## About

I am a final year Mathematics Ph.D. student at the University of Chicago.

I mainly work in ergodic theory and dynamical systems.

My advisor is Alex Eskin.

## Selected Papers:

**Random Walk on Homogeneous spaces with nondiscrete quotients**.

Abstract ±
- in preparation (available upon request)
We study the problem of classifying stationary measures on homogeneous spaces of the form G/H, where G is a connected
real Lie group, and H is a closed unimodular subgroup of G. Under an assumption of relative uniform expansion, we show
that the stationary measures can be decomposed into homogeneous parts and Bernoulli convolutions.

The main tools used are a relative version of the technique of Eskin-Lindenstrauss, and a measure classification result
of linear action on real vector spaces based on a result of Bougerol.

**Stationary measures and orbit closures of random walks on Veech surfaces**.

Abstract ±
- in preparation (available upon request)
We study the problem of measure rigidity, equidistribution and orbit closure classification for the action of subsemigroups of the
affine diffeomorphism group on a given Veech surface. In particular, we show that if the linear part of the acting semigroup
generates a non-elementary subgroup of SL(2, R), then the orbits are finite or dense, and the only nonatomic stationary measure is
induced by the given 1-form on the Veech surface.

**Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces**.
arxiv

Abstract ±
slides
recording
We study the problem of classifying stationary measures and orbit closures for non-abelian action
on a surface with a given smooth invariant measure. Using a result of Brown and Rodriguez Hertz,
we show that under a certain finite verifiable average growth condition, the only nonatomic stationary measure
is the given smooth invariant measure, and every orbit closure is either finite or dense.
Moreover, every point with infinite orbit equidistributes on the surface with respect to the smooth
invariant measure. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the
homogeneous setting, and the result of Eskin-Mirzakhani in the setting of moduli spaces of translation
surfaces. We then apply this result to two concrete settings, namely discrete perturbation of the standard
map and Out(F_2)-action on a certain character variety. We verify the growth condition analytically in
the former setting, and verify numerically in the latter setting.

**Fast, Uniform Scalar Multiplication for Genus 2 Jacobians with Fast Kummers** (joint with B. Smith and C. Costello).
conference

*International Conference on Selected Areas in Cryptography* (2016), 465-481.

Abstract ±
slides
We give one- and two-dimensional scalar multiplication algorithms for Jacobians of genus 2 curves that operate by projecting to
Kummer surfaces, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper “signed” output back on the
Jacobian. This extends the work of L´opez and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional scalar
multiplication. The technique is especially interesting in genus 2, because
Kummer surfaces can outperform comparable elliptic curve systems.

**Bounded gaps between products of special primes** (joint with S. Li).
journal

*Mathematics* 2(1) (2014), 37-52.

Abstract ±
slides
In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps
between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of
square-free numbers with r prime factors for any r ≥ 2, all of which are in a given set of primes. His results
yield applications to the divisibility of class numbers and the triviality of ranks of elliptic curves. In this
paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach.
We then revisit Thorne’s applications and give a better bound in each case.

**On the c-strong coloring of t-intersecting hypergraphs**.
journal

*Discrete Mathematics* 313 (2013), 1063-1069.

Abstract ±
For a fixed c≥2, a c-strong coloring of the hypergraph G is a vertex coloring such that each edge e
of G covers vertices with at least min{c,|e|} distinct colors. A hypergraph is t-intersecting if the
intersection of any two of its edges contains at least t vertices. This paper addresses the question:
what is the minimum number of colors which suffices to c-strong color any t-intersecting hypergraph?
We first show that the number of colors required to c-strong color a hypergraph of size n is O(n).
Then we prove that we can use finitely many colors to 3-strong color any 2-intersecting hypergraph.
Finally, we show that 2c−1 colors are enough to c-strong color any shifted (c−1)-intersecting hypergraph,
and 2c−2 colors are enough to c-strong color any shifted t-intersecting hypergraph for t≥c. Both chromatic
numbers are optimal and match conjectured statements in which the shifted condition is removed.

**Isoperimetric pentagonal tilings** (joint with F. Morgan et al. ).
journal

*Notices Amer. Math. Soc.* 59 (2012), 632-640.

Abstract ±
We identify least-perimeter unit-area tilings of the plane by convex pentagons, namely
tilings by Cairo and Prismatic pentagons, find infinitely many, and prove that they
minimize perimeter among tilings by convex polygons with at most five sides.

(see Remark 2.3 for the main reason I was part of the collaboration when I was a freshman.)

## Notes:

Other published work can be found in Google scholar.

## Contact Info ±