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.

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