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Oberwolfach Reports
The Mathematisches Forschungsinstitut Oberwolfach regularly organizes workshops in all fields of Mathematics. Their aim is to offer 45–48 experts, invited by the Institute's Director, the opportunity to present recent research results, especially new methods, and to initiate future research projects. A great number of important papers have been initiated at Oberwolfach due to informal talks and discussions in small groups. Contrary to the typically large conferences all over the world, workshops at Oberwolfach emphasize active research. The Oberwolfach Reports are meant to capture, in an informal manner, the characteristic ideas and discussions of these workshops. As a service to the community, they are now offered by the Institute, at a nominal price, and allow the public to partake in the lively and stimulating atmosphere of these meetings. While the peerreviewed results will appear elsewhere, the Oberwolfach Reports will keep the reader abreast of current developments and open problems, and serve as an indispensable source of information for the active mathematician.
In [K], a convexgeometric algorithm was introduced for building new analogs of Gelfand–Zetlin polytopes for arbitrary reductive groups. Conjecturally, these polytopes coincide with the Newton–Okounkov polytopes of flag varieties for a geometric valuation. I outline an algorithm (geometric mitosis) for finding collec tion of faces in these polytopes that represent a given Schubert cycle. For GL_n and Gelfand–Zetlin polytopes, this algorithm reduces to a geometric version of Knutson–Miller mitosis introduced in [KST].
This article covers possible ways of implementing intersubject relations in foreign language class as a reflection of modern trends for integration of science and practice and a way to fulfill the tasks set by modern education concepts.
This article consider The project of the scientific and educational Center for integration of multimedia technologies in science, education and culture, as spacetechnological environment for the implementation of innovative scientific and educational projects of the 21st century, which should become the support for the master's programs, especially interdisciplinary; at the intersection of science, art and information technologies, and implementation of innovative scientific and commercial projects, which are to become a master's thesis.
The three already traditional volumes of the WDS Proceedings you are holding in the hands are composed of the contributions which have been presented during the 21st Annual Conference of Doctoral Students that was held in Prague, at Charles University, Faculty of Mathematics and Physics from May 29 to June 1, 2012. In this year, 100 student manuscripts were submitted to publishing and 88 were accepted after the review process.

The paper discusses in detail the scale of translation of primary points scored by school graduates in the unified state exam in mathematics, used from 2013 to the present time. Based on the analysis of the dynamics of these scales, a conclusion is made about the annual increase in the "average" 100point result, as well as the presence of a significant increase in the final grade compared with the linear scale. Additionally, the authors describe the effect of reducing the value of primary points as they approach the maximum.
This article presents the results of a pilot study assessing the level of formation of a stochastic competence among teachers of mathematics. Besides, the indicators that reflect the competence of formation of stochastic students are identified and ranked in order of importance. Different instruments (questionnaires, tests, assignments) have been used to solve the problem under study.
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 crosssection of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a crosssection 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 crosssection 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 crosssection 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 Wequivariant map T   >G/T where T is a maximal torus of G and W the Weyl group.