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22. Analyze "outstanding issues" of your science.

1) equality of P and NP classes: In the theory, the class P consists of all those decision problems that can be solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input; Of all those decision problems of a positive solutions can be verified in polynomial time given the right information, or equivalently, whose solution can be found in polynomial time on a non-deterministic machine. The biggest open question in theoretical computer science concerns the relationship between those Two classes:

Is P = NP?

Most people think that the answer is probably "no"; some people believe the question may be undecidable from the currently accepted axioms. A $ 1,000,000 prize has been offered for a correct solution.

    2) In mathematics, the Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture said that certain de Rham cohomology Classes are algebraic, that are are sums of Poincaré duals of the homology classes of subvarieties. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology To include extra structure that is present in the case of complex algebraic varieties.

3) In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers andcomplex numbers with real part 1/2. It was proposed by Bernhard Riemann (1859), after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

4) Yang-Mills theory is a gauge theory based on the SU (N) group, or more generally any compact, semi-simple Lie group. Yang-Mills theory seeks to describe the behavior of elementary particles using these non- And is at the core of the unification of the electromagnetic and weak forces (ie U (1) × SU (2)) as well as quantum chromodynamics, the theory of the strong force (based on SU (3)). Thus it forms The basis of our understanding of the Standard Model of particle physics.

5) The Navier-Stokes harmony and smoothness problem concerns the mathematical properties of solutions to the Navier-Stokes equations, one of the pillars of fluid mechanics (such as with turbulence). These equations describe the motion of a fluid (that is, a Liquid or a gas) in space. Solutions to the Navier-Stokes equations are used in many practical applications.

6) In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to idealized an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. The Conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, which had offered a $ 1,000,000 prize for the first correct proof

7) In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. As of 2016, the Poincaré conjecture is the only solvedMillennium problem.