Upcoming meeting:
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Meeting #993
Alexander Zubkov (Sobolev Institute of Mathematics)
"Quasi-reductive supergroups with small even part"
Abstract. We describe supergroups G whose largest even supersubgroup are isomorphic to GL_2, SL_2, or PSL_2. Such supergroups arise when describing the centralizers of certain tori in an arbitrary quasi-reductive supergroup. This result may occur useful in the structure theory of quasi-reductive supergroups and their root systems.
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Meeting #992
Alexander Rybalov (Sobolev Institute of Mathematics)
"On the Diophantine problem related to power circuits"
Abstract. In 2010 Miasnikov, Ushakov and Won introduced so-called power circuits to prove the polynomial-time solvability of the word problem in the Baumslag group, which has a non-elementary Dehn function. Power circuits are the simplest programs for computing over integers, in which at each step the operations of addition or the operation of multiplication by a power of two (x,y) = x 2^y are applied either to the input variable or to what was calculated in the previous step. Thus, power circuits compute the values of terms of the algebraic structure N* =
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Meeting #991
Alexandre Grishkov (San Paulo University, Brazil), Marina Rasskazova (São Paulo, Federal University of ABC, Brazil)
"J-nilpotent Mal'tsev algebras"
Abstract. We describe the Mal'tsev algebras in the minimal variety J4 of Mal'tsev algebras containing all Lie algebras. The main result consists of partitioning the variety J4 into four parts. The main difficulty is proving that each part is nonempty.
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Meeting #990
Alexei Miasnikov (Stevens Institute of Technology)
"On the Diophantine Problem in Wreath Products of Groups"
Abstract. In the talk I will show that the problem of solvability of equations with coefficients in wreath products of Abelian groups is unsolvable in almost all cases.
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Meeting #989
Alexander Rybalov (Sobolev Institute of Mathematics)
"P=/=NP problem and universal algebraic geometry"
Abstract. The study of computational complexity in various algebraic structures was initiated by S. Smale, L. Blum and M. Shub in 1989, when they introduced analogues of the basic concepts of the classical theory of computability and computational complexity for fields of real and complex numbers. In particular, they defined analogues of the classes P and NP and posed problems on the coincidence of these classes for fields R and C. These problems are still open. However, later it was possible to prove the inequality P=/=NP for other algebraic systems: the additive group of real numbers
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Meeting #988
Matvei Kotov (Sobolev Institute of Mathematics)
"Problems in algebra inspired by tropical cryptography"
Abstract. In 2011, Grigoriev and Shpilrain proposed using tropical algebraic structures in cryptography. In recent years, many different protocols using tropical and similar structures have been proposed, as well as many attacks on some of these protocols. As a result of the cryptanalysis, problems of independent interest arise. In this talk, we will give an overview of the results in this direction and will discuss new results on the knapsack problem and related problems for matrix tropical structures. This talk is based on joint work with I. Buchinskiy and A. Treyer.
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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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