Thursday, December 6, 2012

Building the Foundations

Here I am at the end of an intense study term, and I'm diving under for a big feat: the hardest exam I've ever written, for the hardest course I've ever taken. Writing this post - long overdue - was one of my decided break activities in my planned 12-hour work day. (Day one of 5 :S)

Studying mathematics rigorously has really taught me a lot about myself, mostly about how I think (or fail to). If not for a deep love and fascination for the topic, I would have jumped ship long ago. I'm forever humbled by (and envious of) the prodigies whose natural aptitude lies in mathematical reasoning. Here comes me the writer trying to piece together the story that the beautiful logic makes, and it's quite a mess.

I've realized amidst this, though, that I am a writer at heart. As November passed and I revived my story-telling habits, I realized that, had I to chose one thing to give my life to, it would be writing.

Meanwhile, though, there's so much to learn in mathematics, and by that I don't mean theorems and proofs and calculation techniques. Learning how to organize my thoughts has been an important sort of "mental yoga", an often unpleasant experience of wrestling down faulty logic so that what is true and sound comes out on top. The best part of this: it's transfered to my writing, my daily life, and through it I feel I am growing as a person.

Alright. Break's over. The nice thing is when this is done - December 10th - I will be launching full-swing on the novel I've started. Boy, I can't wait. I aim to finish the manuscript (or at least get to 50,000 words) by January. But I must confess, after I reach my 5000 word goal for each day, I'll be doing some math problems. After all, I might be a writer at heart, but my love for math will never be far away. In fact, on this road I'm coming to know as my life, they make good companions.

Saturday, October 20, 2012

Formula for Success

Einstein once said, "If A equals success, then the formula is A = X + Y + Z.  X is work.  Y is play. Z is keep your mouth shut."

This more or less reflects the advice I was given when I met to consult on the work I've been doing on mathematical topics that are far over my head.  It is good to be enthusiastic about some of the difficult problems, but so many people have been working on them, many who are very gifted and resourceful, and they still remain unsolved.  While it is true that I have come up with a unique approach, the formality of checking to see if it leads anywhere is far beyond my present ability.

What is within my reach, though, is a wonderful opportunity I was given last week.  The professor I work with supported me to come and research full-time next summer on the congruent number problem and related topics.  I presented my results to him and he suggested that we look into getting a grant.  It was a straightforward process and within the day we were putting the application in.  This grant is not as competitive as NSERC, and as I qualified for NSERC, we expect there is a very good chance I will get it.  I have my fingers crossed!

Meanwhile, all my energy is on my studies.  This is a formal second year, with all of the honours math topics, and it is teaching me that my talent is mediocre.  Real analysis is a lot of work, particularly because it forces one to part with lazy thinking habits.  There is no hand-waving.  Mathematics is a language of exactness and precision.  Sometimes this leads to a headache, but I suppose its no different than the cramps I remember feelings when I trained to run in the marathon.

Needless to say, I'm working very hard, and making sure to balance my week out with a bit of exercise, some fun on the weekend, and of course, writing.  Please be sure to check out my Twitter feed, as I tweet on a mathematician every day.

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.

Wednesday, August 29, 2012

August 29, 2012: Onward to Modularity

After a month of independent work, I had a meeting with the professor who has offered to help me with my research.  This morning I got on the wrong bus, so ended up taking an hour to get to the university.  It meant I arrived right on time.  It also meant I managed to get through the rest of Chahal and Osserman's paper about the Riemann Hypothesis for Elliptic curves.

Because I am an undergraduate with a deep curiosity for particular topics (RH, BSD conjecture), I am presented with the challenge of grappling with concepts that are beyond what I have (formally) learned.  It means I have to teach myself, and that I get to constantly upgrade what I'm doing as I learn ways to do it better.  I am very grateful to Dr. Padmanabhan for his time, as it gives me an idea where to look to learn the concepts I need to.  I have also started adding to my daily word research routine the task of looking up one mathematician a day.  Usually, this leads to a whole bunch of mischief.  Yesterday, for instance, reading about David Hilbert led to researching the meaning of transfinite numbers and how proof theory complements model theory, recursion theory and axiomatic set theory to form the four pillars of modern mathematics.  I have a notebook devoted to mathematicians' names, and usually get 5 to 10 for every mathematician I look up (let's not mention the two pages I filled when I stumbled on the list of those who have won the Fields Medal and the Wolf Prize).  This new part of my routine is very exciting, because it is something I can build upon daily.  With Godel on my radar for today, who can say what new things I will know about math at bedtime?  As the short-lived Niels Heinrik Abel put it, "Study the masters, not their pupils".

For those who follow my writing blog (Graeme Brown Winnipeg fantasy writer), you are likely aware of how I have wrestled with to fit writing - a passion that pulls at my heart - in with another passion that I want to devote my life to.  The best advice I have ever heard from someone was, "Do both."  With the start of my publishing career last month, I can no longer say writing is just a hobby.  So, as the month of August has passed, I've been devoting a lot of time to the learning curve involved with establishing an online routine that is sufficient.  My number one purpose of blogging is to provide interaction with those who are interested in the work I do, and because I do different kinds of work, I have made different blogs to suit the context.  This means a lot of maintenance, and it means a need for efficiency.

As the term resumes, I look forward to having a well-balanced diet of writing and study.  My work on the Riemann Hypothesis and congruent number problem continues, and, as always, I am looking every day to take my mathematical toolbag to a new edge.  I was happy to hear that one of my proposed directions of research on the congruent number problem looks promising, so that will be my task for now.  Unfortunately, the problem of understanding the Riemann Hypothesis is one I cannot part with, and this means a neverending kick at the can.  My current intuition on how to approach the problem has evolved from an idea that came to me during a particularly bad sermon at church.  At the time I was taking an algebraic approach, breaking Euler's product expansion into chunks and developing a notation for a combinatorial function (then analyzing the holomorphic functions comprising the numerator and denominator), and though I found some interesting convergence properties, there was nothing telling.  My idea was to look for a modularity pattern in the ring 2pi as the primes evolved, but I could not find a useful connection between the natural logarithm and a prime to a given exponent, even though intuition would say that e and pi's relation should be linked through some form of trigonometric analysis.  Later, I analyzed its general term for various values of the complex coefficient as a composition of three functions, but this did not lead to the node of convergence I expected.  As I was working through the paper by Chahal and Osserman, I returned to Ulam's Spiral (which kick-started my interest in this new approach to RH), looking for a connection between limited sums of squares.  This, too, led to nothing useful.  Then, yesterday morning, while reading and enjoying coffee, I had a new idea to move from squares to circles in the geometric model (which has an interesting link to the Bernoulli numbers) of the proof, and I devised a four-step research strategy.  Inevitably, I will explore this one as well, and if it leads to a wall, I'm sure there will come another cup of coffee and a new idea.  There's always more coffee when it comes to math.  That's why I have a mug that has pi on it.

Now, if you are a mathematician and want a cool mathematical mug, I've made something special and it's available at Cafepress: Coffee cup for math lovers: relationship between pi, e and i

Monday, August 20, 2012

August 20, 2012: Mixing Math and Hobbies

Are you passionate about Math?  You might want to check out the latest project I've started.  Math Milestones: from Pi to Euler to the Imaginary is a shop I will be building (about one product a month) which will feature a design based on one of the mathematicians or mathematical topics I've looked up (I do one a day).  For the launch, I have featured the logo, depicting the relationship of Euler's constant, pi and the imaginary symbol, three milestones of math that are as might as Gauss, Euler and Reimann (though I say mightier).

The next (potential) project will be based on the fractal Sierpinski curve.  I am a vector graphics artist, and the idea for this project has been evolving over the last several months as a way to lateralize what I already do, while also tieing into my studies.  As my university doesn't have a history of math program, I have taken it upon myself to research mathematicians and the relationships between topics in mathematics to build the picture for myself.  I hope that in five years time, doing this daily, I will have somewhat adequate knowledge to tackle some of the deep mathematical problems that often wake me up early in the morning.

Thursday, August 16, 2012

August 16, 2012: Groups and Elliptic Curves

As my first post here, I think it would be fitting to tell you how I got to where I am today, so here goes.

My journey with math took at exciting turn in the fall of 2011 when I took a number theory course, after a few years away from school.  I did extremely well in the course and maintained a connection with the professor, meeting biweekly to consult on a problem he gave me.  Little did I know, this was in fact the Birch-Swinnerton-Dyer conjecture.

I have had a deep love for prime numbers for the last several years, but now that I have studied number theory and taken a number of higher-level math courses, I realize this is a true passion that will lead me deeper down the rabbit hole every day.  And it's a good rabbit hole.

As the summer began, I put the congruent number problem aside, realizing I needed more formal training in groups and abstract algebra.  I studied computer programming, and was acquainted with Ulam's spiral.  On my early morning walks to work at Starbucks, I contemplated this spiral and some of the patterns it exhibited, realizing there was a deep connection between prime-generating polynomials and, ultimately, the Riemann Hypothesis.  I did a great deal of work on this before setting it aside.  Of note was a modified Ulam Spiral that evolved along the Riemann line (square root sequence), but I was unable to formalize my intuition that fractional modifications of this addition would lead to an infinite set of convergent areas.

After a meeting with my professor, we discussed Hasse's Theorem and I realized, after going full circle, that the kernel to gaining further insight into the Riemann Hypothesis lay in first understanding Birch-Swinnerton-Dyer, which meant tackling groups and modular forms of elliptic curves on my own.  I have spent the last week and a half of my summer working on this, and it is going quite well.  I feel like a mathematical teenager going through growth spurts, the type that sometimes have me waking up at 1am and working until dawn (though only occasionally, and with the help of coffee).

So, here I am.  I look forward to sharing, now and then.