Karl Winsor

Karl Winsor


Email: kwinsor at math dot harvard dot edu

I am a fifth-year graduate student in mathematics at Harvard, advised by Curt McMullen. Previously, I was an undergraduate student at the University of Michigan.

Beginning Fall 2022, I will be a Simons Postdoctoral Fellow at the Fields Institute in Toronto. Beginning Fall 2023, I will be a Simons Instructor (research postdoc) at Stony Brook University.

My main research interests include group actions and foliations on moduli spaces, mapping class groups, and dynamics of billiards and interval exchanges.

Papers and preprints:

(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 a connected stratum of holomorphic 1-forms with at least 2 distinct zeros, and we give an explicit full measure set of dense leaves. We also address the question of when two holomorphic 1-forms in a stratum, with the same absolute periods, can be connected by a path in the stratum along which the absolute periods are constant.

(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. The proof is based on a general density criterion for SL(2,R)-orbit closures, which 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) Uniqueness of the Veech 14-gon, in preparation.

We show that the Veech 14-gon generates the unique algebraically primitive Teichmüller curve in its stratum component.

Selected Talks:

AMS Special Session on Geometry and Dynamics in Moduli Spaces of Abelian Differentials, January 2023

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 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

Under Construction