arXiv subjects

João Camarneiro

Publications and source records attributed to João Camarneiro.

3 recordsLinked to original sources

arXiv2026

Toric real loci via moment polytopes

Toric real loci are distinguished lagrangian submanifolds of toric symplectic manifolds. We develop a polyhedral model for toric real loci, called a kaleidoscope, based on the restriction of the moment map. This construction provides a direct geometric description of the topology of toric real loci and leads to simple criteria for orientability, together with transparent formulas for the Euler characteristic. We illustrate the theory with numerous examples and count, in every complex dimension up to 9, the smooth toric Fano varieties whose real loci are orientable. The kaleidoscope construction offers a simple, visual, and effective framework for understanding the topology of toric real loci through the combinatorics of their moment polytopes.

math.SG↗

arXiv2026

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗

arXiv2022

Convex bodies and asymptotic invariants for powers of monomial ideals

Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomial ideal by means of a more easily computable invariant, which we introduce under the name of naive Waldschmidt constant.

math.AC↗