## Kathryn LindseyEmail: klindsey [at] math [dot] uchicago
[dot] eduOffice: 416 Eckhart Hall, University
of Chicago Postdoc advisor: Amie WilkinsonCVResearch Statement |

I am an L. E. Dickson Instructor and N.S.F. MSPRF postdoc in the Department of Mathematics at the University of Chicago. You can find out about my research, teaching, and other mathematical activities below.

**Publications:**

- Fekete
polynomials and shapes of Julia sets (with Malik Younsi). (Submitted, preprint on arxiv.org.) [Abstract: We prove that a nonempty,
proper subset S of the complex plane can be approximated in a strong
sense by filled Julia sets of polynomials of degree at least two if and
only if S is bounded and its interior has connected complement. The
proof that such a set is approximable by filled Julia sets is
constructive and relies on Fekete polynomials. Illustrative
examples are presented. We also prove an estimate for the ratoe
or approximation in terms of geometric and potential theoretic
quantities.]

- Convex shapes and harmonic caps
(with Laura
DeMarco). (To appear in Arnold J. Math. for special issue in honor of IMS XXV. Preprint on Arxiv.org.) [Abstract:
Any planar shape P can be embedded isometrically as part of a convex
surface S in R^3 such that the boundary of P supports the positive
curvature of S. Of particular interest is the case when P is a
filled polynomial Julia set and the curvature is proportional to the
measure of maximal entropy. The (flat) surface Q = S \ P is the
associated cap. In this article, we study the cap construction
when the curvature is harmonic measure on the boundary of (\hat{C} \ P,
\infty).

**Horocycle flow orbits and lattice surface characterizations**(with Jon Chaika). (To appear in Ergodic Theory & Dynamical Systems, preprint on arxiv). [Abstract: The orbit closure of any translation surface under the horocycle flow in almost any direction equals its SL2R orbit closure. This result gives rise to new characterizations of lattice surfaces in terms of the horocycle flow.]

**Counting Invariant Components of Hyperelliptic Translation Surfaces**(I*srael J. Math.*, 210 (2015), 125-146). Link. [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.

**Shapes of Polynomial Julia Sets**(*Ergodic Theory & Dynamical Systems*, vol 35, 06, 2015). Link. [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**.]

**Resilient Universal Cellular Automata on Quasiperiodic Tilings**(with D. Bailey). [Abstract: We present a natural method of embedding cellular automata within any tiling that is the result of the multigrid projection technique. The logic of the automaton is logically executed on the periodic feature of a higher dimensional space and then projected onto the aperiodic tiling. We give as a primary example the embedding of Conway's Game of Life in Penrose's aperiodic tilings by rhombuses, and we argue that such an embedding is particularly natural.]

**Flat Surface Models of Ergodic Systems**(with R. Trevino) (To appear in Discrete and Continuous Dynamical Systems - A) [Abstract: We explore connections between translation flows on flat surfaces, adic transformations defined on Bratteli diagrams, and cutting and stacking transformations. We do so by presenting a general technique which takes an adic transformation and constructs a flat surface whose vertical translation flow admits a cross section for which the first return map is measurably isomorphic to the adic transformation. Any finite entropy, measure-preserving flow on a Lebesgue space is measurably isomorphic to the translation flow on a flat surface obtained through our technique. We give a criterion for unique ergodicity for these systems and apply this criterion to several examples, as well as describing specific examples of infinite type flat surfaces on which the translation flow exhibits dynamical properties not possible for finite type flat surfaces.

**Measurable Sensitivity**(with James, Koberda, Silva, Speh), (Proc. Amer. Math. Soc. 136 (2008), 3549-3559.) Link. [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.) Link. [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.

**Families of Dynamical Systems Associated to Translation Surfaces**, Ph.D. dissertation, Cornell University, 2014.

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

**Projects/papers currently in
progress:**

- Polynomial interpolation on multiply connected domains via harmonic measure.

- Invariant components of hyperelliptic quadratic differentials, with Paul Apisa.
We classify how invariant components for the translation flow on
Riemann surfaces equipped with quadratic differentials sit next to each
other.

**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, Harry Baik and Dylan Thurston.

- Exploring the bending locus of the convex bodies defined in "Convex shapes and harmonic caps," with L. DeMarco.

- Spring
2017: Analysis in R^n III (accelerated) - Math 20500

- Summer 2016: U. Chicago REU: dynamical systems

- Spring 2016: Analysis in R^n I - Math 20300
- Autumn 2015: Calculus III - Math 15300
- Summer 2015: U. Chicago REU: ergodic theory.

**Summer 2012:**Instructor for Math 1110 (Calculus I). Course website.**Spring 2009:**TA for Math 2130 (Calculus III)- Fall 2008: TA for Math 1910 (Calculus for Engineers)

At Williams College:

**Fall 2004:**TA for Math 315 (Groups & Characters)

**Summers 2003-2013**: Mate or deckhand on the*SSV Corwith Cramer*, teaching nautical science and navigation to graduate, college, and high school students. Offshore programs one to six weeks in duration.

Video clip

- * Arithmetic Dynamics special session of the Mathematical Congress of the Americas, July 23-28, 2017, Montreal, Canada.
- * AMS Special Session on Discrete Structures in Conformal Dynamics and Geometry, Indiana University Bloomington, Apr 1-2, 2017.
- Teichmüller Space, Polygonal Billiard, Interval Exchanges. CIRM, Luminy, France. Feb 13-17, 2017.

- * Rutgers University, Jan. 26-27, 2017.

- * Boston College, Dec. 5, 2016.

- * Washington University in St. Louis, Dec. 1, 2016.

- * University of Wisconsin-Madison, Nov. 16, 2016.

- * 2016 Midwest
Dynamical Systems Conference, Indiana
University-Purdue University Indianapolis, Nov. 4-6 2016.

- * Complex Analysis Seminar, U. Michigan, Ann Arbor, Oct. 31, 2016.

- * 2016
Midwest Workshop on Asymptotic Analysis, Indiana University-Purdue
University Fort Wayne, Oct. 7-9, 2016.

- * Cycles
on Moduli Spaces, Geometric Invariant Theory, and Dynamics,
I.C.E.R.M., Aug. 1-5, 2016.

- * BIRS CMO Workshop on Flat Surfaces and Dynamics of Moduli Space, Oaxaca, Mexico, May 8 - 13 2016. Video.
- * Dynamics Seminar, SUNY Stony Brook, Apr. 22, 2016.
- * Math Club, U. Chicago, Apr. 12, 2015.

- * Dynamics Seminar, Tufts University, Apr. 5, 2016.

- * British
Mathematics Colloquium, Ergodic Theory Special Session, Bristol
University, Mar. 21-24, 2016.

- * Seminar, Bristol University, Mar. 15 2016.

- * Dynamical
Systems Special Session, 2016 Spring Topology and Dynamics Conference,
Baylor University, Mar. 10-13, 2016.

- * Dynamics
Seminar, University of Maryland, College Park, Feb. 18, 2016.

- RTG Workshop in Arithmetic Dynamics, U. Michigan, Ann Arbor, Dec. 3-6 2015.
- * Teichmueller Theory Seminar, Indiana University Bloomington, Nov. 6, 2015.
- * Dynamical Developments: a conference in Complex Dynamics and Teichmuller theory, in honor of John Hubbard's 70th birthday, Jacobs University, Bremen, Aug. 17-25, 2015.
- IMS XXV, Stony Brook, May 8-12, 2015.
- Dynamics on Moduli Spaces, M.S.R.I., April 13-17, 2015.
- * Seminar talk , Indiana University-Purdue University Indianapolis, Feb. 11, 2015.
- * 3rd Annual Midwest Women in Mathematics Symposium, Dominican University, March 7, 2015.
- * Geometry and Topology Seminar, Stanford University, Nov. 14, 2014.
- Midwest Dynamical Systems Meeting, Nov. 2014.
- * Math Dept. Seminar, Beloit College, Oct. 10, 2014.
- * Wasatch Topology Conference, U. Utah, Aug. 22-24, 2014.
- What's Next?
The legacy of Bill Thurston, Cornell University, June 23-27, 2014.

- * Workshop on Dynamical Systems and Related Topics, U. Maryland, April 11-13, 2014.
- * Dynamical
Systems Seminar, Northwestern University, April 8, 2014

- * Geometry Seminar, University of Michigan, March 28, 2014
- * Joint Math Meetings, Special Session on Complex Dynamics, Baltimore, January 2014.
- ICERM Semester Program on "Low-dimensional Topology, Geometry, and Dynamics," ICERM, Fall Semester 2013.
- ~ Postdoc and grad student seminar, ICERM semester program Fall
2013.

- * International
Conference and Workshop on Surfaces of Infinite Type, Morelia,
Mexico, August 2013. Video.

- Complex Dynamics, AMS Research Community, Snowbird, Utah, June 2013.
- Complex Dynamics (and Arithmetic Geometry), University of Illinois at Chicago, June 2013.
- Dynamics on Parameter Spaces 2013, Sde Boker, Israel, January 2013.
- Ecole de geometrie algebrique, Roscoff, France, September 2012.
- The Horocycle Flow in Different Situations, CIRM, Luminy, France, April 2012.
- Moduli Spaces Associated to Dynamical Systems, ICERM, April 2012.
- * Informal
Seminar: Dynamics and Geometry, Harvard University, March
2012.

- * "Truth Values" Panel Discussion,
Williams College, February 2012.

- * Action Now Wandering Seminar, Ben Gurion University, Israel, April 2011.
- * New York Regional Graduate Math. Conference, Syracuse University, April 2011.
- * Oxtoby Centennial Conference, Bryn Mawr College, November 2010.
- Geometry
and Dynamics of Teichmuller Spaces Conference, HIM, Bonn, Germany,
June 2010.

- Dynamics and Geometry of Teichmuller Space, CIRM, Luminy, France, June 2009

- What's Next?
The legacy of Bill Thurston, Cornell University, June 23-27, 2014.
Local organizing committee.

I had fun making the poster.

I co-organize U. Chicago's Dynamics Seminar. The seminar is Mondays at 3 p.m. in Eckhart 206. Email me if you would like to be added to the seminar email list, or if you or your guest would be interested in speaking in the seminar.

The Math
Explorers Club is a collection of materials designed to give middle
school and high school students an introduction to interesting and
advanced topics in mathematics. I wrote the module *An
Introduction to Tilings*.