## About

I am a fifth-year Mathematics Ph.D. student at the University of Chicago. I mainly work in dynamical systems.

My advisor is Alex Eskin.

## Papers ±

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

Abstract ± slidesWe 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 ± slidesWe 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 ± slidesIn 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:

**Stationary measures on vector spaces**. pdf

Abstract ±We study the classification of stationary measures for linear actions on vector spaces.

Other published work can be found in Google scholar.

## Contact Info ±

Email: briancpn (followed by @math.uchicago.edu)

Mail: Department of Mathematics

University of Chicago

5734 S. University Avenue

Chicago, IL, 60637

Office: Eckhart

copyright © 2020 Ping Ngai (Brian) Chung