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.

Grad School Decision: Caltech vs. UPenn

Several months ago, I was making a declaration of survival with my "I'm Alive!" post. Sad to say, the worst had not yet obtained at that point. Graduate admissions results began with an optimistic waitlisting from Caltech and was then followed by embarrassing rejection after rejection. Initially enthusiastic response from Berkeley and Chicago was only later met with personal emails filled with "Sorry to inform you"'s and the like. After several nauseating months were over and the dust settled, I had been accepted to Caltech, UPenn, UCSD, and UMD. The week after visiting England for spring break I spent 3 days each at Caltech and Penn meditating on my graduate school decision.

First Caltech. This is the second time that I have seriously played with the idea of attending Caltech - and both times I've rejected their offer. I hope this doesn't catch up with me come job hunting, because I really admire Caltech as a prestigious research institute that draws an elect undergraduate body definitely rivaling MIT in quality. I love the idea of living in California - although LA seems less exciting to me than SF/Berkeley does - the campus is gorgeous, it has the same sort of quirk that I've grown to love about MIT, but there are two problems:

1. Size.
2. Focus of the (few) faculty members in the math department.

The first problem is the exact same problem I had when I toured as a high school senior - The lack of people makes for a peaceful, meditative atmosphere that can drive you insane. I think on the level of the entire campus I could get used to, and grow to like, the size, but the math department seemed suffocatingly small to me. Here is roughly how my visit went:

On arrival, I checked into the math office and was handed a schedule for my 3-day visit. I have maybe 3 hours filled in. "Hopefully we can fill that in for you" says the secretary. We then try to make appointments with as many of the faculty that I think I will be interested in: Alexei Borodin, Nikolai Makarov, Eric Rains, Barry Simon, Tom Graber and the only geometer on the faculty, Danny Calegari.

Other prospectives - "prospies" - drift in, awkward conversations ensue. There are at most four of us that I meet on the first day and I'm having serious trouble getting along with almost everyone I meet. Later on some seemingly well-adjusted people from Columbia, Chicago, MIT and Toronto show up, but the interaction is short-lived. For the first two days most of my time is spent waiting for meetings to happen. Here is a standard scene. Some details are exaggerated:

[Four Prospies, two graduate students are standing and sitting around in the tea room, desparately trying to execute normal social behavior. A balding, un-identified mathematician is sitting on a couch, muttering to himself]
Grad Student: So What do you think you want to study?
Me: Geometry, Topology, Mathematical Physics. Stuff like that.
Grad Student: [Snickers] Oh -- I see. Can you be more specific.
Other Grad Student: Yeah, why didn't you just say "Math"? [Imagined high-fives going around. I'm the weeny getting towel-whipped in the locker room by T-bone and his goonies.]
Other Prospective: Yeah, I want to study counting and coloring - Combinatorics.
[There are a few forced laughs, and then everyone goes silent. The breaks in conversation become unbearable, so I high-tail it to the tea and without hesitation sling back a styrofoam cup-full of hot, scalding liquid. I scream internally, "Oh Thank God for this searing pain! Please spare me from this agonizing social situation."

I wander back over to my position in the circle. Either there is no conversation, or we squeeze out forced explanations from graduate students their thoughts on Caltech's math department and what sort of research they're doing, or other prospies interject with dick-measuring comments in an effort to discover weaknesses and strengths in the fellow prospies' math backgrounds. This scene is then broken every 10 minutes or so when a Caltech regular enters the tea room, stops in his or her tracks, drops their jaws in an uncontrolled moment of surprise and stammers out "There...There...There are sooo many people in here!" Evidently the presence of four other people in the Sloan Math wing is enough to initiate this sort of response.]

Thankfully, Sasha arrived on the second evening and a physics professor treated us to an amazing dinner in Old Pasadena. A physics grad student comes along, who is pretty sociable and easy to talk to. I get the sense that even the math department is viewed as a slight oddity at Caltech. I am recovering well from my lonely exploits and by Day 3, I am starting to warm-up considerably to Caltech.

By this point, I've met a handful of faculty and have been impressed by everyone. The faculty know there stuff and they exude this almost scary intensity and passion for whatever they are doing. There is only one problem - none of it seems to lie ahead of my current projected path through mathematical space. It is either too analytical, as in the case of Makarov, Simon, and Rains. Or it is too algebraic as in the case of Tom Graber and Danny Calegari. There is also this design to the math department - one faculty person to one niche - there seems to be little collaboration between faculty and almost no interaction. I start to come to terms with all of these concerns as I sit in Tom Graber's class and am easily impressed, but then I have to duck out early to meet Jerrold Marsden.

I wander across the idyllic Caltech campus, smell the flowers carried on the cool California breeze, and eventually find the Control and Dynamical Systems department where Jerry's office is. I knock an open door, and an older man, slouched slightly in an office chair, clicking away with his one-button mouse on his 30 inch Apple Cinema Display, turns to look at me. I introduce myself and he smiles - there is a spark in his eye and a sort of immediate recognition, not of me particularly, but of a wandering soul who has come to seek out advice and answers. He motions for me to come in and sit down.

Within a few minutes our conversation ramps up and takes on speed. He reminds me very much of my own personal hero at MIT: Gerald Jay Sussman. He has this remarkable ability to discover what I'm interested in hearing and then goes ahead and delivers an off-the-cuff speech that pulls on a few central insights made during his life. He tells me about Dirac structures, an invention of Ted Courant's, named by Marsden, conducted work at Berkeley in the late 80s. The basic idea seems to be that in Lagrangian and Hamiltonian systems you have maps from tangent to co-tangent bundles and vice versa, but if you consider the graph of this map as sitting inside T*+T, this gives you in some ways a more fundamental picture than the Lagrangian and Hamiltonian pictures do on their own. Although this stood as beautiful un-applied mathematics for a while, somebody figured out that electrical circuits, which are incompletely characterized by the Lagrangian or Hamiltonian viewpoint is best captured by this Dirac structure picture instead. He gave me a copy of one of his new papers (Reports of Mathematical Physics Vol. 60 2007, No. 3 "Reduction of Dirac Structures and the Hamilton-Pontryagin Principle"), which had some of the most exciting research I encountered all week. Jerry keeps rapping on about the "wild blue wonder of pure mathematics," showing me the latest edition of his Hamiltonian Reduction by Stages book. I'm grooving on all of this, when suddenly Jerrold Marsden shatters my world:

-He doesn't advise pure math students anymore.
-The Pure math job market sucks.
-Ted Courant, a PhD from Berkeley who has whole whopping fields of mathematics named after him, is teaching high school for a living.

Jerry has shifted gears on me, and goes onto lament the woes of pure math research and the government funding today. The "good ol' days at Berkeley" where all Jerry and Alan Weinstein had to do was chat to the Navy once a month to secure funding, has been replaced by "flashy powerpoint presentations, presenting practical solutions to real-world problems." The CDS department at Caltech apparently consists of a whole bunch of converted pure mathematicians who are "not afraid of deep, powerful mathematics" and can "calculate the curvature of the connection on a principal bundle" if needed to, are all preaching the way of using beautiful mathematics to get a handle on applied, engineering and science problems.

For nearly an two hours, my head is spinning - Am I not going to get a job as a pure mathematician? Should I come to Caltech so I can pursue a more applied PhD in the CDS department, working on numerical techniques for symplectic integration? Oh God! What should I do?

I go and see Jerry again. I let him know that he's blown my world apart and he smiles. I try to place the Penn-Caltech decision in perspective, letting him know that if I intend on pursuing pure mathematics for my PhD, Penn is much more suited to my research interests than Caltech is. I ask the more dramatic question, Should I join the choir of the Converted - the ranks of John Doyle, another Berkeley complex analysis PhD who left pure math for control theory, and now works on my past love of complex systems? Jerry smiles and says that I should do whatever makes me happiest, saying that being a pure mathematician is like being a poet, you might be poor at times, but if you truly love what you're doing it won't matter. He then encourages me to see one of his students give a lecture that afternoon on integration techniques, discrete differential forms and the like.

Before the lecture, I go and see Barry Simon. He is friendly and formal. I wait in his secretary's office and eventually I go in. I'm still reeling from my two conversations with Marsden, so I ask realistically, what the job market is like for pure mathematics. Barry goes on to dismiss most of Jerry's concerns, explaining to me that the old generation of space-race faculty hires are all retiring or dying, and the job market has been climbing steadily since it's early 90s slump when people like Ted Courant were looking for a job. He then goes on to compare Caltech and Penn, pulling on US News Rankings and says "If you want to do Analysis, Caltech is the clear choice."

The meeting comes to an end, and then I visit Eric Rains for a little bit. He talks rapidly, connecting random terms that I know here and there with work on orthogonal polynomials and then seems to work on a level which is down and dirty, but also deep. I struggle to follow what he says as he walks me towards the lecture. I am impressed that a professor would do such a thing. We shake hands and say good bye.

Finally, I am in the lecture hall were Jerry and his student are. People filter in and sit down. The talk gets rolling. The speaker is intelligent and has the social graces of a state school graduate. He is talking about discrete differential forms and their application in variational integrators, which is used to simulate physics and respect the geometry by focusing on conserved quantities (symmetries ala Noether's principle). At some point an audience-member asked whether you could balance the tolerance of conserving energy in the integrator with say time or momentum or what have you. The student then replied "Yeah... Well it is actually a theorem that... I think you [pointing towards Jerry] may have proved that..." Jerry sort of nodded sheepishly and everyone laughed. It was at that point that I realized I wanted to be a theorem-prover like Jerry instead of a code-monkey like his student. It was at that moment, that I realized Jerry's pure mathematical work, theorem-proving - "the stuff he does for fun on Sundays" - really was the spark that made both Jerry and I light up with excitement. It was at that moment that I realized and remembered why I fell in love with mathematics in the first place, I wasn't content to be a code monkey, I wanted to be a theorem-prover.

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Suffice it to say that the Penn Prospective's Weekend went well. As I later wrote to my research advisor, Aliaa Barakat, in an email:

Penn seems to be my clear choice for graduate school. I had lots of fun at the weekend and met many people I would like to work with in addition to Tony and his group. The graduate students have fun and seem very sociable and generally happy... If I were to go [to Caltech] I would be pretty much selling myself on geometric group theory, which may or may not actually interest me. Tom Graber (Caltech, from Berkeley) is also a great algebraic geometer, but I don't know if his work on enumeration will interest me.

One of the things I realized in the past week is that I have not seriously studied algebraic geometry, and it is not clear at this point whether or not it will interest me once I study it. Ron Donagi's class in Complex Algebraic Geometry seemed really interesting but I understood very little this late into the semester. Algebraic Geometry seems to be at the core of Ron Donagi, Tony Pantev, and Antonella Grassi's work. Although, outside of this domain, I had several really great conversations with Wolfgang Ziller and his work on "Exotic Spheres" and other positive curvature examples and counterexamples to some open problems. His arsenal of tools primarily consists of Lie Groups, Spectral Sequences, Chern Characteristic Classes... all things which directly appeal to me given what little surface reading I've done. He also takes his students to Brazil occasionally :-). Christopher Croke is also a really friendly differential geometer doing more negative curvature things. I met Jonathan Block and one of his students, so there is also interesting work there. Alexandre Kirillov is also pretty notable and has his own symplectic form named after him.

Overall the story of my Penn visit is a pretty enjoyable one. There is a much larger math department and both graduate students and faculty were much, much easier to talk to because they seemed genuinely excited about their subject. There is plenty of funding (5 years, two years no teaching duties, 3 paid summers) and lorry-loads of potential advisors doing mathematics that interests me. The graduate students even threw a party for the prospectives that was fun and social, featuring good tunes (Radiohead and an overall well-DJ'ed mix) and I could converse about anything from topoi and differential geometry to traveling and hiking around the world.

After all the visits were said and done, and my choice was seemingly obvious, I still had a hard time letting go of an opportunity to head West and join that Other Institute of Technology. Sasha had lots of trouble too, and making a choice between Berkeley, MIT and Princeton was an emotional roller coaster that I hope neither of us have to relive. Eventually she realized how important our relationship was to her, and she decided to attend Princeton, which is only 45 mins by car away from Philadelphia and the Penn campus. I look forward to the many years ahead of us, and am confident that the Philadelphia, Tri-State area will be a good home for the next couple of years, even though a part of us will always be eager to head out Californey' way to find the American dream.