Specialized Lab · Dedicated ViewPure Mathematics & Foundational Conjectures

The 7 Millennium Prize Problems (Clay Mathematics Institute)

The greatest unsolved mathematical conjectures: P vs NP, Riemann, Yang-Mills, Navier-Stokes

Rigorous survey of the seven mathematical challenges formulated in 2000 by CMI. Precise statements, computational and physical ramifications, and current status of resolution attempts.

Fundamental Law
$$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = 0 \implies \text{Re}(s) = \frac{1}{2}$$
The Pinnacle of Modern Mathematics

The 7 Millennium Prize Problems (Clay Institute)

In the year 2000, the Clay Mathematics Institute designated seven foundational unsolved mathematical conjectures with a $1M bounty each, standing at the extreme frontier of modern theoretical knowledge.

#1Open · Differential Equations

Navier-Stokes Existence and Smoothness

Do smooth, physically reasonable solutions exist globally in time for the three-dimensional incompressible Navier-Stokes equations, or can singularities form in finite time?

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#2Open · Computer Science

P versus NP

Determines whether every problem whose solution can be efficiently verified in polynomial time ($NP$) can also be solved in polynomial time ($P$).

#3Open · Analytic Number Theory

The Riemann Hypothesis

Asserts that all non-trivial zeros of the Riemann zeta function $\zeta(s)$ have real part equal to $1/2$. Dictates the asymptotic distribution of prime numbers.

#4Solved (Perelman, 2003) · Topology

Poincaré Conjecture

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. Proven by Grigori Perelman using Ricci flow with surgery.

#5Open · Quantum Field Theory

Yang-Mills and Mass Gap

Rigorously construct non-Abelian quantum Yang-Mills gauge theory in four spacetime dimensions $\mathbb{R}^4$ and prove the existence of a positive mass gap $\Delta > 0$.

#6Open · Algebraic Geometry

The Hodge Conjecture

On non-singular complex projective algebraic varieties, certain De Rham cohomology classes are rational linear combinations of cohomology classes of algebraic cycles.

#7Open · Elliptic Curves

Birch and Swinnerton-Dyer Conjecture

Relates the rank of the group of rational points on an elliptic curve to the behavior of its associated $L$-function at the critical point $s = 1$.