Knotty edukashun problems

 Alongside the Poincaré Conjecture, which has been solved, there are six other Millennium Prize Problems—each carrying a $1 million reward from the Clay Mathematics Institute for a correct solution:

  1. Riemann Hypothesis Predicts a deep pattern in the distribution of prime numbers, tied to the zeros of the Riemann zeta function.

  2. P vs NP Problem Asks whether every problem whose solution can be quickly verified can also be quickly solved—central to computer science and cryptography.

  3. Navier–Stokes Existence and Smoothness Concerns whether solutions to the equations governing fluid motion always exist and behave nicely in three dimensions.

  4. Yang–Mills Existence and Mass Gap Seeks a rigorous mathematical foundation for quantum field theories that describe fundamental forces, particularly the existence of a mass gap.

  5. Hodge Conjecture A question in algebraic geometry about which types of shapes (Hodge classes) arise from geometric objects.

  6. Birch and Swinnerton-Dyer Conjecture Relates to the number of rational solutions on elliptic curves and their connection to a complex function called the L-function.

Only the Poincaré Conjecture has been solved—by Grigori Perelman in the early 2000s, though he famously declined the prize. The rest remain open, tantalising challenges at the frontier of mathematics.


🧠 Mathematics

  • Hilbert’s 23 Problems (1900): A legendary list that shaped 20th-century math. Some, like the Riemann Hypothesis, remain unsolved.

  • Landau’s Problems (1912): Four deceptively simple number theory questions—none fully resolved.

  • Erdős Problems: Over 900 open problems posed by Paul Erdős, many with cash prizes attached.

  • Smale’s Problems (1998): Eighteen problems for the 21st century, spanning chaos theory, computation, and geometry.

🔬 Science & Physics

  • The 18 Unsolved Mysteries in Physics: Includes dark matter, dark energy, quantum gravity, and the arrow of time.

  • The Hard Problem of Consciousness: How and why do physical processes in the brain give rise to subjective experience?

  • Protein Folding Problem: Despite AI advances, the full mechanism of how proteins fold so rapidly remains elusive.

🧪 Computer Science & Logic

  • P vs NP Problem: Can every problem whose solution can be verified quickly also be solved quickly?

  • The Halting Problem: Proven undecidable—no algorithm can determine whether every program halts or loops forever.

  • Collatz Conjecture: A simple arithmetic sequence that no one can prove always ends in 1.

🧭 Cross-Disciplinary Notebooks

  • Kourovka Notebook: A living document of open problems in group theory.

  • Dniester and Sverdlovsk Notebooks: Collections of unsolved problems in algebra and semigroup theory.

Comments

Post a Comment

Popular posts from this blog

Policy #5: Development not Over-Development

As MP I will halt the development boom of Concrete Kent in Thanet. We are overbuilding far beyond South East targets as mere property speculation for the construction industry. I will insist construction is refocused on renovating both public housing and empty properties: over 3,000 empty properties in Thanet - for years - before one brick is laid on new build. The idea of a 4th Town at Westwood Cross should also be ripped up as property and Planning overkill given vacant properties, an ageing population and a failed economy. And on prime farmland when Climate Change is a major issue and Food Security in preserving our farmland to reduce food imports. And of course major derelict brownfield sites such as Manston airport, Richboro: derelict longer than in operation, and many more. Councils are notorious for being focused on property and construction where the main parties get their kickbacks for handing out Planning permissions. Far greater scrutiny of the Planning system is required. A...
Read more

David Cockburn KCC: Parkway and council corruption

(email to David Cockburn KCC CEO 3/3/21 cc Parliament etc as below) Hi David I and others were surprised at an access road for Kent Parkway rail station planned for 15th March without notice/consultation. As you know from KCC's own research the Parkway was overwhelmingly rejected by the public and councillors. And any rail improvements were less than 90 seconds or so(!). Yes, seconds. It seems a massive KCC civil servant white elephant tarmac scheme again: £20M in Whitehall funds were spent on documents? And is it a leftover of the secret KCC attempts to develop Manston as London International airport with KCC quango funds for the Virginia flights that bellyflopped and resulted in the sacking of your predecessor? Can you halt any works at the site for a call-in review and please provide FOI/EIR documents/emails etc of costs/plans etc of this utterly ludicrous project. For clarity the East Kent aquifer for drinking water for 150k people is under the Manston/Parkway r...
Read more

Tobias MP and Yellow Peril 2.0 for China and 77th Brigade?

The Tobias Ellwood MP article in last week’s Mail on Sunday (and a full page no less) was chilling for several reasons. https://www.dailymail.co.uk/debate/article-7592503/TOBIAS-ELLWOOD-China-winning-war-West-cyber-space-dark-moon.html The blaring headline of: "From cyber space to the dark side of the moon, China is waging war on the West - and it’s winning" is concerning not just for Pink Floyd fans worried that China’s counterfeit music industry extended from their back catalogue to the Moon itself. But also that a former Defence Minister was peddling such Yellow Peril 2.0 nonsense in the 21st century. And all the more concerning with Tobias not just as a Colonel Blimp eccentric MP hyping up the military and its tax budgets, but somehow extending a minor captaincy in the Army into a Lieutenant-Colonel role in the 77th Brigade, the military psyops function. No doubt Bill Cash MP or the fightin’ bank manager reservist Mark Francois MP, or Nigel Farage already...
Read more