Subword Complexes in Algebra, Combinatorics, and Geometry

Subword complexes are simplicial complexes introduced by Knutson and Miller as a tool to study Gröbner geometry of Schubert polynomials. In this talk, I will summarize some relevant appearances of these objects in algebra, combinatorics, and geometry. In particular, I will focus on two new applications:

• a decisive result about denominator vectors in cluster algebras of finite type, and
• a Hopf algebra structure on subword complexes.