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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
Preprint: 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 branched covers of translation surfaces
Preprint: arXiv:2606.05492, 2026
Abstract
Starting with a finite-area translation surface whose vertical flow is uniquely ergodic, we construct branched cyclic covers by gluing copies of the surface crosswise along a slit. We establish a geometric criterion ensuring unique ergodicity of the lifted flow and show that it applies to almost every choice of slit endpoint under a natural geometric condition on the Teichmüller orbit. We also give sufficient conditions in terms of cylinder geometry, introducing the notion of pipe cylinders. Joint with Elizaveta Shuvaeva.
Recommended citation: Polina Baron and Elizaveta Shuvaeva. (2026). "Unique ergodicity of branched covers of translation surfaces." arXiv:2606.05492.
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service
Tutorial Coordinator — Mathematics REU
University of Chicago · Summer 2022
Coordinated the REU tutorial program, developed materials, and supported tutorial leaders.
Organizer — Warm-Up Program for New Graduate Students
University of Chicago · September 2024
Organized a week-long academic orientation program for incoming mathematics graduate students, including scheduling sessions and coordinating presenters.
Dynamics Pre-Seminar Organizer
University of Chicago · 2025–2026
Co-organized a graduate pre-seminar in dynamical systems, including selecting topics and coordinating speakers.
Phoenix STEM Collaborative Learning Coordinator
University of Chicago · 2025–2026
Coordinated half of the tutorial sections for the 400-student MATH 131–132 calculus sequence. Observed tutorial instructors and provided pedagogical feedback, co-developed tutorial materials, and completed training in collaborative-learning pedagogy.
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
University of Chicago · 2021–2022
Courses included Real Analysis I, Inquiry-Based Learning in Real Analysis II, Ordinary Differential Equations, and Brownian Motion. Led problem-solving sessions, held office hours, and graded coursework.
Senior Mentor — Mathematics REU
University of Chicago · Summer 2022
Mentored undergraduate researchers, advised the development of their projects, and supported their final presentations.
Lecturer — MATH 131–133: Elementary Functions and Calculus I–III
University of Chicago · 2022–2023
Taught approximately 30 students per quarter in a coordinated, multi-section, three-quarter sequence. Prepared and delivered lectures following the shared syllabus; held office hours; and administered and graded common assessments.
Course Instructor — MATH 151–153: Calculus I–III
University of Chicago · 2023–2024
Independently designed and taught all three courses in the sequence to approximately 30 students per quarter. Responsible for lectures, course materials, assessments, office hours, and grading.
Course Instructor — MATH 112: Studies in Mathematics I (Introduction to Number Theory)
University of Chicago · Fall 2024
Designed and taught a 20-student introduction to number theory, including all lectures, course materials, assignments, and assessments.
Course Instructor — MAT244H1F: Introduction to Ordinary Differential Equations
University of Toronto · Fall 2026
Independently design and deliver three hours of lectures per week for approximately 200 students. Prepare original lecture notes and course materials for every class; hold office hours; collaborate with other instructors on assessment design and course administration.
Course Instructor — MAT136H1S: Calculus II
University of Toronto · Winter 2027 (scheduled)
Scheduled to teach a lecture section in a coordinated, multi-section course.