arXiv subjects

Polina Baron

Publications and source records attributed to Polina Baron.

4 recordsLinked to original sources

arXiv2026

Unique ergodicity of branched covers of translation surfaces

Let $X$ be a finite-area translation surface whose vertical flow is uniquely ergodic. Given a slit joining two nonsingular points of $X$, one can form a branched cyclic cover by gluing $\mathrm{N}$ copies of $X$ crosswise along the slit. We study when the vertical flow on the resulting cover is uniquely ergodic. We first prove a geometric criterion for unique ergodicity of the branched cover. We show that if, for a sequence of times along the Teichm\"uller geodesic, one endpoint of the slit is contained in embedded Euclidean disks of uniformly positive radius that avoid the other endpoint, then the branched cover is uniquely ergodic. The proof uses the special symmetry of the cover together with an analysis of forward and backward generic points for the vertical flow. We then show that this criterion applies for Lebesgue-almost every choice of slit endpoint under a natural geometric hypothesis on the Teichm\"uller orbit of $X$, namely a uniform lower bound for the embedded radius along a subsequence. Finally, we give sufficient conditions for such a lower bound in terms of the cylinder geometry of $g_tX$, introducing the notion of pipe cylinders and proving that embedded disks of definite size must exist. As a consequence, for the class of uniquely ergodic translation surfaces, almost every slit produces a uniquely ergodic branched $\mathrm{N}$-cover.

math.DS↗

arXiv2026

QEDBENCH: Quantifying the Alignment Gap in Automated Evaluation of University-Level Mathematical Proofs

As Large Language Models (LLMs) saturate elementary benchmarks, the research frontier has shifted from generation to the reliability of automated evaluation. We demonstrate that standard "LLM-as-a-Judge" protocols suffer from a systematic Alignment Gap when applied to upper-undergraduate to early graduate level mathematics. To quantify this, we introduce QEDBench, the first large-scale dual-rubric alignment benchmark to systematically measure alignment with human experts on university-level math proofs by contrasting course-specific rubrics against expert common knowledge criteria. By deploying a dual-evaluation matrix (7 judges x 5 solvers) against 1,000+ hours of human evaluation, we reveal that certain frontier evaluators like Claude Opus 4.5, DeepSeek-V3, Qwen 2.5 Max, and Llama 4 Maverick exhibit significant positive bias (up to +0.18, +0.20, +0.30, +0.36 mean score inflation, respectively). Furthermore, we uncover a critical reasoning gap in the discrete domain: while Gemini 3.0 Pro achieves state-of-the-art performance (0.91 average human evaluation score), other reasoning models like GPT-5 Pro and Claude Sonnet 4.5 see their performance significantly degrade in discrete domains. Specifically, their average human evaluation scores drop to 0.72 and 0.63 in Discrete Math, and to 0.74 and 0.50 in Graph Theory. In addition to these research results, we also release QEDBench as a public benchmark for evaluating and improving AI judges. Our benchmark is publicly published at https://github.com/qqliu/Yale-QEDBench.

cs.LG↗

arXiv2024

The Neumann-Moser dynamical system and the Korteweg-de Vries hierarchy

At the focus of the paper are applications of the well-known Moser transformation of the C. Neumann dynamical system. It yields us a new quadratic integrable dynamical system on $\mathbb{C}^{3n+1}$, which we call the Neumann-Moser dynamical system. We present an explicit formula of the inverse of the Moser transformation. Consequently, we obtain explicitly an invertible transformation of the Uhlenbeck-Devaney integrals of the Neumann system into the integrals of our system. One of the main results of the paper is the recurrent 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.

nlin.SI↗

arXiv2024

The Mumford dynamical system and the Gelfand-Dikii recursion

In his paper "The Mumford Dynamical System and Hyperelliptic Kleinian Functions" (see arXiv:2402.09218), 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 his paper. In the present paper, we further develop the theory of (P,Q)-recursion and describe its connections to the Korteweg-de Vries hierarchy, the Lenard operator, and the Gelfand-Dikii recursion.

nlin.SI↗