Algebra, Geometry, and Physics in the 21st Century
This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim’s vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim’s heart.
This paper focuses on the algebraic theory underlying the study of the complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We unify, simplify, and extend parts of the three approaches that have been developed to study the CSP over finite templates - absorption theory that was used to characterize CSPs solvable by local consistency methods (JACM'14), and Bulatov's and Zhuk's theories that were used for two independent proofs of the CSP Dichotomy Theorem (FOCS'17, JACM'20). As the first contribution we present an elementary theorem about primitive positive definability and use it to obtain the starting points of Bulatov's and Zhuk's proofs as corollaries. As the second contribution we propose and initiate a systematic study of minimal Taylor algebras. This class of algebras is broad enough so that it suffices to verify the CSP Dichotomy Theorem on this class only, but still is unusually well behaved. In particular, many concepts from the three approaches coincide in the class, which is in striking contrast with the general setting. We believe that the theory initiated in this paper will eventually result in a simple and more natural proof of the Dichotomy Theorem that employs a simpler and more efficient algorithm, and will help in attacking complexity questions in other CSP-related problems.
Article analyzes the methods of study of complex systems. A general approach to the design of complex systems based on the use of the theory of tensor analysis. The expediency of transition from continuous to discrete space. The necessity of the transition to the new geometry that is different from the analytic or projective. We introduce the notion of Petri net structure or NP-structure. Introduces the concept of the reference model NP-structures. We introduce the concept of coordinate transformation NP structures. We introduce the notion of the invariant under tensor analysis of NP-structures. A general model of the use of tensor methods and detailing the study of superconducting structures
The article is devoted to the problem of interpreting the concept of time in the context of some concepts of modern physics. This fundamental concept inevitably arises in physical theories, but there is still no adequate description of it in the philosophy of science. In the theory of relativity, quantum field theory, the standard model of elementary particle physics, the theory of loop quantum gravity, superstring theory, and other recent theories, the idea of time is explicitly or implicitly present. Sometimes, as, for example, in the special theory of relativity, it plays a pronounced role, sometimes not, but somehow it is and is implied by the content of the theory and in some cases by its mathematical apparatus. Especially important for solving the problem is the fundamental difference between the space-time processes of the microcosm and the macrocosm. In this connection, there is a need to understand time as it appears in modern physics, its description in the language of philosophy (there is also no satisfactory mathematical apparatus for describing time). This will make it possible to get closer to the answer to the question about the properties of time and, if not answered, then formulate the correct question about what it is. To this end, this study analyzes the key concepts of modern physics, taking into account the historical-scientific and historical-philosophical perspectives, which in some cases allows us to detect the continuity of ideas related to the understanding of time, their development, as well as the emergence of fundamentally new ones. In the course of the analysis, the characteristics of time that are correct from the point of view of physical theory are formulated, and an attempt is made to raise the question of the nature of time. Based on the work done, conclusions are drawn about the current state of the problem and the prospects for its solution.
In this, the third paper of the series, we construct a large family of representations of the quantum toroidal gl(1)-algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application We obtain a Gelfand-Zetlin-type basis for a class of irreducible lowest weight gl(infinity)-modules.
We consider the “Matthew effect” in the citation process which leads to reallocation (or misallocation) of the citations received by scientific papers within the same journals. The case when such reallocation correlates with a country where an author works is investigated. Russian papers in chemistry and physics published abroad were examined. We found that in both disciplines in about 60% of journals Russian papers are cited less than average ones. However, if we consider each discipline as a whole, citedness of a Russian paper in physics will be on the average level, while chemistry publications receive about 16% citations less than one may expect from the citedness of the journals where they appear. Moreover, Russian chemistry papers mostly become undercited in the leading journals of the field. Characteristics of a “Matthew index” indicator and its significance for scientometric studies are also discussed.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
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.