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.
