Research

Preprints

  1. Geometry-Aware MCTS for Extremal Problems in Combinatorial Geometry With Luoning Zhang, Tianhao Wang, and Nathan Kaplan. Submitted to Transactions on Machine Learning Research (TMLR).

    Abstract
    We study certain extremal problems in combinatorial geometry that ask about configurations of points in an \(n \times n\) grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits. To overcome these bottlenecks, we propose a Geometry-Aware Monte Carlo Tree Search (MCTS) framework. Our approach strictly enforces geometric constraints through incremental updates to the feasible action space. For constraints about collections of collinear points, like those that occur in the classic No-Three-in-Line problem (Max-N3IL), this mechanism reduces the constraint checking complexity from \(O(n^3)\) to \(O(n^2)\). To improve search efficiency, we exploit geometric symmetries in two ways: canonical pruning during node expansion to reduce the branching factor, and symmetric batch transitions to accelerate the discovery of promising configurations. We perform extensive experiments and establish new best-known computational results on five out of six of the problems that we considered. Notably, for Max-N3IL we find configurations of size roughly \(1.8n\) for grids of size \(82 \le n \le 119\). For the Smallest Complete Set problem, we find configurations of size roughly \(0.95n\), providing new upper bounds within the tested grids. This work establishes Geometry-Aware MCTS as a highly adaptable framework for discovering novel configurations in combinatorial geometry.

Publications

  1. The Jacobian of a Sixth-Root-of-Unity Matroid With Matthew Baker and Changxin Ding. Annals of Combinatorics.
    Abstract
    The Jacobian group (also called the sandpile group, Picard group, or critical group) of a graph or, more generally, of a regular matroid has been well studied. Sixth-root-of-unity matroids, also called complex unimodular matroids, are generalizations of regular matroids. This paper provides a definition, and establishes some basic properties, of the Jacobian group of a sixth-root-of-unity matroid.

  2. Machine Learning and LLM-Boost Symbolic Regression for Predicting \(\mathbb{Q}\)-Gonality of Modular Curves With Xiaokang Wang, Yuxiang Yao, Po-Chu Hsu, and Peikai Qi. Accepted at the 2nd AI4MATH@ICML 2025 workshop.

    Abstract
    We aim to predict the \(\mathbb{Q}\)-gonality of modular curves, an invariant measuring the minimal degree of a nonconstant rational map to \(\mathbb{P}^1\). Three machine-learning architectures—Extrem gradient-boosted trees, feedforward neural networks, and transformer-based models—achieve over 90\% exact-match accuracy on existing curves, with more than 89\% of predictions falling within known theoretical bounds. To improve interpretability, we employ an LLM-guided boost symbolic regression pipeline that proposes nonlinear feature combinations and uncovers concise analytic formulas. These expressions match the predictive power of our models while revealing how core arithmetic invariants interact. Our results highlight the effectiveness of combining data-driven prediction with LLM-enhanced symbolic discovery in arithmetic geometry.