Karl Winsor

Karl Winsor

Winsor_websitephoto

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:

The most up-to-date versions are posted here.

(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. 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.

Selected Talks:

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

Under Construction