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Researcher profile

Salvatore Mercuri

Salvatore Mercuri contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.NTArtificial IntelligenceMachine Learning

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Published work

4 published item(s)

preprint2026arXiv

Formal Conjectures: An Open and Evolving Benchmark for Verified Discovery in Mathematics

As automated reasoning systems advance rapidly, there is a growing need for research-level formal mathematical problems to accurately evaluate their capabilities. To address this, we present Formal Conjectures, an evolving benchmark of currently 2615 mathematical problem statements formalized in Lean 4. Sourced from areas of active mathematical research, the dataset features 1029 open research conjectures providing a zero-contamination benchmark for mathematical proof discovery, and 836 solved problems for proof autoformalization. Notably, the repository provides a structured interface connecting mathematicians who formalize and clarify problems with the AI systems and humans attempting to solve them. Demonstrating its immediate utility, the benchmark has already been leveraged to make new mathematical discoveries, including the resolution of open research conjectures. We describe our approach to ensuring the correctness of these formalizations in a collaborative open-source project where contributions stem from an active community. In this framework, AI-generated proofs and disproofs serve as a valuable auditing mechanism to iteratively improve the fidelity of the benchmark. Finally, we provide a standardized evaluation setup and report baseline results on frozen evaluation subsets, demonstrating a climbable signal that measures the current frontier of automated reasoning on research-level mathematics.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

An Introduction to Machine Unlearning

Removing the influence of a specified subset of training data from a machine learning model may be required to address issues such as privacy, fairness, and data quality. Retraining the model from scratch on the remaining data after removal of the subset is an effective but often infeasible option, due to its computational expense. The past few years have therefore seen several novel approaches towards efficient removal, forming the field of "machine unlearning", however, many aspects of the literature published thus far are disparate and lack consensus. In this paper, we summarise and compare seven state-of-the-art machine unlearning algorithms, consolidate definitions of core concepts used in the field, reconcile different approaches for evaluating algorithms, and discuss issues related to applying machine unlearning in practice.

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preprint2020arXiv

$p$-adic $L$-functions on metaplectic groups

With respect to the analytic-algebraic dichotomy, the theory of Siegel modular forms of half-integral weight is lopsided; the analytic theory is strong whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture -- the $p$-adic $L$-function obtained by interpolating the complex $L$-function at special values. This is achieved through the Rankin-Selberg method and the explicit Fourier expansion of non-holomorphic Siegel Eisenstein series. The construction of the $p$-stabilisation in this setting is also of independent interest.

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preprint2020arXiv

Algebraicity of metaplectic $L$-functions

Notable results on the special values of $L$-functions of Siegel modular forms were obtained by J. Sturm in the case when the degree $n$ is even and the weight $k$ is an integer. In this paper we extend this method to half-integer weights $k$ and arbitrary degree $n$, determining the algebraic field in which they lie. This method hinges on the Rankin-Selberg method; our extension of this is aided by the theory of half-integral modular forms developed by G. Shimura. In the second half, an analogue of P. B. Garrett's conjecture is proved in this setting, a result that is of independent interest but that bears direct applications to our first results. It determines exactly how the decomposition of modular forms into cusp forms and Eisenstein series preserves algebraicity and, ultimately, the full range of special values.

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