A divisor on a Riemann surface is a choice of finitely many points. The degree of a divisor is the number of points in it. It is denoted by “deg.” A priori, a divisor depends simply on topological information, as does the degree of the divisor. Once a Riemann surface is equipped with a complex structure, however—in other words, when it is considered a complex manifold—then one may ask questions about that complex structure, such as whether there is a meromorphic function whose zeros or poles agree with a given divisor. This leads to a version of the classical Mittag-Leffler problem. One of the simplest invariants that are determined by the complex structure is the holomorphic Euler characteristic of a divisor, which, in a way, is a correction term for the Mittag-Leffler problem on this particular Riemann surface. For a divisor ℒ, its holomorphic Euler characteristic is denoted by χ(ℒ).

I find this equation beautiful because it’s novel and surprising, and information clearly flows from the right-hand term (RHT) to the left-hand term (LHT). Indeed, the LHT is manifestly geometric, while the RHT is purely algebraic. Furthermore, because the RHT is calculable, this equation gives a classification of manifolds, an enterprise of intense study. Lastly, this result was a first of its kind, whose proof techniques prompted creative new approaches to and perspectives in many areas of math. —David Ayala, Montana State University

A set Σ whose points satisfy the inequalities in the picture (they are hidden, but there are two!) is called Reifenberg flat. As usual, the symbol ε in mathematics represents a small quantity; we are essentially saying that in every ball B(x,r), at every scale and location, there exists a plane P(x,r), whose distance from our set is very tiny (εr).

My favorite equation is (¹/_p) + (¹/_q) + (¹/_r) = 1. Its only solutions in positive integers are p = q = r = 3; p = 2, q = r = 4; and p = 2, q = 3, r = 6. This equation pertains to a beautiful class of groups called Coxeter, or reflection, groups and marks the crucial boundary between finite and infinite groups of this kind. Its solutions are intimately related to the only ways of tiling the plane by regular polygons, namely by equilateral triangles, squares or hexagons. —Monty McGovern, University of Washington