Project Updates

Hello all,

Given some of the comments to my last post and a meeting I had today with my ASU adviser Dr Paupert, I thought I should take this time to explain some further nuances and updates to my project.

My project will cover and focus on classifying Mobius transformations by symmetry. What this means is looking at what properties are kept constant under the group action of Mobius transformations. For example, to demonstrate symmetry around the Unit Circle, one must find a transformation that maps everything in the interval [0,1] to another element in [0,1], so though all the other points on the sphere may be dislocated, this circle is preserved.

This is a picture I took from a Rutgers lecture, if the shaded area were to be preserved, then when the mobius transformation acts on the Riemannian Sphere, it stays put

However, though I intend on classifying these Mobius transformations, I also intend on using other resources such as Indra's Pearls: The Vision of Felix Klein to examine other objects, such as fractals (which are based in symmetry, free groups, Mobius maps, and Kleinian groups). Hopefully, I become adept at using math programming resources like Maple to visually compute examples (fortunately there are resources out there to aid me).

This is great for you! It means you get to see pretty pictures, and lots of them. To demonstrate, please take a look at some of these beautiful pictures taken from Indra's Pearls (found by googling "Indra's Pearls" in google images):



The University of Arkansas describes further these fractals, and cites three more beautiful pictures, which I think would make a good picture to have at the top of my blog.

"Kleinian Group fractals are fractals based on 2 pairs of Mobius transformations and allow you to produce Quasifuchsian, Single Cusp, and Double Cusp, Two-Generator Group fractals described in the book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright"




Please comment if you have any questions, and I'd be happy to go further in detail about this as much as I can.

Best,

Connor

Introduction to the Research Project

Hello world!

I would like to introduce you to both my math blog and myself. Probably the least interesting between the two, I am Connor Davis, senior at BASIS Scottsdale, guitarist, baseball fan (go Dbacks!), debater, Redditor, aspiring mathematician, lover of puns and silly jokes, and, if nothing else, a man who doesn't know how to end his lists succinctly.

Officially starting in February, this blog will chronicle my math research at ASU with Dr. Julien Paupert in group theory, complex analysis, and my attempts to classify Mobius Transformations by symmetry. Additionally, I will work very closely with the amazing Mrs. Bailey of our very own BASIS Scottsdale in researching various concepts. Specifically, I will spend much time in ASU's libraries, computing examples trying to develop conjectures so I can subsequently try to prove them.

Already, I have been working with Mrs. Bailey for a few weeks, and I have discovered some relevant mathematical properties, for instance, that the Klein-4 group (D4 intersect A4) preserves the cross ratio for 4 element transpositions. Currently, I am trying to determine when, where, and how lines and circles in the complex plane are mapped to lines and circles on the Riemannian Sphere.

A link to my SRP proposal can be found by clicking on that link, but it is also on my side bar.

Finally, I was bored this last week and wrote a sonnet inspired by abstract math. Maybe I shouldn't have chosen math, and should have considered giving Shakespeare a run for his money...

Had I not searched for you like proofs untold,

Weren’t it one-to-one, where you’re for me,

Did you not make me feel like the manifold,

Then I’d be torsor and have no 'dentity.

My love for you is of an indeterminate size

And our love’s fundamental, set of closed loops,

As conjugating in C, you made me realize

That coming together we form abelian group.

I'll follow you always, the if to my then,

And h divide y, we both saw the sine

That's why I’ve for you a ring like Z mod n

Because we go together like Borel and Heine
They say that math’s a hunt for basic truth
QED, that’s why I love you.

See you in February,

Connor