Seven Millennium Problems

Grigori Perelman

- Russian Mathematician - He has made contributions to Riemannian geometry and geometric topology. - Graduated from the St. Petersburg State University. - He has proved the soul conjecture, Thurston's Geometrization conjecture, and the Poincare Conjecture.

P vs NP Problem

Computer Science. It asks whether every problem whose solution can be quickly verified can also be solved quickly. It is considered by many to be the most important open problem in computer science. In 1972, Richard Karp formulated 21 problems that are known to be NP(nondeterministic polynomial time)-complete. These are known as Karp's 21 NP-complete problems. They include problems such as the Integer programming problem, which applies linear programming techniques to the integers, the knapsack problem, or the vertex cover problem.

Poincaré Conjecture https://www.youtube.com/watch?v=GItmC9lxeco&t=330s

Essentially the first conjecture ever made in topology; it asserts that a 3-dimensional manifold is the same as the 3-dimensional sphere precisely when a certain algebraic condition is satisfied. The conjecture was formulated by Henri Poincare around the turn of the 20th century. Summarize it is that you can have an object with no holes, is finite, and can be made into a sphere in a variety of dimensions. In 1982, Michael Freedman proved the Poincaré conjecture in four dimensions.

Who solve one of these impossible problem!?!?!

Grigori Perelman.

What are they???

In 2000, the Clay Mathematics Institute of Cambridge, Massachusetts laid out seven of the most challenging problems mathematicians were grappling with at the time and offered a cool $1 million reward to anyone who could solve one. These problems represent the deepest mysteries in the field of mathematics. Some of them point to extremely useful practical applications, like engineering better spaceships, more effective drug treatments, and tougher cybersecurity encryption standards. Others seem to have no practical applications whatsoever, and simply offer the human race a more detailed look at how the universe works. One 1 of the 7 problems has been solved.

Riemann Hypothesis https://www.youtube.com/watch?v=d6c6uIyieoo&t=394s

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function(ζ(x) = 1 + 2−x + 3−x + 4−x + ⋯.) has its zeros only at the negative even integers and complex numbers with a real part. Many consider it to be the most important unsolved problem in pure mathematics. It was first proposed by Bernhard Riemann in 1859. In 2018, Sir Michael Atiyah claimed to have solved it, although this is still under speculation.

Birch and Swinnerton-Dyer Conjecture

In the early 1960s in England, British mathematicians Bryan Birch and Peter Swinnerton-Dyer used the EDSAC (Electronic Delay Storage Automatic Calculator) computer at the University of Cambridge to do numerical investigations of elliptic curves. Based on these numerical results, they made their famous conjecture. It basically asserts that the size of the group of rational points is related to the behavior of an associated zeta function ζ(s) near the point s=1.

Hodge Conjecture

Major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. The conjecture was first formulated by British mathematician William Hodge in 1941, though it received little attention before he presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Mass., U.S. It asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles.

Navier-Stokes Equation https://www.youtube.com/watch?v=ERBVFcutl3M

Named after Claude-Louis Navier and George Gabriel Stokes, describing the motion of viscous fluid substances first made in 1822. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier-Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations, they can be used to model and study magnetohydrodynamics. The question at hand is to make sense of the equations.

Yang-Mills and Mass Gap

The question is to prove that for any compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists on {\displaystyle \mathbb {R} ^{4}} and has a mass gap Δ > 0. To answer the question, the answer must take into consideration: Yang-Mills theory exists and satisfies the standard of rigor that characterizes contemporary mathematical physics, in particular, constructive quantum field theory,[2][3] and The mass of the least massive particle of the force field predicted by the theory is strictly positive.