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# Top 7 Unsolved Million Dollar Problems

The Millennium Prize Problems are seven of the most well-known and important unsolved problems in mathematics. The Clay Mathematics Institute, a private nonprofit foundation devoted to mathematical research, famously challenged the mathematical community in 2000 to solve these seven problems, and established a US $1,000,000 reward for the solvers of each. One of the seven problems has been solved, and the other six are the subject of a great deal of current research. Let's take a look. ** **

**Poincare Conjecture:** The
only millennium problem that has been solved to date is Poincare Conjecture.
This problem was posed in 1904 about the topology of objects called manifolds. The
most basic problem in topology is to determine when two topological spaces are
the same that is they can be identified with one another in a continuous way. The
conjecture is essentially the first conjecture ever made in topology, it
asserts that a three dimensional manifolds is the same as the 3D sphere
precisely when a certain algebraic condition is satisfied. The conjecture was
formulated by Poincare around the turn of 20^{th} century.

**P vs. NP:** The
problem of determining whether P equal to NP is the most important open problem
in theoretical computer science. The question asks whether computation on
problems whose solutions can be verified quickly can also be solved quickly. The
consensus of most experts in the field is that this is not true in general that
is P is not equal to NP but there is very little progress to other proof.

**Hodge Conjecture:** Hodge
Conjecture is a statement about geometric shapes cut out by polynomial equation
over the complex numbers; these are called complex algebraic varieties. An extremely
useful tool in the study of these varieties was the construction of groups
called Co Homology Groups which contains the information about the structure of
varieties. The Hodge Conjecture states that every Hodge class on a projective
complex manifold is algebraic.

**Riemann Hypothesis:** The
Riemann Hypothesis is perhaps the most famous unsolved problems in mathematics.
It concerns the non-trivial zeros of the Riemann Zeta Function. The generalized
Riemann Hypothesis is a statement about the zeros of certain functions known as
L functions defined by Barclay series which are
the generalization on Riemann Zeta Functions.

**Yang Mills Existence & Mass
Gap:** Yang Mill theory in quantum
physics is a generalization of Maxwell’s work on electromagnetic forces to the
strong and weak nuclear forces. It is so key ingredient in the so called
standard model of particle physics. The Standard Model provides a framework for
explaining electromagnetic and nuclear forces classifying subatomic particles. It
has so far proved to be consistent with experimental evidence but questions
remain about its internal consistency.

**Navier Stokes Existence &
Smoothness:** The Navier-Stokes equations are
partial differential equations modeling the motion of liquids or gases. The fluid
is acted on by forces including pressure, viscous stress and a specified
external force. The Navier-Stokes are the results of writing down Newton’s
second law for the fluid with respect to these forces in terms of partial
derivatives of velocity of the fluid as the function of position and time.

**Birch-Swinnertoner conjecture:** In mathematics,
the Birch and Swinnerton-Dyer conjecture describes the set of rational
solutions to equations defining 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 has
offered a $1,000,000 prize for the first correct proof. It is named after mathematicians Bryan Birch
and Peter Swinnerton-Dyer, who developed the
conjecture during the first half of the 1960s with the help of machine
computation. As of 2021, only special cases of the conjecture have been proven.
Mordell
(1922) proved Mordell's theorem: the group of rational
points on an elliptic curve has a finite basis. This means that for any elliptic
curve there is a finite subset of the rational points on the curve, from which
all further rational points may be generated. If the number of rational points
on a curve is infinite then some point in a finite basis must
have infinite order. The number of independent basis points with infinite order
is called the rank of the curve, and is an important invariant property of an elliptic curve. If
the rank of an elliptic curve is 0, then the curve has only a finite number of
rational points. On the other hand, if the rank of the curve is greater than 0,
then the curve has an infinite number of rational points. Although Mordell's
theorem shows that the rank of an elliptic curve is always finite, it does not
give an effective method for calculating the rank of every curve. The rank of
certain elliptic curves can be calculated using numerical methods but (in the
current state of knowledge) it is unknown if these methods handle all curves. An
L-function L(E, s) can be defined for an elliptic curve E by constructing
an Euler product
from the number of points on the curve modulo each prime
p. This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary
quadratic
form. It is a special case of a Hasse–Weil L-function.

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