Polina Baron

Polina Baron

Math PhD Candidate, University of Chicago. Research in Teichmüller dynamics, flat/translation surfaces, ergodic theory, integrable systems.

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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>

Slides (PDF)

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.

talks

teaching

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.