Mark O.-S. Sing


Contact

Office: Kassar 012
Email: mark_sing@brown.edu


Introduction

I am a student of Joe Silverman, currently in my fourth year at Brown University.

My research is in number theory and arithmetic dynamics. I'm interested in dynamical Galois representations and dynamics over local fields, and especially the overlap between the two. For instance: what can we say about (higher) ramification in dynamical extensions of fields?

Another collection of interests I have are anabelian questions in arithmetic dynamics: to what extent can a dynamical system be recovered from associated Galois representations?
For instance, under dynamically interesting hypotheses one can recover pieces of tropicalizations associated to the dynamical system. What other information is encoded in these representations?


Research

  1. A Dynamical Analogue of Sen's Theorem. International Mathematics Research Notices, to appear.

    Abstract: We study the higher ramification structure of dynamical branch extensions, and propose a connection between the natural dynamical filtration and the filtration arising from the higher ramification groups: each member of the former should, after a linear change of index, coincide with a member of the latter. This is an analogue of Sen's theorem on ramification in $p$-adic Lie extensions. By explicitly calculating the Hasse-Herbrand functions of such branch extensions, we are able to show that this description is accurate for some families of polynomials, in particular post-critically bounded polynomials of $p$-power degree. We apply our results to give a partial answer to a question of Berger and a partial answer to a question about wild ramification in arboreal extensions of number fields raised by Aitken, Hajir, and Maire and also by Bridy, Ingram, Jones, Juul, Levy, Manes, Rubinstein-Salzedo, and Silverman.

  2. (unpublished) Conditions for Solvability in Chemical Reaction Networks at Quasi-Steady-State. arXiv:1712.05533

    Note: This paper was written in 2017 based on work at an REU hosted Texas A&M, mentored by Anne Shiu. In the intervening years, I have simplified some of the proofs, though these improvements are not available online. To my knowledge, the main questions and conjectures remain open. Interestingly, the tree-like reaction networks considered in this paper give rise to extensions not unlike the arboreal extensions of arithmetic dynamics, though at the time I was not aware of the latter.


Teaching

Spring 2022:TF for MATH 420 (Number Theory) course website
Fall 2021:TF for MATH 50 (Calculus I, first half) course website
Summer 2021:TA for MATH 100 (Calculus II)
Spring 2021:TA for MATH 180 (Calculus III for physics/engineering)
Spring 2020:TA for MATH 100 (Calculus II)
Fall 2019:TA for MATH 100 (Calculus II)

Personal

My husband and I live in Providence with our bunnies, Cujo and Lula, who have a great passion for mathematics and loves to nibble unguarded books!

As an undergraduate, I attended the University of Rochester, where I earned separate degrees in Chemical Engineering (BSc) and Mathematics (Honors BSc).

Headshot of me. Cujo, my rabbit.