This book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided.

We begin with a brief but fairly complete course on the theory of holomorphic, meromorphic, and harmonic functions. We then present a uniformization theory, and discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as a factor space of a contractible space by a discrete group. Next, we consider compact Riemann surfaces and prove the classical theorems of Riemann-Roch, Abel, Weierstrass, etc. We also construct theta functions that are very important for a range of applications.

After that, we turn to modern applications of this theory. First, we build the (important for mathematics and mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at important solutions to these differential equations. We subsequently use the theory of harmonic functions and the theory of differential hierarchies to explicitly construct a conformal mapping that translates an arbitrary contractible domain into a standard disk – a classical problem that has important applications in hydrodynamics, gas dynamics, etc.

The book is based on numerous lecture courses given by the author at the Independent University of Moscow and at the Mathematics Department of the Higher School of Economics.

In the last 30 years a new pattern of interaction between mathematics and physics emerged, in which the latter catalyzed the creation of new mathematical theories. Most notable examples of this kind of interaction can be found in the theory of moduli spaces. In algebraic geometry the theory of moduli spaces goes back at least to Riemann, but they were first rigorously constructed by Mumford only in the 1960s. The theory has experienced an extraordinary development in recent decades, finding an increasing number of connections with other fields of mathematics and physics. In particular, moduli spaces of different objects (sheaves, instantons, curves, stable maps, etc.) have been used to construct invariants (such as Donaldson, Seiberg-Witten, Gromov-Witten, Donaldson-Thomas invariants) that solve longstanding, difficult enumerative problems. These invariants are related to the partition functions and expectation values of quantum field and string theories. In recent years, developments in both fields have led to an unprecedented cross-fertilization between geometry and physics. These striking interactions between geometry and physics were the theme of the CIME School Geometric Representation Theory and Gauge Theory. The School took place at the Grand Hotel San Michele, Cetraro, Italy, in June, Monday 25 to Friday 29, 2018. The present volume is a collection of notes of the lectures delivered at the school. It consists of three articles from Alexander Braverman and Michael Finkelberg, Andrei Negut, and Alexei Oblomkov, respectively.

The paper provides findings of the research work and scientific discussions under the “Global Sustainability Strategy Forum” (GSSF) that aims to develop evidence-informed judgments on challenges and solutions. It views attaining sustainability as a set of closely-coupled societal and environmental challenges and opportunities that require integration of multiple disciplines, new research methods, and new knowledge sources with sensitivity to regional and cultural diversities. The project is designed to produce innovative insights and strategies to support effective governance of transitions to sustainability of our complex global social-ecological system within its inherent resource limitations, and to develop sustainable lifestyles that are practical and appealing in the different regions and cultures of the world. 

The global climate change is one of the most dangerous threats to human society in the 21st Century. The dramatic losses have already been observed, and the risks are rising over time. CEECCA region experiences many negative impacts of global warming, which is faster and stronger than the world average. Numerous adaptation and resilience measures are required to protect people, but regional governments often underestimate and ignore the social implications of climate policies.This paper explores what are the priority challenges for CEECCA countries and how to address them effectively.

We propose a novel machine-learning-based approach to detect bid leakage in first-price sealed-bid auctions. We extract and analyze the data on more than 1.4 million Russian procurement auctions between 2014 and 2018. As bid leakage in each particular auction is tacit, the direct classification is impossible. Instead, we reduce the problem of bid leakage detection to Positive-Unlabeled Classification. The key idea is to regard the losing participants as fair and the winners as possibly corrupted. This allows us to estimate the prior probability of bid leakage in the sample, as well as the posterior probability of bid leakage for each specific auction. We find that at least 16% of auctions are exposed to bid leakage. Bid leakage is more likely in auctions with a higher reserve price, lower number of bidders and lower price fall, and where the winning bid is received in the last hour before the deadline.

The International Workshop on Enterprise and Organizational Modeling and Simulation (EOMAS) represents a forum where researchers and practitioners exchange and mutually enrich their views, approaches, and obtain results in the field of enterprise engineering and enterprise architecture. The most valuable asset of every conference and workshop is its community. The community of EOMAS is small, but it consists of founding members, long-term contributors, and every year it attracts new innovative participants. This year, EOMAS reached its 15th edition and took place in Rome, Italy, during June 3–4, 2019. Traditionally, we can offer a balanced assortment of papers addressing formal foundations of enterprise modeling and simulation, conceptual modeling approaches, higher-level insights and applications bringing novel ideas to traditional approaches, as well as new emerging trends. Out of 24 submitted papers, 12 were accepted for publication as full papers and for oral presentation, and each paper was carefully selected, reviewed, and revised. In additional to this we reflected on the interest of last year’s invited workshop on usability and invited the experts to make a sequel. You can find a short report in this issue. This year, we included a novel outlet of Master and Doctoral Consortium, which attracted young talent to present their work. The presented work was then discussed, and feedback, advice, and encouragement was given. We were really surprised by the relevance, methodological quality, and results of their work – you may find their contributions on our website https://eomas-workshop.org. We would like to express our sincere thanks to the entire EOMAS community: the authors, the Program Committee and the CAiSE organizers, the chairs for their enthusiasm and devotion, as well as all participants for their contributions. We look forward to the 16th edition of EOMAS!

Workshop concentrates on an interdisciplinary approach to modelling human behavior incorporating data mining and expert knowledge from behavioral sciences. Data analysis results extracted from clean data of laboratory experiments will be compared with noisy industrial datasets 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.

This volume contains the refereed proceedings of the 8th International Conference on Analysis of Images, Social Networks, and Texts (AIST 2019). The previous conferences during 2012–2018 attracted a significant number of data scientists – students, researchers, academics, and engineers working on interdisciplinary data analysis of images, texts, and social networks.

This is an advanced text on ordinary differential equations (ODES) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It yields the concise exposition of the fundamentals with the fast, but rigorous and systematic transition to the up-fronts of modern research in linear and nonlinear partial and pseudo-differential equations, general kinetic equations and fractional evolutions. The level of generality is chosen to be suitable for the study of the most important nonlinear equations of mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov, Landau-Fokker-Planck, Cahn-Hilliard, Hamilton-Jacobi-Bellman, nonlinear Schroedinger, McKean-Vlasov diffusions and their nonlocal extensions, mass-action-law kinetics from chemistry. It also covers nonlinear evolutions arising in evolutionary biology and mean-field games, optimization theory, epidemics and system biology, in general models of interacting particles or agents describing splitting and merging, collisions and breakage, mutations and the preferential-attachment growth on networks. The book is meant for final year undergraduate and postgraduate students and researchers in differential equations and their applications. A significant amount of attention is paid to the interconnections between various topics revealing where and how a particular result is used in other chapters or may be used in other contexts, as well as to the clarification of the links between the languages of pseudo-differential operators, generalized functions, operator theory, abstract linear spaces, fractional calculus and path integrals.

This volume constitutes the refereed proceedings of the 4th International Conference on Digital Transformation and Global Society, DTGS 2019, held in St. Petersburg, Russia, in June 2019.

The 56 revised full papers and 9 short papers presented in the volume were carefully reviewed and selected from 194 submissions. The papers are organized in topical sections on ​e-polity: governance; e-polity: politics online; e-city: smart cities and urban planning; e-economy: online consumers and solutions; e-society: computational social science; e-society: humanities and education; international workshop on internet psychology; international workshop on computational linguistics.

Provides an overview of the developments and advances in the field of network clustering and blockmodeling over the last 10 years

This book offers an integrated treatment of network clustering and blockmodeling, covering all of the newest approaches and methods that have been developed over the last decade. Presented in a comprehensive manner, it offers the foundations for understanding network structures and processes, and features a wide variety of new techniques addressing issues that occur during the partitioning of networks across multiple disciplines such as community detection, blockmodeling of valued networks, role assignment, and stochastic blockmodeling.

Written by a team of international experts in the field, Advances in Network Clustering and Blockmodeling offers a plethora of diverse perspectives covering topics such as: bibliometric analyses of the network clustering literature; clustering approaches to networks; label propagation for clustering; and treating missing network data before partitioning. It also examines the partitioning of signed networks, multimode networks, and linked networks. A chapter on structured networks and coarsegrained descriptions is presented, along with another on scientific coauthorship networks. The book finishes with a section covering conclusions and directions for future work. In addition, the editors provide numerous tables, figures, case studies, examples, datasets, and more.

Offers a clear and insightful look at the state of the art in network clustering and blockmodeling Provides an excellent mix of mathematical rigor and practical application in a comprehensive manner Presents a suite of new methods, procedures, algorithms for partitioning networks, as well as new techniques for visualizing matrix arrays Features numerous examples throughout, enabling readers to gain a better understanding of research methods and to conduct their own research effectively Written by leading contributors in the field of spatial networks analysis

Advances in Network Clustering and Blockmodeling is an ideal book for graduate and undergraduate students taking courses on network analysis or working with networks using real data. It will also benefit researchers and practitioners interested in network analysis.

This edition of Procedia Computer Science represents the proceedings of the 23rd International Conference on Knowledge - Based and Intelligent Information & Engineering Systems (KES 2019), organised by KES International and held at the Danubius Health Spa Resort, Budapest over 4-6 September 2019. KES 2019 was the 23rd event in a series of broad-spectrum intelligent systems conferences first held in Adelaide, Australia in 1997. The main aim of this KES conference series is to provide an internationally respected forum for the dissemination of research results and the discussion of issues relating to the theory, technologies and applications of intelligent engineering and information systems. This truly international conference attracted submissions from a substantial number of researchers and practitioners from all over the world, who submitted their papers to three general tracks, one thematic track and 34 special sessions on specific topics. A large number of submissions was received and each paper was peer reviewed by at least two members of the International Program Committee. From them, 274 high-quality papers were accepted for oral presentation and publication in Procedia Computer Science, submitted for indexing in Conference Proceedings Citation Index (CPCI) and Scopus. The conference chairs would like to express their gratitude to the Keynote Speakers: Prof Dana Barry, Clarkson University, USA, title of talk: 'STEM and ICT Education in Intelligent Environments'; Dr Carlos Toro, ARTC (Advanced Remanufacturing and Technology Centre) - A*Star, Singapore, title of talk: 'Smart Manufacturing coming of age'; Prof Katsutoshi Yada, Kansai University, Japan, title of talk: 'Sensor Marketing and Data Mining'; Prof Cecilia Zanni-Merk, INSA Rouen Normandie / LITIS Laboratory, France, title of talk 'On the need of an Explainable Artificial Intelligence'; and Prof Sergey Zykov, National Research University Higher School of Economics, Russia, title of talk: 'IT Crisisology: the New Discipline for Managing Software Development in Crisis'. We would like to acknowledge also the Programme Co-Chairs, the General Track Chairs, the International Programme Committee members and reviewers for their valuable efforts in the review process, helping us to guarantee the highest quality possible for the conference. We would also like to thank the organisers and chairs of the special sessions which make an essential contribution to the success of the conference. Lastly, we would like to thank all the authors, presenters and delegates for their valuable contribution in making this an extraordinary event. KES International hopes and intends that KES2019 will make a significant contribution to international research collaboration and understanding, an essential task for the promotion of scientific joint work and excellence.

The goal of this International Roadmap for Devices and Systems (IRDS) chapter is to survey, catalog, and assess the status of technologies in the areas of cryogenic electronics and quantum information processing. Application drivers are identified for sufficiently developed technologies and application needs are mapped as a function of time against projected capabilities to identify challenges requiring research and development effort. Cryogenic electronics (also referred to as low-temperature electronics or cold electronics) is defined by operation at cryogenic temperatures (below −150 °C or 123.15 K) and includes devices and circuits made from a variety of materials including insulators, conductors, semiconductors, superconductors, or topological materials. Existing and emerging applications are driving development of novel cryogenic electronic technologies. Information processing refers to the input, transmission, storage, manipulation or processing, and output of data. Information processing systems to accomplish a specific function, in general, require several different interactive layers of technology. A top-down list of these layers begins with the required application or system function, leading to system architecture, micro- or nano-architecture, circuits, devices, and materials. A fundamental unit of information (e.g., a bit) is represented by a computational state variable, for example, the position of a bead in the ancient abacus calculator or the voltage (or charge) state of a node capacitance in CMOS logic. A binary computational state variable serves as the foundation for von Neumann computational system architectures that dominated conventional computing. Quantum information processing is different in that it uses qubits, two-state quantum-mechanical systems that can be in coherent superpositions of both states at the same time, which can have computational advantages. Measurement of a qubit in a given basis causes it to collapse to one of the basis states. Technology categories covered in this report include: • Superconductor electronics (SCE) • Cryogenic semiconductor electronics (Cryo-Semi) • Quantum information processing (QIP)

This book constitutes the post-conference proceedings of the 8th International Conference on Analysis of Images, Social Networks and Texts, AIST 2019, held in Kazan, Russia, in July 2019.

The 27 full and 8 short papers were carefully reviewed and selected from 134 submissions (of which 21 papers were automatically rejected without being reviewed). The papers are organized in topical sections on general topics of data analysis; natural language processing; social network analysis; analysis of images and video; optimization problems on graphs and network structures; and analysis of dynamic behavior through event data.

We study the Maximum Happy Vertices and Maximum Happy Edges problems. The former problem is a variant of clusterization, where some vertices have already been assigned to clusters. The second problem gives a natural generalization of Multiway Uncut, which is the complement of the classical Multiway Cut problem. Due to their fundamental role in theory and practice, clusterization and cut problems has always attracted a lot of attention. We establish a new connection between these two classes of problems by providing a reduction between Maximum Happy Vertices and Node Multiway Cut. Moreover, we study structural and distance to triviality parameterizations of Maximum Happy Vertices and Maximum Happy Edges. Obtained results in these directions answer questions explicitly asked in four works: Agrawal ’17, Aravind et al. ’16, Choudhari and Reddy ’18, Misra and Reddy ’17.

For the first time, the method of bioclimatic modeling was used to determine the boundaries of the range of occurrence of a potential polemochoral species Avenella flexuosa (L.) Drejer. The simulation was performed by the method of maximum entropy (MaxEnt). It is shown that the conditions for the growth of this species are initially suitable (the probability of finding 0.54 and higher) in the territory of Central Russia. For comparison, the same bioclimatic modeling approach was applied for the occurrence data of Carex brizoides L. species, the polemochoral origin of which is not in doubt. It is shown that its growing places are beyond the limits of the climatic optimum (the probability of finding 0.08) in the territory of Central Russia.

The definition of recreational sustainability in bogs is becoming an urgent task in Russia as a result of ecological tourism developed in Protected Areas. In this study, we assessed the impact of recreational nature management on wetland plant communities of the Plavnitskoye Boloto ecological path in the Polistovsky State Nature Reserve solving the following tasks: 1) to determine the threshold of permissible anthropogenic impact on wetland sites with different types of plant communities; 2) to evaluate the recovery of vegetation cover; 3) to determine the relative tolerance of bog plant communities to trampling. Modelling of the direct anthropogenic impact on bog plant communities with different load values has made it possible to assess the damage on the vegetation cover visually and to reveal their resistance and ability to subsequent recover. We demonstrated that the Phragmites-Sphagnum community (Phragmites australis + Eriophorum vaginatum - Sphagnum fallax + Sphagnum angustifolium) was the least resistant to trampling, while the shrub-Carex-Sphagnum plant community (Chamaedaphne calyculata + Oxycoccus palustris + Menyanthes trifoliata + Eriophorum vaginatum - Sphagnum fallax) was the most resistant. After the removal of anthropogenic load on the studied sites, the damaged vegetation cover recovered to different degrees. In the first year, the plant communities of the shrub-Carex-Sphagnum bog had the fastest recovery, where relatively hygrophilous species of Sphagnum mosses predominate under mesotrophic conditions and a higher level of groundwater. The Eriophorum-Sphagnum community (Pinus sylvestris (f. willkomii) - Andromeda polifolia + Oxycoccus palustris + Eriophorum vaginatum + Rhynchospora alba - Sphagnum magellanicum) had the best long-term resilience for three years, and, as a result, it turned out to be the most tolerant to trampling. In some parts of the Phragmites-Sphagnum mesotrophic bog, the original plant community did not recover after three study years. Thus, this type of bog is the most vulnerable to recreational impact.

We consider the two-dimensional bimolecular recombination of charge carriers in amorphous organic semiconductors having the lamellar structure. We calculate the dependence of the effective recombination rate constant on the carrier density taking into account the correlated nature of the energetic disorder typical for organic semiconductors. The resulting recombination kinetics demonstrates a very rich variety of behaviors depending on the correlation properties of the particular semiconductor and relevant charge density range.

This work is devoted to the study of application of new topologies in the design of networks‑on‑chip (NoCs). It is proposed to use two‑dimensional optimal circulant topologies for NoC design, and it is developed an optimized routing algorithm with the decreased memory usage. The proposed routing algorithm was compared with Table routing, Clockwise routing, and Adaptive routing algorithms, previously developed for ring circulant topologies, and specialized routing algorithm for multiplicative circulants. The results of synthesis of routers implementing proposed routing algorithms are presented. The cost of ALM and register resources for the implementation of communication subsystems in NoCs with circulant topologies is estimated.

2-Azidomethyl-5-ethynylfuran, a new ambivalent compound with both azide and alkyne moieties that can be used as a self-clickable monomer, is synthesized starting directly from renewable biomass. The reactivity of the azide group linked to furfural is tested via the efficient preparation of a broad range of furfural-containing triazoles in good to excellent yields using a ‘green’ copper(I)-catalyzed azide–alkyne cycloaddition procedure. Access to new bio-based chemicals and oligomeric materials via a click-chemistry approach is also demonstrated using this bio-derived building block.

Oxidative esterification of biomass-derived 5-(hydroxymethyl) furfural (HMF) and furfural and their derivatives has been performed using a simple MnO2/NaCN system. The developed method allows the selective one-pot transformation of HMF to dimethyl furan-2,5-dicarboxylate (FDME) in 83% isolated yield without the formation of a free acid. Simplification of FDME production provides the missing link for manufacturing sustainable value-added materials from biomass. Addition of water to the oxidative system allows finetuning of reaction selectivity to obtain the previously difficult-to-access pure methyl 5-(hydroxylmethyl) furan-2-carboxylate in one step directly from the unprotected HMF without chromatographic separation.

  The purpose of this paper is   to answer two questions left open in [B. Durand, A. Shen, and N. Vereshchagin, Descriptive Complexity of Computable Sequences, Theoretical Computer Science 171 (2001), pp. 47--58].   Namely, we consider   the following two   complexities of an infinite computable 0-1-sequence $\alpha$:   $C^{0'}(\alpha )$, defined as   the minimal length of a program with oracle $0'$ that prints   $\alpha$,   and  $\MM(\alpha)$,  defined as $\limsup C(\alpha_{1:n}|n)$, where   $\alpha_{1:n}$ denotes the length-$n$ prefix of $\alpha$   and $C(x|y)$ stands for conditional Kolmogorov complexity.   We show that $C^{0'}(\alpha )\le \MM(\alpha)+O(1)$ and   $\MM(\alpha)$ is not bounded by any computable function of $C^{0'}(\alpha )$,   even   on the domain of computable sequences.

Independent domination is one of the rare problems for which the complexity of weighted and unweighted versions is known to be different in some classes of graphs. Trying to better understand the gap between the two versions of the problem, in the present paper we prove two complexity results. First, we extend NP-hardness of the weighted version in a certain class to the unweighted case. Second, we strengthen polynomial-time solvability of the unweighted version in the class of P2+P3-free graphs to the weighted case. This result is tight in the sense that both versions are NP-hard in the class of P3+P3-free graphs, i.e. P3+P3 is a minimal graph forbidding of which produces an NP-hard case for both versions of the problem.

We consider the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology rings of $X_h$ are described using GKM-theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and regular semisimple Hessenberg varieties well known in algebraic geometry. Both spaces $X_h$ and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families which can be generalized to other submanifolds of the flag variety.

Learning models with discrete latent variables using stochastic gradient descent remains a challenge due to the high variance of gradients. Modern variance reduction techniques mostly consider categorical distributions and have limited applicability when the number of possible outcomes becomes large. In this work, we consider models with latent permutations and propose control variates for the Plackett-Luce distribution. Our proof-of-concept experiment recasts optimization over permutations as a variational optimization w.r.t. the Plackett-Luce distribution and solves it using stochastic gradient descent.

We consider quadrangles of perimeter 2 in the plane with marked directed edge. To such quadrangle  Q two-dimensional plane  P with orthonormal base is corresponded. Orthoganal plane defines  a plane quadrangle R of perimeter 2 and with marked directed edge. This quadrangle is definded uniquely ( up to rotation and symmetry ). Quadrangles Q and R will be called dual to each ather. The following properties of duality are prooved: a)duality preserves qonvexity, non convexity and self-intersection;b) duality preserves the length of diagonals; c) the sum of length of corresponding edges in Q and R is 1.l

Presburger Arithmetic PrA is the true theory of natural numbers with addition. We consider linear orderings interpretable in Presburger Arithmetic and establish various necessary and sucient conditions for interpretability depending on dimension n of interpretation. We note this problem is relevant to the interpretations of Presburger Arithmetic in itself, as well as the characterization of automatic orderings. For n = 2 we obtain the complete criterion of interpretability.

We consider quadrangles of perimeter 2 in the plane with marked directed edge. To such quadrangle $Q$ a two-dimensional plane $\Pi\in\mathbb{R}^4$ with orthonormal base is corresponded. Orthogonal plane $\Pi^bot$ defines a plane quadrangle $Q^\circ$ of perimeter 2 and with marked directed edge. This quadrangle is defined uniquely (up to rotation and symmetry). Quadrangles $Q$ and $Q^\circ$ will be called dual to each other. The following properties of duality are proved: a) duality preserves convexity, non convexity and self-intersection; b) duality preserves the length of diagonals; c) the sum of lengths of corresponding edges in $Q$ and $Q^\circ$ is 1.

Interactions between vortices in thin superconducting films are investigated in the crossover (intertype) regime between superconductivity types I and II. We consider two main factors responsible for this crossover: a) changes in the material characteristics of the film and b) variations of the film thickness controlling the effect of the stray magnetic fields outside superconducting sample. The analysis is done within the formalism that combines the perturbation expansion of the microscopic equations to one order beyond the Ginzburg-Landau theory with the leading contribution of the stray fields. It is shown that the latter gives rise to qualitatively different spatial profile and temperature dependence of the vortex interaction potential, as compared to bulk vortex interactions. The resulting interaction is long-range repulsive while exhibiting complex competition of attraction and repulsion at small and intermediate separations of vortices. This explains the appearance of vortex chains reported earlier for superconducting films.

Buchstaber and Terzic introduced a notion of universal space of parameters F for a manifold M^2n, which has an effective action of compact torus T^k , k≤n with some additional properties. with special properties. This space is needed to construction of factor M^2n/T^k. Buchstaber and Terzic constructed the universal space of parameters for G_5,2. In this work we construct universal space of parameters for complex Grassmann manifold G_q+1,2. Our construction is based on the construction of moduli space of stable curves of genus zero with q+1 marked points due to Salamon, McDuff and Hofer.

Let G be a complex Lie group acting on a compact complex Hermitian manifold M by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott-Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.

Motivated by the Chern-Weil theory, we prove that for a given vector bundle E on a smooth scheme X over a field k of any characteristic, the Chern classes of E in the Hodge cohomology can be recovered from the Atiyah class. Although this problem was solved by Illusie in \cite{i}, we present another proof by means of derived algebraic geometry.  Also, for a scheme X over a field k of characteristic p with a vector bundle E we construct elements c^cris_n(E,α(E))∈H^2n_dR(X) using an obstruction α(E) to a lifting of F∗E to a crystal modulo p2 and prove that c^cris_n(E,α(E))=n!⋅c^dR_n(E), where cdRn(E)are the Chern classes of E in the de Rham cohomology and F is the Frobenius map.

Sasakian manifolds are odd-dimensional counterpart to Kahler manifolds. They can be defined as contact manifolds equipped with an invariant Kahler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition on Vaisman and Sasakian manifolds. We construct a Lie superalgebra associated to a Sasakian manifold in the same way as the Kahler supersymmetry algebra is associated to a Kahler manifold. We use this construction to produce a self-contained, coordinate-free proof of the results by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and cohomology of Sasakian and Vaisman manifolds. In the last section, we compute the supersymmetry algebra of Sasakian manifolds explicitly.

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup with the category of GL(N−1,ℂ[[t]])-equivariant perverse sheaves on the affine Grassmannian of GL_N. We explain how our equivalences fit into a more general conjectural framework proposed by D. Gaiotto.

We classify smooth Fano threefolds with infinite automorphism groups.

We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.

We classify smooth Fano weighted complete intersections of large codimension.

We show that automorphism groups of Hopf and Kodaira surfaces have unbounded finite subgroups. For elliptic fibrations on Hopf, Kodaira, bielliptic, and K3 surfaces, we make some observations on finite groups acting along the fibers and on the base of such a fibration.

We prove that a finite group acting by birational automorphisms of a non-trivial Severi-Brauer surface over a field of characteristic zero contains a normal abelian subgroup of index at most 3. Also, we find an explicit bound for orders of such finite groups in the case when the base field contains all roots of 1.

Given a holomorphic conic bundle without sections, we show that finite groups acting by its fiberwise bimeromorphic transformations are bounded. This provides an analog of a similar result obtained by T.Bandman and Yu.Zarhin for quasi-projective conic bundles.