Upcoming meeting:
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Meeting #964
Alexander Rybalov (Sobolev Institute of Mathematics)
"Generic complexity of the word problem in some semigroups."
Abstract. Generic algorithms decide problems for almost all inputs and ignore remaining rare inputs. I.Kapovich, A.Myasnikov, P.Schupp and V.Shpilrain in 2003 suggested a generic algorithm, which decides the word problem in a wide class of finitely generated groups, including classical groups with undecidable word problem. D. Won in 2008 proposed a generic algorithm for finitely defined semigroups with so called balanced presentation. In particular, classical Markov-Post, Tseitin and Makanin semigroups with undecidable word problems, all have balanced representation. In 2019 M. Volkov at the Sverdlovsk semigroup seminar asked me about generic decidability of the word problem in every semigroup with one relation. For the classical decidability this is a well known and still open Adjan problem. In this talk I will present a generic algorithm for the word problem in a wide class of finitely generated semigroups, including balanced semigroups and semigroups with one relation.
Latest announcements:
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Meeting #963
Alexander Kornev (Universidade Federal do ABC)
"Embedding of Malcev and alternative algebras."
Abstract. We continue studying associative f-representations, which was begun in A. I. Kornev, I.P. Shestakov, On associative representations of non-associative algebras, J. Algebra Appl.,17, No. 3, 1850051 (2018). We are interested in finding identities that an algebra satisfies if it has a faithful f-representation. We introduce the notions of g-associative and g-Lie algebras and we prove that any alternative (Malcev) algebra can be embedded into some g-associative (g-Lie) algebra. We show that any Malcev algebra can be embedded as a commutator subalgebra in some g-associative for two different polynomials g.
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Meeting #962 (in Russian)
Artem Shevlyakov (Sobolev Institute of Mathematics)
"Varieties of equationally Noetherian semigroups and Plotkin`s problem."
Abstract. B.Plotkin posed a problem: find all varieties V of groups, where each G \in V is equationally Noetherian (i.e. any infinite system of equations is equivalent over G to a finite subsystem). In our talk we solve this problem for
a) semigroups
b) equations with constants
It will be obtained the description of such varieties by algebraic (structures of semigroups) and logical (set of identities) approaches. -
Meeting #961 (in Russian)
Irina Zubareva (Sobolev Institute of Mathematics)
"Abnormal extremals of abnormal sub-Finsler quasimetrics on four-dimensional Lie groups with three-dimensional generating distributions."
Abstract. All three-dimensional generating subspaces q of all four-dimensional real Lie algebras are found. Exact formulas for abnormal extremals are found for an arbitrary left-invariant sub-Finsler quasimetric d on any four-dimensional connected Lie group G with Lie algebra g defined by the seminorm F on q. On the basis of the structure constants of the Lie algebra g and the seminorm dual to F on g*, a criterion is established for the (non)strict abnormality of these extremals.
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Meeting #960 (in Russian)
Alexei Miasnikov (Stevens Institute of Technology) (Sobolev Institute of Mathematics)
"General algebraic schemes, non-standard groups, and first-order classification."
Abstract. In this talk I will discuss a new notion of an algebraic group scheme and the related class of “new” algebraic groups (which, of course, contains the classical ones). This leads to some interesting results on the first-order classification problems and sheds new light on first-order rigidity and quasi-finite axiomatization. In another direction I will touch on non-standard models of groups (aka non-standard analysis), especially on non-standard models of the finitely generated ones with decidable or recursively enumerable word problems.