Algebra seminars

Starting with a cyclic A-infinity category, this talk constructs enumerative invariants with both open and closed sectors. The approach extends Costello's closed-sector construction through appropriate differential graded Lie algebras and trivializations of circle actions. Applications include a unital, cyclic version of the Bogomolov–Tian–Todorov theorem and a construction of quantum A-infinity algebras. The presentation describes ongoing joint work with Lino Amorim.

Operational definitions identify subsystems through the transformations an agent can perform. This talk classifies complementary, bicommutant-closed subsystems using unitary channels, whose disregard for global phase permits possibilities beyond conventional operator subalgebras. The classification involves graded subalgebras. Some resulting subsystems are nontrivial yet support no measurements, raising a foundational choice between excluding them and accepting agents that act without being able to observe.

Poisson sigma models connect geometric deformation with quantum field theory. This talk extends them into higher-dimensional theories with holomorphic and topological directions and relates the construction to quantization of factorization algebras. It develops interfaces, boundaries and defects, then uses Koszul duality to connect the models with quantum-group structures, including examples involving Lie bialgebras, W-algebras and Yangians.

A refinement of the product replacement algorithm aims to draw nearly uniform elements from a finite group. This talk studies how many steps are needed for the output distribution to approach uniformity, using spectral-gap estimates and computer-assisted calculations in joint work with Michał Marcinkowski.

A unitary-group analogue of Aldous' spectral-gap problem randomizes coordinate subspaces associated with weighted hyperedges. Its spectrum contains those of related particle processes and symmetric-group walks. This talk reduces the search for the gap to zero-weight sectors of balanced mixed tensor powers, then proposes equality with a two-particle KMP gap. Proofs cover mean-field and codimension-one cases.

This introduction connects Kazhdan's property T with algebraic spectral gaps for a group Laplacian. Numerical searches for sum-of-squares expressions can suggest proofs, while certification converts approximate solutions into exact mathematical conclusions. Examples include new proofs for integer special linear groups and results for automorphism groups of free groups.