This is reproduced in its entireity from a note I wrote in late 2008. I wrote it as if someone would want to read it; an informal statement of purpose.
A small, informal note-to-self sort of thing with a short description of my achievements and interests.
Some academic background
My interests have changed over the years and thought an outline of their evolution over time would be useful. My first academic “achievement” of note was a bronze medal in the International Chemistry Olympiad. It was a pretty tough selection process just to be able to represent the country: my school encouraged me to take the screening exams (three or four rounds of theory and labs) and four were selected from 104 applicants. I also got the “Best Experimental Work” award in the Indian National Olympiad selection camp.
I really enjoyed hands on, building-stuff-in-the-kitchen sort of work then and joined the mechanical engineering program at IIT Madras. I worked with Prof. Ramesh in the IC Engines lab on some interesting old-school hardcore engineering stuff in the first couple of years. I learnt a little electromagnetic theory, designed and built a solenoid-type linear actuator to move the control rack of a pretty large fuel injection pump, and wrote the control software (very basic PID kind of stuff) for the engine governor. This was a defense project for the diesel engine governing system on the “Arjun” (weird coincidence, no?) main battle tank for the Indian Army. It was great fun, especially as a freshman, to control this huge engine with just a little computer.
My interests started moving more towards theoretical problems and mathematics during my last year at IITM. I worked with Prof. Sujith in the Aerospace department on a model problem in gas dynamics. We obtained an asymptotic solution to the Euler equations for a particular type of inhomogeneity in the PDEs. Of course, being undergraduate work, it was neither very novel nor very sophisticated; but I did present the work in an AIAA conference here in the states and it was a good experience. However, I really enjoyed myself then and learnt a lot about nonlinear wave equations in general. My first introduction to dynamical systems and chaos was in my senior year as well – Prof. Balakrishnan taught the course and I kind of fell in love with the material. He covered an insane amount of ground in the course, Linear Algebra, Group Theory, nonlinear ODEs… the works. He was one of the best teachers I’ve ever had. You might have read his foreword (for the Indian edition, at least) to “Feynman’s Lectures in Physics” …
At the University of Michigan, I worked with Prof. Bogdan Epureanu on a biological problem. We modeled the motion of motor proteins: these are little agents of transportation that carry cargo from one place in the cell to another. Since their size is of the order of nanometers, the Reynolds number of the flow around it is really low (Purcell, Nobel laureate, inventor of NMR, gave a beautiful talk on the movement of bacteria that is oft cited in the literature). This means their motion is overdamped and inertia plays no part in the dynamics. One might think that this makes their dynamics less interesting, but the motor protein’s small size implies that it is considerably affected by “thermal fluctuations” or collisions with the molecules in the media and works in very random environment. It turns out, that these fluctuations are essential to its operation (like Feynman’s Brownian ratchets). All this stuff makes this a really interesting problem physically and mathematically. So I taught myself stochastics – it’s not my adviser’s field of expertise – and did some interesting analysis on the problem. My adviser presented at a paper at a conference, and I’m writing another right now. I’ve attached my thesis and the conference paper if you’re interested. Oh, yeah, almost forgot – The College of Engineering awarded me the 2008 William Mirsky Memorial Award for outstanding achievements in research and academics.
Comment in 2015: In fact, it’s a very interesting question: given a Brownian particle moving in a periodic potential, what is its effective speed of travel. This is a question answered by homogenization theory in dimension one. By some strange quirk of fate, I study this sort of thing now.
A year into my masters program, I knew I wouldn’t stay on for a PhD working the current problem I was working on then and told my adviser that I would quit in a year. I also realized that I lacked formal training in many fields of mathematics although I’ve picked up enough bits and pieces over time to apply it fairly competently in most fields of engineering and physics.
I decided I wanted to move to an applied mathematics program where I could consolidate my mathematical knowledge in a formal setting, even though my adviser expressed his interest in continuing to work with me and gently cautioned me of the impracticality (stupidity) of my choice. I would like to work on a hard, physically inspired mathematical problem in an interdisciplinary kind of environment. That is, I’d ideally like to have a one adviser in mathematics along with another in a science department – this, I believe, would be the ideal situation for a person with my skills and interests.
My fields of interest include complex and dynamical systems, pattern formation and emergent phenomena in such systems (very little background in this), wave equations and fluid dynamics, biological problems and probability. I do admit that this is pretty broad and I do not know what specific problem I would like to work on. However, I do believe that I should be able to figure this out in the first couple of semesters of the PhD program.