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Posts
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portfolio
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publications
Mumford’s dynamical system and Gelfand–Dikii recursion
Published in Funct. Anal. & Its Appl. 57(4), 2023
Abstract
In his paper “The Mumford dynamical system and hyperelliptic Kleinian functions” (*Funct. Anal. & Its Appl.* 57:4, 27–45, 2023), Victor Buchstaber developed the differential-algebraic theory of the Mumford dynamical system. The key object of this theory is the (P,Q)-recursion introduced in that paper. In the present work, we further develop the theory of the (P,Q)-recursion and describe its connections to the Korteweg–de Vries (KdV) hierarchy, the Lenard operator, and the Gelfand–Dikii recursion.
Recommended citation: Polina Baron. (2023). "Mumford’s dynamical system and Gelfand–Dikii recursion." Funct. Anal. & Its Appl. 57(4).
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The Neumann–Moser dynamical system and the Korteweg–de Vries hierarchy
Published in arXiv:2402.18079, 2024
Abstract
At the focus of the paper are applications of the well-known Moser transformation of the C. Neumann dynamical system. It yields a new quadratic integrable dynamical system on \( \mathbb{C}^{3n+1} \), which we call the Neumann–Moser dynamical system. We present an explicit formula for the inverse of the Moser transformation. Consequently, we obtain an explicit invertible transformation sending the Uhlenbeck–Devaney integrals of the Neumann system to the integrals of our system. One of the main results is a recurrence for solutions of the Neumann–Moser system. We show that every solution of our system solves the Mumford dynamical system, and vice versa. Every solution of the Neumann–Moser system is proven to solve the stationary Korteweg–de Vries hierarchy. As a corollary, we construct explicit solutions of the Neumann–Moser system in hyperelliptic Kleinian functions.
Recommended citation: Polina Baron. (2024). "The Neumann–Moser dynamical system and the Korteweg–de Vries hierarchy." arXiv:2402.18079.
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Semisimplicity and Rigidity of the Kontsevich–Zorich cocycle for products of strata
In preparation, 2025
Abstract
The moduli space of translation surfaces carries an action of \( \mathrm{SL}_2\mathbb{R} \) that is central to understanding dynamics on individual surfaces. We generalize this to actions on products of translation surfaces by a subgroup \( G \le \prod_{i=1}^n \mathrm{SL}_2\mathbb{R} \). Using variations of Hodge structure, we show that invariant subbundles for the \(G\)-action respect the Hodge structure, yielding semisimplicity results. Our key result is a joint polynomiality theorem: for any affine invariant manifold \( \mathcal{M} \) inside a product of strata, the \(G\)-equivariant orthogonal projector onto a flat invariant subbundle (and its tensors) has, in period/affine coordinates \( (x,y) \), entries rational in \( (x,y) \) with a fixed quadratic denominator \( A(x,y) \); equivalently, the entries are homogeneous of degree \(0\) in \(x,y,1/A\).
Recommended citation: Polina Baron. (2025). "Semisimplicity and Rigidity of the Kontsevich–Zorich cocycle for products of strata." In preparation.
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Unique ergodicity of translation surfaces under branched n-covers
In preparation, 2025
Abstract
We present a new construction on translation surfaces called the branched slit-induced \(n\)-cover: on a uniquely ergodic \(X\), pick a slit \(s=[P,Q]\); take n copies and switch sheets \(i \mapsto i+1 \pmod n\) each time the vertical flow hits \(s\) (i.e., glue the copies together). Unique ergodicity is shown to be robust for such covers under fairly weak constraints. Moreover, the conditions are geometric despite the measure-theoretic core of the problem—particularly notable because the varied parameter here is not the flow direction (as is standard in the field) but the new surface construction itself. Joint with Elizaveta/i> </details>
Recommended citation: Polina Baron and Elizaveta Shuvaeva. (2025). "Unique ergodicity of translation surfaces under branched n-covers." In preparation.
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service
Mathematics REU — Tutorial Coordinator
Service role (Tutorial Coordinator), University of Chicago, Department of Mathematics, 2022
Coordinated tutorials for summer REU; aligned topics with research groups; managed schedules.
Warm-Up Program for New Graduate Students — Organizer
Service role (Organizer), University of Chicago, Department of Mathematics, 2024
Planned and ran the department’s one-week onboarding program for incoming PhD students.
Dynamics Pre-Seminar — Organizer
Service role (Organizer), University of Chicago, Department of Mathematics, 2025
Ran a weekly pre-seminar ahead of the Dynamics Seminar.
Phoenix STEM — Collaborative Learning Coordinator
Service role (Coordinator), University of Chicago, Department of Mathematics, 2025
Led collaborative teaching sessions; coached and observed 12 junior tutors; curated problem bank and rubrics.
talks
Talk 1 on Relevant Topic in Your Field
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Conference Proceeding talk 3 on Relevant Topic in Your Field
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teaching
Teaching Assistant — Multiple undergraduate courses
Teaching Assistant, University of Chicago, Department of Mathematics, 2021
Led discussion sections, office hours, grading, and exam review sessions; coordinated with instructors to align problem sets and rubrics.
Senior Mentor — Mathematics REU
Research mentorship (Senior Mentor), University of Chicago, Department of Mathematics, 2022
Mentored undergraduate research groups; weekly seminars on reading math, problem selection, and communicating results; individualized project guidance.
Lecturer — Calculus 131–132
Undergraduate course (Lecturer), University of Chicago, Department of Mathematics, 2022
Taught the standard Calculus I–III sequence; emphasized conceptual understanding, modeling, and scaffolded problem-solving.
Lecturer — Calculus 153 (Honors Calculus sequence)
Undergraduate course (Lecturer), University of Chicago, Department of Mathematics, 2023
Led Calculus III: integration and series, multivariable calculus; created problem banks and weekly conceptual quizzes.
Lecturer — Studies in Mathematics I 112: Introduction to Number Theory
Undergraduate course (Lecturer), University of Chicago, Department of Mathematics, 2024
Designed and taught an intro to number theory: divisibility, modular arithmetic, Fermat/Euler, and Diophantine problems; weekly proof-writing workshops and problem sets.