# Unsolved — Open Problems in Mathematics Canonical URL: https://shipslides.com/d/catalog-math-unsolved Raw viewer URL: https://content.shipslides.com/d/catalog-math-unsolved/raw Category: Mathematics Slides: 13 Updated: 2026-05-17T20:55:51.210Z Tags: catalog, math, unsolved ## Summary Some questions sit on the board for centuries. They look simple. They aren't. Key sections include: Unsolved; Why open problems matter; The Millennium Prize Problems; P vs NP; The Riemann Hypothesis; Yang–Mills & the Mass Gap; Navier–Stokes Existence & Smoothness; Birch & Swinnerton-Dyer; The Hodge Conjecture; The Twin Prime Conjecture. ## Slide Outline 1. Unsolved 2. Why open problems matter 3. The Millennium Prize Problems 4. P vs NP 5. The Riemann Hypothesis 6. Yang–Mills & the Mass Gap 7. Navier–Stokes Existence & Smoothness 8. Birch & Swinnerton-Dyer 9. The Hodge Conjecture 10. The Twin Prime Conjecture 11. The Collatz Conjecture 12. Goldbach's Conjecture 13. Where to read & watch ## Slide Transcript ### Slide 1: Unsolved - Lecture · Chalkboard Notes - Open problems in mathematics - Some questions sit on the board for centuries. They look simple. They aren't. ### Slide 2: I. - Why open problems matter - Mathematics doesn't grow outward from facts. It grows around the things it cannot prove. - An open problem is a tiny crack — and entire fields rush in to fill it. - Fermat's Last Theorem (open 358 years) gave us modern algebraic number theory. - The four-color theorem dragged proofs into the age of computers. - Failed attempts often become more useful than a hypothetical solution. - A good problem is a generator. A solved problem is a souvenir. ### Slide 3: II. - The Millennium Prize Problems - Posted by the Clay Mathematics Institute in May 2000. Seven problems. $1,000,000 each. - P vs NP - Computation's deepest divide. - Riemann - Zeros of the zeta function. - Yang–Mills - Mass gap in gauge theory. - Navier–Stokes - Smooth flow, forever? - Birch & S-D - Elliptic curves & L-functions. - Hodge - Algebra meets topology. - The Poincaré Conjecture was solved by Grigori Perelman in 2003. He declined both the prize and the Fields Medal. Six remain. ### Slide 4: III. - P vs NP - Is P = NP ? - If a solution can be checked in polynomial time, can it also be found in polynomial time? - Sudoku puzzles, protein folding, route optimization — easy to verify, brutal to solve. - Most mathematicians believe P ≠ NP. No one can prove it. - A proof either way would reshape cryptography, biology, AI. - Cook–Levin (1971) made it formal. Half a century later: nothing. ### Slide 5: IV. - The Riemann Hypothesis - ζ(s) = Σ 1 / ns - All non-trivial zeros of ζ(s) lie on the critical line Re(s) = 1/2. - Proposed by Bernhard Riemann in 1859. - Equivalent to deep statements about how primes are distributed. - Verified for the first 1013+ zeros — none stray off the line. - "If I were to awaken after a thousand years, my first question would be: has the Riemann Hypothesis been proven?" — David Hilbert ### Slide 6: V. - Yang–Mills & the Mass Gap - A question stitched between mathematics and physics. - Quantum Yang–Mills theory underlies the Standard Model of particle physics. - Experiments imply force-carrying particles like gluons should have a positive minimum energy — a "mass gap." - The challenge: build a mathematically rigorous theory and prove the gap exists. - No one even knows how to formally define the relevant quantum field on R4. - In short: physicists use it daily; mathematicians can't agree it exists. ### Slide 7: VI. - Navier–Stokes Existence & Smoothness - The equations that describe every fluid you've ever seen — water, air, blood, weather. - ∂u/∂t + (u·∇)u = −∇p + ν Δu - Given smooth initial conditions in 3D, do solutions stay smooth — or can a vortex blow up to infinity in finite time? - Engineers compute with these equations daily, trusting them implicitly. - If singularities can form, our model of turbulence is incomplete at its very foundation. - A river of equations. Nobody knows whether it floods. ### Slide 8: VII. - Birch & Swinnerton-Dyer - y2 = x3 + ax + b - The number of rational points on an elliptic curve is governed by the behavior of an associated L-function at s = 1. - Conjectured in the 1960s using early computer experiments at Cambridge. - Connects three worlds: arithmetic, complex analysis, geometry. - Elliptic curves powered Wiles' proof of Fermat's Last Theorem and underlie modern cryptography. ### Slide 9: VIII. - The Hodge Conjecture - The most abstract of the seven. The hardest to even state casually. - On a smooth complex projective variety, certain topological features (Hodge classes) should always come from algebraic geometry. - Translation: shapes you can see with topology should always be cut out by polynomial equations. - It is a bridge — promising that two languages describe the same continent. - Known in low dimensions and for specific cases. Open in general. - If true: a deep unity. If false: a strange and useful asymmetry. ### Slide 10: IX. - The Twin Prime Conjecture - Are there infinitely many primes p such that p + 2 is also prime? - (3, 5), (5, 7), (11, 13), (17, 19), … (1016 + 1, …) ? - Easy to state. Proposed in some form by de Polignac in 1849. - 2013 — Yitang Zhang, an unknown lecturer, proved infinitely many prime pairs differ by at most 70,000,000. - Within months, the Polymath project shrank the bound to 246. - From 70 million to 246 in a year. From 246 to 2: still nobody. ### Slide 11: X. - The Collatz Conjecture - if n is even: n → n/2 · - if n is odd: n → 3n + 1 - Pick any positive integer. Apply the rule. Repeat. - Conjecture: every starting n eventually reaches 1. - Verified for every n < 2.95 × 1020. - Paul Erdős: "Mathematics may not be ready for such problems." - Looks like a homework exercise. Has resisted proof for 90 years. ### Slide 12: XI. - Goldbach's Conjecture - Every even integer greater than 2 is the sum of two primes. - 4 = 2 + 2 · 6 = 3 + 3 · 100 = 3 + 97 · 1018 = … - Proposed in a 1742 letter from Christian Goldbach to Leonhard Euler. - Verified by computer up to 4 × 1018. - Helfgott (2013) proved the weak Goldbach: every odd integer > 5 is a sum of three primes. - The strong version — sum of two primes — has stood untouched for nearly three centuries. ### Slide 13: XII. - Where to read & watch - A few directions if any of these problems caught you. - The Music of the Primes — Marcus du Sautoy - Prime Obsession — John Derbyshire (the Riemann Hypothesis) - The Millennium Problems — Keith Devlin - Clay Mathematics Institute — official problem statements (claymath.org) - Quanta Magazine — superb reporting on number theory breakthroughs - YouTube: Riemann Hypothesis explained → - YouTube: P vs NP problem → - "In mathematics the art of asking questions is more valuable than solving them." — Cantor ## Related Decks - [Calculus — The Mathematics of Change](https://shipslides.com/d/catalog-math-calculus) - [Mathematical Cryptography](https://shipslides.com/d/catalog-math-cryptography) - [Game Theory — Strategy when others strategize too](https://shipslides.com/d/catalog-math-game-theory) - [Geometry — A Drafting Table Deck](https://shipslides.com/d/catalog-math-geometry)