Kathryn Lindsey  Ph.D. candidate Department of Mathematics Cornell University Email: klindsey [at] math [dot] cornell [dot] edu Office: 114 Malott Hall Department webpage:  http://www.math.cornell.edu/People/Grads/lindsey.html Advisor: John Smillie CV

I'm a graduate student in the Department of Mathematics at Cornell University. You can find out about my research, teaching, and other mathematical activities below.

Research

Primary research interests: the moduli space of flat surfaces, complex dynamics

Papers:

• Counting Invariant Components of Hyperelliptic Translation Surfaces (submitted, preprint on arxiv).  [Abstract: The flow in a fixed direction on a translation surface S determines a decomposition of S into closed invariant sets, each of which is either periodic or minimal.  We study this decomposition for translation surfaces in the hyperelliptic connected components $\mathcal{H}^{hyp}(2g-2)$ and $\mathcal{H}^{hyp}(g-1,g-1)$ of the corresponding strata of the moduli space of translation surfaces.  Specifically, we characterize the pairs of nonnegative integers (p,m) for which there exists a translation surface in $\mathcal{H}^{hyp}(2g-2)$ or $\mathcal{H}^{hyp}(g-1,g-1)$ with precisely p periodic components and m minimal components. This extends results by Naveh ([Naveh08]), who obtained tight upper bounds on the numbers of minimal components and invariant components in a translation surface in any given stratum may have.  Analogous results for the other connected components of moduli space are forthcoming.]

• Shapes of Polynomial Julia Sets (submitted, preprint on arxiv).  [Abstract: Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Finite collections of disjoint Jordan domains can be approximated by basins of attraction of rational maps.] [Read about this result in Scientific American.]

• Measurable Sensitivity (with James, Koberda, Silva, Speh), Proc. Amer. Math. Soc. 136 (2008), 3549-3559.[Abstract: We introduce the notions of measurable and strong measurable sensitivity, which are measure-theoretic versions of the conditions of sensitive dependence on initial conditions and strong sensitive dependence on initial conditions, respectively.  Strong measurable sensitivity is a consequence of light mixing, implies that a transformation has only finitely many eigenvalues, and does not exist in the infinite measure-preserving case.  Unlike the tradiational notions of sensitive dependence, measurable and strong measurable sensitivity carry up to measure-theoretic isomorphism, thus ignoring the behavior of the transformation on null sets and eliminating dependence on the choice of metric. 

• On Ergodic Transformations that are Both Weakly Mixing and Uniformly Rigid (with James, Koberda, Silva, Speh), New York Journal of Math. 15 (2009), 393-403. [Abstract: We examine some of the properties of uniformly rigid transformations, and analyze the compatibility of uniform rigidity and (measurable) weak mixing along with some of their asymptotic convergence properties.  We show that on Cantor space, there does not exist a finite measure-preserving, totally ergodic, uniformly rigid transformation.  We briefly discuss general group actions and show that (measurable) weak mixing and uniform rigidity can coexist in a more general setting.  

• Descriptive Dynamics of Borel Endomorphisms and Group Actions, senior thesis in mathematics, Williams College, 2007.
  
  

Projects/papers currently in progress:

• Virtual flow.  I am using a technique called the "virtual flow" on the moduli space of translation surfaces to study orbit closures under the horocycle flow.  The virtual flow is closely connected to the "rel flow" studied by Smillie and Weiss.
  
  
• Degree-d invariant laminations.  I am working to complete an unfinished manuscript by William Thurston that develops a dynamical theory of laminations.  This theory models the behavior of iterated degree-d complex polynomials.  This is a joint project with Tan Lei, Gao Yan and Harry Baik. 
  
  
• Flat surfaces of infinite type and rank r cutting and stacking transformations.  In this paper we explore connections between rank r maps on the unit interval and translation flows on flat surfaces of infinite type.  Joint with Rodrigo Trevino.  
  
  
• Counting invariant components of non-hyperelliptic translation surfaces.  I finished a paper on this topic for hyperelliptic translation surfaces; this paper will present analogous results for the non-hyperelliptic connected components of moduli space.  
  
  
• A descriptive version of the Birkhoff ergodic theorem.  An outgrowth of my undergraduate honors thesis at Williams College.  Joint with Cesar Silva.  
  
  
• A Game of Life on Penrose tilings.  John Conway's "Game of Life" is played on a square grid; we present rules for a resilient "game of life" played on Penrose Tilings and other quasi-periodic tilings of the plane.  This project began while I was an undergrad at Williams College.  Joint with Duane Bailey.
  
  
• Convex bodies and measures of maximal entropy.  The filled Julia set of a complex polynomial can be embedded isometrically as part of the boundary of a convex set in R^3.  This convex body is obtained by interpreting the measure of maximal entropy as the curvature of a conformal metric on P^1.  William Thurston devised a simple way to compute these shapes, and together he and I made some models. Link to a few pictures.

Teaching

I am supported by NSF GRFP and DoD NDSEG fellowships so I am not teaching this semester. 

Past teaching:

Video clip - teaching my "watch" how to launch a life raft in an abandon ship scenario: click here

Video clip - calling the striking and furling of the mains'l: click here