arXiv subjects

Ekaterina S. Ivshina

Publications and source records attributed to Ekaterina S. Ivshina.

5 recordsLinked to original sources

arXiv2026

Neural Point-Forms

Point cloud learning often rests on the premise that observed samples are noisy traces of an underlying geometric object, such as a manifold embedded in a high-dimensional feature space. Yet much of this geometry is not captured directly by coordinates, pairwise distances, or learned graph neighborhoods alone. In the smooth setting, differential forms are devices to encode higher order tangency information. In this work, we introduce a new family of principled learnable geometric features for point clouds called neural point-forms (NPFs). In the absence of a natural tangency structure, we instead use Laplacian-based techniques from Diffusion Geometry to build a discrete model for comparing differential forms on point clouds via inner products. In the continuum, submanifolds of a shared ambient feature space are represented as comparison matrices, whose entries describe how pairs of feature forms interact with extrinsic tangency information. We make this intuition precise by proving the long-run consistency of comparison matrices under standard sampling, bandwidth, density, and manifold-hypothesis assumptions. This yields a compact, efficient and permutation-invariant neural layer whose output is a learned form-comparison matrix. Across synthetic and biologically relevant experiments, we show that NPFs provide a competitive, and interpretable representation, with the strongest benefits appearing when labels depend on sampling density, manifold-like structure, or response-relevant population geometry.

cs.LG↗

arXiv2025

Doughnut or Mickey Mouse? Detecting Toroidal Structure in Data through Persistent Cup-Length

Understanding the structure of high-dimensional data is fundamental to neuroscience and other data-intensive scientific fields. While persistent homology effectively identifies basic topological features such as "holes," it lacks the ability to reliably detect more complex topologies, particularly toroidal structures, despite previous heuristic attempts. To address this limitation, recent work introduced persistent cup-length, a novel topological invariant derived from persistent cohomology. In this paper, we present the first implementation of this method and demonstrate its practical effectiveness in detecting toroidal structures, uncovering topological features beyond the reach of persistent homology alone. Our implementation overcomes computational bottlenecks through strategic optimization, efficient integration with the Ripser library, and the application of landmark subsampling techniques. By applying our method to grid cell population activity data, we demonstrate that persistent cup-length effectively identifies toroidal structures in neural manifolds. Our approach offers a powerful new tool for analyzing high-dimensional data, advancing existing topological methods.

math.AT↗

arXiv2025

HARDMath2: A Benchmark for Applied Mathematics Built by Students as Part of a Graduate Class

Large language models (LLMs) have shown remarkable progress in mathematical problem-solving, but evaluation has largely focused on problems that have exact analytical solutions or involve formal proofs, often overlooking approximation-based problems ubiquitous in applied science and engineering. To fill this gap, we build on prior work and present HARDMath2, a dataset of 211 original problems covering the core topics in an introductory graduate applied math class, including boundary-layer analysis, WKB methods, asymptotic solutions of nonlinear partial differential equations, and the asymptotics of oscillatory integrals. This dataset was designed and verified by the students and instructors of a core graduate applied mathematics course at Harvard. We build the dataset through a novel collaborative environment that challenges students to write and refine difficult problems consistent with the class syllabus, peer-validate solutions, test different models, and automatically check LLM-generated solutions against their own answers and numerical ground truths. Evaluation results show that leading frontier models still struggle with many of the problems in the dataset, highlighting a gap in the mathematical reasoning skills of current LLMs. Importantly, students identified strategies to create increasingly difficult problems by interacting with the models and exploiting common failure modes. This back-and-forth with the models not only resulted in a richer and more challenging benchmark but also led to qualitative improvements in the students' understanding of the course material, which is increasingly important as we enter an age where state-of-the-art language models can solve many challenging problems across a wide domain of fields.

cs.LG↗

arXiv2023

Patterns in Knot Floer Homology

Based on the data of 12-17-crossing knots, we establish three new conjectures about the hyperbolic volume and knot cohomology: (1) There exists a constant $a \in R_{>0}$ such that the percentage of knots for which the following inequality holds converges to 1 as the crossing number $c \to \infty$: $\log r(K) < a \cdot Vol(K)$ for a knot $K$ where $r(K)$ is the total rank of knot Floer homology (KFH) of $K$ and $Vol(K)$ is the hyperbolic volume of $K$. (2) There exist constants $a,b\in R$ such that the percentage of knots for which the following inequality holds converges to 1 as the crossing number $c \to \infty$: $\log \det(K) < a \cdot Vol(K) + b$ for a knot $K$ where $\det(K)$ is the knot determinant of $K$. (3) Fix a small cut-off value $d$ of the total rank of KFH and let $f(x)$ be defined as the fraction of knots whose total rank of knot Floer homology is less than $d$ among the knots whose hyperbolic volume is less than $x$. Then for sufficiently large crossing numbers, the following inequality holds: $f(x)<\frac{L}{1+\exp{(-k \cdot (x-x_0))}} + b$ where $L, x_0, k, b$ are constants.

math.GT↗

arXiv2022

TESS Transit Timing of Hundreds of Hot Jupiters

We provide a database of transit times and updated ephemerides for 382 planets based on data from the NASA Transiting Exoplanet Survey Satellite (TESS) and previously reported transit times which were scraped from the literature in a semi-automated fashion. In total, our database contains 8,667 transit timing measurements for 382 systems. About 240 planets in the catalog are hot Jupiters (i.e. planets with mass $>$0.3$M_{\rm Jup}$ and period $<$10 days) that have been observed by TESS. The new ephemerides are useful for scheduling follow-up observations and searching for long-term period changes. WASP-12 remains the only system for which a period change is securely detected. We remark on other cases of interest, such as a few systems with suggestive (but not yet convincing) evidence for period changes, and the detection of a second transiting planet in the NGTS-11 system. The compilation of light curves, transit times, ephemerides, and timing residuals are made available online, along with the Python code that generated them (visit https://transit-timing.github.io).

astro-ph.EP↗