Maurer–Cartan Moduli and Theorems of Riemann–Hilbert Type

We study Maurer–Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion that has not been studied before. We prove, in several different contexts, Schlessinger–Stasheff type theorems comparing the notions of homotopy and gauge equivalence for Maurer–Cartan elements as well as their categorified versions. As an application, we re-prove and generalize Block–Smith’s higher Riemann–Hilbert correspondence, and develop its analogue for simplicial complexes and topological spaces.

Algorithms in A ∞-algebras

Based on Kadeishvili’s original theorem inducing{A_{\infty}}-algebra structures on the homology of dg-algebras, several directions of algorithmic research in{A_{\infty}}-algebras have been pursued. In this paper, we survey the work done on calculating explicit{A_{\infty}}-algebra structures from homotopy retractions, in group cohomology and in persistent homology.

Calabi–Yau properties of nontrivial Noetherian DG down-up algebras

In this paper, we introduce and study differential graded (DG) down–up algebras. In brief, a DG down–up algebra [Formula: see text] is a connected cochain DG algebra such that its underlying graded algebra [Formula: see text] is a graded down–up algebra. We describe all possible differential structures on Noetherian DG down–up algebras. For those Noetherian DG down-up algebras with nonzero differential, we compute their DG automorphism groups; study their isomorphism problems; and show that they are all Calabi–Yau DG algebras.