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## The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem

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.

We use (non-)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Ext's in a suitable bisheaf category. We compare our approach to various definitions present in the literature.

The aim of this work is to construct a complex which through its higher structure directly controlls deformations of general prestacks, building on the work of Gerstenhaber and Schack for presheaves of algebras. In defining a Gerstenhaber–Schack complex for an arbitrary prestack , we have to introduce a differential with an infinite sequence of components instead of just two as in the presheaf case. If denotes the Grothendieck construction of , which is a -graded category, we explicitly construct inverse quasi-isomorphisms and between and the Hochschild complex , as well as a concrete homotopy , which had not been obtained even in the presheaf case. As a consequence, by applying the Homotopy Transfer Theorem, one can transfer the dg Lie structure present on the Hochschild complex in order to obtain an -structure on , which controlls the higher deformation theory of the prestack . This answers the open problem about the higher structure on the Gerstenhaber–Schack complex at once in the general prestack case.

We define and study the Hochschild (co)homology of the second kind (known also as the Borel-Moore Hochschild homology and the compactly supported Hochschild cohomology) for curved DG categories. An isomorphism between the Hochschild (co)homology of the second kind of a CDG-category B and the same of the DG category C of right CDG-modules over B, projective and finitely generated as graded B-modules, is constructed. Sufficient conditions for an isomorphism of the two kinds of Hochschild (co)homology of a DG-category are formulated in terms of the two kinds of derived categories of DG-modules over it. In particular, a kind of “resolution of the diagonal” condition for the diagonal CDG-bimodule B over a CDG-category B guarantees an isomorphism of the two kinds of Hochschild (co)homology of the corresponding DG-category C. Several classes of examples are discussed. In particular, we show that the two kinds of Hochschild (co)homology are isomorphic for the DG-category of matrix factorizations of a regular function on a smooth affine variety over a perfect field provided that the function has no other critical values but zero.

The article considers the Views of L. N. Tolstoy not only as a representative, but also as a accomplisher of the Enlightenment. A comparison of his philosophy with the ideas of Spinoza and Diderot made it possible to clarify some aspects of the transition to the unique Tolstoy’s religious and philosophical doctrine. The comparison of General and specific features of the three philosophers was subjected to a special analysis. Special attention is paid to the way of thinking, the relation to science and the specifics of the worldview by Tolstoy and Diderot. An important aspect is researched the contradiction between the way of thinking and the way of life of the three philosophers.

Tolstoy's transition from rational perception of life to its religious and existential bases is shown. Tolstoy gradually moves away from the idea of a natural man to the idea of a man, who living the commandments of Christ. Starting from the educational worldview, Tolstoy ended by creation of religious and philosophical doctrine, which were relevant for the 20th century.

This important new book offers the first full-length interpretation of the thought of Martin Heidegger with respect to irony. In a radical reading of Heidegger's major works (from *Being and **Time* through the ‘Rector's Address' and the ‘Letter on Humanism' to ‘The Origin of the Work of Art' and the *Spiegel* interview), Andrew Haas does not claim that Heidegger is simply being ironic. Rather he argues that Heidegger's writings make such an interpretation possible - perhaps even necessary.

Heidegger begins *Being and Time* with a quote from Plato, a thinker famous for his insistence upon Socratic irony. *The Irony of Heidegger *takes seriously the apparently curious decision to introduce the threat of irony even as philosophy begins in earnest to raise the question of the meaning of being. Through a detailed and thorough reading of Heidegger's major texts and the fundamental questions they raise, Haas reveals that one of the most important philosophers of the 20th century can be read with as much irony as earnestness.* The Irony of Heidegger* attempts to show that the essence of this irony lies in uncertainty, and that the entire project of onto-heno-chrono-phenomenology, therefore needs to be called into question.

The article is concerned with the notions of technology in essays of Ernst and Friedrich Georg Jünger. The special problem of the connection between technology and freedom is discussed in the broader context of the criticism of culture and technocracy discussion in the German intellectual history of the first half of the 20th century.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.