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## Koszulity of cohomology = K(π,1)-ness + quasi-formality

This paper is a greatly expanded version of [37, Section 9.11]. A series of definitions and results illustrating the thesis in the title (where quasi-formality means vanishing of a certain kind of Massey multiplications in the cohomology) is presented. In particular, we include a categorical interpretation of the “Koszulity implies K(π,1)$\u201d\; claim,\; discuss\; the\; differences\; between\; two\; versions\; of\; Massey\; operations,\; and\; apply\; the\; derived\; nonhomogeneous\; Koszul\; duality\; theory\; in\; order\; to\; deduce\; the\; main\; theorem.\; In\; the\; end\; we\; demonstrate\; a\; counterexample\; providing\; a\; negative\; answer\; to\; a\; question\; of\; Hopkins\; and\; Wickelgren\; about\; formality\; of\; the\; cochain\; DG-algebras\; of\; absolute\; Galois\; groups,\; thus\; showing\; that\; quasi-formality\; cannot\; be\; strengthened\; to\; formality\; in\; the\; title\; assertion.$

We prove that the Milnor ring of any (one-dimensional) local or global field K modulo a prime number l is a Koszul algebra over Z/l. Under mild assumptions that are only needed in the case l=2, we also prove various module Koszulity properties of this algebra. This provides evidence in support of Koszulity conjectures that were proposed in our previous papers. The proofs are based on the Class FIeld Theory and computations with quadratic commutative Groebner bases (commutative PBW-bases).

This is an introduction to: (1) the enumerative geometry of rational curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer Institute.

The aim of this paper is to systematize the variety of logical hylomorphism according to different types of formal relations. Various explications of substantial and dynamic formality will be sketched. My larger purpose is to discuss demarcation principles for the bounds of logic as formal ontology and formal deontology.

Recently there has appeared an increased necessity in publishing research results in the English language to share knowledge and experience. International scientific journals put certain requirements to the quality of the language; therefore, there arise problems concerning the standards of English as a means of scientific publication. It is known that academic style is characterized by formality, logical, coherent and cohesive presentation of arguments, abstraction, nominality, accuracy and the objective attitude of the author to facts stated. These characteristics can be achieved through various techniques of the English language, in particular, through lexical, morphological and syntactic features, which are discussed in the article. Awareness of the English lexical-grammatical system and stylistic peculiarities is considered as a necessary condition for successful inclusion in the scientific world community.

We construct and study universal spaces for birational invariants of algebraic varieties over algebraic closures of finite fields.

This paper addresses Ludwig Wittgenstein’s claim that “there can never be surprises in logic” (TLP 6.1251) from a perspective of the distinction between substantial and dynamic models of formality. It attempts to provide an interpretation of this claim as stressing the dynamic formality of logic. Focusing on interactive interpretation of compositionality as dynamic formality, it argues for the advantages of dynamic, i.e., game-theoretical approach to some binary semantical phenomena. Firstly, model-theoretical and game-theoretical interpretations of binary quantifiers are compared. Secondly, the paper offers an analysis ofWittgenstein’s idea that mixed colours (e.g., bluish green, reddish yellow, etc.) possess logical structures. To answer some experimental challenges, it provides a game-theoretical interpretation of the colours opponency in Payoff Independence (PI) logic. Comparing Nikolay Vasiliev’s logical principles and Wittgenstein’s internal properties and relations, Wittgenstein’s approach is argued for as an attempt of modelling a balance between logic and the empirical.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

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

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.