Email: kgwinsor at gmail dot com
I am an NSF Postdoctoral Fellow and a Simons Instructor (research postdoc) at Stony Brook University.
Previously, I was a graduate student in mathematics at Harvard, advised by Curt McMullen. Before that, I was an undergraduate student in math and computer science at the University of Michigan.
My main research interests include group actions and foliations on moduli spaces, mapping class groups, and dynamics of billiards and interval exchanges.
(1) Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms (2021), preprint
We establish the ergodicity of the absolute period foliation of the area-1 locus of most stratum components of holomorphic 1-forms, and we give explicit full measure sets of dense leaves. We obtain similar results for subspaces of stratum components defined by constraints on the absolute periods. We also address the question of when two isoperiodic 1-forms in a stratum component can be connected by a path in the stratum along which the absolute periods are constant. Our dynamical results are a consequence of a classification of stratum components in which a typical space of isoperiodic forms is connected.
(2) Dense real Rel flow orbits and absolute period leaves (2022), preprint
We show the existence of a dense orbit for real Rel flows on the area-1 locus of every connected component of every stratum of holomorphic 1-forms with at least 2 distinct zeros. For this purpose, we establish a general density criterion for SL(2,R)-orbit closures, based on finding an orbit of a real Rel flow whose closure contains a horocycle. This criterion can be verified using explicit constructions of holomorphic 1-forms with a periodic horizontal foliation. Our constructions also provide explicit examples of dense leaves of the absolute period foliation and many subfoliations of these loci.
(3) Saturated orbit closures in the Hodge bundle (2022), preprint
We give a new proof of the classification of GL(2,R)-orbit closures that are saturated for the absolute period foliation of the Hodge bundle. As a consequence, we obtain a short proof of the classification of closures of leaves of the absolute period foliation of the Hodge bundle. Our approach is based on a method for classifying GL(2,R)-orbit closures using deformations of flat pairs of pants.
(4) Uniqueness of the Veech 14-gon (2022), preprint, Sage code forthcoming
We show that the Veech 14-gon generates the unique algebraically primitive Teichmüller curve in its stratum component.
(5) Complex relative period geodesics in moduli space, in preparation.
We classify the closures of leaves of the absolute period foliation of most strata, outside a proper algebraic subvariety, and in many cases we obtain a complete classification of closures of leaves.
CMS Special Session on Geometry Topology, Pseudo-Anosov Maps and Complex Dynamics, June 2023
University of Maryland, Dynamics Seminar, April 2023
University of California San Diego, Group Actions Seminar, January 2023
AMS Special Session on Geometry and Dynamics in Moduli Spaces of Abelian Differentials, January 2023
Stony Brook University, Dynamics Seminar, November 2022
University of Toronto, Hyperbolic Lunch, November 2022
Indiana University Bloomington, Geometry Seminar, September 2022
The Circle at Infinity, PechaKucha, June 2022
University of Glasgow, Geometry and Topology Seminar, January 2022
University of Texas at Austin, Groups and Dynamics Seminar, January 2022
Brown University, Geometry and Topology Seminar, November 2021
University of Michigan, Geometry Seminar, November 2021
Boston College, Geometry Topology and Dynamics Seminar, April 2021
Billiards and Surfaces à la Teichmüller and Riemann, Online (BiSTRO), October 2020
Harvard Informal Geometry and Dynamics Seminar, March 2020
Harvard Informal Geometry and Dynamics Seminar, September 2019