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

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