Ophelia Adams
Introduction
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 polynomials of prime-power degree, 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:
- What do Galois representations on iterated monodromy groups, dynamically natural quotients of the full étale fundamental group, look like?
- What can we say about (higher) ramification in dynamical extensions of fields?
- How rigid are dynamical representations: to what extent can a dynamical system or dynamical information be recovered from associated Galois representation?
- A dynamical system on \(\mathbb P^1_K\) gives rise to a \(\Gamma_K\)-equivariant dynamical system on particular étale fundamental groups – what information is contained in this representation? How are these related to arboreal representations?
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?
Articles
Semiconjugacy and Self-Similar Subgroups of pfIMGs. August 2025.
Preprint: arXiv:2508.12122.
The profinite iterated monodromy group (pfIMG) is a self-similar group associated to dynamical systems. We show that its proper open self-similar subgroups correspond to highly rigid semiconjugacies, which we partly classify in general. For polynomials, we show that only the twisted Chebyshev maps can arise. Next, we define and construct self-similar closures of subgroups of pfIMGs, and show that this preserves many group-theoretic properties of the original subgroup. As a consequence, we conclude that pfIMGs with open subgroups satisfying certain properties (e.g. prosolvable or pronilpotent) either satisfy that property themselves, or arise from one of these exceptional semiconjugacies.
This is applied to answer some questions posed in [BGJT25] about open Frattini subgroups of pfIMGs: unicritical polynomials of composite degree do not have an open Frattini subgroup, and a polynomial with an open Frattini subgroup is often pro-\(p\).(joint with Trevor Hyde) Profinite Iterated Monodromy Groups of Unicritical Polynomials. April 2025.
Preprint: arXiv:2504.13028. (submitted)
Let \(f(x)=ax^d + b \in K[x]\) be a unicritical polynomial with degree \(d\geq\) which is coprime to \(\operatorname{char} K\). We provide an explicit presentation for the profinite iterated monodromy group of \(f\), analyze the structure of this group, and use this analysis to determine the constant field extension in \(K(f^{-\infty}(t))/K(t)\).A Dynamical Analogue of the Criterion of Néron-Ogg-Shafarevich. July 2022.
Preprint: arXiv:2208.00359. (submitted)
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.A Dynamical Analogue of Sen’s Theorem.
International Mathematics Research Notices, Volume 2023, Issue 9, pages 7502-7540.
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 -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.Conditions for Solvability in Chemical Reaction Networks at Quasi-Steady-State.
Preprint: arXiv:1712.05533. (undergraduate, currently unpublished)
The quasi-steady-state assumption (QSSA) is an approximation that is widely used in chemistry and chemical engineering to simplify reaction mechanisms. The key step in the method requires a solution by radicals of a system of multivariate polynomials. However, Pantea, Gupta, Rawlings, and Craciun showed that there exist mechanisms for which the associated polynomials are not solvable by radicals, and hence reduction by QSSA is not possible. In practice, however, reduction by QSSA always succeeds. To provide some explanation for this phenomenon, we prove that solvability is guaranteed for a class of common chemical reaction networks. In the course of establishing this result, we examine the question of when it is possible to ensure that there are finitely many (quasi) steady states. We also apply our results to several examples, in particular demonstrating the minimality of the nonsolvable example presented by Pantea, Gupta, Rawlings, and Craciun.
Translation
I am working part-time on a master’s degree in literary translation, advised by Stella Wang (University of Rochester; WSAP). I am translating the poetry of 朱淑真 (Zhū Shúzhēn) as collected in 斷腸詩詞 (Duàncháng Shī Cí). Part of the project is archival: some of her work is difficult to access in convenient electronic form, and making transcriptions available digitally is important to me.
Related work:
- Ongoing: Transcribing, glossing, and translating 朱淑真.
- Spring 2025: Organizing book club with MALTS students.
- Spring 2025: (with Stella Wang) Speaking at ARTS + Change Conference about interdisplinary research and translation.
I am broadly curious about back-translation and re-translation as an analogue of monodromy.
Teaching
The University of Rochester (postdoc)
- Spring 2025: Math 218 (Mathematical Models in the Life Sciences)
- Spring 2025: Math 141 (Calculus I, first half)
- Fall 2024: Math 280 (Numerical Methods, redesigned)
- Fall 2024: Math 141 (Calculus I, first half)
- 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)
As a postdoc at the University of Rochester, I redesigned Math 280: Introduction to Numerical Analysis, the core course for the applied mathematics major, to be more project-oriented to better serve student goals and interests. This work was supported by a combination of grants: the University Library’s Open Educational Resources Grant, as well the Teaching Center’s College Course Development Fellowship (Summer ’24) and Student Course Deveopment Project, in addition to departmental support for initial technology expenses.
Brown University (grad)
TF = Teaching Fellow; instructor of record
- 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) and course material; everything was taught from my own notes. When teaching Math 50-60: Calculus I at Brown, I took a slightly nonstandard approach to the definition of the derivative for 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. Each semester, their questions and comments greatly improved the notes for my next students. These notes will appear here soon. Questions and comments are extremely welcome.
Personal
I live in Rochester, NY. I have a bunny, Mochi, who is quite interested in treats and once-punctured elliptic curves. In my free time, I read and translate. I’m interested in the work of the late poet Kent Johnson, and the questions he and his “broader circle” raise about the nature of authorship. Other poets of great interest to me include Emily Dickinson, Jack Spicer, and Leslie Scalapino.
As an undergraduate, I attended the University of Rochester, where I earned separate degrees in Chemical Engineering (BSc) and Mathematics (Honors BSc).
