Vladimir Dotsenko

Title: Three guises of toric varieties of Loday's associahedra and related
algebraic structures

Abstract: 
Associahedra are remarkable CW-complexes introduced by Stasheff in
1960s to encode a homotopically coherent notion of associativity. They
have been realised as polytopes with integer coordinates in several
different ways over the past few decades. I shall explain that the
realisations of associahedra due to Loday lead to toric varieties of
particular merit. These varieties have been already identified with
"brick manifolds" arising when studying subword complexes for Coxeter
groups (Escobar, 2014). It turns out that they also arise as
"wonderful models" in the sense of de Concini and Procesi for certain
subspace arrangements. Guided by that geometric picture, I shall argue
that in some sense these varieties give a "noncommutative version" of
Deligne-Mumford compactifications of moduli spaces of genus zero
curves with marked points, in that they give rise to remarkable
algebraic structures resembling cohomological field theories of
Kontsevich and Manin. This is a joint work with Sergey Shadrin and
Bruno Vallette.