Maryam Mirzakhani is known for her work on moduli spaces of Riemann surfaces.  Some of her most cited work looks at the moduli space of a genus $g$ Riemann surface with $n$ geodesic boundary components.  In two of her papers, she computes the volume of these moduli spaces, with respect to the Weil-Petersson metric (see below).  In another, she provides a means for counting the number of simple closed geodesics of length at most $L$. Mirzakhani is also known for her work on billiards (see the review of her paper with Eskin and Mohammadi below), a subject closely related to moduli space questions.  Teichmüller theory and the geometry of moduli spaces are famously deep subjects.  Making progress requires mastering large areas of analysis, dynamical systems, differential geometry, algebraic geometry, and topology.  I can only appreciate Mirzakhani’s work superficially, as I have not mastered those subjects.   Instead, some reviews of her work are reproduced below.   Continue reading →

Karen Smith is a mathematician at the University of Michigan, which is where she also did her Ph.D.  Her thesis was on tight closure, an important topic in commutative algebra.  There is, of course,  a lot of overlap between commutative algebra and algebraic geometry, and Smith’s publications demonstrate this mix with about three quarters of them being in commutative algebra and a quarter in algebraic geometry.  Her publications are also a good demonstration of the collaborative nature of mathematics.  Continue reading →