Mathematical Education in Universities in the Soviet Union and Modern Russia
Traditions of mathematical education in Russia on both school and university level, research done by Russian scientists and its impact on the development of mathematics is considered by many a unique and valuable part of the world cultural heritage. In the present paper, we describe the development of mathematical education in Russian universities after 1955 — a period that proved to be most fruitful.
The competence of mathematical modelling is well conceptualized, thought a much-debated question is how to develop it in schools. International achievement test PISA consider mathematics achievements from the modelling perspective (formulating, employing and interpreting), and provides us with comprehensive data to analyze school factors in math results from the comparative perspective. The goal of our study was to estimate the effect of teaching practices on students’ achievements in different PISA mathematical processes while controlling prior achievements (TIMSS).
It is widely known that Soviet school of exact sciences, was among the strongest in the world, particularly in terms of physics and mathematics. Why? This is the question we would like to address in this paper by collecting and summarizing different viewpoints on this issue expressed by prominent mathematicians. Many of them witnessed the most fruitful period, the “golden years” of Soviet science and played a major role in the subsequent development of Soviet/Russian mathematics. There is little controversy in the explanations provided by different people; the only essential differences are in the emphases. Thus the list of factors may be regarded as precisely determined. This paper simply aims at communicating them to a non-mathematical community interested in issues of science and education.
The significance of the education in the field of philosophy of mathematics as the part of both philosohpy and mathematics at the universities is the subject of the article.
Proceedings of the Moscow regional conference of The International Group for the Psychology of Mathematics Education (PME) and Yandex are presented.
Math in Moscow (MiM) is the name of a short-term (1-2 semesters) study abroad program offered in English jointly by the Independent University of Moscow (IUM), National Research University Higher School of Economics (HSE), and Moscow Center for Continuous Mathematical Education (MCCME). It was first launched in spring 2001 by IUM. Along with courses in mathematics and computer science, students can study Russian language, Russian literature, history of mathematics and science, and history of Russia. All MiM courses are credited to the students at their home institutions.
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.