This volume collects the referred papers based on plenary, invited, and oral talks, as well on the posters presented at the Third International Conference on Computer Simulations in Physics and beyond (CSP2018), which took place September 24-27, 2018 in Moscow. The Conference continues the tradition started by an inaugural conference in 2015. It took place on the campus of A.N. Tikhonov Moscow Institute of Electronics and Mathematics in Strogino, was jointly organized by the National Research University Higher School of Economics, the Landau Institute for Theoretical Physics and Science Center in Chernogolovka.

The Conference is a multidisciplinary meeting, with a focus on computational physics and related subjects. Indeed, methods of computational physics prove useful in a broad spectrum of research in multiple branches of natural sciences, and this volume provides a sample.

We hope that this volume will interest readers, and we are already looking forward to the next conference in the series.

Moscow, Russia

November, 2018

CSP2018 Conference Chair and Volume Editor

Lev Shchur

This book covers the classical theory of Markov chains on general state-spaces as well as many recent developments. The theoretical results are illustrated by simple examples, many of which are taken from Markov Chain Monte Carlo methods. The book is self-contained, while all the results are carefully and concisely proven. Bibliographical notes are added at the end of each chapter to provide an overview of the literature.

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

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.

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 the proceedings of the 7th International Conference on Analysis of Images, Social Networks and Texts, AIST 2018, held in Moscow, Russia, in July 2018.

The 29 full papers were carefully reviewed and selected from 107 submissions (of which 26 papers were rejected without being reviewed). The papers are organized in topical sections on natural language processing; analysis of images and video; general topics of data analysis; analysis of dynamic behavior through event data; optimization problems on graphs and network structures; and innovative systems.

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.

There is an ongoing evolution involving a new approach to large-scale optimisations based on co-evolutionary searches using interacting heterogeneous agent-processes via the implementation of synchronised genetic algorithms with local populations. The individualisation of heuristic operators at the level of agent-processes that implement independent evolutionary searches facilitate the improved likelihood of obtaining the best solutions in the fastest time. Based on this property, a parallel multi-agent single-objective real-coded genetic algorithm for large-scale constrained black-box single-objective optimisations (LSOPs ) is proposed. This facilitates the effective frequency exchange of the best potential decisions between interacting agent-processes with individual parameters, such as types of crossover and mutation operators with their own characteristics. We have improved the quality of both solutions and the time-efficiency of a multi-agent real-coded genetic algorithm (MA−RCGA ). A novel framework was developed that represents the aggregation of MA−RCGA with simulation models by implementing a set of objective functions for real-world large-scale optimisation problems such as the simulation model of the ecological-economics system implemented in the AnyLogic tool.

The Riccati equation with coefficients expandable in convergent power series in a neigh- borhood of infinity are considered. Extendable solutions of such equations are studied. Methods of power geometry are used to obtain conditions for convergent series expansions of these solutions. An algorithm for deriving such series is given.

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic. By results of Kiwi, any dendritic polynomial is semiconjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the ``pinched disk'' model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus.

A fixed set of n agents share a random object: the distribution μ of the profile of utilities is IID across periods, but arbitrary across agents. We consider a class of online division rules that learn the realized utility profile, and only know from μ the individual expected utilities. They have no record from past realized utilities, and do not know either if and how many new objects will appear in the future. We call such rules prior-independent. 

A rule is fair if each agent, ex ante, expects at least 1/n-th of his utility for the object if it is a good, at most 1/n-th of his disutility for it if it is a bad. Among fair prior-independent rules to divide goods (bads) we uncover those collecting the largest (lowest) total expected (dis)utility. There is exactly one fair rule for bads that is optimal in this sense. But for goods, the set of optimal fair rules is one dimensional. Both in the worst case and in the asymptotic sense, our optimal rules perform much better than the natural Proportional rule (for goods or for bads), and not much worse than the optimal fair prior-dependent rule that knows the full distribution μ in addition to realized utilities.

Three Lagrangian invariants are shown to exist for flows in the equatorial region in the β - plane approximation.

They extend the Cauchy invariants to a non-rotating fluid. The relationship between these generalized invariants

and the results following from Kelvin's and Ertel's theorems is ascertained. Explicit expressions of the invariants

for equatorially trapped waves and equatorial Gerstner waves are presented.

In this manuscript, we study the electrically induced breathing of Metal-Organic Framework (MOF) within a 2D lattice model. The Helmholtz free energy of the MOF in electric eld consists of two parts: the electrostatic energy of the dielectric body in the external electric eld and elastic energy of the framework. The rst contribution is calculated from the rst principles of statistical mechanics with an account of MOF symmetry. By minimizing the obtained free energy and solving the resulting system of equations, we obtain the local electric eld and the parameter of the unit cell (angle ). The paper also studies the cross-section area of the unit cell and the polarization as functions of the external electric eld. We obtain the hysteresis in the region of the structural transition of the framework. Our results are in qualitative agreement with the literature data of the molecular dynamics (MD) simulation of MIL-53(Cr).

The main problems and features of combined approach to the complex objects control and management stability analysis are investigated in the paper. Analytical-simulation scenarios and scenarios of intelligent models and systems execution for complex objects control and management stability analysis are given. The investigations have shown successful possibility of risks evaluation by the combined implementation of the analytical-simulation models and algorithms, and ANFIS method – the method of hybrid neuralfuzzy modelling.

The mathematical model describing the dynamics of HIV in the human body is a nonlinear system of differential equations. This model takes into account the effect of drugs on the body. Thus, it is possible to obtain ”optimal” treatment regimens for patients, which cause minimal harm to the body. In the work for constructing suboptimal control of the supply of drugs, the method of ”extended linearization” is used, which makes it possible to switch from a nonlinear model to a linear model, but with parameters that depend on the state. To solve the resulting equation Riccati and search for control actions, a method is proposed for the formation of optimization algorithms for nonlinear control systems based on the application of functions of admissible values of control actions.

Matrix multiplication is one of the core operations in many areas of scientific computing. We present the results of the experiments with the matrix multiplication of the big size comparable with the big size of the onboard memory, which is 1.5 terabyte in our case. We run experiments on the computing board with two sockets and with two Intel Xeon Platinum 8164 processors, each with 26 cores and with multi-threading. The most interesting result of our study is the observation of the perfect scalability law of the matrix multiplication, and of the universality of this law.

Bachet's game is a variant of the game of Nim. There are n objects in one pile. Two players make moves one after another. On every move, a player is allowed to take any positive number of objects not exceeding some fixed number m. The player who takes the last object loses. We consider a variant of Bachet's game in which each move is a lottery over set {1,2,…,m}. Outcome of a lottery is the number of objects that player takes from the pile. We show that under some nondegenericity assumptions on the set of available lotteries the probability that the first player wins in subgame perfect Nash equilibrium converges to 1/2 as n tends to infinity.

We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment-complex $\mathcal{Z_K}$. Namely, we say that a simplicial complex K realises an iterated higher Whitehead product w if w is a nontrivial element of the homotopy group $\pi_*(\mathcal{Z_K})$. The combinatorial approach to the question above uses the operation of substitution of simplicial complexes: for any iterated higher Whitehead product w we describe a simplicial complex $\partial\Delta_w$ that realises $w$. Furthermore, for a particular form of brackets inside w, we prove that $\partial\Delta_w$ is the smallest complex that realises $w$. We also give a combinatorial criterion for the nontriviality of the product w. In the proof of nontriviality we use the Hurewicz image of w in the cellular chains of $\mathcal{Z_K}$ and the description of the cohomology product of $\mathcal{Z_K}$. The second approach is algebraic: we use the coalgebraic versions of the Koszul complex and the Taylor resolution of the face coalgebra of $\mathcal{K}$ to describe the canonical cycles corresponding to iterated higher Whitehead products $w$. This gives another criterion for realisability of $w$.

We investigate the effect of the proportional hazards assumption on prognostic and predictive models of the survival time of patients suffering from amyotrophic lateral sclerosis (ALS). We theoretically compare the underlying model formulations of several variants of survival forests and implementations thereof, including random forests for survival, conditional inference forests, Ranger, and survival forests with L1 splitting, with two novel variants, namely distributional and transformation survival forests. Theoretical considerations explain the low power of log-rank-based splitting in detecting patterns in non-proportional hazards situations in survival trees and corresponding forests. This limitation can potentially be overcome by the alternative split procedures suggested herein. We empirically investigated this effect using simulation experiments and a re-analysis of the PRO-ACT database of ALS survival, giving special emphasis to both prognostic and predictive models.

Tensor products of irreducible representations of a simple Lie algebra define two natural measures on the space of weights: one comes from the weight multiplicities and another is induced by the decomposition of the tensor product into irreducible components. Our first result is an explicit description of density of the limiting measure for weight multiplicities when the numbers of irreducible factors linearly tend to infinity. We use the derived density to obtain the limit of the measures coming from the decomposition into irreducibles. The density of the resulting measure in type A coincides with the density of the famous GUE joint eigenvalues density restricted to the the traceless matrices. We thus generalize Kerov's theorem and prove the Postnova-Reshetikhin formula.

We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel-Weil theorem for the flat irreducible locus.

A description of rational Newton maps in terms of the partial fraction decomposition of rational functions is obtained. Dynamics on parabolic immediate basins for rational Newton maps of entire functions have been studied. It is proved that every parabolic immediate basin contains invariant accesses to the parabolic fixed point at infinity. Moreover, among these accesses there exists a unique dynamically defined access where dynamics are attracted towards the parabolic fixed point, whereas for other accesses, if there is any, the dynamics are repelled.

It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf--Ness Theorem,  arXiv:1811.07195v1 (17 Nov 2018) is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems.

Under certain initial conditions, we prove the existence of set-valued selectors of univariate compact-valued multifunctions of bounded (Jordan) variation when the notion of variation is defined taking into account only the Pompeiu asymmetric excess between compact sets from the target metric space. For this, we study subtle properties of the directional variations. We show by examples that all assumptions in the main existence result are essential. As an application, we establish the existence of set-valued solutions $X(t)$ of bounded variation to the functional inclusion of the form $X(t)\subset F(t,X(t))$ satisfying the initial condition $X(t_0)=X_0$.  

This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12–16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant versions. We discuss several topics inspired by this exploration, including the problem of classifying a class of affine algebraic groups that are naturally singled out in studying the conjugacy problem for algebraic subgroups of the Cremona groups.

We look at how evolution method deforms, when one considers Khovanov polynomials instead of Jones polynomials. We do this for the figure-eight-like knots (also known as 'double braid' knots, see arXiv:1306.3197) -- a two-parametric family of knots which "grows" from the figure-eight knot and contains both two-strand torus knots and twist knots. We prove that parameter space splits into four chambers, each with its own evolution, and two isolated points. Remarkably, the evolution in the Khovanov case features an extra eigenvalue, which drops out in the Jones (t -> -1) limit.

Estimation of the relationship between DNA sequences is one of the most important problems in genomics. Understanding these relationships is central to de- mographic inference, correction of population structure in GWAS, identifying signals of selection etc. The data structure containing the full information about sample genealogy is called the ancestral recombination graph (ARG). However, ARG inference is a very dicult problem, not least due to a very complex state space. In this work we describe a new approach for fast and scalable generation of local tree topologies relating large numbers of haplotypes. Our method is closely related to the estimation of ARG, and captures both local and global properties of an ARG. It is based on a data structure which we call tree consistent PBWT , a modi cation of PBWT data structure intro- duced by R. Durbin (2014). We also explore some methods to estimate the quality of the generated tree topologies and to make inferences based on them. At the end we discuss a probabilistic model which could potentially lead to the estimation of ARG node times.

The article demonstrates rather general approach to problems of discrete geometry: treat them as global optimization problems to be solved by one of general purpose solver implementing branch-and-bound algorithm (B&B). This approach may be used for various types of problems, i.e. Tammes problems, Thomson problems, search of minimal potential energy of micro-clusters, etc. Here we consider a problem of densest packing of equal circles in special geometrical object, so called square flat torus ℝ2/ℤ2 with the induced metric. It is formulated as Mixed-Integer Nonlinear Problem with linear and non-convex quadratic constraints.  The open-source B&B-solver SCIP, this http URL, and its parallel implementation ParaSCIP, this http URL, had been used in computing experiments to find "very good" approximations of optimal arrangements. The main result is a confirmation of the conjecture on optimal packing for N=9 that was published in 2012 by O. Musin and A. Nikitenko. To do that, ParaSCIP took about 2000 CPU*hours (16 hours x 128 CPUs) of cluster HPC4/HPC5, National Research Centre "Kurchatov Institute", this http URL

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.