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.