Pro Answers > Claymath Millennium Problems > The Hodge Conjecture
The Hodge Conjecture
Introduction
Manipulation of abstract mathematical objects has created a class so abstracted that the actual calculations are yet to be fully understood.
Conjecture
The Hodge conjecture uses visualisation to investigate large amounts of mathematical results and associated functions which are studied as discrete objects.
Projection
We project the results of our equations creating a geometric shape or pattern and study the structure of the resulting object.
Blocks
Imagine gluing together geometric building blocks that become greater and greater in size.
Viable
This approach was widely generalised and it eventually led to the creation of a class of objects that were mathematically viable, but were not necessarily geometric as they were originally intended to be.
Top of Page
Interpretation
Some of these objects were so complex when represented by their equations that there was no apparent geometric interpretation.
Include
In some cases it was necessary to include these new objects as parts of an investigation, even though nobody could be certain whether they were able to be interpreted geometrically.
Projective
Hodge stated that for certain types of objects called projective algebraic varieties, there is the possibility that the pieces known as Hodge cycles may actually be (rational linear) projections of geometric pieces known as algebraic cycles.
Cycles
The term cycles refers to Hodge's suggestion that all objects may ultimately be built up from smaller parts being repeatedly projected.
Summary
The Hodge Conjecture suggests that all mathematical objects formed in this way, no matter how abstracted and large in relative size, can be built up from smaller geometric pieces.
The Millennium Problem
The Millennium problem is to fully explain the Hodge Conjecture and to prove whether it is true or false.
For the exact problem description please refer to Claymath.org
The Answer
Conjecture
The Hodge conjecture presents us with two options, either these objects can be created using repeatedly projected Hodge Cycles or they cannot, and asks us to choose one of them.
Shapes
All shapes and objects, whether geometric or not, may form in three ways.
Top of Page
Forms
There are three criteria for all forms of shape or pattern.
Criteria
1. A curve.
2. A line.
3. A combination of both.
Unknown
If any unknown object meets any of these three criteria, it can be built up by using projective algebraic cycles and therefore Hodge would be correct.
Opposite
My point is that the opposite and neutral potentials to Hodge's conjecture are also simultaneously possible.
Infinite
We could theoretically create a shape or object that is infinite in its relative "size" but no matter how far forward we choose to go, we are still ultimately repeating the same three original and simple potentials.
Top of Page
Repeating
1. A curve.
2. A line.
3. A combination of both.
Over
Over and over and over again.
Geometric
All geometric objects may exist as a curve, a line, and a combination of both simultaneously.
Relative
The "details" and "size" of our projections are strictly relative, the size of our objects may be so enormous as to defy accurate and adequate representation by equations but in order to understand what we have created, regardless of size or detail, we need to realise that we are infinitely repeating ourselves.
Can we create these objects using Hodge Cycles?
1. We can create these objects using Hodge Cycles.
2. We cannot create these objects using Hodge Cycles.
3. Hodge Cycles are neutral in relation to the creation of these objects.
Simultaneously.
Am I wrong?
I simultaneously oppose, agree with, and neutralise all criticism ad infinitum.
My point is literal.
There is no point creating a theory of everything that doesn't work.
NEXT > The Navier-Stokes Equations
Interested?
If you're interested in a positive discussion about everything the whole of protheory.com is mirrored on my Theory Of Everything Forum