I am not specialized in mathematics but I love maths as it is my favorite subject
I think that I can solve Millennium Problems so I want someone to tell me how to learn mathematics and be perfect in it by telling me about recommended books, websites and so on
This is a very good question.
A good place to start in answering this question is
J. Carlson, A. Jaffe, and A. Wiles, Eds., The Millennium Prize Problems:
http://www.claymath.org/library/monographs/MPPc.pdf
This book gives a complete overview of famous problems in mathematics.
Birch and Swinnerton-Dyer Conjecture: The Taylor expansion of an L-series at s = 1 has a particular form (p. 32).
Hodge Conjecture: ... any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles (p. 46).
Navier–Stokes Equation: The Euler and Navier–Stokes equations describe the motion of a fluid in R^n (n = 2 or 3). A fundamental problem in analysis is to decide whether such smooth, physically reasonable solutions exist for the Navier–Stokes equations (p. 58). Four such problems are given.
Poincaré Conjecture: If a compact three-dimensional manifold M^3 has the property
that every simple closed curve within the manifold can be deformed continuously
to a point, does it follow that M^3 is homeomorphic to the sphere S^3? This problem (from 1904) has captured the imagination of many mathematicians, beginning with Poincaré himself, who stated a false theorem (p. 72):
False Theorem. Every compact polyhedral manifold with the homology of an n-dimensional sphere is actually homeomorphic to the n-dimensional sphere.
But Poincaré's 1904 paper provided a beautiful counterexample to this claim,
based on the concept of fundamental group. Stephen Smale announced a proof of the Poincaré's Conjecture in high dimensions in 1960 (p. 74).
The Thurston Geometrization Conjecture: This conjecture asserts that every compact orientable three-dimensional manifold can be cut up along 2-spheres and tori so as to decompose into essentially unique pieces, each of which has a simple geometrical structure (p. 80).
P versus NP Problem: The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time (p. 88).
Riemann Hypothesis: The nontrivial zeros of ζ(s) have real part equal to 1/2 (p. 109).
Quantum Yang–Mills Theory: Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R^4 and has a mass gap ∆ > 0 (p. 142).
In this book, Poincaré's conjecture and the Yang-Mills non-trivial quantum theory problem receive the most attention.
Dear Peters, thanks for the good info. However, I think Grisha Perelman solved the Poincaré conjecture. Maybe it would be a good warm up for the poser to study the solution?
Dear Patrick Solé,
Yes, many thanks for pointing this out. To summarize, here is
Poincaré conjecture (1904). Every smooth, compact, simply connected three-dimensional manifold is homeomorphic (or diffeomorphic) to a three-dimensional sphere S^3.
Here is an overview of G. Perelman's proof:
Terence Tao, Univ. of California,Perelman’s proof of the Poincaré conjecture:
https://terrytao.files.wordpress.com/2009/09/poincare1.pdf
For a great introduction to Perelman's work and proof the Poincaré conjecture, see
Graham P. Collins, The Shapes of Space:
http://www2.ic.uff.br/~aconci/poincare.pdf
Thanks to Perelman's work, all possible 3D manifolds are classified.
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