Abstract: In this talk we give a homotopical interpretation of a result of Panov and Veryovkin on the commutator subgroup of a graph product of discrete groups, using the language of simplicial groups. It turns out that the commutator subgroup of a graph product of simplicial groups gives a model of the loop spaces of certain polyhedral products. The Pontryagin algebra of these loop spaces will also be discussed. We will also introduce some questions that require further work.

Abstract: Heron's formula gives the area of a triangle in terms of the lengths of its sides. More generally, the volume of any simplex is determined by its edge-lengths via a Cayley-Menger determinant.

In work-in-progress with Seth Asante and Michelle Hatzel, we ask

*Which other sets of volumes of faces of an n-simplex, when known, determine the remaining unknown face-volumes?*.

An answer to this question is encoded in the algebraic matroid of the Heron variety. Moreover, we ask

*When are these unknown volumes recoverable via formulae in the known volumes?*

We answer these questions for n<5 by combining techniques in computational group theory, computer algebra, field theory, and numerical algebraic geometry. Of particular focus is the problem of recovering the 10 edge lengths of a 4-simplex from its 10 triangular face areas, a problem motivated by applications in theoretical physics.