Publications

You can also find my articles on my Google Scholar profile.

In progress

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.
Download Slides

Preprints

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.
Download Paper | Download Slides

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.
Download Paper | Download Slides

Peer-reviewed articles

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).
Download Paper