### Book

## Методика изучения содержательной линии "Мера и мощность числовых множеств" в магистерских программах педагогических вузов

In the monography we consider theoretic and methodic origins of fundamental notions in the theory of functions of real variable. The text is designed for future and active school math teachers.

In this note we interpret Voevodsky's Univalence Axiom in the language of (abstract) model categories. We then show that any posetal locally Cartesian closed model category Qt in which the mapping Hom(w)(Z B;C) : Qt --> Sets is functorial in Z and represented in Qt satises our homotopy version of the Univalence Axiom, albeit in a rather trivial way. This work was motivated by a question reported in [Gar11], asking for a model of the Univalence Axiom not equivalent to the standard one.

We propose a new construction of surface measures on infinite-dimensional spaces

The text introduces all main subjects of "naive" (nonaxiomatic) set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals. Included are discussions and proofs of the Cantor-Bernstein Theorem, Cantor's diagonal method, Zorn's Lemma, Zermelo's Theorem, and Hamel bases. With over 150 problems, the book is a complete and accessible introduction to the subject.

We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of niteness, countability and innite equi-cardinality. We argue that from the homotopy theoretic point of view our construction is essentially automatic following basic existing methods, and so is (almost all) the verication that the construction works.

We use the posetal model category to introduce homotopy-theoretic intu- itions to set theory. Our main observation is that the homotopy invariant version of cardinality is the covering number of Shelah's PCF theory, and that other combinatorial objects, such as Shelah's revised power function - the cardinal function featuring in Shelah's revised GCH theorem | can be obtained using similar tools. We include a small \dictionary" for set theory in QtNaamen, hoping it will help in nding more meaningful homotopy-theoretic intuitions in set theory.

This book gives a systematic presentation of modern measure theory as it has developed over the past century. It includes material for a standard graduate course, advanced material not covered by the standard course but necessary in order to read research literature in the area, and extensive additional information on the most diverse aspects of measure theory and its connections with other fields. Over 850 exercises with detailed hints or references are given. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided. The subject index includes more than 1000 items. The book is intended for graduate students, instructors of courses in measure and integration theory, and researchers in all fields of mathematics; it may serve as either a textbook, a source for a variety of advanced courses, or a reference work.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

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.

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.