The Reciprocity Project
Tutorials in Computational Number Theory
Tutorials
Tutorial No. 1
Topics: Elliptic curves, Hecke operators and Hecke eigenforms, q-expansions, Eichler-Shimura congruence relation
Tutorial No. 2
Topics: Completions, Cauchy sequences, Ostrowski's theorem,
p-adic distance and valuation, Euclidean models
Tutorial No. 3
Topics: Riemann zeta-function, distribution of primes, elliptic curve solutions mod p, motives, Hasse-Weil zeta-function, Eisenstein series, cusp forms, adeles, automorphic L-functions
Bonus Tutorial
Topics: abc conjecture, Hodge theaters, labels, non-scheme-theoretic mappings, multiplicative and additive structures of schemes
The Importance of Computational Examples
Computational mathematics is increasingly focused on automated theorem-proving, formalization, and verification. However, theorems are just mathematical statements, and automated theorem-proving can be done by exhaustive pattern-matching over such statements: it doesn't really involve "doing mathematics". Actual mathematical work involves more than just manipulation of formal statements; it encompasses all the calculations and computations that produce examples, build intuition, and explore the behavior and dynamics of mathematical data and objects. In other words, examples and computations constitute the 'actual mathematics' so often captured succinctly in mathematical statements.
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I think one reason why math formalization has become so popular is that so much of contemporary mathematics, in Bourbaki style, is represented in highly compressed notation. Even mathematical objects that lend themselves to explicit computations are just shown symbolically. Formalizing notation in something like Lean allows one to verify statements written in notation, but it still keeps the actual calculations and behavior of such objects hidden. Bourbaki style has really eliminated the particulars from mathematical expression in favor of generality. I look at Lean as, essentially, a way to deal with the increasing untenability of Bourbaki-style math. I don't think it's the last frontier of computational mathematics; rather, I think computational number theory in particular should produce rich examples.
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For this reason, the I've started the Reciprocity Project: a repository of resources exploring prominent topics in number theory via explicit calculations and examples.
The Reciprocity Project is created by James Douglas Boyd. James was formerly the CEO of the Wolfram Institute, a computational mathematics research center he cofounded with Stephen Wolfram, and was previously a project manager and Stephen Wolfram's research assistant at Wolfram Research. Since leaving the Institute, he has spent time at the Instituto Nacional de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro; the Research Institute for Mathematical Sciences (RIMS) in Kyoto; the Nesin Matematik Köyü in Åžirince; the Rotman Institute for Philosophy in London (Ontario); and the Astera Institute in Berkeley.
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