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Unsolved — Open Problems in Mathematics

Some questions sit on the board for centuries. They look simple. They aren't.

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This Shipslides page presents Unsolved — Open Problems in Mathematics as an interactive HTML presentation deck in the Mathematics catalog with 13 slides. The share page keeps the uploaded deck sandboxed while exposing readable context, topics, and a slide outline for viewers and search engines.

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.

Key sections

  • 01Unsolved
  • 02Why open problems matter
  • 03The Millennium Prize Problems
  • 04P vs NP
  • 05The Riemann Hypothesis
  • 06Yang–Mills & the Mass Gap
  • 07Navier–Stokes Existence & Smoothness
  • 08Birch & Swinnerton-Dyer
  • 09The Hodge Conjecture
  • 10The Twin Prime Conjecture
  • 11The Collatz Conjecture
  • 12Goldbach's Conjecture
  • 13Where to read & watch

Topics covered

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Slide outline
  1. 01Unsolved
  2. 02Why open problems matter
  3. 03The Millennium Prize Problems
  4. 04P vs NP
  5. 05The Riemann Hypothesis
  6. 06Yang–Mills & the Mass Gap
  7. 07Navier–Stokes Existence & Smoothness
  8. 08Birch & Swinnerton-Dyer
  9. 09The Hodge Conjecture
  10. 10The Twin Prime Conjecture
  11. 11The Collatz Conjecture
  12. 12Goldbach's Conjecture
  13. 13Where to read & watch
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Updated
2026-05-17
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Presentation Transcript

Detailed slide-by-slide text content extracted from this presentation.

Slide 01

Unsolved

  • Lecture · Chalkboard Notes
  • Open problems in mathematics
  • Some questions sit on the board for centuries. They look simple. They aren't.
Slide 02

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 03

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 04

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 05

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 06

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 07

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 08

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 09

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
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