# Clay Millennium Problems

**Type:** Overview  
**Domain:** Named Results  
**Posed:** 2000, Clay Mathematics Institute  
**Status:** 6 of 7 OPEN (only Poincaré Conjecture solved)  
**Codex URL:** /mathematics/named-results/clay-millennium-problems/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Seven $1M problems named in 2000, modelled on Hilbert's 1900 list. Only Poincaré Conjecture (2003) solved — Perelman declined both the prize and his Fields Medal.

## Hodge Conjecture (1950, W.V.D. Hodge)
Do certain topological classes on algebraic varieties always come from algebraic cycles? Known true for surfaces (Lefschetz (1,1) theorem, 1924); open in general.

## Birch and Swinnerton-Dyer Conjecture (1965)
Connects the number of rational points on an elliptic curve to its L-function's behavior at s=1. Proven for rank 0/1 curves (Kolyvagin, Gross, Zagier, late 1980s); open in general.

## Yang-Mills Existence and Mass Gap (formal statement 2000)
Yang-Mills theory (1954) underlies the Standard Model of particle physics, used successfully daily by physicists — yet never rigorously proven to mathematically exist in 4D, nor its "mass gap" prediction proven.

## Other Millennium Problems (own Codex entries)
Riemann Hypothesis, P vs NP, Navier-Stokes Existence and Smoothness.

## Sources
### Tier 1
- Hodge, W.V.D. (1950). Proceedings of the ICM.
- Birch, B.J. and Swinnerton-Dyer, H.P.F. (1965).
- Jaffe, A. and Witten, E. (2000). Clay Mathematics Institute.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*