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Since forever, mathematicians are interested in a question to describe all x, y, z whole number solutions to algebraic equations such as x which Euclid managed to solve. In particular, we don’t have any general methods to determine when such equations have a whole-number solution. When the solutions come out as the points of an abelian variety, it is asserted by the Birch and Swinnerton-Dyer conjecture that the size of a rational point group corresponds to the behavior of an associated zeta function ζ(s) near where s=1.But it’s extremely difficult to solve more complex equations. In explanation, according to this conjecture, if ζ(1)=0, then there are infinitely rational points (aka solutions), otherwise, if ζ(1) does not equal 0, there is a finite number of solutions.
This technique was so useful that it was generalized in various ways, and finally gave mathematicians powerful tools enabling them to further catalog the variety of objects that they encountered during their investigations.
However, in this generalization, the geometric origins of this procedure got obscured.
While the Poincaré Conjecture and Riemann Hypothesis were solved, the others still remain challenging. NP, in the world of theoretical computer science, is like a mythical unicorn, which has become infamous for being unsolved.
Essentially, the questions to answer here is: If it is easy to check that a solution to a problem is correct, is it also easy to solve that problem?
About 50 years ago, Yang and Mills unveiled a new framework where they use geometric structures to describe elementary particles.
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Today, the theory of Quantum Yang-Mills is the foundation for most elementary particle theory.It is obvious that P problems are NP problems, which means if the computer can solve a problem, it can check that too.If we can prove that P equals NP, it would mean difficult problems in the real world are easy for computers to solve.The list included Birch and Swinner-Dyer Conjecture, Poincaré Conjecture, Hoghe Conjecture, Navier-Stokes Equation, Yang-Mills and Mass Gap, Riemann Hypothesis, and P vs.NP Problem, representing the deepest mysteries in mathematics.The watermelons need packing into boxes, each of which can hold 20 kilograms at most.She wants to know if 10 boxes are enough to carry all the melons to the market.It is hosted by the Universities of the Witwatersrand and Western Cape, the African Population and Health Research Centre and the Nigerian Academy of Science.The Bill & Melinda Gates Foundation is a Strategic Partner.Sometimes, it was critical to add pieces without geometric interpretation.It was claimed by the Hodge Conjecture that in projective algebraic varieties, Hodge cycles are, in fact, are combinations of algebraic cycles. Matiyasevich, in 1970, indicated that it’s impossible to solve Hilbert’s tenth problem.