Title: A Combinatorial Approach to Homology of Algebraic Systems
Speaker: Viktor Lopatkin
Institution: Higher School of Economics
Date & Time: April 6, 2024, 16:00–18:00
Location: Zoom ID: 904 645 6677, Password: 2024
Many fundamental algebras, groups, and semigroups are defined via generators and relations. This naturally raises questions about "higher-order" relations—relations among relations, and so on—which lead to the concept of
resolutions in homological algebra. In this talk, we explore a powerful and universal method for constructing such resolutions, known as
Anick resolutions. This approach relies on two key foundations: the theory of Gröbner–Shirshov bases and algebraic discrete Morse theory. We illustrate how this method works through explicit examples, such as the symmetric group
S3, and discuss the comultiplication of Anick complexes, which induces a multiplication on cohomology.
Viktor Lopatkin is an Associate Professor at HSE University (Higher School of Economics) in Moscow, Russia. His research focuses on Anick resolutions and Gröbner–Shirshov bases theory, with additional interests in knot theory, vertex algebra theory, and Lie algebra theory. He contributes to the development of combinatorial and homological methods for analyzing algebraic structures, bridging pure algebra with geometric and topological perspectives.
