leios
'Time'& 'Symbols' meet 'eigen-atlas' - (I)ntroduction and (M)otivation - draft
For introductions check my previous issue:
https://github.com/leios/SoME_Topics/issues/236
This is my best try with my available time to create a better explanation of the motivation of previous posts
Summary with some raw-formulas explaining the motivation of the previously published videos:
https://github.com/fbunc/eigen_atlas_raw/blob/master/000_README_EigenAtlas.ipynb
A video showing the important stuff with cool music to take notes:
https://www.youtube.com/watch?v=UaLlIp4doLg
Related links:
MS QC intro:
https://www.youtube.com/watch?v=F_Riqjdh2oM
Simon Cowel- (text from Quora) about Hodge theaters:
Hodge theater is a mathematical model. Now to understand a glimpse of it, you need to start what modeling in math really is.
So start from the set of numbers 0,1,2,3,4,5,6. What can you say about it? First of all the object in scope is irrelevant, it could be number of cows, planets, solution of the equation. Yet they all behave in the same manner. There is element 0 which if added to any other element changes nothing, there is element 1 that allows you to walk around elements in order. You can add element to element. With that you formed a notion of group. As much as number is not connected to an object it describes, group is not connected to a set or operation within that set. You can define instead of 0 or 1, for example empty set and set with one element. If your operation is union of sets with these three you have something that behaves the same way as 0,1,2,3,4,5…
What mathematicians have realized is that these meta-object can be extended probably to infinity, we can always group things ignoring what individual representation they have, call them something and move on.
Now, soon they realized that geometry, even though it appears as a completely different mathematical notion has some properties that reflect those that discrete sets have. Take a simple equation x^3=y^2. You can ask if it has integer, real, complex solutions, or you can define some completely different operations that are replacement for multiplication and ask if there is a solution with some again different set and that definition of multiplication. And it turned out that all these things are connected. Integer solutions depend on real and this one depends on complex and so on. In some sense they are all just a projection of one and the same thing. But if you start from our basic understanding x^3=y^2 is a strangely looking curve, for example you can find the first derivative at each point. You cannot find first derivative if you look integer solutions, right? It is too jumpy.
It took some time for mathematicians to translate all our common analytical tools like integration and differentiation into this world of combined curves and numbers. And they got another level of understanding things. From that perspective so many of these shared the same properties. Now only to get to this level you need to understand around 300–500 definitions, without completely being sure at first why they are chosen to be like that. Notice that even if it would take you only a day to understand each, you would need a year to be able to walk around the vocabulary. And this is about 10–20 years away from Mochizuki at the moment. Just to give you a glimpse of how many light years this is from the current math. 10–20 years is not that much in scientific sense, but at the moment it is that far away.
Mochizuki kept on. His objects are now generalizations of another high level group of objects. Now, it is still all about addition and multiplication of a specific kind and it is still about geometry - how curves behave and arithmetic - how number behaves. Notice that in all these modeling we always lose something. For example we can decide to compare equation not based on their solutions, rather based on the number of solutions. Sometimes it happens that these more universal objects can be transformed into each other as long as they look the same by that new criteria, with some maybe not that important details lost.
What his theory wants is to find a reconstruction system. A trivial example, if you know the solution of an equations are (1,2),(3,4),(5,6)… what equation is this? Only his reconstruction is about symmetries. If any object has a symmetry of rotation of 120 degrees, which objects we might be talking about? Well equilateral triangle, right? All we have is this symmetry yet we are able to talk about a concrete group of object.
So, it was some task to group all these properties and make them universal regarding these possible symmetries. Notice that although it is still a version of addition and multiplication, their generalization in this case is light years away from our normal understanding of these operations, yet one is still arithmetic and another geometric. So, first of all they are split, separated and observed through two different lenses.
And now Hodge theatre is this object that is so general that it encompasses a good portion of math as we know it. Math that we know you can call Hodge theatre H0,0. Yes, that is how far it is. If you change additive property, it will change multiplication but then you want to know how much is addition change because of this. So essentially you walk around these new math objects H0,1 H0,2 H1,0 H2,0 where you have defined what steps and what deformation means to jump from H0,0 to H0,1 and so on. And now within this lattice structure you ask what happens if I change multiplication, how far addition will be stretched.
So, yes, it is really strange that this much of filigree work was necessary to cover up one statement like abc conjecture. Is it really possible that we had to create an object that encompasses math to that extent (where it turns out that our math is just one of the possible historical paths we took)? It is not that strange, we have done that before. However, it is strange that multiplication and addition are so natural to our understanding yet in theory they are that far apart that you have to exit all what we know, and even could know if you stay in 0,0 forever, and create object that are universal not to the math we know, but to many other maths that exist.
So Hodge theatre is a mathematical Universe.
This is what is putting off way too many mathematicians. It is difficult to climb that ladder all along having nothing but trust that Mochizuki knew what he was doing. And once you are there, suddenly it is just you on the Mount Everest and everybody else not even thinking to start.
Quite discouraging indeed.
