Upcoming meeting:
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Meeting #982
Ivan Chesnokov (Sobolev Institute of Mathematics)
"On the description of the centroid of CT-groups"
Abstract. In 2005, A. G. Myasnikov and S. Lyutikov introduced the concept of the centroid of an arbitrary group as the ring of all normal quasi-endomorphisms of this group. The study of this structure led them to a result on the rigidity of finitely generated non-abelian free nilpotent groups. The study of centroids is important for the theory of exponential groups (MR-groups), since the centroid of a group is the maximal ring of scalars of this group.
The talk will present joint results of A. V. Treyer (Sobolev Institute of Mathematics) and the present author. A criterion was obtained for decomposing the centroid of an arbitrary group into a direct product of the centroids of centralizers. Based on this result, explicit descriptions of the centroids of free soluble groups and the lamplighter group were obtained.
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Meeting #981
Artem Iljev (Sobolev Institute of Mathematics)
"On the bound of clustering complexity of graph in the Cluster Deletion problem with clusters of bounded size"
Abstract. In the research, the graph clustering problem with bounded size s of clusters is considered. In the Cluster Deletion problem, for a given graph G, one has to find a nearest cluster graph on the same vertex set and the edge set of cluster graph is a subset of edge set of the original graph G. Using a polynomial time approximation algorithm for solving this problem for arbitrary s>3, an upper bound of clustering complexity of graph is proved by algebraic methods.
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Meeting #980
Savva Voloshin (Sobolev Institute of Mathematics)
"Generalized operations of discrete differentiation and integration"
Abstract. The talk will be devoted to generalized operations of discrete differentiation and integration, which have not been used in serious research so far. V.A. Romankov proposed applications in cryptography and information security, especially in light of the discovered vulnerabilities of most common protocols. In particular, it is planned to deal with the version of the Massey-Omura protocol, with several points about application to data storage, with the lemma on the choice of invertible matrices, to which there is a reference, and, perhaps, with how Peter M. Neumann implicitly proves the existence of a discrete integral in his paper.
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Meeting #979
Alexandre Grishkov (Sobolev Institute of Mathematics)
"Algebraic diassociative loops and cubic forms"
Abstract. In 1955 A.Malcev introduce the notion of analitic (local) diassociative loop, where every two elements gerate a subgroup. In 1963 Yu.Manin note that cubic surface has a structure of algebraic (local) diassociative loop. There exists big difference between those definition. In this talk we discuss the results on analitic and algebraic diassocitive loops obtained in last years.
Alexander Zubkov (Sobolev Institute of Mathematics)
"Approaches to the description of (super)groups of symmetries of the Duflo-Serganova functor"
Abstract. The talk considers two approaches to describing the symmetry (super)groups of the Duflo-Serganova functor. In the original definition, this functor associates with each supermodule \(M\) over a given Lie superalgebra \(g\) its cohomology group with respect to the action of an odd element \(x\) of \(g\) such that \([x, x]=0\), that is, \(M_x=ker(x)/Im(x)\). It is easy to see that \(M_x\) is a supermodule over the Lie superalgebra \(g_x=cent_g(x)/[g, x]\). A natural question arises: if \(g\) is a Lie superalgebra of some supergroup \(G\), then is it possible to "lift" (integrate) the action of \(g_x\) to the action of the supergroup naturally associated with \(G\)?
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Meeting #978
José Victor Gomes Teixeira (USP/UFRN)
"The Group of Outer Automorphisms of the Category of Finitely Generated Free Non-associative Nilpotent of Degree n Algebras"
Abstract. The abstract in the attached file.
Abstract
Latest announcements:
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Meeting #977
Matvei Kotov (Sobolev Institute of Mathematics)
"Tropical Approach in Cryptography: Old and New Results"
Abstract. Grigoriev and Shpilrain proposed a key-exchange protocol based on a min-plus matrix algebra in 2011. Several new protocols based on tropical algebra and attacks on some of them have been proposed for the last few years. In this talk, I recall some old and new results in tropical cryptography. The main part of this talk will be devoted to the problem of solving one-sided systems of tropical polynomial equations of degree two. The worst-case complexity and the generic-case complexity of this problem will be discussed. This talk based on joint work with I. Buchinskiy and A. Treier.
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Meeting #976
Pavel Gvozdevsky (Bar-Ilan University, Israel)
"On countable isotypic structures"
Abstract. Two structures in the same first order language are isotypic if for any natural number n the sets of n-types that are realised in these structures coincide. We discuss several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct certain countable isotypic but not isomorphic structures: totally ordered sets, fields, and groups. This answers an old question by B. Plotkin for groups.
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Meeting #975
Alexander Treyer (Sobolev Institute of Mathematics)
"Quasivarieties of colored graphs"
Abstract. The report will provide an overview of the main results and problems associated with colorings of graphs and metric spaces, and will also demonstrate how the apparatus of universal algebraic geometry can be used to address this problem.
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Meeting #974
Denis Solomatin (Omsk State Pedagogical University)
"Comparison of outerplanarity and generalized outerplanarity properties of Cayley graphs of planar semigroups"
Abstract. The report provides an overview of the results obtained for direct products of cyclic semigroups (monoids, semigroups with zero), free partially commutative nilpotent semigroups, ordinal sums of rectangular semigroups admitting outerplanar or generalized outerplanar Cayley graphs, and found a new series of semigroups with generalized outerplanarity property of Cayley graphs are given which are equivalent to their property of being planar, but not to being outerplanar, and series of semigroups whose property of generalized outerplanarity of Cayley graphs is equivalent to their property of being outerplanar, but not equivalent to being planar.
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Meeting #973
Ivan Buchinskiy (Sobolev Institute of Mathematics)
"Equationally Noethericity property for predicate algebraic systems"
Abstract. Predicate algebraic systems, or predicate structures, are algebraic systems over languages consisting only of predicate symbols, and which can be extended by constant symbols. Well-known representatives of such structures are, for example, graphs, hypergraphs, partially ordered sets. An algebraic system \( A \) is called equationally Noetherian if each system of equations considered over \( A \) is equivalent to some of its finite subsystems. This property occupies an important place in universal algebraic geometry, a branch of mathematics that studies equations over various algebraic systems. Previously, the results of studies of the property of equationally Noethericity for some specific predicate structures have already been obtained. In our report we will present a brief overview of the results of this direction of research and will formulate a general criterion for equationally Noethericity for an arbitrary predicate algebraic system, obtained jointly with M.V. Kotov and A.V. Treier.
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Meeting #972
Alexander Prolubnikov (Dostoevsky Omsk State University)
"Solving some problems on graphs using methods for solving systems of linear equations"
Abstract. Сombinatorial problems on graphs may be stated as computational problems of linear algebra. In such formulations, many graph problems can be considered. Along with combinatorial approaches to constructing algorithms for solving problems on graphs, this approach is quite traditional and allows to obtain computational algorithms that make the most efficient use of the capabilities provided by modern computer architectures for implementing parallel computing and optimizing memory management. The algorithms we propose for solving some problems on graphs are based on matching a graph with a system of linear algebraic equations with a graph adjacency matrix and a specially selected right-hand side. Among the problems we consider are the following: the problem of checking isomorphism of graphs and related problems, including the problem of restoring a graph from a set of its subgraphs, the problem of traversing the graph.
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Meeting #971
Alexander Rybalov (Sobolev Institute of Mathematics)
"On complexity of the word problem in semigroups with homogeneous relations"
Abstract. We study the computational complexity of the word problem in semigroups with the condition of homogeneity of the defining relations. These are finitely defined semigroups, in which for each defining relation the lengths of the left and right parts are equal. The word problem for such semigroups is decidable, but known algorithms require exponential time and memory. We prove that this problem belongs to the class PSPACE, consisting of algorithmic problems that are solved by Turing machines using space (memory cells) bounded polynomially. This improves the upper bound on the space complexity known before. On the other hand, we prove that there exists a semigroup with the condition of homogeneity of defining relations, in which the equality problem is complete in the class PSPACE with respect to polynomial reducibility. It is assumed (although not proven) that the class PSPACE is wider than the class NP and, even more so, the class P.
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Meeting #970
Andrey Mironov (Sobolev Institute of Mathematics)
"Rational functions on algebraic curves and commuting difference operators"
Abstract. In the report I will tell you how, using a certain set of rational functions on an algebraic curve, you can construct commuting difference operators. It will also be described how to obtain ordinary commuting differential operators using passage to the limit from commuting difference operators.
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Meeting #969
Alexander Prolubnikov (Omsk State University)
"An approach to finding connected components of graphs using perturbations of adjacency matrices."
Abstract. We consider the problem of finding connected components of a graph. The natural approach to the problem is the breadth-first search (BFS) that is implemented in the software libraries which are using the benefits of modern computer architectures. We present the approach that uses perturbations of adjacency matrix of a graph to implement the connectivity test. To perform the test, we compare solutions of the two systems of linear algebraic equations (SLAE) which we associate with the unperturbed adjacency matrix and with the perturbed one. Such approach allows to use computationally effective implementations of numerical methods for solving SLAE to solve connectivity problems on graphs. Iterations of iterative methods for solving SLAE may be considered as implementations of graph traversals which, in general, are not equivalent to the traversal that is implemented by BFS. We present the algorithm to find connected components of a graph that uses such a traversal. For every instance of the problem, the computational complexity of the algorithm is not greater than the one of BFS, and it is lower than that in most instances.
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Meeting #968
Alexey Nikitin (Deeplay, Omsk)
"Computations in universal classes with applications for graph and order models."
Abstract. In the talk I will present results concerning algebraic geometry over partially ordered sets:
1. Complexity of algorithms for solving systems of equations over partial orders;
2. Noethericity by equations for partial orders;
3. Decidability of theories over partial orders;
4. Construction of radicals and coordinate partial orders for systems of equations over partial orders and strict partially ordered sets. -
Meeting #967
Vitaly Roman'kov (Sobolev Institute of Mathematics)
"On the embedding of free nilpotent groups in partially commutative nilpotent groups."
Abstract. An algorithm is presented that allows one to calculate the maximum rank of a free nilpotent group of class k into a given partially commutative nilpotent group of class k.
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Meeting #966
Marina Rasskazova (Omsk State Technical University)
"Binary Lie algebras and its application to the theory of Binary Lie superalgebras."
Abstract. The tropical structures min-plus and max-plus are well-known and have many interesting applications.
By definition an algebra \(B\) is binary-Lie algebra iff any two elements \(a,b\in B\) generate a Lie subalgebra. A \({\bf Z}_2-\)graded algebra \(B=B_0\oplus B_1\) is a binary-Lie superalgebra iff \(B\times \Gamma=B_0\otimes \Gamma_0\oplus B_1\otimes \Gamma_1\) is a binary Lie algebra, where \(\Gamma=\Gamma_0\oplus \Gamma_1\) is a Grassman algebra with natural gradation. We apply the theory of binary-Lie algebras for proving the following result.
Theorem. Let \(B=B_0\oplus B_1\) be a simple binary-Lie superalgebra finite dimensional over the field \({\bf C}\) of complex numbers. Then \(B_0\) is a solvable algebra or \(B\) is a Lie superalgebra.
This theorem reduced the problem (open yet!) of classification of a simple binary-Lie superalgebra finite dimensional over the field \({\bf C}\) to the case where subalgebra \(B_0\) is solvable.
This talk is based on the joint paper with A.Grishkov and I.Shestakov.