Structure and computation in nilpotent algebras

This is a project in universal algebra, a part of foundations of mathematics with connections to classical algebra and computer science. Universal algebra generally studies algebraic structures, the mathematical framework for computations, solving equations, etc. This project investigates various notions of nilpotence, specific finiteness conditions on algebras, and how they can be used to develop efficient algorithms. It is motivated by questions that arise in computer science when manipulating relations, in the structure theory of classical algebras, and in the classification of non-standard general algebras. The goal is to combine and extend recent independent advances in these areas to develop a general algebraic toolkit for specifying and analyzing nilpotent algebras, and to apply it to solving key open problems in all these areas. Developed from the classical notion in group theory, commutator theory in universal algebra is one of the main tools for investigating the structure of algebras via properties of their congruences. The principal investigator will study and compare the distinct notions of nilpotence and supernilpotence that arise from binary and higher term condition commutators. The first goal is a more precise classification of central extensions in congruence modular varieties using the new concept of clonoids. The investigator will then use this to study to what extent known results in the supernilpotent setting generalize to the nilpotent. Specific projects include the Finite Basis Problem, which concerns nilpotent algebras having finitely axiomatizable equational theories, and the development of efficient algorithms for computing in direct powers of algebras with cube terms, which reduces to central extensions based on the investigator's previous work. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.