Here’s something I came across the other day and found super interesting, even though it broke my brain. While “controversial” in mathematics usually refers to unsolved problems or conjectures, here are 12 notable and often-debated mathematical concepts: (all links will open in a new window)

While some of these problems are unsolved, others are well-established results with ongoing discussions about their implications or

• 1. The Riemann Hypothesis:This famous conjecture proposes a specific relationship between the distribution of prime numbers and the behavior of a complex function. It remains unsolved despite extensive research. 
• 2. The Goldbach Conjecture:This states that every even integer greater than 2 can be expressed as the sum of two prime numbers. It’s been verified for large numbers, but a general proof remains elusive. 
• 3. The P vs NP Problem:This problem asks whether every problem whose solution can be quickly verified can also be solved quickly. A positive answer would have significant implications for computer science and cryptography. 
• 4. The Four Color Theorem:This theorem states that any map can be colored with only four colors so that no two adjacent regions have the same color. It was famously proven with the aid of computer analysis. 
• 5. Fermat’s Last Theorem:This theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. It was proven centuries after its formulation by Andrew Wiles. 
• 6. The Monty Hall Problem:This probability puzzle involves a contestant choosing a door with a prize behind it, and the host then revealing a door with no prize. Should the contestant switch their choice, or stay with their initial selection? The counterintuitive answer is that switching increases the odds of winning. 
• 7. The Twin Prime Conjecture:This conjecture suggests that there are infinitely many pairs of prime numbers that differ by 2. While there’s no proof, extensive research suggests it’s true. 
• 8. The Poincaré Conjecture:This conjecture, proven by Grigoriy Perelman, states that any simply-connected, closed 3-manifold is homeomorphic to the 3-sphere. It was one of the Millennium Prize Problems. 
• 9. The Collatz Conjecture:This conjecture, also known as the “100 problem”, posits that starting with any positive integer and repeatedly applying the rule (n is even, n = n/2; n is odd, n = 3n+1), you will eventually reach the number 1. 
• 10. The Birch and Swinnerton-Dyer Conjecture:This conjecture is a deep result in number theory concerning the number of rational points on algebraic curves. It remains unsolved. 
• 11. The Hodge Conjecture:This conjecture relates the algebraic structure of a complex projective variety to its topological structure. It’s a major unsolved problem in algebraic geometry. 
• 12. The Navier-Stokes Equations and Yang-Mills Existence and Mass Gap:These two problems are fundamental in mathematical physics and fluid dynamics. They concern the existence and properties of solutions to certain differential equations.