Heron's formula, attributed to Heron of Alexandria (an ancient Greek), explicitly describes the area of a triangle as a function of its sidelengths. Brahmagupta's formula, named after a 7th century Indian mathematician, does the same for convex quadrilaterals that are cyclic, i.e. inscribed in a circle. In a 1995 American Mathematical Monthly paper, the number theorist David P. Robbins described a sequence of "Heron polynomials" that generalize these formulas to arbitrary cyclic polygons, and he made three conjectures about these polynomials' properties. Robbins' conjectures were proved by several different authors during a short period of intense activity in the early 2000's, around the time of his tragic death due to pancreatic cancer.
In this talk I'll go into some detail about the Heron polynomials and Robbins' conjectures. Then I'll present analogs of the Heron and Brahmagupta formulas, which were respectively proved in 1969 and 2014 (!!), for polygons in the hyperbolic plane. The nature of the corresponding analogs to Heron polynomials is a wide open question. I worked on this a bit with Lucy Newman, a Pitt undergraduate, a few years ago, and I'll briefly mention a few things that we found.