7 missions · 4 completed
The mathematics of finite and discrete structures — counting the arrangements of a set, deciding when a configuration meeting prescribed constraints can exist, and characterizing the patterns such structures are forced to contain. It encompasses enumerative and extremal combinatorics, graph theory, design theory, and additive combinatorics, with deep ties to algebra, probability, and computer science.
Determine the exact second-order term in the least possible largest factor in a factorization of $n!$ into distinct integers exceeding $n$, with the proposed rational constant $4029639598/25970038185$.
That the prime numbers, thinning out as they climb yet never quite vanishing, should nonetheless contain arithmetic progressions of every finite length is one of the most celebrated discoveries of twenty-first-century mathematics. Ben Green and Terence Tao proved it in 2004 (published in the Annals of Mathematics in 2008), resolving a question whose roots reach back to Lagrange and Waring around 1770 and which had crystallized in the Erdős–Turán conjecture. The primes have density zero, so Szemerédi's theorem — which guarantees long progressions only in positive-density sets — does not apply directly; the genius of the proof was a transference principle extending Szemerédi's theorem to sets sitting densely inside a 'pseudorandom' host, built from the sieve ideas of Goldston, Pintz, and Yıldırım. The result was a centerpiece of the citation for Tao's 2006 Fields Medal and opened a whole industry, including the Tao–Ziegler extension to polynomial progressions. Unusually for a headline problem, this theorem is already proved — which makes it an ideal flagship formalization mission: a deep, decomposable argument whose pieces, from Szemerédi's theorem to the transference principle, the community can rebuild and verify in Lean.
A Hadamard matrix is a square array of +1s and −1s whose rows are mutually orthogonal — equivalently, one whose determinant attains the absolute maximum that Jacques Hadamard proved in 1893 any ±1 matrix can reach. The story opens earlier, with James Joseph Sylvester's 1867 doubling construction producing such matrices in every power-of-two order; Hadamard himself added orders 12 and 20. The conjecture bearing his name asserts that a Hadamard matrix exists for every order divisible by four. Raymond Paley's 1933 construction from finite fields settled vast new families, and computer searches filled stubborn gaps — beginning with order 92 at JPL in 1962 and reaching order 428 only in 2005, after which 668 became the smallest order whose existence is still unknown. Far from a curiosity, these matrices are workhorses of applied mathematics, underpinning error-correcting codes (the Reed–Muller code that sharpened Mariner spacecraft imagery), spread-spectrum and CDMA signal design, optimal statistical designs of experiments, and coded-aperture spectroscopy. Settling the conjecture would close a 130-year-old gap where combinatorics, number theory, and design theory meet.