Mathematics and Art -- Expression

So my friend Paul -- a fellow math grad student at Penn who does combinatorics -- sent me the following email:

I was talking to someone last night, and they said they loved math because it was "beautiful". This maybe always struck me as a rather odd, if widespread, idea; it at least seemed odd then. My idea of "beautiful", aesthetically or artistically speaking, involves expressiveness, or at least something to do with human emotions, some human connection. Maybe that's *just* my idea, but in any case mathematics has nothing whatsoever *expressive* about it, so it doesn't fit at least my idea of beautiful in any artistic sense (yeah, a sunset is pretty, but, I wouldn't go see a movie that was just a sunset). This got me thinking about what I do find, if not exactly beautiful, at least appealing, about some mathematical work. I posted it because I think it's not the same thing a lot of my peers find appealing, and, I was curious as to their reaction.

Basically, I think of mathematics as a bag of tautologies--statements that are trivially true, and thus in a sense devoid of content, not really saying anything at all, like "If we're going to the movies, either we'll see 'Star Trek' or something else". (Or "If you take a bunch of apples and put them in 10 boxes, at most one to a box, you have 10 boxes you can put the first one in, and then only 9 boxes you can put the second one in. Because it's one to a box. Can't put the second one in the same box as the first one. Ten minus one is nine. Etc."). That's why I don't find mathematical theorems, as such, very interesting. Theorems in the physical sciences have content, they describe how the world works or might work. Mathematical theorems have no content, they're theorizing about nothing, a bag of tautologies; they're trivialities in disguise. Who cares?

That's (certainly) not to say they're trivially true to me, or to anyone else necessarily, but, to the extent I, or everyone else, *don't* think of a mathematical result as tautological, I believe it reflects on a failure of my personal, or our collective, understanding. Overall, mathematical language doesn't seem like something we're terribly well-suited to, as a species. We have quirky and limited human brains that are good at other things, like figuring out how to bite things or passing immediate judgment on the hotness of people we just met. So in one sense, mathematical results don't say anything; in another sense, they say something trivial that we can't recognize as trivial because we're idiots about that kind of thing. The interesting thing about the triviality in disguise is the disguise, not the triviality, and when I work on math, I'm much more motivated by frustration over not being able to see through the disguise than any particular interest in the triviality. This is a "problem-solver"'s attitude, I suppose, and a "theorist" would be more interested in the triviality (or, more likely, characterize the whole thing differently).

What I like in a mathematical argument, and maybe this is what people think of as "beautiful", is when it manages to make the essentially trivial or tautological nature of the result apparent to our quirky and limited human brains. That's why I don't like long or messy arguments and why I like visual proofs of combinatorial identities and why I feel compelled to redo certain proofs--if they "work", but don't make the essentially tautological nature of the result clear, then a "new" result may be established, but the basic failure of understanding--the recognition that the result *isn't saying anything*--remains, and it's that basic failure of understanding that interests me. But if the essentially tautological nature of the result *is* made clear, you can think of it as making our quirky and limited human brains a little more universal and a little less limited, which provides some human connection to the enterprise. It doesn't feed a single starving child, but, it has an appeal, as activities go. Is it something to spend 80 years doing until you die of old age? YMMV.

This conjured up lots of possible responses and this wasn't something meant to stir debate, but I felt like I had to be a voice for "beauty" in mathematics. This is a subject I have thought about for quite some time, but I eventually settled on the following response:
___________________________________________________________

Well articulated Paul, but I disagree on two primary issues:

1. Mathematics is not expressive -- I will argue that mathematics is the pinnacle of human expressiveness.
2. Mathematics does not connect with human emotion. -- Mathematics is both something to be appreciated passively and carried out actively. The first expresses satisfaction in certainty, elegance and other archetypical emotions. The second is the consequence of curiosity and mystery.

First I will address the second issue because it refers directly to your use of the word "expressive". Then I will move to broaden the definition.

There are several emotions that mathematics appeals to. The ones that come most readily to mind are: elegance, permanence, purity and certainty. Beauty is a multi-faceted sense but I would argue that several of the aforementioned components do account for a classic sense of beauty. The entire pursuit of art is to express feelings of forlorning, hope, despair, love, abdonment, achievement, failure and the whole palette of human experience.

What emotions are conjured when appreciating the marble statues of Greco-Roman past? Although their world bears little resemblance to our own they were familiar with much the same range of emotion that we are. Their art expresses archetypes and reflects universal stories or ideals that humans spread across millenia can appreciate. Art has the same purpose of philosophy -- it is meant to make explicit these emotions so that one may stand back and reflect. It is an attempt to purify a given aspect of human experience and abstract away irrelevant details. Art accomplishes this through physical instantiation, whereas philosophy and thus mathematics (since the two were always part of the same enterprise and only recently have grown separate) sets aside the object for study inside of one's own mind so that introspection may be applied. One cannot say that one method is better than another just because the philosopher and mathematician frames his or her art in words and symbols. If one can appreciate the marble rapture of two bodies intertwined, but not Aristotle's words "Love is composed of a single soul inhabiting two bodies" then I concede the issue at hand.

On a meta-level the age of statues and words alike -- their permanence -- evokes a sense of solidity and foundation -- a rock that even the most tortured souls can cling to. Mathematics is without a doubt the most permanent human enterprise (meaning: as long as there are humans to support its meme pool). When a theorem is proved it is acknowledged as true for the rest of time. It is changeless. It echoes Zeno and other members of the Parmenidean school's belief that reality is one, change is impossible, and existence is timeless, uniform, and unchanging.

This way of thinking is spawned out of a desire for certainty. This is perhaps the most primordial human emotion. It is the impetus for all mythology, all religions and all science. Religion and Science are manifestations of a desire to narrate human existence. They both advocate methodologies for how to construct this narrative, but ultimately they are just developments of the more basic role of mythology -- coherent story-telling. Mathematics is the most coherent form of story-telling yet devised by human beings. One might argue that this is like comparing apples to modules, but the distinction is superficial. In one, the scene and characters are put in place and the governing dynamics are given by the personalities and tempers. In the other, a background universe and cast of definitions are put in place with logic and inspiration for what might be true determining the dynamics at hand. The success of physics since the time of Archimedes has been the application of abstraction to human perception to create a model with initial conditions and letting mathematics provide a story that unfolds in logical, clock-work fashion. The simple models our ancestors had pale in comparison with our current (and probably still over-simplified) model of the universe as a fiber bundle with local sections equipped with SU(3)xSU(2)xU(1) (or E8) symmetry group. Perhaps faith provides the input but reason creates the output. The individual steps may be tautologies, but the unfolding pattern is the most beautiful, spectacular, psychadelic creation humankind has ever conceived.

What is it that makes this patterned way of thinking beautiful? It is the sucess of expression unparalled by any other human endeavor. The example of putting apples in boxes may lack expressiveness because the initial conditions and structure of the problem are transparent. The action of solving the problem is just a process of deduction (or of building a general algorithm and then deducing the answer). What really makes mathematics expressive is the process of identifying features of system or processes and then naming them. This resonates with the old Christian/Greek notion that naming something gives one power over that thing. The tautological statement that "plugging in numbers into this polynomial in different ways leaves the result unchanged" seems like a silly thing to say, but when the correct notion of a Galois group is harnessed suddenly problems of the ancients come tumbling down (trisecting the angle, doubling the cube, squaring the circle). The ability to transplant one problem from a completely unrelated domain into another is at the core of human intelligence. More fundamentally, meaning comes from the isomorphism discovered between two things. Reasoning by analogy -- a method perfected in mathematics -- is all that expression is and can ever be. It is all literature, psychology and art has ever done. One must take some personal internal state that is largely unspeakable, unnameable, undrawable, irrational and point to some isomorphic system sitting outside of ourselves and say "That is me!" We are ourselves manifolds, only locally similar to anything that is understood. Although antithetical in nature, I believe that Riemann would have agreed with Emerson, a contemporary on a different continent using different methods to grasp at transcendental truths, when he said

"Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth."

A Brighter Future

Hello Everyone!

Quick Update:

1. I placed out of the core Complex Analysis course at Penn, freeing me up to do a reading course on Symplectic Topology with Prof. Block.

2. I have been neglecting my reading course because I have been occupied with the details of the Thom-Pontryagin Construction which relates homotopy classes of maps $f: M^n\to S^p$ (the p-sphere) and framed cobordism classes of closed $n-p$-dimensional manifolds. It was fun but I wasn't planning on the extra labor.

3. I just gave the pizza seminar talk on "Soliton Solutions of Integrable Systems and Hirota's Method" Friday that went really well.

4. I just looked at Penn Math Courses for Fall 2009 and there is an awesome selection!

Notable entries include:

- P. Freyd's "Calculus of Variations" Course.

- Jonathan Block will continue the 3 course sequence on Algebraic Topology with 618.

- M. Ballard's "Complex Algebraic Geometry" Course: It should be interesting to see fresh blood teach this staple of Penn.

- Joachim Krieger is teaching a course on PDEs: A subject I hope to take as a minor oral topic. Calabi Fellow Joachim Kriger was recently cited five times Terry Tao's AMS Bulletin (Jan. 09) article on "Why Are Solitons Stable?"

- Ron Donagi is teaching a course on "Mathematical Foundations of Theoretical Physics" a course that enticed me into coming to Penn, but hasn't been taught in several years.

- The Eugenio Calabi is teaching a topics course on Differential Geometry backed with W. Wylie's Ricci Flow madness.

Next year is going to be awesome!

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.

Consciousness and Intelligence: Be Here Now!

So I received an email from someone who had just finished watching Curran's and my GEB video lectures. I'm always glad to hear that OCW is impacting peoples lives in positive ways.

Anyways, in this email I was asked several questions about my thoughts on consciousness and intelligence, the differences between the brilliant and the ordinary, and so on. Since it is the most I've written for a while, I figured I'd share my response below:

First I have to say that I am very flattered by your message -- I've had very few people contact me directly about the lectures and only a few blogs seem to have anything to say about them.

I would also like to thank you for donating to OCW -- It really is a great cause and is based on the belief that knowledge and education should be free to whoever is willing to try to access them. Thanks again for that!

Now I will try to answer some of your questions:

Thoughts on Consciousness & Intelligence and what I'm reading:

This is a big one and I am not even sure how to answer it -- I should first say that there are probably a lot of people who understand both of these subjects better than me. I gave up pursuing some of these questions because I began to doubt I would ever see the answers in my lifetime.

First, I have to admit that I'm not sure if I buy the basic thesis of GEB -- that there is an isomorphism between the physical world and formal systems and that the "I" represents the same sort of twisted-back-on-itself strange loop that we see in Godel's statement G that says of itself "G is not provable".

Funnily enough I think that Hofstadter doesn't believe this either. If you look for his Scientific American column collection -- Metamagical Themas -- he does a great piece on "Who shoves whom around inside the Careenium?" Here he introduces the "symball" view of thought and consciousness, which I believe to be a very good approximation of what is correct. The basic idea is that thought is an emergent process of a lot of small unconscious agents, but more importantly these different levels -- the higher emergent level and the lower physico-mechanical/biological level -- push around each other and interact in strange and beautiful ways on different temporal and spatial scales. I would highly recommend reading that piece.

Also, I have come to the opinion that Hofstadter's new book I am a Strange Loop gets to the point a lot faster and a lot easier than GEB does -- in almost half the number of pages as well! You should check out the new paperback edition on amazon, which appears to have a good review by the washington post.

I would recommend Marvin Minsky's books Society of Mind and The Emotion Machine -- consciousness has this very mystical and apparently point-like quality that resists reduction, but Minsky does an incredible job in analyzing seemingly simple actions, say crossing a street, and unraveling all the routines and sub-routines the mind must employ to accomplish this task. Thinking like a computer scientist offers a lot of powerful inferences into the sort of psychological structures we all have.

2. What separates the brilliant from the ordinary?

Not a lot in terms of hardware, but maybe a good amount in terms of software, but then differences in software are able to affect and change the hardware, adapting it to its needs. The brain's plasticity is a wonderful thing. Put in other words, I think that differences in hardware are not significant for distinguishing genius from the ordinary, but that the environmental conditions in which we are raised dictate a great deal of whether we are "brilliant" or "ordinary".

Also, try to define what what it is meant by these two terms. If you define "brilliant" as "good at math" then you are going to have a very biased view of intelligence. I propose that we abandon these terms all together and focus on what is really important -- self-actualization.

I doubt that classic examples of "genius" -- famous artists, writers, thinkers and so on -- share some common feature in terms of intellectual capacity, whatever that might mean. In terms of psychological well-being, however, they probably all shared some sort of feature of self-actualized/actualizing human beings. (Please explore the writings of Abraham Maslow for more information.)

Basically, once you have secured your physiological, safety, love/belonging, esteem needs, you are then in the position to pursue aspects of morality, creativity, spontaneity and problem-solving. See Abraham Maslow's Hierarchy of Needs.


Of course this begs the question "Well most middle/upper class people have (arguably) secured the bottom levels in the Hierarchy of Needs, so why aren't all of these people geniuses?" The answer is that most of these people probably are pretty bright -- geniuses on a certain coarse scale -- but differences in work-ethic and subjective taste dictate what is commonly perceived as genius. This goes back to the hiking-around-the-world-in-pursuit-of-enlightenment vs. toiling-for-hours-in-a-lab-in-pursuit-of-a-Nobel-Prize view of genius. One or both might make you famous, but how do you evaluate genius without a good definition of its properties?

Suppose you take the best students from all the high-schools in the world and you put them in one place, how do you rate genius among them? There is a scaling effect whenever you restrict your population size that seems to result in a normal distribution with different mean.

I also want to emphasize the role of hard work in the development of genius and excellence. The difference in genius is almost identical to the difference in athletic prowess -- What do you think separates the people trying out at the Olympic time-trials and the general populus? What separates the person who wins the 800m and goes to Beijing and the person who comes in fourth? We will remember one of them, but not the other.

A similar phenomenon is observed at MIT. Coming from a place like MIT, which in many ways represents the cream of the crop in terms of intelligence, the difference between the average student and the best student can seem like miles, yet to the average person in the public, every MIT student seems like they're miles ahead. Whether you "win" of not is a large mix of chance, training and genetics, and chance/training dictates a much larger percentage than innate genetic ability. I don't think "winning" should matter, but maybe just being at that level of performance is important. I would say that two things mark the difference between the best MIT student from the average MIT student or any top performer from the average person in general.

1. There is a catalytic moment that I call the "Ahaa!" moment that is completely personal, emotional, and irrational. This is a sort of "peak experience" as described by Maslow that is often interpreted as spiritual in nature. It can be the sudden or gradual realization of a deep curiosity or desire to pursue a subject of study, a vision, an art, an invention, or what have you.

This provides the impetus and energy to pursue:

2. The commitment of blood, sweat, tears, and other hard labor in developing a skill, art, trade, knowledge or ability to actualize and instantiate the creative vision attained in the first part.

I don't think I was born good at mathematics. I think I did have a seed of curiosity (like many many children) that was remarkably not stamped out by my upbringing. My parents were wise and kind enough to not push me into anything, but rather made available resources for me to exercise my curiosity. I didn't realize that I really loved math and wanted to work hard at it until the end of my sophomore year of high school. I then began the long road of studying and investing time in my new interest. At several times I lost my way, thinking that I was going to pursue other subjects, but then again after my sophomore year in college I realized my love for pure mathematics. I don't think I am really better than anyone, but the way in which I responded to challenge is different from some people. If you respond to challenge with a desire to conquer regardless of how long it takes, then you will eventually overcome, but you should pick your battles carefully. The brain is a plastic organ that can be trained like a muscle -- if you wish to train it through study or meditation than it will become stronger in that regard, but it can also be over-trained, fatigue or injure. Avoid extremes.

Living a balanced life is very important and sometimes brilliant people work too hard for their own good and their intelligence deteriorates along with their emotional well-being. Enjoy every day as if it were your last because you might get run over by a bus today or tomorrow. Work hard only if that gives you pleasure. Don't suffer only for the hope of better things to come.

Be Here Now!

Graduation and the Road Ahead, Behind, Sideways...

Many months have passed since my last blog entry. I apologize for that. My last semester at MIT was an enjoyable one. I got to take Victor Guillemin's course, 18.952 - Theory of Differential Forms, with a friend of mine and I always love chatting about courses with people.

I have put together a Googlepages page summarizing my academic work and projects from the past four years. It can be accessed here:

Justin M. Curry - The Collected Works of a Philosopher-Mathematician

I know it sounds depressing, but I, along with many of those graduating in 2008, have really struggled with "letting go" and leaving my undergraduate career behind and entering the "real world." Granted the real world for the next 5 years will be graduate school at the University of Pennsylvania, working with any of the fine mathematicians there, but still, the idyllic times of college have come to an end. You can see my proof of purchase below:





Sasha and I had a wonderful time decorating our mortarboards - a quirky re-capturing of an otherwise boring ceremony - making it an individualized expression of coming to terms with our frenetic time at MIT. I discovered that one of the best ways to make a three-dimensional "8" is to first make a mobius strip and then pinch the middle together. From the photo you can see that Sasha has spelled "MIT" using Feynman Diagrams and I have contented myself with a knockoff of one of the many variants on the MIT logo.

How did I get here from my last blog post on January 20, 2008? Over 5 months have passed and no letter to my internet home? What gives? I will hope to answer some of these questions in the next couple of posts. Most of these recent posts were written around the end of June and the beginning of July, but I have altered the dates to present them in a given order. This post I'm dating June 6th, 2008 to reflect my graduation date. The one below I've dated April 15th to symbolize the common notification deadline for graduate schools in the U.S. I hope you enjoy my re-telling of these stories!

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.

------------------------------------------


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.

Structure and Geometry Govern Interaction: Follow Up

In an older post - Structure and Geometry Govern Interaction - I introduced an argument about how Orange County purposely designed its neighborhoods to have a lack of public spaces and opportunities for interaction. Every time I cite this argument, people ask for my references and I always feel bad to come up empty-handed.

Well, Matt has helped me address this grievance once and for all --

Lisa McGirr's "Suburban Warriors: The Origins of the New American Right", published by the Princeton University Press, contains the much sought after reference. Pages 41 through 44 or so are the most condemning with the most powerful passage on page 42:

Convenience, privacy, and decentralization were the keys to the master plan, with few central public spaces except those dominated by consumption. Irvine executives, with a good sense for business, conciously created solidly middle-class neighborhoods. They preferred to forgo federal subsidies that would have required them to open their developments to poorer residents, and they did not incorporate open-housing provisions into their master plan. Their desire to build high-priced homes helped to reinforce an already existing social homogeneity in Orange County. The result of development along these lines, of both the corporate and the free-market models, was spatial isolation and an abscence of community, which, in a complicated way, helped to reinforce a conservative ethos.

Page 41 introduces the company that was responsible for designing the Irvine area and there are plenty of other interesting things said later as well. In particular, I found the "NIMBY" political movement ("Not In My Back Yard") referenced on page 43 to provide another powerful defense of the idea that home ownership, with a focus on cultivating your own private landscape, encouraged a protective attitude towards private property and a dislike of communal spaces.

I must confess that the idea of simply mulching your own flower beds and garden (something my grandmother so enjoyed in her previous large suburban home) on the weekends hardly seems like an act of conservatism, but it does seem like a selfish act compared to donating your time to working on the community play ground or garden, something to be built and enjoyed by many.

Page 42 is not by default part of the limited preview of the relevant section, but if you use google books to search for the phrase "open spaces" in the book, the first result is on page 42, allowing you to view the missing page.