Module 4 · The Theorems
Key Theorems & Themes
A selection of award-winning results, with the bare statement and a sentence on why it mattered. The mathematics is genuinely deep; these notes are signposts.
1. AtiyahâSinger Index Theorem (1963)
For an elliptic differential operator \(D\) on a compact manifold \(M\), the analytic index \(\mathrm{ind}(D) = \dim\ker D - \dim\mathrm{coker}\,D\)equals a topological invariant of \(M\) and the symbol of \(D\). Unifies the RiemannâRoch, Hirzebruch signature, and GaussâBonnet theorems and undergirds gauge theory and parts of string theory. Atiyah & Singer received the 2004 Abel Prize.
2. Weil Conjectures â Deligne (1974)
For a smooth projective variety over a finite field \(\mathbb{F}_q\), the zeta function satisfies a rationality, functional equation, and âRiemann hypothesisâ with eigenvalues of Frobenius having modulus \(q^{i/2}\) on the i-th cohomology. Grothendieck developed Ă©tale cohomology to attack the conjectures; Deligne finished the proof. The 1973 paper âLa conjecture de Weil Iâ in IHES Publications is one of the most influential mathematical papers of the 20th century. Deligne won the Fields Medal (1978) and the Abel Prize (2013).
3. Fermatâs Last Theorem â Wiles (1995)
No three positive integers \(a, b, c\) satisfy \(a^n + b^n = c^n\) for any integer \(n > 2\). Conjectured by Fermat in 1637 in the margin of his copy of Diophantus; proved by Andrew Wiles by establishing the modularity of semistable elliptic curves over the rationals (with a critical lemma due to Richard Taylor and Wiles, 1995). The argument is a tour de force of modern number theory. Wiles received the 2016 Abel Prize. He was older than 40 at the time of completion and so did not qualify for the Fields.
4. Geometrisation of 3-Manifolds â Perelman (2002â3)
Every closed 3-manifold can be canonically decomposed into pieces, each of which admits one of eight Thurston model geometries. As a special case the PoincarĂ© conjecture â every simply connected closed 3-manifold is homeomorphic to \(S^3\) â follows. Perelmanâs preprints used Hamiltonâs Ricci flow with surgery, controlling singularity formation by an entropy monotonicity argument. He posted the proofs to arXiv (2002, 2003), declined to publish them, and declined both the Fields Medal (2006) and the $1 million Clay Prize (2010).
5. Mirzakhani â Moduli of Riemann Surfaces
Maryam Mirzakhani computed the volumes of moduli spaces of Riemann surfaces with respect to the WeilâPetersson metric, found the recursive structure of these volumes (Mirzakhaniâs recursion), and proved deep results on geodesic counts on hyperbolic surfaces and on dynamics on the moduli space. Her work connects geometric topology, dynamics, and algebraic geometry. She received the Fields Medal in 2014 â the first woman, the first Iranian. She died of breast cancer in 2017 at age 40.
6. Perfectoid Spaces â Scholze (2012)
Peter Scholze introduced perfectoid spaces â a kind of analytic space in characteristic 0 with a controlled lift to characteristic p â that allow transfer of techniques between the two settings. The theory underlies a sweeping programme connecting p-adic Hodge theory, the local Langlands programme, and algebraic topology. Scholze received the Fields Medal in 2018, at age 30.
7. Sphere Packing in 8 and 24 Dimensions â Viazovska (2016)
Maryna Viazovska proved that the \(E_8\) root lattice gives the densest sphere packing in 8 dimensions, and (with Cohn, Kumar, Miller, and Radchenko) the Leech lattice in 24. The proof uses a single ingenious modular form as a âmagic functionâ whose zeros and Fourier coefficients produce the matching upper bound. Viazovska received the Fields Medal in 2022 â the second woman.
8. The Langlands Programme
In a 1967 letter to AndrĂ© Weil, Robert Langlands proposed a vast web of conjectures relating Galois representations to automorphic forms, generalising both class-field theory and the EichlerâShimuraâDeligne theorem. The programme is now the central organising principle of much of modern number theory; many Fields-level theorems (Lafforgue 2002 on function-field GLn; NgĂŽ BĂĄo ChĂąu 2010 on the fundamental lemma; Scholze on p-adic Langlands) are pieces of it. Langlands himself received the Abel in 2018.