Wednesday, September 26, 2012

September 26, 2012: The Riemann Adventure

Ever since I was introduced to Ulam's Spiral in May, the Riemann Hypothesis, a problem that fascinated me (but which I had decided was too hard to bother with), has had me in its grip.  Perhaps it was the confidence I gained after working on the congruent number problem with another professor that made me realize if I really wanted to, undergratuate or not, I can learn what I need to and have some fun.  So that's what I've been doing.

My work on the Riemann Hypothesis has come in many growth spurts.  Each time, it's felt like I am getting closer and closer.  The original exploration came from a unique choice I made in how to represent the sum, that later an attempt to understand it geometrically (in images that could be explained to a classroom) led to another interesting connection.  At the beginning of this month, I had a conjecture, based on some of the data I had gathered, but it wasn't solid enough.

The term has been very busy, so both the RH and my other explorations have only come out now and then - usually when I have an idea that I simply cannot ignore.  Last night - or should I say this morning - was one of those times.

Prior to this, I had been exploring some of the properties or composite expansions that naturally cropped up in the unique way I had written the zeta function.  I noticed that there were certain types of series that could be written to span all the numbers - much like the proof of Euler's infinite product series.

It was 1:30am, and the idea came to me like a dream.  For a while I did math in my sleep, then I realized I was awake, and the moment was now.  Long story short, by 6am I had an outline for how to turn that idea into a proof, and have been exploring it on and off today, in between study sessions for my courses.  I had a chance to run my reasoning by a professor who was familiar with series analysis, and he encouraged me to follow up on it.

This is very exciting!  However, I can't be sure that the correct series can be found easily (if at all).  I have some rigorous (and careful) expansions to do, some trial and modification, and perhaps some more surprises from intuition, but regardless of where this goes, there's no doubt it's going to be fun. 

And, I must add, it's keeping my trigonometry and algebra skills sharp!

Tuesday, September 18, 2012

September 18, 2012: In the Number Lab

Research into the congruent number problem has taken on a fun spin after being given incentive to present my findings in a seminar this spring.

I'm starting to develop a sense of how to organize and conduct my research, and this has led to the construction of a "number lab" of sorts.  As a student, I have found that organization is my saving grace.  By nature, I work chaotically, so I found a way to label my pages, tabbing them for quick access; that allows me to move between problems and go with new ideas without losing myself. 

I've applied this to my research.  Work on the Riemann Hypothesis has its own place, and work on the congruent numbers has another, and as I am conducting an investigation over fields, I've tried to organize everything into tables.  So far, I've found and proven three lemmas en route to looking for some sort of theory to explain the pattern of solutions that might be extrapolated to the whole field of integers.  It's very much like being a scientist in a lab.  When I got to the field of size 17, I took the time to write a computer program to generate larger fields (and verify calculations in others).  This was very helpful, because I was able to quickly work on the next field, over 19, which showed there was an exception to a fourth lemma that I had struggled to prove (and now I know why, and am working on proving it prove another angle).

This is wonderful fun!  I'm studying four math topics, and as I learn, not only am I enjoying the new challenges and techniques I assimilate, but the opportunity to tackle my hobby research from new angles.  The best part is, I can bring my lab with me anywhere, and I don't have to worry about anything blowing up...

Sunday, September 9, 2012

September 9, 2012: The Second Year Regimen

The study term has started and I simply cannot contain the excitement.  This second year is chock-full of some important building blocks that I hope will help me with my independent work on both the Riemann hypothesis and the congruent number problem.  I have four courses, but it is the third year course, introduction to topology, which interests me the most.  I am hopeful that some of the concepts of metric spaces will assist in the geometric proof I have been trying to put together since May.

Meanwhile, I began an investigation into modularity patterns for the elliptic curves associated with the congruent numbers, and have thus far found and proven three lemmas that have half-formulated a theorem about the relationship between cubic and quadratic residues.  I have analysed curves for all n up to the field formed by 13 elements, and hope to find some useful generalization that extends between fields as they increase (thus approximating the integers), with the hypothesis that there will be some pattern emerging for values of n that correspond to congruent numbers.

It is all a little bit of undergraduate fun, which I enjoy.  Not only do I get to learn lots of neat and useful things, but my innate curiosity for problem-solving is satisfied.  There's nothing like taking abstract concepts and finding a practical use for them.  After all, that's what mathematicians do.