Revision as of 11:19, 19 July 2010

I'm Jamie Weigandt, I am graduate student in the department of mathematics specializing in Algorithmic Number Theory, Arithmetic Algebraic Geometry, and Arithmetic Statistics.

Note on this page

For the time being I will use LaTeX code freely when editing this page.

Random Thoughts About Rhea as I use it

• Can we add LaTeX functionality with jsmath, at least for the pages relevant to mathematicians?
• Can we add the option to "Open Poor editior in a new window"? The sidebar gets too big when I increase the font size to see in safari.

The Bigfoot Project

As a motivating project for learning a lot of background material I am engaged in what I consider a mythical quest to find an elliptic curve over $\Bbb Q$ with torsion subgroup $Z_2 \times Z_8$ and Mordell-Weil rank at least 4. Such a curve is affectionally referred to by my friends and I as "The Bigfoot." This nomenclature is somewhat misleading, such a curve, should it exist is not by any stretch of the imagination expected to be unique. I hope to expound on the status of this project at a later date.

For now I will be motivated in my development of this page by 3 facts:

• My bank account is suffering from conference fatigue.
• There is an essay contest for which I can win $100.
• I'd like to stop eating at Taco Bell.

That being said I'll get right to this following section:

Why do I "math"?

Note about the Development of this Essay

As this is a wiki page, I will take the liberty to develop my essay freely on this page. That being said, there will initially be a lot of material that is unclear. I won't make sense to anyone but myself, or perhaps not even myself. Hopefully, since a record of these edits will remain, it will provide insight anyone trying to write a similar essay. This seems quite similar to one of those "Statement of Purpose" type questions that anyone wanting to go to grad school will have to write about.

A Remark About the Question

This questions is somewhat general and open to interpretation. This is good, it provides me with some freedom to see where the ideas take me, and then decide my interpretation of the question based on the answer I come up with. This is something that I think mathematicians do a lot. The example that comes to mind is Andrew Wiles proof of Fermat's Last Theorem. Wiles was trying to prove the modularity of all elliptic curves, something that is of great mathematical interest. He was unable to complete this task, but was able to prove that a large enough class of elliptic curves were "modular" that the proof of Fermat's Last Theorem followed from the work of Ribet. The story he told was determined by what he was able to do. So the story I tell will depend on how much I develop this essay.

Focus

I want to muse about 3 fundamentally important parts of the mathematical experience.

• Open Questions
• The Beauty of Mathematical Discovery
• Telling Stories

I think mathematics is highly misunderstood in the general populous, even the college educated populous. Some of my ramblings will (at least temporarily) be in the form of questions and answers, where the question will come from some generic member of the populous, who I may occasionally address as Charlie. Charlie being a ficticious everyman character developed by my friends and I at Purdue Improv Club.

Open Questions

I think that when many people learn mathematics, they get a skewed perspective. They get the impression that everything about mathematics is known, and that a mathematician's job is to pass this perfect knowledge down to the next generation. While I appreciate teaching as an important and fulfilling part of being a mathematician, if this were all there was to it, I would eventually find it really boring, and do something better with my life.

Thankfully this is not the case. We, as mathematicians, have questions to which we don't know the answer by any stretch of the imagination. Fortunately for the exposition, I happen to spend most of my time thinking about number theory, which, roughly, is the branch of mathematics that asks questions about the integers ...,-2,-1,0,1,2,...

In this field, we have the fortune of having many easily understood questions to which the answer is unknown.

"... number theory has an annoying habit: the field produces, without effort, innumerable problems which have a sweet, innocent air about them, tempting flowers; and yet ... the quests for the solutions to these problems have been known to lead to the creation (from nothing) of theories which spread their light on all mathematics, have been known to goad mathematicians on to achieve major unifications of their science, have been known to entail painful exertion in other branches of mathematics to make those branches serviceable. Number theory swarms with bugs, waiting to bit the tempted flower-lovers who, once bitten, are inspired to excesses of effort!" - Barry Mazur, Number Theory as Gadfly

What are some of these "tempting flowers"?

• Can every even number greater than 2 be expressed as the sum of two prime numbers?
• Are there infinitely many pairs of prime numbers $ p $ and $ q $ such that $ q = p + 2 $?

It is believed that the answer to both of these questions is: Yes! These assertions are known as Goldbach's conjecture and the twin primes conjecture respectively.