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Ignacio Tejeda

Publications and source records attributed to Ignacio Tejeda.

4 recordsLinked to original sources
arXiv2026

Semantic Search over 9 Million Mathematical Theorems

Searching for mathematical results remains difficult: most existing tools retrieve entire papers, while mathematicians and theorem-proving agents often seek a specific theorem, lemma, or proposition that answers a query. While semantic search has seen rapid progress, its behavior on large, highly technical corpora such as research-level mathematical theorems remains poorly understood. In this work, we introduce and study semantic theorem retrieval at scale over a unified corpus of $9.2$ million theorem statements extracted from arXiv and seven other sources, representing the largest publicly available corpus of human-authored, research-level theorems. We represent each theorem with a short natural-language description as a retrieval representation and systematically analyze how representation context, language model choice, embedding model, and prompting strategy affect retrieval quality. On a curated evaluation set of theorem-search queries written by professional mathematicians, our approach substantially improves both theorem-level and paper-level retrieval compared to existing baselines, demonstrating that semantic theorem search is feasible and effective at web scale. The project page, search tool, dataset, REST API, and MCP server are available at theoremsearch.com.

cs.IR↗
arXiv2025

On the geometry of measures with density bounds in a H\"older anisotropic setting

We study the regularity of the support of a Radon measure $\mu$ on $\mathbb R^{n+1}$ for which anisotropic versions of its $n$-dimensional density ratio and its doubling character are assumed to converge with H\"older rate. We show that in either case, if the support of $\mu$ is flat enough, then it is a $C^{1,\gamma}$ $n$-dimensional submanifold of $\mathbb R^{n+1}$, for some $\gamma\in (0,1)$. If the flatness assumption is dropped, then the support of $\mu$ is the union of a $C^{1,\gamma}$ $n$-dimensional submanifold of $\mathbb R^{n+1}$ and a closed singular set that is either empty if $n\leq 2$, or has Hausdorff dimension at most $n-3$ if $n\geq 3$.

math.AP↗
arXiv2019

Spectral properties of Landau Hamiltonians with non-local potentials

We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({\mathbb R^2})$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues $Λ_q$, $q \in {\mathbb Z}_+$. We perturb $H_0$ by a non-local potential written as a bounded pseudo-differential operator ${\rm Op}^{\rm w}({\mathcal V})$ with real-valued Weyl symbol ${\mathcal V}$, such that ${\rm Op}^{\rm w}({\mathcal V}) H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{\mathcal V} = H_0 + {\rm Op}^{\rm w}({\mathcal V})$. First, we construct symbols ${\mathcal V}$, possessing a suitable symmetry, such that the operator $H_{\mathcal V}$ admits an explicit eigenbasis in $L^2({\mathbb R^2})$, and calculate the corresponding eigenvalues. Moreover, for ${\mathcal V}$ which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of $H_{\mathcal V}$ adjoining any given $Λ_q$. We find that the effective Hamiltonian in this context is the Toeplitz operator ${\mathcal T}_q({\mathcal V}) = p_q {\rm Op}^{\rm w}({\mathcal V}) p_q$, where $p_q$ is the orthogonal projection onto ${\rm Ker}(H_0 - Λ_q I)$, and investigate its spectral asymptotics.

math-ph↗