Salvatore Mercuri
Knowledge Transfer Associate
School of Mathematics, The University of Edinburgh
Email: smercuri (at) ed.ac.uk
Address:
School of Mathematics, University of Edinburgh
James Clerk Maxwell Building
The King's Buildings
Peter Guthrie Tait Road Edinburgh
EH9 3FD, United Kingdom
Office: 5422
I am currently a Knowledge Transfer Associate in Digital Mathematics Assessment in the School of Mathematics at the University of Edinburgh, working in partnership with IDEMS International CIC, as well as a Research Scientist at Google DeepMind.
My current primary research area encompasses digitisation of mathematics research and education, with a focus in developing the STACK computer-aided assessment system to automate assessment of proof questions as well as to broaden applications to machine learning and data science. I am also interested in formal proof verification for mathematical research and AI, with a focus on the Lean theorem prover. I was recently involved with training AlphaProof at Google DeepMind, which is an AI system that got enough marks at the International Mathematical Olympiad in 2024 for a silver medal.
My mathematical research background is in the intersection of algebraic and analytic number theory, with research focusing on the algebraicity of metaplectic automorphic forms and the broader context of the Langlands program. Since then I have been a researcher in machine learning in scientific and financial industries, leading to my interests circling back to the application of computational and statistical methods to enhance mathematics research and education.
Employment
Research Scientist at Google DeepMind (via Adecco)
2024 - Present
Research Data Scientist at NatWest Group
2021 - 2023
Mathematical Researcher at ThinkTank Maths
2020 - 2021
Education
PhD in Mathematics
2015 - 2019
Department of Mathematics, Durham University
Advisor: Thanasis Bouganis
Thesis: The arithmetic of metaplectic modular forms
MSc. in Pure Mathematics
2014 - 2015
Imperial College London
Advisor: Kevin Buzzard
BSc. in Mathematics
2010 - 2014
University of Edinburgh
Publications
- Formalising the Local Compactness of the Adele Ring, preprint (2024).
- (with D. Batra, R. Khraishi) Conformal Predictions for Longitudinal Data, preprint (2023).
- (with G. Cowan, R. Khraishi) Modelling customer lifetime value in the retail banking industry, preprint (2023).
- (with R. Khraishi, R. Okhrati, D. Batra, C. Hamill, T. Ghasempour, A. Nowlan) An Introduction to Machine Unlearning, (2022).
- Algebraicity of metaplectic L-functions, (arXiv link), J. Number Theory 219 (2021), 109-161.
- (with T. Bouganis) On the Rankin–Selberg method for vector-valued Siegel modular forms, (arXiv link), Int. J. Number Theory 17 (2020), no. 5, 1207-1242.
- p-adic L-functions on metaplectic groups (arXiv link), J. London Math. Soc., 102 (2020), 229-256.
- The p-adic L-function for half-integral weight modular forms, manuscripta math., 161 (2020), 61-91.
Talks
- Proof assessment in STACK with Parsons problems, MERGE Seminar, University of Trieste, Italy (Mar. 2024).
- Proof assessment in STACK using SortableJS, International Meeting of the STACK Community 2024, OTH Amberg-Weiden, Amberg, Germany (Mar. 2024).
- Garrett’s conjecture for metaplectic Eisenstein series, Young Researchers in Algebraic Number Theory 2018, The University of Sheffield, UK (Nov. 2018).
Conferences and workshops attended
- The 2nd African STACK Conference for Undergraduate Mathematics, Technical University of Kenya, Nairobi, Kenya (Aug. 2024)
- Open source mathematics curriculum and assessment tools, Maseno University, Kisumu, Kenya (Aug. 2024)
- International Meeting of the STACK Community 2024, OTH Amberg-Weiden, Amberg, Germany (Mar. 2024).
- Young Researchers in Algebraic Number Theory 2018, The University of Sheffield, UK (Nov. 2018).
- 32nd Automorphic Forms Workshop, Tufts University, Medford, Massachusetts, USA (Mar. 2018).
- Iwasawa 2017, The University of Tokyo, Tokyo, Japan (Jul. 2017).
Resources
STACK
STACK is open-source software that is used for automated digital assessment of mathematics within universities and schools globally. To get get started writing questions, install STACK and then use the author quick-start guide.
There are increasing ways to assess proof questions in STACK and my work so far has helped enable assessing proofs as drag-and-drop reordering questions through a parsons block. This was released in Dec. 2023 in STACK v4.5.0. See the documentation for how to write and assess these types of questions in STACK.
Lean
The Lean theorem prover is an automated theorem prover that has grown rapidly as a tool for research and for education in both mathematics and AI. Mathlib is an open-source library for Lean containing a large and growing body of mathematical theory in Lean.
A great place to start learning is the Natural Number Game. After that, a good next step would be to install Lean and follow Imperial College London's "Formalising Mathematics 2024" course on GitHub, as well as reading Mathematics in Lean.
At Google DeepMind, I helped to train an AI using Lean which went on to get enough marks for a silver medal at the International Mathematical Olympiad in July 2024.
Here is a paper, GitHub repository and documentation which formalises the proof of the local compactness of the adele ring of a number field in the Lean theorem prover.
FLINT
The images below and on the home page were made using the C library Arb, which is now part of FLINT (Fast Library for Number Theory). They show the complex plots of modular forms, including the half-integral weight Dedekind eta function, and other elliptic functions as well as L-functions. The images use colour to plot the phase of the output complex number of the function for each complex input, being forgetful of magnitude.
As well as facilitating the production of high resolution images of intricate complex objects through the use of ball arithmetic, FLINT has been used for computational research purposes such as the highest-known (as of January 2024) verification of the Riemann Hypothesis, Dave Platt & Tim Trudgian (2020).
