The Most Valuable Unsolved Math Problems You Should Know
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Chapter 1: Introduction to Unsolved Math Problems
During my high school years, I had an exceptional geometry instructor who administered a challenging test every Friday. At the conclusion of these assessments, he would include an exceptionally tough geometry puzzle, rewarding anyone who solved it with a chocolate bar the following Monday. While many of my peers were anxious about the test, I eagerly anticipated the challenge.
This idea of incentivizing difficult math problems wasn't unique to my teacher. In the year 2000, the Clay Mathematics Institute (CMI) unveiled a list of some of the most perplexing unsolved math problems at a Paris conference, offering a staggering one million dollar reward for each.
A century earlier, at another Parisian gathering, the esteemed mathematician David Hilbert presented a list of 23 carefully selected math problems, inviting mathematicians to engage in a grand challenge. This call to action was met with enthusiasm, and the solutions to those problems significantly influenced mathematics throughout the 20th century. You can find Hilbert's famous problems documented here.
As of now, 22 years after CMI's announcement, only one of the seven presented problems has been solved. The Poincare Conjecture was addressed by Russian mathematician Grigori Perelman, who notably declined the one million dollar reward.
Despite the passage of time, six problems remain unsolved:
- Riemann Hypothesis
- P vs NP Problem
- Hodge Conjecture
- Yang-Mills Theory
- Navier-Stokes Equations
- Birch and Swinnerton-Dyer Conjecture
These challenges represent some of the most difficult paths to earning a million dollars.
Section 1.1: Riemann Hypothesis
The Riemann Hypothesis, proposed by the distinguished mathematician Bernhard Riemann in 1859, has stumped mathematicians for over a century. Even the legendary Euler found it perplexing while working on functions. This hypothesis is so formidable that Hilbert famously remarked, "If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?"
The Riemann Zeta function can be defined as follows:
For any positive value of s, the solution ζ(s) remains elusive. Riemann introduced the idea of substituting a complex number (a + bi) for s, claiming that the zeta function has its zeros exclusively at the negative even integers and at complex numbers with a real part of ½. However, despite over 250 million zeros confirming this assertion, a formal proof that all zeros satisfy this condition is still pending.
If someone were to prove the Riemann Hypothesis true, it would revolutionize mathematics by unveiling the secrets of prime numbers. But what if a function could be devised to define prime numbers?
The first video explores the unsolved math problem that could potentially be worth a billion dollars.
Section 1.2: P vs NP Problem
The P vs NP problem, formulated by computer scientist Stephen Cook in 1971, questions whether every problem that can be quickly verified can also be solved quickly.
To illustrate, consider the difficulty of factoring a large number. While it may take considerable time to find the factors, verifying that the factors multiply back to the original number is a straightforward task. This verification can be completed in polynomial time, which is considered efficient.
In computational terms, an algorithm that needs solving is denoted as P, while problems that can be verified quickly fall under NP. If we can solve P in polynomial time, does that imply we can also solve NP in polynomial time?
This leads to a vital question: can we achieve polynomial time for every NP problem? The day someone proves P = NP could dramatically disrupt the job market for mathematicians, as it would equate the processes of proving and verifying mathematical theorems. This would have profound implications, including potentially jeopardizing banking systems since cracking passwords could become instantaneous.
The second video discusses the simplest math problem that remains unsolved: the Collatz Conjecture.
Chapter 2: Additional Unsolved Problems
Section 2.1: Hodge Conjecture
The Hodge Conjecture, the third unsolved problem, investigates how intricate mathematical structures can be formed from simpler components. Essentially, it seeks to relate two mathematical concepts in a meaningful way.
In the 20th century, mathematicians developed a powerful method to analyze complex objects by assembling increasingly larger entities to approximate the original shape. Unfortunately, this approach led to a loss of the geometric origins, as the essential concept became abstracted from its geometric foundations.
Section 2.2: Quantum Yang-Mills Theory
Quantum Yang-Mills Theory, the fourth unsolved problem, is rooted in physics and describes particles through mathematical symmetry. While physicists utilize Yang-Mills theory to characterize the fundamental forces of nature, it remains uncertain if solutions to Yang-Mills equations exist or if they possess a "mass gap" that explains the inability to isolate quarks. Currently, mathematicians lack a method or model to tackle this problem.
Section 2.4: Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture, established in the 1960s, focuses on rational points on a curve where both x and y are rational. This theorem is significant in cryptography and plays a vital role in solving challenges like Fermat's Last Theorem. Mathematicians employ an equation known as the L series to analyze these curves.
The conjecture posits that if an elliptic curve has infinite solutions, it will equate to zero at specific points of the L series.
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