Upcoming meeting:
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Meeting #987
Alexander Zubkov (Sobolev Institute of Mathematics)
"Almost-simple supergroups"
Abstract. In the category of algebraic groups over an algebraically closed field (of arbitrary characteristic), it is proved that a connected, smooth, semisimple group is quasi-isomorphic to a direct product of almost-simple algebraic groups. An almost-simple group is a connected, smooth, non-commutative group in which every connected, smooth, normal, proper subgroup is trivial. Almost-simple groups are uniquely determined by their root systems, i.e., by their indecomposable Dynkin diagrams. An equivalent definition of an almost-simple group is that every proper normal subgroup of it is finite. Passing to the category of algebraic supergroups, one can notice that the two above definitions of almost-simplicity are no longer equivalent if the ground field has nonzero characteristic. Moreover, the first definition gives a wider class of supergroups than the second. The problem of classifying almost-simple supergroups of both types arises, similar to the problem of classifying simple Lie superalgebras. The report will discuss the author's recent results related to this problem.
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Meeting #986
Ivan Buchinskiy (Sobolev Institute of Mathematics)
"Centroids of partially commutative two-step nilpotent groups"
Abstract. The talk will present a description of the centroid (according to S. Lioutikov and A. Myasnikov) of an arbitrary finitely generated partially commutative two-step nilpotent R-exponential group, where R is an arbitrary binomial ring. The resulting description completely depends on the structure of the commutativity graph. In addition, based on the resulting description, it is proved that if the non-commutativity graph consists of exactly one connected component, then the group is rigid. As an intermediate result, a description of the centralizers of block elements in a finitely generated partially commutative two-step nilpotent R-exponential group will be given.
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Meeting #985
Arkady Tsurkov (UFRN)
"Categories and functors of universal algebraic geometry. Automorphic equivalence of algebras"
Abstract. The abstract in the attached file.
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Meeting #984
Alexander Zubkov (Sobolev Institute of Mathematics)
"On Duflo-Serganova functor in positive characteristic"
Abstract. In my talk I am going to discuss the following properties of Duflo-Serganova functor in positive characteristic:
1. The symmetry supergroups of this functor, minimal and extended ones. They will be explicitly calculated for the supergroups GL(m|n), Q(n) and some square-zero odd elements of their Lie superalgebras.
2. Vanishing of Duflo-Serganova functor on injective supermodules. For a specific class of supergroups we formulate a criterion, when a supermodule is injective, that involves the vanishing of this functor.
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Meeting #983
Ivan Buchinskiy (Sobolev Institute of Mathematics)
"Equationally Noetherian partially commutative two-step nilpotent groups"
Abstract. The talk will present the equivalence of the properties of equational Noethericity of a partially commutative (graph) two-step nilpotent group G and equational Noethericity of its commutativity graph \(\Gamma_G\) considered in the category of graphs with loops. In particular, it will be shown that to study equational Noethericity of a partially commutative two-step nilpotent group, it suffices to consider equations in one variable. Using the concept of residuality of groups, it is proved that an arbitrary partially commutative two-step nilpotent group is embedded in a countable direct power of a free two-step nilpotent group of rank 2. In addition, earlier in the works of A.J. Duncan, I.V. Kazachkov and V.N. Remeslennikov it was shown that the centralizer dimension of each finitely generated free partially commutative group coincides with the height of the lattice of canonical centralizers of this group. In this talk we will present some results relating the centralizer dimension and the height of the lattice of canonical centralizers for the case of an infinitely generated free partially commutative group, and we will generalize a similar result from the work of V. Blatherwick to the case of an infinitely generated partially commutative two-step nilpotent group.
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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.