Monodromy in Topology and Geometric Group Theory award

Topology and algebraic geometry are two subjects of central importance in contemporary mathematics. Topology is the study of spatial structures (e.g. the surface of a donut, the large-scale shape of the cosmos, a tangled strand of DNA, a cluster of points in a massive data set), while algebraic geometry studies the mathematics of the ubiquitous polynomial equation. These disciplines have many and diverse points of contact; the principal investigator will study one in particular: families of Riemann surfaces. A surface is a topological object like the two-dimensional surface of a coffee cup, possibly one with lots of extra handles. A Riemann surface gives a special way of describing such an object as the solution to a polynomial equation, much as we learn in high school algebra that some equations describe circles, others ellipses, and others far more complicated shapes. A family of Riemann surfaces arises by varying the equations used to define the Riemann surface - one can imagine 'turning a knob' to stretch and distort the shapes of the surfaces. Families of Riemann surfaces arise in many parts of mathematics and also play an important role in theoretical physics. The principal investigator will apply tools from topology and the related discipline of geometric group theory in order to better understand some of the most important families of Riemann surfaces that mathematicians today are interested in. Broader impacts include the establishment of a new chapter of the national Directed Reading Project network.

The project has two components. The first will investigate the topology of strata of Abelian differentials (translation surfaces). These families have been intensively studied by dynamicists and geometers, but there are foundational topological questions that still remain. In particular, the (orbifold) fundamental groups of strata are still highly mysterious. This can be effectively probed by way of the monodromy representation, a map into the mapping class group. The principal investigator will continue his work describing these monodromy representations, and will develop new tools to understand the monodromy kernel and so further develop the theory of fundamental groups of strata. Central to this endeavor will be an elucidation of the precise relationship between Artin groups and fundamental groups of strata. The second component of this project concerns families constructed via branched covers. These families have served as an important source of examples in topology, algebraic geometry, and group theory. The principal investigator will investigate the monodromy of these families with the objective of further synthesizing the topological aspects (braid groups, mapping class groups) with the algebraic (arithmetic groups and generalizations, Burau-like representations, primitive homology).

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.