Related papers: Murmurations: a case study in AI-assisted mathemat…
We prove the murmuration phenomenon, which is a correlation between signs of functional equations and Fourier coefficients, in the case of modular forms in the weight aspect. We in particular improve the range of visibility of murmurations…
We introduce a new method for studying murmurations, based on random matrix theory. With this method, we exhibit murmurations or similar phenomena: assuming ratios conjectures, for elliptic curves ordered by height, quadratic twists of a…
We argue how AI can assist mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and theoretical physics in 2017, we summarize how this emerging field has…
Can machine learning help discover new mathematical structures? In this article we discuss an approach to doing this which one can call "mathematical data science". In this paradigm, one studies mathematical objects collectively rather than…
In the words of the esteemed mathematician Paul Erd\"os, the mathematician's task is to \emph{prove and conjecture}. These two processes form the bedrock of all mathematical endeavours, and in the recent years, the mathematical community…
In this paper, we conduct a data-scientific investigation of Maass forms. We find that averaging the Fourier coefficients of Maass forms with the same Fricke sign reveals patterns analogous to the recently discovered "murmuration"…
We report on two machine learning experiments in search of statistical relationships between Dirichlet coefficients and root numbers or analytic ranks of certain low-degree $L$-functions. The first experiment is to construct interpretable…
The paper reflects on the future role of AI in scientific research, with a special focus on turbulence studies, and examines the evolution of AI, particularly through Diffusion Models rooted in non-equilibrium statistical mechanics. It…
This overview article highlights the critical role of mathematics in artificial intelligence (AI), emphasizing that mathematics provides tools to better understand and enhance AI systems. Conversely, AI raises new problems and drives the…
Recently, we showed that global root numbers of modular forms are biased toward +1. Together with Pharis, we also showed an initial bias of Fourier coefficients towards the sign of the root number. First, we prove analogous results with…
Conjecturing formulas and other symbolic relations occurs frequently in number theory and combinatorics. If we could automate conjecturing, we could benefit not only from speeding up, but also from finding conjectures previously out of our…
It has recently been argued that AI models' representations are becoming aligned as their scale and performance increase. Empirical analyses have been designed to support this idea and conjecture the possible alignment of different…
This paper presents a comprehensive overview on the applications of artificial intelligence (AI) in mathematical research, highlighting the transformative role AI has begun to play in this domain. Traditionally, AI advancements have heavily…
Can artificial intelligence truly contribute to creative mathematical research, or does it merely automate routine calculations while introducing risks of error? We provide empirical evidence through a detailed case study: the discovery of…
Recent years have seen the dramatic rise of the usage of AI algorithms in pure mathematics and fundamental sciences such as theoretical physics. This is perhaps counter-intuitive since mathematical sciences require the rigorous definitions,…
Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving…
The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically…
This book provides a comprehensive and accessible introduction to the emerging field of AI for mathematics. It covers the core principles and diverse applications of using artificial intelligence to advance mathematical research. Through…
Recent progress in artificial intelligence (AI) is unlocking transformative capabilities for mathematics. There is great hope that AI will help solve major open problems and autonomously discover new mathematical concepts. In this essay, we…
Recently, there is an increasing interest in understanding the performance of AI systems in solving math problems. A multitude of tests have been performed, with mixed conclusions. In this paper we discuss an experiment we have made in the…