Hello, everybody.

Are you the person who wants a scenic background rain shower in the backyard of your dream house? Or a four sided Lego wall for full potential for creativity? Or a Batman Tumbler just to beat the rush hour traffic?

If no, then you are not that good with rhetorical questions.

Anyway, obviously all that would be possible if *sighs* if you had a million dollars.Right?.

Well. We just happen to know a way to do that. Yup, you read it right, and we promise this way does not include clicking on ads and taking surveys on social networking sites.

So, cutting to the chase.

All you gotta do is solve any of these seven problems. Oh and I shit you not, these problem do have million dollar reward over them. Solve any one or maybe you are a wee bit greedy and/or bored solve more.

- P vs NP problem
- Riemann Hypothesis.
- Poincare Conjecture.
- Navier-Stokes Equation.
- Hodge Conjecture.
- Birch and Swinnerton-Dyer Conjecture.
- Yang–Mills and Mass Gap.

Let’s begin by learning what each problem is basically. Cool?

**P vs NP problem**

I’ll break it down for ya all. It asks “If a solution is easy to check, is it easy to solve? And vice versa“ During the Retro age, computer scientists were finding out faster alternative solutions to various mathematical problems ,well they did succeed somewhat in doing so, but there were some problems like the Travelling Salesman Problem, they just couldn’t find a better albeit faster way to solve such problems. They divided all problems into two teams: P and NP. P problems were the ones that could be solved in reasonable amount of time whereas NP are the problems which can be checked in reasonable amount of time. The whole P vs NP issue is that whether a problem that can be checked (or comprehended or whose solutions can be verified) in reasonable time can be also solved in reasonable time?

*Mind-Blown alert* If this question is answered it will also imply that if one can understand a symphony, he’ll be equivalent to Mozart. (If he can comprehend; he can create).

**Riemann Hypothesis**

This involves Mathematics. (Alert)

Prime numbers also have the annoying habit of not following any pattern. 3,137 is a prime and the next one after that is not until 3,163, but then 3,167 and 3,169 suddenly appear in quick succession, followed by another gap until 3,187. If you find one prime number, there is no way to tell where the next one is without checking all the numbers as you go.

One possible way to get a handle on how primes are spaced is to calculate, for any number, how many primes there are smaller than it. This is exactly what Riemann did in 1859: he found a formula that would calculate how many primes there are below any given threshold.

So, Riemann had this famous function known as the zeta function.

What this means is zeta function gives you a sum of infinite series.

Suppose you enter s = 2. What zeta function will return is and so on till infinity and beyond….(Beyoncé alert!) actually till infinity only.

People later realised that they can put anything in the zeta function, any goddamn number except 1 and zeta function will return a unique number. This is where it gets interesting by any number I mean ANY NUMBER even complex numbers.

The fun part is at what values will the zeta function return a 0? Eh?

And the Riemann Hypothesis states that all complex numbers whose real part is 0.5 will make the zeta function’s value 0. This hypothesis actually solves the problem of how many primes number are distributed between some limit, which is a topic from a completely different branch of mathematics. So, math nerds …get going.

**Poincare Conjecture**.

This belongs to a branch called ‘Topology’ which is basically the stream that studies shapes and bending of shapes. In topology, you cannot bend a shape with one hole into a shape with no hole. A coffee cup and a doughnut are transformable into one other because both have one hole hence the joke “Topologists can’t tell the difference between their coffee cup and their doughnut. (N.B I agree it is not funny. What can we do?)

The conjecture says that any shape that is closed and has no holes can be transformed into a sphere. (….wait for it) and this is valid for any higher dimensions too. Easier said than done.

[Sorry to say but a Russian already proved this one aaaaaaaaand he refused the money *like a boss*. It took over 10 days for a group of American mathematicians to actually understand his proof]

**Navier-Stokes Equation.**

Navier-Stokes Equations are like the pillars of fluid mechanics. A large amount of important works and results in physics as well as engineering are based on Navier-Stokes Equation. It governs the motion of a fluid (liquid or gas). So far so good? Okay, so, where’s the million dollar catch?

Here it is. Turbulence. It is a time dependent chaotic behaviour of fluids which cannot be accounted in the Navier-Stokes equation. The solution of N-S equations of turbulent flow are very difficult or sometimes unstable.

All you gotta do is successfully explain Turbulence in Navier-Stokes equation, and Ka-ching!!

**Hodge Conjecture.**

In order to get a grip on Hodge’s Conjecture-and claim your million dollars, you should be good at thinking about doughnuts. I promise I’ll stop talking on this the moment it gets too complex.[There is a picture of doughnuts for you,don’t worry]

Ok, here goes. Earlier, there were two groups of people studying two different things, Algebra and Geometry. Algebraists dealt with equations and Geometrists dealt with shapes.

Later, both the parties realised that what if both subjects are the same and can be connected *shockingly gasps* This is why your high school teacher taught you y=mx+c (algebra) is actually line (geometry). So, they teamed up to solve greater problems, save the world, etc. Sometimes, algebra would fail to explain something then geometry would do it for algebra and obviously, vice versa.

*awwww, I so ship them*

Mathematicians didn’t stop at lines and circles they moved on to complicated entities called “shapes” which could be explained using even higher algebraic entities known as “algebraic cycles” If an algebraic cycle was a smooth and nice shape then it was called “manifold”.

Now, they went on looking what happens if you draw “shapes” on “manifold”. Imagine a chocolate triangle drawn over a nice smooth glistening doughnut and another doughnut with a pentagon instead of a triangle. Are they the same? Yes.

The algebraists did something different. They kept on adding new equations to their existing algebraic cycle to produce a new algebraic cycle over existing manifold.

Soon, people realised that what both the guys did were actually the same thing. The difficulty was nobody had any idea how to prove that if you could draw a shape –possibly nasty- on a manifold, then you could stretch it or bend it into a less complicated shape that can be explained by an algebraic cycle.

William Hodge had a great idea, but could never prove it. If you can prove it then prize is yours. Hodge’s conjecture actually uses complex co-ordinates and complex spatial dimensions, and yes as I had promised, this is where I stop.

**Birch Swinnerton-Dyer conjecture.**

Once upon a time, there was a man named Diophantus. He had a spark for algebra and because of his works in mathematics, he is now known as the Father of Algebra. He had put forth certain problems, known as Diophantine Equations or problems, some of them were solved in the coming ages by heroes such as Fermat and some other mathematicians. The unsolved ones were passed onto generations of future mathematicians.

Precisely, three problems included something known as the elliptical curve which looks like this.

Now if you know what an ellipse looks like you would probably go “Wtf dude, Ain’t no ellipse in here!”

You are right elliptical curves have nothing to do with ellipses, then why the name? Eh, because they are great deal in elliptical integration which is not our concern right now. So about these elliptical curves which have the equation,

What’s the big fuss?

Elliptic curve E. The problem is to find all the solutions (x, y) which satisfy the equation from which our E is made. If we call this set of points a family then we are asking to find a way to obtain all families or rather number of families. In mathematics this number of families of solutions is known as rank of an elliptical curve. Rank (E).

For that we have the L-function who is a close cousin of the Riemann zeta function we saw earlier. But difference is L-function gives us the points on the elliptical curve. Isn’t that what we want? Eh? Happy?

What Birch and Swinnerton-Dyer said was very simple. An elliptical curve has infinitely many solutions if and only if L-function’s value is 0. This has never been proved but has been used in various breakthroughs of mathematics. There. All yours, folks.

**Yang-Mills existence and Mass gap.**

The fun part of having a gigantic pile of unexplainable stuff is that there is equally large scope for theories to explain it. Choose a topic that you want to explain, start from scratch, make a hypothesis, test it with experiments and VOILA! You have your own theory. And that is what people have been trying to do since centuries about the phenomena of the Universe like dark matter, dark energy, neutrino oscillations or quantum gravity.

Yang-Mills theory is similar to those theories. So, why is it the elephant in the room? Because it claims to explain everything except gravitation. Quantum Electrodynamics (which is a specialised study of how light and matter interact), Strong and Weak Forces, The Standard Model in particle physics (of what everything is made up).

Yang-Mills theory explains everything. Not going into the details, I’ll just say Yang-Mills theory unifies all previous theories and shows that there exist a special type of symmetry (gauge symmetry) between everything.

One theory to rule them all. One theory to explain them all. One theory and I am the Master of the Universe!

Alas, this theory is yet to be proved. The experiments are being carried out, you know, the ones where two nasty particles go bangity bang against each other in that ginormous, underground tunnel in Geneva.

** **

***Phew***

So that’s it, we are done. Seven ways that can make you a millionaire. (Well, technically, six).

If anybody out there reading this actually solves one of these and filles his/her pockets with loads of money, please, please don’t forget us. We would be eternally grateful.

That’s all folks. Stay classy and thank you for stopping by.

-Ajinkya Gawali

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**Email**: captainknowledge1@gmail.com

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