mathematical language, legal language?

A friend of mine made the recent comment (on someone else's private journal; I won't quote it here in detail) that a "pure" subject is suspect to her, and she believes that all fields of study are linked together in intricate ways. Her opinion make sense ..

.. but I wonder whether "pure mathematics" falls into that category or not. Most of the time? Some of the time?

Off and on I've discussed matters with scientists (or rather, grad students in the hards sciences) and on less occasions, folks in the social sciences and humanities. As pure mathematicians we get our share of flak for not doing much interdisciplinary work. Maybe they think that we mathematicians are arrogant, that we don't need them, but they need us. Maybe I'm imagining things.

There are the occasional hotspots in research: dynamical systems are useful for scientific models and more recently, computational biology. Algebra and combinatorics become useful in encryption and search algorithms. The nascent theory of wavelets has grown in parallel to needs of image and signal processing. I'm confident I could find more examples of this sort.

However, I believe that most of pure mathematics is not immediately applicable and to the rest of the world, "current worthless." Maybe different areas will have untold worldly uses in the future, but maybe they won't. I get the impression that past and present administrations of American government fund mathematical research because of hope towards the former, not because of any noble purpose on their part.

If our mathematics helps the rest of the world, fine; if not, then as Archimedes said, don't disturb our circles.

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Me, I like to think of mathematics as logic, or even philosophy, applied to very specific situations, often to geometry or the structure and interaction of numbers and their abstractions. It's a convenient way of saying that it's the study of abstract objects using abstract tools.

Many modern problems have a tendency towards abstraction and take a long time to ponder; perhaps this is the fault of logical formalism and rigor, which adds apparent volume and mass to a little monster, so we fear it more and make much of it.

If you ask a mathematician about a problem, (s)he may tell you the full regalia of the issue and possibly give a few definitions. If you ask a mathematician about his/her ideas, then more likely than not, you'll get a concrete example and possibly heuristical reasons and intuition in the form of pictures and diagrams.

So why the logical formalism? Why the rigor and fearsome-looking definitions? Why a formulaic outline when assessing and/or solving a problem whose words are hard to understand?

It's the price we pay for consistency and accuracy, and it plays a similar role as legal language. In some ways, legal documents and mathematical papers are similar. For example, the first section is full of definitions. In a union contract, how do you define "Employee?" and on some apartment leases, what exactly is a "tenant?"

If you believe your method is correct and it accurately addresses the problem at hand, then you must "fill out the legal paperwork" .. in short, write out precisely what you mean in a consistent way that can be checked logically.

On the other hand, when reading mathematical work it takes a careful mind, often a mathematician's, to decipher the full strength (or weakness) of a new theorem. Is it argued correctly, any logical mistakes? Does it address even the pathological cases, if the definition allows such possibilities? For example, it's a bit like hiring a lawyer for when you're closing on a house: the last thing you want is to hear that sinkholes aren't covered by a "good condition" policy.

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Perhaps a problem with mathematics and interdisciplinary work is, oddly enough, a language barrier. Another problem is that mathematicians are attached to their own problems, just like any other type of academian. Combining the two, sometimes it seems like a one-sided relationship with other disciplines.

A mathematician might be helpful to a think tank of physicists or economists or biologists, but only after (s)he processes the situation and question in a way (s)he can understand, and then asks enough questions so that (s)he understands what can be done and what cannot: maybe it's an issue of conservation of matter, or that negative profit is bad. If this is successful, then (s)he might suggest an idea, and with any luck (s)he can explain the rationale apart from any tendency towards excessive terminology. Maybe it helps, who knows?

I remain wary of a think tank of mathematicians discussing a common (open) problem to a physicist, economist, or biologist. It may take too long to explain, and even with a good dose of patience, it may be too many new concepts to juggle -- and this is not an insult to non-mathematicians, but a matter of technical details. It's a lot to ask someone to think about something away from their area of expertise, and with many issues to consider.

However, I do see the value in mathematicians asking non-mathematicians how they think about their own problems, because it could help the mathematician learn how to think about the mathematical problem in a new way. It could be as simple as analogy: the notion of extremal length, also called modulus of curve families, is mysterious unless you describe it in terms of electro-magnetism and current through a condenser. It doesn't proves your theorems, but it helps you write your proofs.

But there remains the problem: how did we know to think of extremal length in terms of electromagnetic current? Someone else has to answer that one, though my guess is complicated and needs some explanation [1].

I don't claim to know the answers. If non-mathematicians had a tried-and-true method that could help mathematicians with their own research problems, then great! Sign me up! But every time I explain what I'm studying to the non-mathematician, we mutually give up and it leads to little added progress, though by explaining I might understand the issues better.

When from experience others are of little help, we tend to ask less often and continue on by ourselves. After a while you forget to ask, and expect confusion.

What can I say?

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[1] I think it has to do with the relation of extremal length and a notion called "capacity" which, physically speaking, describes the least kinetic energy under a given arrangement of position. If this relationship is true, then there may be an optimal arrangement of the magnetic field induced by the current, and the resulting field lines may give some comparison, as curves, to the given curve family.

Conservation of Paranoia / Planning Ahead.

So I met with my advisor yesterday for the first time in .. wait, six weeks?

I'm still inclined towards disbelief, but between my advisor's one month holiday with his family and the travelling-planning issues for a week before and after, I suppose it must be right. Wow.

At any rate, we met and I showed sufficient progress, but apparently I was working on the wrong metric space. Instead of Euclidean space Rⁿ, the results were meant for its 1-point compactification, or if one wishes concreteness, on the n-sphere Sⁿ in Rⁿ⁺¹.

Thus for more than a week, I was worried about a issue that is, in fact, a non-issue: the (local) Sobolev regularity of the inversion map x → |x|^-2 x is problematic near the origin in Rⁿ, but the equivalent mapping is a diffeomorphism on Sⁿ, so I was worried over nothing ..

.. well, sort of.
Something else is now suspect.

For those mathematically-inclined, I'm trying to prove an extension theorem for a particular type of homeomorphism on particular types of domains. In my efforts to resolve the (non)issue, I began a systematic examination in how "large" a sub-domain can the old and new homeomorphisms agree in value.

This in turn led to the study of the functions' behaviors for points approaching infinity, and rather interesting limits, at that! Apparently these same limits might cause trouble for another function in a different part of the proof. It would have been fine were we in Rⁿ, but it may go wrong in Sⁿ.

I suppose some difficulties never go away, do they? (;

---

On a less technical note, I gave my advisor a preprint of an article of my co-author's and mine, and he didn't seem too displeased by it. Then again, he didn't get much time to read it during our meeting, so I suppose we'll wait and see ..

We also talked a little about the future, and my previous guess now seems true: after this term I'll have gotten all my course requirements completed (both cognate and distributional), and tentatively my oral Prelim Exam is set at the end of the Fall 2005 Term.

If I pass, it would mean that I'll be a Ph.D. candidate by January. Zounds! "Candidate" has such a nice ring to it, too .. q:

a month passes ..

.. and my advisor has returned from .. where he was, earlier.

  (I never asked where he went, and he never mentioned it)

Again I am accountable for my mathematical deeds. Now all I need to do is find him and allow myself to be held accountable.

I guess it hasn't been too unproductive of a month .. but there are few results to speak of. Earlier in June, co-author and I ran into some technical troubles in the final stage of our proof ..

  (for those keeping score at home, this is the second project, not the first)

.. so now the project has been put back on indefinite hold. It only adds to my deep-rooted suspicion that as much as I like geometric problems, I'm likely little good at them and should stick to more analytical topics, such as PDE.

I did have an idea to circumvent that technical difficulty, the other day, but I suppose it has to wait until the weekend. Thesis work comes first.

Update Alert: my new mathematical webpage!

With any luck, the Fox Trot people won't sue me for copyright infringement .. but I just couldn't resist using that particular comic strip for an opening .. q:

Main changes: I've tweaked around the page layout a bit and altered the color scheme. Also, I designed a new "logo" which you can find on the upper-right corner here: it's meant as the infinitesmal picture of quasiconformality, treated as colored spheres and ellipsoids under the light of a distant light source.

Other changes: The list of UM Mathematics Seminars is still there, but since I predict that I won't be doing much traveling in the next few years (I do have a problem to solve and a thesis to write, after all), I've removed the Events / Conferences page from the list of Links.

Anyways, back to work. Enjoy the site!

Putting my money where my mouth is.

No work done today, but looking at my precedent of work hours and hours in the office, one could make the argument that "the night's still young" and "the day ain't over yet."

I continued with playing host to two friends of mine from high school, and then I played soccer with friends from the Department. Now it feels like time to work, but I can't identify whether that is my conscience urging me to work my status quo routine of hours, or whether it is something larger in scope ..

.. such as accepting and meeting my current fate head-on: having been handed a research problem by my advisor, to review the context and what is known, then to plan a strategy and execute it, baby step by baby step. I'm sorry to say that I'm still somewhat overwhelmed, and it's unclear where to start first:

• Do I read the background material more in depth? That way I can be aware of the trouble spots early on, and when they occur then I am prepared to combat them.
  
  
• Do I close the gap, and get some preliminary results done? It's not like I don't understand the problem nor misunderstand the broad strokes of its shape. Doing something might suggest to me where I should start my study of the background material ..

It's probably a convex combination of the two, or likely a feedback loop of repeating one and then the other .. and I should get to it! Now if I can only get myself to accomplish that something ..

I should keep my damned mouth shut ..

.. and stop this proclivity of wasting time on the Internet, particularly blogging and commenting on blog posts of others. In my hierarchy of must-do's, I should be:

1. Start work on my advisor-given thesis problem. Oh yes: I have a problem now, and I'm in the state of affairs where I actually understand the problem at hand, but apart from making parallels from previous results in the field, I'm at a loss for how even to approach it.
   
   Then again, if I knew how to solve it, then it would no longer be a problem, would it? q: Still, the inherent and newfound uncertainty of the matter is still unnerving to me.
   
   
2. Continue reading my advisor's book; I still feel like I don't know enough to start work in the Analysis on Metric Spaces. [1] Then again, half the days of the week I wake up and wonder if I'll ever feel ready to work in this field.
   
   
3. LaTeX my results on previous research. There are still details to check, but if I don't do them now then I'll be swamped by more immediate work: for example, the earlier items on this list!

Instead, over the last few days I've been posting on my blog, doing work for the last item on the Must-Do List, and consistently replying to a blog post about matters of Linear Algebra.

Below is the link, if you're really that keen on reading what I have to say. My comments are under the username "grey_ghost."

[from a LiveJournal mathematics community]

[1] Lately I've prefered to refer to the subject as "metric analysis" rather than "analysis on metric spaces." Less words make a single, comprehensible mouthful.

Rants after a Bad Day of Reading.

I feel like I'm in a minor rut. I was able to do some actual work today, though I gained no ground or progress. To sum it up, there seem to be three stages to reading a mathematics text:

1. A lot of smiling and nodding and page-turning, because everything makes sense. Usually no paper or pen is involved at this stage.
   
   
2. A lot of frowning and confusion, because what made sense before no longer does. Usually paper and pen have manifested themselves, and in particular, there are many pages covered with hasty diagrams and scrawls .. all of which contribute nothing to one's understanding (but seemed like good ideas at the time).
   
   
3. A lot of groaning and self-incrimination, because everything does, in fact, make sense. Usually by now extraneous ideas are set aside and one peers into the heart of the matter. Furthermore, two sub-cases occur: (a) the result is clear and you wonder why you didn't see it sooner, or (b) you were right all along but your paranoia caused you doubts and in particular, your counter-example is not a counter-example, after all.

I'm currently on Chapter 4 and Stage 2: what fun and joy, I tell you. q:

However, thinking about these various stages of learning and understanding, it makes me wonder how any of us got into the mathematics business. Are we mental masochists? Fanatics? Or is it that we have nothing better to do, and mathematics is the least of all evils?

Don't mind me; those previous questions were meant to be rhetorical. \:

---

As alluded in a comment to my previous post, I believe that I entered mathematics because I couldn't leave "well enough alone" and certain problems in analysis and geometry were so tantalizing that I couldn't let them go. Now I'm trying to make a career out of it.

I don't know about other students, but this inclination of mine causes untold frustration when I'm learning something, which at this stage of the game is always abstract theory. I begin to wonder how this or that definition came to be, and why anyone would ever think of such a thing. Textbooks often emanate an illusion that as presented, this theory is fact and for lack of a better term, "God-given." It has been written and taken this form. Understand it so.

It's easy to forget that books are written by people who have specific perspectives and ideas on how to view a certain phenomenon in mathematics, whether it is the structure of a particular space subject to its metric, or the nature of functions in accordance to their regularity. It's equally easy to forget that these ideas and intuitions take different shapes, from how they appear in print and under the full regalia of logical rigor. If there is one meta-lesson I've learned from reading mathematical texts, it is this:

Always draw a picture or work out an example, but be careful not to trust the picture or example too much.

I'm lucky to be in a department where my professors are willing to explain to me the motivations of how these theories were designed: as frameworks to overcome difficulties and issues in solving problems. They patiently draw pictures for me and show me how these matters reduce in simple cases, and my nerves settle.

All of this lead me to wonder: Will I ever be able to do this for myself? Moreover, if I want to be in the business of solving problems, then such skills are key, and I must be able to do this for myself.

I suppose there is no underestimating the power of negative thinking. \:

Looking Back (EDITED)

I was playing hopscotch with mathematicial blogs today and I stumbled upon an article written by a professor at UC Berkeley. It concerns undergraduate education for those math majors who do not proceed to graduate school.

(The article can be found here in PDF format.)

Mathematics without graduate school?
Heresy, you say?

I admit, that was my first inclination and now I fear I've been fully indoctrinated by academia. But that's fine, because I'm not arguing about that today. In fact, I'm not arguing anything at all: just reminiscing.


my education. Most of you already know the story of how I fell into the field of mathematics, so I won't retell it. But as for those years of enrolling in classes, I've realized something.

My early study of mathematics was hardly circumspect. As an undergrad I took exactly one abstract algebra course. I never took topology or number theory or logic / set theory, and if you asked me what Galois theory or representation theory or cohomology was, you'd receive a blank look in return.

The more I think about it, the more I realize that I learned very specific topics for a very specific purpose. I was still dabbling in the prerequisites in my sophomore year of college, stuff like Advanced Calculus I & II, Linear Algebra, ODE, and Abstract Algebra.

It was later that summer that my Linear Algebra prof offered me a research project and threw me a problem. Read this paper and learn what you need to understand it, he suggested. Ask me questions, and read. When you're ready, we'll test the technique on another case. Maybe we'll prove something new.

Forget the vector bundle approach (as if I knew then what it was). I was introduced to Riemannian Geometry from the concrete point of view of matrix theory and inner products, then the far newer subject of sub-Riemannian geometry from a control-theoretic viewpoint. I never realized geodesics and extremal problems would be that interesting.

The technique never worked, and the problem is still open. But we wouldn't know that until three years later.

My prof left on sabbatical but suggested that we can think about it a year from now, but there was much more to know before we could tackle other methods. When I asked him what sort of topics, he told me and I chose my junior and senior year curricula.

Complex Analysis. Real Analysis I & II. Functional Analysis. PDE I and Topics in PDE. The Calculus of Variations. Differential Geometry. Linear Operators. I read some topology on my own, and tried to understand why Sobolev spaces were the right kind of object to study.

As a result I've always thought of myself as an analyst-in-training and a problem solver. Theory and theory-building don't come natural to me, and every so often I catch myself thinking in terms of computations and technical matters. Now in a different school and a different place, the rules are different and I swear I've learned the mathematics that I missed and relearned all the mathematics that I thought I knew.

I don't think that sort of education was a waste. Every time I studied something new I knew what motivated me to do so .. but later the reasons changed, and I became an academic. Learning for learning's sake.


Why Graduate School? I can't remember if there was a distinct moment when I realized that I wanted to go to graduate school in this field. I wonder if there was ever such a moment. I'm tempted to say that "it just happened."

That's all well and good, but for those of us who've gone through the onslaught of forms and applications, deciding to go to graduate school requires a fair bit of thought and effort. There is the general GRE and the mathematics GRE, the jerryrigging of presenting yourself on paper, and the search for faculty who are willing to aggrandize you to other faculty. We ponder what qualifying exams and oral preliminary exams are like, what research is like, what the job market is like ..

.. but before all that, what five more years of schooling are like.

I guess I was just motivated early on. Someone showed me a seemingly simple problem, how complicated it becomes, and how quickly it does so. Perhaps it is my pride in humanity and its capacity for progress which has led me here: some problems are meant to be solved and someone must do so. If it means that I must become a mathematician, then I shall and so be it.

I wouldn't have it any other way.

Recent Events from my "Life"

• Left town for almost a week to St. Louis, where I attended a very nice conference in honor of Al Baernstein's 65th Birthday [1]. It was much like going to a family reunion: I saw some old friends, my advisor from my undergrad days, and my current research collaborator.
  
  
• Returned to Ann Arbor with my collaborator in tow, and played the occasional role of host. Research work ensues, and some spaghetti sticks to the wall, so to speak: we have results!
  
  
• The Ahlfors-Bers Colloquium begins and on one weekend afternoon I made coffee for over fifty people. It didn't end up too bad. Paradoxically I begin to cut down on my coffee consumption.
  
  I also see old friends, and my acquaintances from separate worlds collide.
  
  
• I continue to play an occasional host, and life almost returns to normal: I scramble to read more for my weekly meetings with my thesis advisor, and during my downtime I think of research ideas that are both doable and interesting.
  
  
• the Near Future: my collaborator leaves town, and my share of the writing begins. Meanwhile I struggle with reading and learning new things, and my advisor might forgive me with my continual shortcomings.

Life isn't so bad. It's nice to come back to the daily grind. Too often during conferences I feel as if I'm not being productive, and listening to others expound on their work, paranoia ensues. I fear that I haven't done enough as a mathematician, that I haven't proven myself, and perhaps I never will.

It's fine, now. A day of steady work took care of that. I've a few new ideas and some goals from before, so it will remain a busy summer. Who knows? Maybe by summer's end I'll actually know what I'm doing! q:

[1] Actually, his birthday had been two weeks before the start of the conference.

Photos of Me, in the Office.

First day of summertime .. well, not really. Today I woke up and it was 43^oF with a high of around 54^o. I mean academic summertime: the semester is over and four months of uninterrupted time await me. Plenty of time for studying and reading and maybe a little writing and exercise, if I budget my time well enough.

I've implemented my plan for working like G.H. Hardy, but instead of 9 am to 1 pm, I opted for a later four-hour block: 11 am to 3 pm. I'm happy to say that the time flew by, and unhappy that I didn't get much done.

Rather than tell you about my day, it's easier if I show you.

Anyways, the night is young: plenty of time for more reading.

I need a better camera.