization problems and metaheuristics.

Control of Discrete-Time Descriptor Systems takes an anisotropy-based approach to the explanation of random input disturbance with an information-theoretic representation. It describes the random input signal more precisely, and the anisotropic norm minimization included in the book enables readers to tune their controllers better through the mathematical methods provided. The book contains numerous examples of practical applications of descriptor systems in various fields, from robotics to economics, and presents an information-theoretic approach to the mathematical description of coloured noise. Anisotropy-based analysis and design for descriptor systems is supplied along with proofs of basic statements, which help readers to understand the algorithms proposed, and to undertake their own numerical simulations.   This book serves as a source of ideas for academic researchers and postgraduate students working in the control of discrete-time systems. The control design procedures outlined are numerically effective and easily implementable in MATLAB®

The materials of The International Scientific – Practical Conference is presented below.

The Conference reflects the modern state of innovation in education, science, industry and social-economic sphere, from the standpoint of introducing new information technologies.

It is interesting for a wide range of researchers, teachers, graduate students and professionals in the field of innovation and information technologies.

This is an advanced guide to optimal stopping and control, focusing on advanced Monte Carlo simulation and its application to finance. Written for quantitative finance practitioners and researchers in academia, the book looks at the classical simulation based algorithms before introducing some of the new, cutting edge approaches under development.

Intended to bridge the gap between the latest methodological developments and cross-cultural research, this interdisciplinary resource presents the latest strategies for analyzing cross-cultural data. Techniques are demonstrated through the use of applications that employ cross-national data sets such as the latest European Social Survey. With an emphasis on the generalized latent variable approach, internationally prominent researchers from a variety of fields explain how the methods work, how to apply them, and how they relate to other methods presented in the book. Syntax and graphical and verbal explanations of the techniques are included. Online resources, available at www.routledge.com/9781138690271, include some of the data sets and syntax commands used in the book.

This edited collection presents a range of methods that can be used to analyse linguistic data quantitatively. A series of case studies of Russian data spanning different aspects of modern linguistics serve as the basis for a discussion of methodological and theoretical issues in linguistic data analysis. The book presents current trends in quantitative linguistics, evaluates methods and presents the advantages and disadvantages of each. The chapters contain introductions to the methods and relevant references for further reading.

The Russian language, despite being one of the most studied in the world, until recently has been little explored quantitatively. After a burst of research activity in the years 1960-1980, quantitative studies of Russian vanished. They are now reappearing in an entirely different context. Today we have large and deeply annotated corpora available for extended quantitative research, such as the Russian National Corpus, ruWac, RuTenTen, to name just a few (websites for these and other resources will be found in a special section in the References). The present volume is intended to fill the lacuna between the available data and the methods that can be applied to studying them.

Our goal is to present current trends in researching Russian quantitative linguistics, to evaluate the research methods vis-à-vis Russian data, and to show both the advantages and the disadvantages of the methods. We especially encouraged our authors to focus on evaluating statistical methods and new models of analysis. New findings concern applicability, evaluation, and the challenges that arise from using quantitative approaches to Russian data.

This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim’s vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim’s heart.

We study typical points with respect to ergofic averaging of a general dynamical system.

The present Yearbook (which is the sixth in the series) is subtitled Economy, Demography, Culture, and Cosmic Civilizations. To some extent it reveals the extraordinary potential of scientific research. The common feature of all our Yearbooks, including the present volume, is the usage of formal methods and social studies methods in their synthesis to analyze different phenomena. In other words, if to borrow Alexander Pushkin's words, ‘to verify the algebra with harmony'. One should note that publishing in a single collection the articles that apply mathematical methods to the study of various epochs and scales - from deep historical reconstruction to the pressing problems of the modern world - reflects our approach to the selection of contributions for the Yearbook. History and Mathematics, Social Studies and formal methods, as previously noted, can bring nontrivial results in the studies of different spheres and epochs. This issue consists of three main sections: (I) Historical and Technological Dimensions includes two papers (the first is about the connection between genes, myths and waves of the peopling of Americas; the second one is devoted to quantitative analysis of innovative activity and competition in technological sphere in the Middle Ages and Modern Period); (II) Economic and Cultural Dimensions (the contributions are mostly focused on modern period); (III) Modeling and Theories includes two papers with interesting models (the first one concerns modeling punctuated equilibria apparent in the macropattern of urbanization over time; in the second one the author attempts to estimate the number of Communicative Civilizations). We hope that this issue will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences.

The book studies the existing and potential connections between Social Network Analysis (SNA) and Formal Concept Analysis (FCA) by showing how standard SNA techniques, usually based on graph theory, can be supplemented by FCA methods, which rely on lattice theory. The book presents contributions to the following areas: acquisition of terminological knowledge from social networks, knowledge communities, individuality computation, other types of FCA-based analysis of bipartite graphs (two-mode networks), multimodal clustering, community detection and description in one-mode and multi-mode networks, adaptation of the dual-projection approach to weighted bipartite graphs, extensions to the Kleinberg's HITS algorithm as well as attributed graph analysis.

This book brings together the latest findings in the area of stochastic analysis and statistics. The individual chapters cover a wide range of topics from limit theorems, Markov processes, nonparametric methods, acturial science, population dynamics, and many others. The volume is dedicated to Valentin Konakov, head of the International Laboratory of Stochastic Analysis and its Applications on the occasion of his 70th birthday. Contributions were prepared by the participants of the international conference of the international conference “Modern problems of stochastic analysis and statistics”, held at the Higher School of Economics in Moscow from May 29 - June 2, 2016. It offers a valuable reference resource for researchers and graduate students interested in modern stochastics.

Workshop concentrates on an interdisciplinary approach to modeling human behavior incorporating data mining and/or expert knowledge from behavioral sciences. Data analysis results extracted from clean data of laboratory experiments can be compared with noisy industrial data-sets from the web e.g.. Insights from behavioral sciences will help data scientists. Behavior scientists will see new inspirations to research from industrial data science. Market leaders in Big Data, as Microsoft, Facebook, and Google, have already realized the importance of experimental economics know-how for their business. In Experimental Economics, although financial rewards restrict subjects preferences in experiments, exclusive application of analytical game theory is not enough to explain the collected data. It calls for the development and evaluation of more sophisticated models. The more data is used for evaluation, the more statistical significance can be achieved. Since large amounts of behavioral data are required to scan for regularities, along with automated agents needed to simulate and intervene in human interactions, Machine Learning is the tool of choice for research in Experimental Economics. This workshop is aimed at bringing together researchers from both Data Analysis and Economics in order to achieve mutually-beneficial results.

 

In this paper, a problem of robust anisotropy-based control with regional pole assignment for descriptor systems with norm-bounded parametric uncertainties is concerned. The goal is to find a state-feedback control law, which guarantees desirable disturbance attenuation level from stochastic input with unknown covariance to controllable output of the closed-loop system, and ensures, that all finite eigenvalues of the closed-loop system belong to the given region inside the unit disk for all uncertainties from the given set.

In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered, assuming that the optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on log R to optimize quasiconvex functions in the ball of radius R using only the comparison oracle. On the other hand, if the optimized function is conic, then we show that there is a polynomial on log R algorithm (the dimension is fixed). We also present an exponential on the dimension lower bound for the oracle complexity of the conic function integer optimization problem. Additionally, we give examples of known problems that can be polynomially reduced to the minimization problem of functions in our classes.

We show that for a large class of measurable vector fields in the sense of N. Weaver (i.e. derivations over the algebra of Lipschitz functions), the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure flows along'' the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in R2. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they enjoy all the natural well-posedness properties, thus allowing for a ‘good calculus’ on nonsmooth level sets. We apply these results to prove an area formula for the intrinsic measure of level sets, along with the corresponding coarea formula

We derive explicit bounds for the computation of normalizing constants Z for log-concave densities \pi=e^{−U}/Z w.r.t. the Lebesgue measure on Rd. Our approach relies on a Gaussian annealing combined with recent and precise bounds on the Unadjusted Langevin Algorithm [15]. Polynomial bounds in the dimension d are obtained with an exponent that depends on the assumptions made on U. The algorithm also provides a theoretically grounded choice of the annealing sequence of variances. A numerical experiment supports our findings. Results of independent interest on the mean squared error of the empirical average of locally Lipschitz functions are established.

The subject of this paper is the M/G/∞ estimation problem: the goal is to estimate the service time distribution G of the M/G/∞ queue from the arrival–departure observations without identification of customers. We develop estimators of G and derive exact non-asymptotic expressions for their mean squared errors. The problem of estimating the service time expectation is addressed as well. We present some numerical results on comparison of different estimators of the service time distribution.

We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras.  Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations.  Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac--Moody (super)algebras are the most known examples.

Modern imaging methods rely strongly on Bayesian inference techniques to solve challenging imaging problems. Currently, the predominant Bayesian computation approach is convex optimization, which scales very efficiently to high-dimensional image models and delivers accurate point estimation results. However, in order to perform more complex analyses, for example, image uncertainty quantification or model selection, it is necessary to use more computationally intensive Bayesian computation techniques such as Markov chain Monte Carlo methods. This paper presents a new and highly efficient Markov chain Monte Carlo methodology to perform Bayesian computation for high-dimensional models that are log-concave and nonsmooth, a class of models that is central in imaging sciences. The methodology is based on a regularized unadjusted Langevin algorithm that exploits tools from convex analysis, namely, Moreau--Yoshida envelopes and proximal operators, to construct Markov chains with favorable convergence properties. In addition to scaling efficiently to high-dimensions, the method is straightforward to apply to models that are currently solved by using proximal optimization algorithms. We provide a detailed theoretical analysis of the proposed methodology, including asymptotic and nonasymptotic convergence results with easily verifiable conditions, and explicit bounds on the convergence rates. The proposed methodology is demonstrated with four experiments related to image deconvolution and tomographic reconstruction with total-variation and $\ell_1$ priors, where we conduct a range of challenging Bayesian analyses related to uncertainty quantification, hypothesis testing, and model selection in the absence of ground truth.

In this article, we consider the problem of sampling from a probability measure π having a density on R d proportional to x↦ e− U (x). The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential U is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings.

Given two small dg categories C,D, defined over a field, we introduce their (non-symmetric) twisted tensor product C⊗∼D. We show that −⊗∼D is left adjoint to the functor Coh(D,−), where Coh(D,E) is the dg category of dg functors D→E and their coherent natural transformations. This adjunction holds in the category of small dg categories (not in the homotopy category of dg categories Hot). We show that for C,D cofibrant, the adjunction descends to the corresponding adjunction in the homotopy category. Then comparison with a result of Toën shows that, for C,D cofibtant, C⊗∼D is isomorphic to C⊗D, as an object of the homotopy category Hot.

We investigate geometry of D-affine varieties. Our main result is that a D-affine uniformly rational projective variety over an algebraically closed field of zero characteristic is a generalised flag variety of a reductive group. This is a partially converse statement for Beilinson-Bernstein Localisation Theorem.

Following the approach of Haiden-Katzarkov-Kontsevich arXiv:1409.8611, to any homologically smooth graded gentle algebra A we associate a triple (Σ_A,Λ_A;η_A), where Σ_A is an oriented smooth surface with non-empty boundary, Λ_A is a set of stops on ∂Σ_A and η_A is a line field on Σ_A, such that the derived category of perfect dg-modules of A is equivalent to the partially wrapped Fukaya category of (Σ_A,Λ_A;η_A). Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of Σ_A on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella-Alaminos-Geiss in math/0607348, as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in arXiv:1705.06023.

We study birational projective models of M_2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of Z-stable curves \bar{M}_2,2(Z) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space \bar{M}_2,2(Z).

Using Auroux's description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of k+1 generic hyperplanes in ℂℙ^n, for k≥n, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of (n+2)-generic hyperplanes in ℂP^n (n-dimensional pair-of-pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety x_1x_2..x_{n+1}=0. By localizing, we deduce that the (fully) wrapped Fukaya category of n-dimensional pants is equivalent to the derived category of x_1x_2...x_{n+1}=0.

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 submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules 𝕃/I(p,n)x where deg x≥p^n−1/(p−1). As a corollary we prove the Syzygies Conjecture of Vishik on the existence of certain free 𝕃-resolutions of Ω∗(X), and show that algebraic cobordism of a smooth surface can be described in terms of K_0 together with a topological filtration.

Generalizing the definition of Cartier, we introduce pn-typical formal group laws over ℤ(p)-algebras. An oriented cohomology theory in the sense of Levin-Morel is called pn-typical if its corresponding formal group law is p^n-typical. The main result of the paper is the construction of 'Chern classes' from the algebraic n-th Morava K-theory to every p^n-typical oriented cohomology theory.  If the coefficient ring of a p^n-typical theory is a free ℤ(p)-module we also prove that these Chern classes freely generate all operations to it. Examples of such theories are algebraic mn-th Morava K-theories K(nm)∗ for all m∈ℕ and CH∗⊗ℤ(p) (operations to Chow groups were studied in a previous paper). The universal pn-typical oriented theory is BP{n}∗=BP∗/(v_j,j∤n) which coefficient ring is also a free ℤ_(p)-module.  Chern classes from the n-th algebraic Morava K-theory K(n)∗ to itself allow us to introduce the gamma filtration on K(n)∗. This is the best approximation to the topological filtration obtained by values of operations and it satisfies properties similar to that of the classical gamma filtration on K0. The major difference from the classical case is that Chern classes from the graded factors gr^i_γ K(n)^∗ to CH^i⊗ℤ_(p) are surjective for i≤p^n. For some projective homogeneous varieties this allows to estimate p-torsion in Chow groups of codimension up to pn.

In the present article we discuss an approach to cohomological invariants of algebraic groups over fields of characteristic zero based on the Morava K-theories, which are generalized oriented cohomology theories in the sense of Levine--Morel.  We show that the second Morava K-theory detects the triviality of the Rost invariant and, more generally, relate the triviality of cohomological invariants and the splitting of Morava motives.  We describe the Morava K-theory of generalized Rost motives, compute the Morava K-theory of some affine varieties, and characterize the powers of the fundamental ideal of the Witt ring with the help of the Morava K-theory. Besides, we obtain new estimates on torsion in Chow groups of codimensions up to 2^n of quadrics from the (n+2)-nd power of the fundamental ideal of the Witt ring. We compute torsion in Chow groups of K(n)-split varieties with respect to a prime p in all codimensions up to p^{n−1}/(p−1) and provide a combinatorial tool to estimate torsion up to codimension p^n. An important role in the proof is played by the gamma filtration on Morava K-theories, which gives a conceptual explanation of the nature of the torsion.  Furthermore, we show that under some conditions the K(n)-motive of a smooth projective variety splits if and only if its K(m)-motive splits for all m≤n.

We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching invariant spanning trees. This procedure uses finite automata associated with the iterated monodromy action.

A cubic polynomial f with a periodic Siegel disk containing an eventual image of a critical point is said to be a \emph{Siegel capture polynomial}. If the Siegel disk is invariant, we call f a \emph{IS-capture polynomial} (or just an IS-capture; IS stands for Invariant Siegel). We study the location of IS-capture polynomials in the parameter space of all cubic polynomials. In particular, we show that any IS-capture is on the boundary of a unique hyperbolic component determined by the rational lamination of the map. We also relate IS-captures to the cubic Principal Hyperbolic Domain and its closure (by definition, the \emph{cubic Principal Hyperbolic Domain} consists of cubic hyperbolic polynomials with Jordan curve Julia sets). We prove that, in the slice of cubic polynomials given by a fixed multiplier at one of the fixed points, the closure of the cubic principal hyperbolic domain cannot have bounded complementary domains containing IS-captures.

Let U be the tautological subbundle on the Grassmannian Gr(k,n). There is a natural morphism Tot(U)→𝔸^n. Using it, we give a semiorthogonal decomposition for the bounded derived category D^b_coh(Tot(U)) into several exceptional objects and several copies of D^b_coh(𝔸n). We also prove a global version of this result: given a vector bundle E with a regular section s, consider a subvariety of the relative Grassmannian Gr(k,E) of those subspaces which contain the value of s. The derived category of this subvariety admits a similar decomposition into copies of the base and the zero locus of s. This may be viewed as a generalization of the blow-up formula of Orlov, which is the case k=1.

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normnalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group Sn+p in characteristic p, where 0≤n≤p−1, and of the Deligne category Rep^ab S_t, where t∈ℕ. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of 𝔰𝔩2. We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of GL(n), SL(n) and PGL(n) in characteristic 2. In the appendix, we classify categorifications of the Grothendieck ring of representations of SO(3) and its truncations.

We give a nontrivial lower bound for global dimension of a spherical fusion category.

An affine algebraic variety X of dimension ≥ 2 is called flexible if the subgroup SAut(X) ⊂ Aut(X) generated by the one-parameter unipotent subgroups acts m-transitively on reg (X) for any m ≥ 1. In a preceding paper ([4]) we proved that any nondegenerate toric affine variety X is flexible. Here we show that if such a toric variety X is smooth in codimension 2 then one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property. In fact, four such subgroups are enough for X = A^n if n ≥ 3, and just three if n = 2.

We prove the conjecture of Berest-Eshmatov-Eshmatov by showing that the group of automorphisms of a product of Calogero-Moser spaces C_{n_i}, where the ni are pairwise distinct, acts m-transitively for each m.

We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety is flexible.

Given two fields k and F, a k-vector space V, an integer r≥1, we study the structure of the F-representation of the projective group PGL(V) in the F-vector space of formal finite linear combinations with coefficients in F of r-dimensional vector subspaces of V.  This gives a series of natural examples of irreducible infinite-dimensional representations of projective groups. These representations are non-smooth when k is a local field.

We show that if we are given a smooth non-isotrivial family of elliptic curves over ℂ with a smooth base B for which the general fiber of the mapping J:B→𝔸^1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on H1(⋅,ℤ) of the fibers) coincides with SL(2,ℤ); if the general fiber has m≥2 connected components, then the monodromy group has index at most 2m in SL(2,ℤ). By contrast, in any family of hyperelliptic curves of genus g≥3, the monodromy group is strictly less than Sp(2g,ℤ).  Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.