Seven Millennium Problems in Mathematics (Part II)
Seven Millennium Problems in Mathematics (Part II)
4. Yang-Mills existence and mass gap
This theory is based on four fundamental sources such as Gravity, Electromagnetism, Strong Nuclear Force, Weak Nuclear Force. Yang–Mills Theory attempts to explain the behavior of elementary particles utilizing these non-abelian Lie groups and is at the core of the fusion of electromagnetic force and weak power.
The problem statement is:
“For any compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists on R4 and has a mass gap Δ > 0. In this statement, a Yang-Mills theory is a non-abelian quantum field theory similar to that underlying the standard model of particle physics; R4 is Euclidean 4-space; the mass gap Δ is the mass of the least massive particle predicted by the theory.”
Yang–Mills ‘s concepts met with general approval in the physics community since Gerard ‘t Hooft’s renormalization in 1972, based on the formulation of the question worked out by his advisor Martinus Veltman.
The field of Yang–Mills theories was included in the Clay Mathematics Institute’s list of “Millennium Prize Problems”. Here the prize-problem consists, in particular, of a proof of the proposition that the lowest excitations of a pure Yang – Mills theory (i.e., without fields of matter) have a finite mass difference with respect to the vacuum condition. Another open question linked with this conjecture is a proof of the principle of containment in the presence of additional Fermion particles.
5. Hodge Conjecture
Official problem statement:
“On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.”
The Hodge conjecture was so complex and so abstract. There was no way to explain it to a layman for those hoping to look it up on the Internet. The official problem statement is a little opaque. to put it lightly but at its core it says something very simple.
Suppose you have a string with jagged edges. The conjecture says that you can smush it into something with smooth edges.
It’s not so hard for the above example. But, what if I gave you a string with a knot? Can you make sure that it’s smooth everywhere without having to untie the knot?
Not so sure, but it gets even worse.
Suppose, you have the same problem but in higher dimensions.
If I gave you this shape, a hole in a hole in a hole, how do you smooth that out? What does smooth even mean here? The point is we have some work to do. The reason the Hodge conjecture is so hard is because it involves so much mathematical machinery. It links to very large fields of modern mathematics Algebraic Geometry and Topology. If the conjecture is true it would open a door between these fields allowing us to pass information freely between them.
The Hodge conjecture says that Given a random shape, when is it homeomorphic to a shape described by polynomials?
In 1982, mathematician Michael Friedman found a counterexample. He called it a Friedman E8 manifold and defined revised Hodge conjecture as,
Find one condition that ensures a shape can be molded into an algebraic set.
This must account for every single shape and every single equation imaginable.
If you can find such condition, you will win a million dollars !
6. Birch and Swinnerton-Dyer Conjecture
Official Problem statement:
“The Taylor expansion of L(C,s) at s=1 has the form
L(C,s)=c(s-1)r + Higher order terms with c≠0 and r = rank(C(Q)).
In particular this conjecture asserts that L(C,1)=0 ó C(Q) is finite.”
“Mathematicians have also been intrigued by the question of representing all solutions in
whole numbers x,y,z to algebraic equations like
x2 + y2 = z2
Euclid has given a complete solution to that equation, but for more complicated equations, it’s going to be extremely difficult. In reality Yu in 1970. V. Matiyasevich proved that Hilbert ‘s tenth problem is unsolvable, i.e. since these equations have a solution in whole numbers, there is no general method to decide. In special cases, however, one can hope to say something. When the solutions are the points of the abelian variety, Birch and Swinnerton-Dyer conjecture argue that the size of the rational point group is related to the behavior of the associated zeta function ζ(s) near the point s=1. Specifically this incredible theorem states that if somehow(1) is equal to 0, then there is an infinite number of rational points (solutions), and vice versa, if somehow(1) is not equal to 0, then there is only a finite number of such items.”
The Conjecture says, an elliptic curve only has an infinite number of rational solutions if S, the constant in the expression of L(1), [where L(s) is its L-function] is equal to zero.
Furthermore, to determine exactly how many ‘starting’ solutions are required to generate all the rest, the slope of the line gained by plotting the solutions modulo primes is equal to the rank.
7. Poincaré Conjecture
Among the seven conjectures, this is the only Conjecture which was solved. Its solution was given by Grigoriy Perelman, announced in preprints posted on ArXiv.org in 2002 and 2003.
Official problem statement:
“When we stretch a rubber band around an apple ‘s surface, then we can compress it to a point by gradually pushing it, without breaking it apart and without causing it to leave the surface. On the other side, if we assume the same rubber band was somehow wrapped around a doughnut in the right direction, then there is no way to shorten it to a point without either damaging the rubber band or the doughnut. We say the apple surface is “just connected” but the doughnut surface is not. Poincaré realized, nearly a hundred years ago, that this property of simple connectivity basically characterizes a two-dimensional sphere, and questioned the same question about the three-dimensional sphere.”
For solution of this Poincaré Conjecture, refer the following link.
Solution link: https://www.claymath.org/millennium-problems-poincar%C3%A9 conjecture/perelmans-solution
Acknowledgements…
I am thankful to the following sources for this blog
- https://www.claymath.org/millennium-problems
- https://www.youtube.com/watch?v=g5C5bwS4p7s&t=76s
- https://www.youtube.com/watch?v=eN0DGaMW3Uk&t=2s
- https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap
- https://www.youtube.com/watch?v=Jqbvat1fhPI
- https://www.youtube.com/watch?v=UNNAoCy3yrk
- https://en.wikipedia.org/wiki/Hodge_conjecture
- https://www.youtube.com/watch?v=R9FKN9MIHlE
- https://www.youtube.com/watch?v=GItmC9lxeco&t=16s
- https://www.youtube.com/watch?v=PwRl5W-whTs
