My Catharsis of a Year Past

It's been many months since I've decided to update this blog. These have been months of success and failure, brief moments of ecstasy framed by doubt, insecurity and regret: I have survived my first semester of graduate school.

This shouldn't sound so dramatic and I realize that in the grand scheme of things, my life is great. I graduated from MIT in June 2008 with a perfect GPA in mathematics and a minor in Philosophy (not perfect in all subjects though). I was accepted by more than one pure mathematics graduate school with full funding. I am currently attending an Ivy League university with 2 years and 3 summers free of any teaching duties and the remaining 3 years supported by teaching. I have a beautiful girlfriend and we have been together over 3 years and she is also attending an Ivy League university less than an hour away and pursuing a PhD in physics. I can imagine that many many people would be envious of my circumstances. So what's the rub?

The rub is that despite my rational decision-making there are parts of me that perpetually fret that I made a bad-decision for my graduate institution. These thoughts are mostly irrational fears based on my own insecurities and adjustment to graduate school. I realize this and part of the symbolic quality of writing this on New Years Day is to fulfill a resolution: To move on with my life at Penn and to actualize my passion and enthusiasm by playing the cards I've been dealt rather than wishing my circumstances were different. Before I can move on I must throw out the old. This is my Catharsis.

Over the summer I was engaged in incredibly rewarding work with Victor Guillemin at MIT. I got to combine my passion for geometry and writing by completely rewriting the core chapter on manifolds in Victor Guillemin's new work-in-progress, a reworked take on the classic "Differential Topology", but with a major focus on multilinear algebra, differential forms and DeRham Cohomology. Those of you who have read the preface of the original classic will know these topics were intentionally left out and all manifold theory occurs in R^n. This book places a focus on doing "Grown-Up" Manifold theory with an emphasis on the previously neglected topics. The praise that Prof. Guillemin handed down to me was both warming and hurtful. The knowledge that had I delayed my graduate applications, taken the math GRE during a less chaotic semester and had the fortune of getting letters of recommendation from both my research advisor and VWG, that I might have fared much better in the admissions game, incenses me from time to time.

Having enjoyed my summer work greatly, it was sad to stop and head off to Philadelphia to start my August lease, away from girlfriend and friends, in short, my support system, so that I could prepare for 6 hours of preliminary exams at the end of August. I knew that my finite group theory was weak and that I had spent my previous year focusing entirely on the beauty of algebraic topology, smooth manifold theory, Riemann surfaces and Integrable Systems (the last two being a brief introduction). I began my review with the excellent "Berkeley Problems in Mathematics" and got my butt kicked, but whenever I turned to past Penn prelims they were usually predictable and elementary. Nevertheless I set to work on these and did many years worth, getting stuck occasionally, but eventually solving several years worth of problems. Nearing exam time there was an exam or two where I could do an entire 3 hour section in less than 45 minutes. The odd thing would be that a question or two would pop up that was much more difficult than the others and would take well over an hour to solve. The pattern, which I only realized later is that these sorts of questions became slightly more frequent in the last year or two. On an exam day I was oddly relaxed and was actually cocky that I might be able to tear through a section in less than hour. What actually happened was that I would recognize a hard question as being related to an older question and spend half an hour trying to remember how the old question went instead of trying to solve the problem in front of me. I became nervous and made several small errors and then later huge conceptual mistakes. After the day was done and discussion ensued, I realized that I had made the same mistakes as many of my classmates, yet instead of simply having common intersection I had the union of say two people's mistakes. I began to worry immensely.

The next three days were filled with incredible nausea and sleeplessness. I had dreams where I professors would yell at me and say "You want to go to Caltech!? Here let me call them and see if they'll take you now!" Immediately my mind fractured into two lives, one real, one imagined. In my imagined life I was at Caltech in Sunny Pasadena, surrounded by an elite community of scientists and engineers, buttressed by an undergraduate community reminiscent of my fond memories as an undergraduate at MIT. Here I had no preliminary exams, no quals, and was never confronted with a feeling of intense inadequacy or failure. My real life was painted with black, an immediate branding as sub-par in my department, and a city filled with poverty, crime, racial and socio-economic tension. My university well known, but not immediately recognized by the layman as elite mathematically as Caltech. After three days I was told that I had failed at the PhD level but had the highest pass at the Master's level. Basically, they had set me as the cut-off, the next person was 6 points ahead of me and another 2 points would have done the trick.

So needless to say I had a rough start to the semester. I was forced to take a remedial course (proseminar) meant to patch-up faults from my undergraduate education (this meant for me supplementing my 8 week course on Groups, Rings, Modules at Cambridge as my only algebra education). This scared me for two reasons. One was that traditionally attrition from people taking Prosem is quite high, meaning I was more likely to drop out of the program. Second was that my intentions on getting ahead on advanced course material and getting into research would be delayed. I initially had planned to place out of the required Geometric Analysis course and take Lie Algebras in its stead, but after having all the fight kicked out of me and a rather difficult placement exam I decided against it. I was also promised new and exciting topics on the Hodge Decomposition Theorem, Gauss Linking Integrals, Calibrated Geometry and all sorts of extra topics I had not been exposed to. Taking the normal first-year courses did not sound so bad, but then I was informed there was a scheduling conflict between this course and the proseminar and I might have to push back my geometry education, which was the whole reason I came to Penn in the first place! Fortunately after some last minute gifts given by the faculty, I was able to meet at another time for prosem and take the normal first year curriculum.

So my courses totaled: Real Analysis (Lebesgue Theory, Frechet/Banach/Hilbert Space theory, etc.) Algebra (Sylow Theorems, Group (Co)Homology, Category Theory, Rings, Modules), Geometric Analysis (Smooth Manifold Theory, Vector Calc, DeRham Cohomology, Frobenius Integrability), Riemannian Geometry (Do Carmo), and proseminar. As I began to recover from my initial setback I was energized to succeed in my courses, yet I found myself perpetually switching back and forth between my real life at Penn and my imagined life at Caltech. Most of my courses were good, but I felt that my Geometric Analysis course moved too slowly on the elementary material and too quickly on the important material. The lectures themselves consisted of mostly intuitive arguments and hardly was a rigorous proof demonstrated in class. To repair this, the instructor would assign problem sets with anywhere from 10 to 20 problems that were either heavily computational and unenlightening or were major results that were routinely in most textbooks and better copied. There would be pretty routinely one problem that was novel and interesting, but would require tremendous effort given the tools at hand or easily done using material not yet studied. There was also an absurd amount of focus on vector calculus that would be interesting for physics undergraduates, but only alienated most of my classmates from the course material. This also resulted in tons of tedious problems that filled my week with what could have been spent on pretty much anything else.

In my parallel life I was receiving stellar lectures filled with exacting detail and rigor. Topics that were "fun" were left to the side and room was made instead for Fundamental groups and covering spaces, homology, cohomology and calculation of homology groups, exact sequences. Fibrations, higher homotopy groups and exact sequences of fibrations, structure of differentiable manifolds, degree theory, de Rham cohomology, elements of Morse theory. Geometry of Riemannian manifolds, covariant derivatives, geodesics, curvature, relations between curvature and topology. (This is the list of topics for Caltech's Ma151 abc) All things which I consider to be simultaneously fun and important...

ENOUGH COMPLAINING! This where I stop my critique of things past.

What I came to realize is that a lot of graduate school is less about how well other people can educate you than it is about how well you can educate yourself.

Most of my internal struggle can be pinpointed to faulty thinking. Part of my insecurity comes from the fact that when you are at a place like MIT or Caltech you have some external verification of your own self worth. Granted it is true that well taught classes and a community of bright, competitive colleagues and professors can help significantly in the training of a mathematician, a lot of work needs to be invested by one's self into one's own study of mathematics (Not to suggest that Penn lacks bright, achieved faculty). Many of my classmates have learned most of their undergraduate mathematics, not because they had top-notch professors that told them what to study when and which problems to solve, but because they put in their own effort and pushed their professors to teach them more.

The bottom line is that a professional mathematician must learn a large chunk of requisite mathematics and this must be done by any means necessary. Beyond the first year or two of material suddenly everything that one needs to learn is self-taught. Success in research ultimately depends on this self-initiative and ability to acquire material independent of the environment around you.

So what about the future? I believe that over all many of my courses at Penn will be quite good. Ironically, based on a cursory glance of some of the problem sets at Caltech and the ones assigned at Penn, I would say that ours were of equivalent if not greater sophistication (Geom/Top not counted since they learn the subjects in opposing order to us). Tony Pantev is a superb instructor and mathematician. I've also heard very good things about Jonathan Block, my start-up advisor, who will most likely be responsible for most of my education here. Also, Peter Storm and Natasa Sesum will be he here next Fall and I expect that I will have the chance to learn much from both of them. Finally, the thing which excites me most about my circumstances is that Helmut Hofer has received a life-long appointment at the IAS starting this year. Given my plans to move to New Jersey and commute next year, I will have some of the best living practicing Symplectic Geometers in my backyard.

I am also pleased to know that I have the option of pursuing symplectic geometry as a major topic for my orals next year. I have had several interesting conversations with other Penn graduate students, and I do believe that I am not the only one with some of my interests. There seems to be several postdocs working on Mirror Symmetry -- a very cool and modern pairing of algebraic and symplectic geometry -- and at least one other graduate student intends to have his dissertation on Homological Mirror Symmetry. Aside from interests, my classmates are overall a fun group and I actually enjoy a social life outside of my studies. All the reasons outlined in my decision post below seem to be borne out, but the true fruits of the dissertation phase are still waiting for the picking.

The past is immutable and the only rational action is to seize the present. I share this story not so much to dwell on the specifics of my life, but so that you might find some common thread in your own life story.