Office: Kassar 012
Email: mark_sing@brown.edu
Pronouns: they/them
I am a student of Joe Silverman, currently in my fifth year at Brown University.
My research is in number theory and arithmetic dynamics. I'm interested in dynamical Galois representations and anabelian geometry, broadly construed, and how the two interact. I am particularly interested in studying these questions over local fields. For instance:
For instance, one of my results exhibits a correspondence between the ramification of a representation and the structure of the critical orbit. A related result connects the higher ramification filtration of the representation to a natural dynamical filtration, and shows that in many cases of interest the ramification ``stabilizes''.
What other information is encoded in these representations?
Abstract: We provide a simple criterion for determining when the arboreal representation of a rational map defined over a local field and having good reduction is infinitely ramified. This can be interpreted as a dynamical analogue of the Néron-Ogg-Shafarevich criterion for an abelian variety to have good reduction at some prime. The results are effective; we work out an example of a post-critically finite rational function. We conclude with some remarks on how the methods of the present paper can be combined with and strengthen previous results of the author.
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.
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.
• | Spring 2023: | TF for MATH 60 (Calculus I, second half) |
• | Fall 2022: | TF for MATH 50 (Calculus I, first half) |
• | Spring 2022: | TF for MATH 420 (Number Theory) |
• | Fall 2021: | TF for MATH 50 (Calculus I, first half) |
• | 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) |
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).