The Hodge Conjecture

On May 24 2000, Clay Mathematics Institute released the millennium prize problems. Seven of the most hardest problems in Science, Technology, Engineering, and Mathematics(S.T.E.M.).

In this post, we are going to discuss or perhaps simplify one of those problems, 'The Hodge Conjecture'.

If you have a string or a rope, you have to make sure that it does not have any sharp edges. This is a one liner simplification of the problem.

https://i.stack.imgur.com/iEob7.png

To understand this in a little more detail, we need to look at 2 very vast fields of Maths.

1.  Algebra: Algebra in short is the study of unknown quantities and how they change over a period of time. An algebraic graph is simply the collection all the points that satisfy those equations.
2. Topology: If you have a rubber sheet which can be stretched as far as you want, topology is comparing shapes drawn on those rubber sheets and trying to understand the relations between the dimensions of this universe. 

So earlier when I was talking about making sharp edges smooth, I was talking about the graphs of polynomial equations. And it has a topological mix to it. If you have a hole, in a hole, in a hole and this goes on infinitely, is there a way to look through all those holes all at once, just like we can look at two holes of an open pipe or telescope?

So that's the 'Hodge Conjecture' in it's most layman language. Of course, I did not cover a lot of it's math and left a a lot of it's details untouched.

To put in simple words, this was just an introduction to anyone interested in it, as I think math is not talked often enough on the internet, and that most people who had the potential for contributing in math end up not liking it at all.

 Mathematics is not just about pure logic. It's not just intuition. It's not a subject of study to understand the mysteries of this hypothetical universe. It's about understanding the universe withing us. It's about playing. It's about making anything possible with spaces, from which one tries to draw logic and other watches and observes, so whom will you call a mathematician?

External Links and References:

Millennium Problems, Topology & physisics | Institute of physics publishing, GRAPH COMPLEMENTS OF CIRCULAR GRAPHS, TOPOLOGY OF THE TROPICAL MODULI SPACES, Hodge-Conjecture official problem-statement, Aleph0 on the Hodge-Conjecture