My name and email have changed a few times. You may have seen "Mark Sweeney" or "Mark Sing".
I am a Visiting Assistant Professor at the University of Rochester. Previously, I was a student of Joe Silverman at Brown University, where I completed my Ph.D. in the Spring of 2023. My dissertation, "Dynamical Galois Representations", explored ramification in dynamical Galois representations: half about higher ramification for prime-power degree polynomials, and half about the relationship between ramification and reduction.
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?
In fact, Confucius (Analects XV:24) has phrased this more succinctly:
(He may or may not have been thinking about number theory and Galois groups.)
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.
Lately, I've returned to some of the questions raised in this REU paper, and more generally the ``arithmetic'' structure of chemical reaction networks and the symmetries of they exhibit. Owing to the discrete nature of mass action kinetics, many questions here involve understanding arithmetic schemes; there are even some parallels to questions I consider in arithmetic dynamics (the interaction of number theory with chemistry is already visible in the appearance of the GCD when balancing reactions!). For example, monodromy methods (homotopy continuation) for calculating equilibria make use of the étale fundamental group.
• | Summer 2023: | MATH 130 (Excursions in Mathematics) |
• | Spring 2024: | MATH 162 (Calculus II) |
• | Spring 2024: | MATH 141 (Calculus I, first half) |
• | Fall 2023: | MATH 230 (Number Theory, proof-based) |
• | Fall 2023: | MATH 141 (Calculus I, first half) |
• | 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 200 (Calculus III for physics/engineering) |
• | Spring 2020: | TA for MATH 100 (Calculus II) |
• | Fall 2019: | TA for MATH 100 (Calculus II) |
During semesters as a Teaching Fellow (TF) at Brown, I was the sole instructor, and was responsible for all aspects of course design, especially the syllabus design (schedule, topic selection, textbook).
When teaching Math 50-60 at Brown, I took a slightly nonstandard approach to the definition of the derivative a few reasons: (1) to better emphasize the way calculus is used as a tool for approximation in engineering and (2) to slightly increase the amount of rigor and (3) to complement the challenges of rigor with many more motivating examples and pieces of intuition. This was made possible by my hard working students and the smaller size of the courses. If you are interested, the (slightly) more polished typed notes are available here. Questions and comments are extremely welcome.
I live in Rochester, NY. I have two wonderful bunnies, Cujo and Lula, who have a great passion for mathematics and loves to nibble unguarded books! In my free time, I like to cook and read, and have lately been studying (Mandarin) Chinese. I'm in some sense a fan of the late poet Kent Johnson, and the questions he and his "broader circle" raise about the nature of authorship. I am currently pursuing an MA in translation.
As an undergraduate, I attended the University of Rochester, where I earned separate degrees in Chemical Engineering (BSc) and Mathematics (Honors BSc).
This website uses MathJax to render mathematics.