Archive for August, 2011

Season: Doing computations.

August 24, 2011

1. What mathematical activities? What level of rigor?

High school math consists of lots of computations and symbol manipulation. Solve this equation, find the roots to that polynomial. Research math is more grounded on concepts and proofs, but computations and symbolic manipulation are essential. Sometimes it seems to me that in order for some collection of ideas to be called math, it needs to have at least a level of conceptual precision and the potential for computation. We need to be able to define symbols and manipulate them. (Some areas of higher category theory that I’ve seen seem to test this.)

In research, I’ve done computations at various times, for various reasons. One is to collect empirical data, on which to base conjectures. Another is for the sake of building intuition about “what’s going on.” A third is in order to construct an object with certain properties. I think most mathematicians do computations almost every day, but, in my experience and from what my advisor has told me, there are also week- or month-long periods of computing. This post will be based mainly on the four-month period in the winter of 2011, when I was constructing a certain mathematical object.

Starting from the way certain homology computations proceed, my goal was to construct an object (in the derived category of a non-Noetherian ring) with certain “nice” properties (periodic homology of certain types). It was a constant back-and-forth between tweaking my object’s construction and computing its properties. Once I got a sense of how the subtleties of the object affected its properties, I could actually build a family of different objects, with a range of nice properties.

My activities mostly consisted of filling pages and pages with symbols, arrows, and lots of subscripts (like this). I would often stand in front of a wall-size whiteboard for hours, covering it, erasing, and covering it again. (For example, as demonstrated in this video.)

There was a small amount of intuitive imagining – thinking of how I might tweak my construction to get it to do what I wanted. But every small new idea required hours of computations, to see the consequences. So the majority of my time was locked into completely rigorous symbol manipulation. Maybe this step-by-step, completely explicit computing is the most rigorous math experience I know.

2. What relevant interactions with other mathematicians?

Every mathematician carries a tool box of problem solving techniques; each tool has a range of applicability, and each subfield has some common tools and some less common ones. For example, my computations in winter 2011 involved using spectral sequences, one of the most elaborate computational tools in algebra.

It was crucial to meet with my advisor, at least once a week. I would often get stuck or make mistakes while trying to use a spectral sequence to compute (the homology of my chain complex). There were lots of tiny, explicit steps, and we could quickly locate the issue and work past it together. But perhaps more importantly, my advisor, an experienced computation-doer, would suggest new tools I didn’t know or didn’t think of.

3. How does it feel, what is the mood?

Doing computations can be fun and satisfying; I think most mathematicians would admit to getting some pleasure out of a long page of scribbled symbols that results in some final correct answer; I think most people, in middle school or high school, experienced this satisfaction.

Over time, however, I started to feel like I was devoting too much time to a question that was too esoteric. I spent months building these nice objects, but they weren’t nice enough to make it intrinsically worth it for me. Computations can be so time-intensive, and so specific, that meaninglessness can sneak in. It felt like I was sewing a family of pretty socks, each with different pretty patterns, while the world around me rushed past towards collapse and/or transcendence. I started to feel empty, and the math stopped being fun; the symbols became meaningless signifiers, cutting me off from real life. (Note: it was also winter in Seattle – a supremely depressing season.)

4. What state of mind? stable vs. chaotic? focused vs. dispersed?

The cognitive strain of doing computations is less than other modes of math, perhaps because the symbols take the concepts out of my head and map them across a whiteboard or chalkboard. My thoughts were closely represented by the string of symbols I wrote, and these were explicit, stable, and linear for the most part. Using the board as an extension of my mind, I could zoom in on any circle of ideas and clarify to any scale I wanted.

When I was working, it felt like the board and my knowledge were two manifestations of the same thing – an exotic landscape of intricately connected concept-symbols, only a subset of which were present at any given time. Each concept-symbol is just a signifier of other relationships with other concept-symbols, in deeper and deeper nested layers.

Since my work was so contingent on this physical representation, it was easy to turn on and off. To start working, I simply had to set myself up in front of a board or pad of paper, and start writing things out. To stop, I just had to pack up and leave. Outside of these work periods, I only rarely would ponder, in a more dispersed mode – where was this headed? what should I try next?

5. What type of self-reflection during the experience, and did it help?

After months of symbol manipulation, I became aware that I was losing interest, succumbing to meaninglessness. What was the point? It wasn’t enough, apparently, to make some nice mathematical object. I was hoping to find some “application” for my constructions. (Eventually I got lucky, and was able to use these objects to answer an open question in the field – a result that a few experts have told me is “interesting.”)

But I’ve always scorned the idea that math should have applications outside of math, are applications within math any different? Wasn’t this math at its purest, math for math’s sake, building nice math because I could? Why did the experience feel so unsatisfying?

I like plants more than animals, shape more than color, form more than function, cosmic gestures more than emotions. These math objects didn’t appeal to my aesthetics; I didn’t think they were “nice,” only “neat.” They were too specific; I felt stuck deep in mud.

Recognizing this as the cause of my lack of enthusiasm, I was able to shift my perspective. On the one hand, I was able to appreciate the specificity of these constructions, and stop worrying about their meaninglessness. I embraced the game of symbols as a game, and one that I was getting better and better at.

On the other hand, I found the cosmic gesture within the mud, as it were. Within the experience of computing homology groups at the whiteboard, I identified a resolution to the mind-body paradox. During that experience, the maps of symbols I wrote on the board were the maps of concepts I experienced in my mind, and vice versa. Approaching my work from this angle, I found renewed enthusiasm and depth.

6. An everyday metaphor for the experience?

Doing computations is like playing a game. The rules are explicit, undebatable, and somewhat arbitrary.

7. An example of a good day and a bad day?

On a good day, I might fill a whiteboard several times with scribbles, occasionally writing down fruitful computations in my notebook. On a bad day, I’d get royally stuck, or find a mistake and have to backtrack, or just not have fun.

8. What did you do when you were stuck?

Most sticking points had to do with small mistakes, which I could hunt down on my own, or misapplications of various computational tools, which I could easily ask my advisor about.

9. When and why did it end?

I stopped when I had succeeded in constructing a family of objects with sufficiently nice properties. I found one application of these computations, answering an open problem in the field. This seemed like a good stopping point. There were more directions I could’ve gone in, but didn’t feel like it.


Flavor: Using my math powers for evil.

August 18, 2011

1. What’s going on mathematically?

Very rarely, when I’m particularly frustrated or angry at some particularly illogical inconvenience, I’ll apply my powers of analytic reasoning with the intent to cause emotional harm.

For example, once while renting a car in Las Vegas, the salesman kept pressuring me to buy a full tank refill at their 10% discounted price. He was aggressive about it. I got flustered, and fired off a terse line of irrefutable-sounding mathematics, pointing out that this only made sense if I returned the car with less than about 10% of the tank remaining, which was not pragmatic. I said it with such damning certainty that he shut up immediately.

More recently, I tried to return a piece of video equipment that was checked out in my roommate’s name. They insisted that I couldn’t just drop it off – my roommate needed to check it back in, for legal reasons. Whereas a reasonable annoyed person would question the logic of this administrivia by lobbing a loose bundle of sense, I shot a dense two-sentence projectile with a certainty and precision that I almost made the poor attendant start crying.

2. What is the emotional and logistical context?

I have to be in a very, very bad mood, or very stressed out. The context is bureaucratic and supremely annoying.

3. What thoughts are there?

There’s the thought that what I’m about to say is aggressive and backed by a mild mental volition to cause harm. I don’t remember really choosing my words to be particularly concise, but they are, and it’s this logical conciseness that characterizes the experience. The logic wells up from my analytic intuition and mathematical training for direct and irrefutable communication.

4. What quality of awareness?

In Tibetan Buddhism, anger is understood as the neurotic manifestation of vajra, the state of clarity; the wisdom behind anger is clarity. And when I’m concocting my aggressively logical and concise retort, there is sharp clarity – the logic lays out clearly and my analytic mind finds the most powerful form in which to yield it.

Throughout the exchange I maintain an awareness of my mental willingness to cause some small level of harm. I’m aware of the small-mindedness of this.

5. What emotions?

It feels good to let myself be angry sometimes; I feel empowered and proud. But only briefly. Then I feel bad. Sometimes I’m amused at the whole thing, since it’s so rare for me to snap at someone like that.

6. What does it resolve to, after how much time?

I can never remember exactly what I said in the heat of the moment, only that it was uncommonly logical and concise. It sort of echoes in my mind, repeating itself in a more spread out form. I often end up pondering my bias towards rationality over other forms of intelligence and knowing.

7. How frequent is this flavor?

I can only remember doing this a few times. Other contexts: shredding apart the logic of unsuspecting Christians, or economists.

8. What are good/bad ways to change or follow it up?

Apologies sometimes (spoken or unspoken), but not always.

Season: Building a theory / Imagining what could be.

August 12, 2011

1. What mathematical activities? What level of rigor?

Last summer, I entered a period of very speculative and theoretical research work. I had some mathematical data before me – information about the homological and cohomological Bousfield classes of certain categories. My goal was to find patterns among that data, to make connections between the homological and cohomological cases. (Hey, don’t give up yet! You don’t have to know what these words mean to understand the story I’m telling.)

Conjectures. Imagining what might be going on. Dreaming up connections that might exist. Constructing relationships, and testing their domain of validity.

The speculation increased significantly over time, because I was forced to make certain assumptions. (Specifically, some of my constructions relied on having a set of cohomological Bousfield classes. But, currently we only know that there is a class of them, which isn’t good enough. However, it is an area of active research, and its possible that we’ll know the answer soon. But, the answer might be in the negative: that there is not a set, only a class.) Rather than dwell on proving my assumptions, I continued building sand castles. For months. They were completely rigorous castles, but they were based on a tenuous hypothesis.

2. What relevant interactions with other mathematicians?

I was meeting regularly with my advisor, but less frequently because we were waiting to see if anything was going to pan out. In my half-hearted attempts to get a sense of how reasonable my assumptions were, I emailed a few experts. The responses were mixed.

3. How does it feel, what is the mood?

This was a very playful time. I was pushing my imagination, trying to read the tea leaves. It was summer, so there was a lot of dispersed pondering during hikes and climbing trips.

It felt like I was creating math. Of course, we don’t really create math – we pursue the logical consequences of our conceptual frameworks. But we do create perspectives. We decide where to look, and how to look. No one had ever looked at this puzzle, and so I was inventing a new way of seeing some poorly understood math. This was really fun, exhilarating even. Especially when things stuck together, when I found connections.

4. What state of mind? stable vs. chaotic? focused vs. dispersed?

It was a creative state of mind – very grounded in intuition, pre-conscious or post-conscious. I was conscious of structuring my math process to be very open and free. It didn’t feel chaotic, but everything was very ambiguous and intentionally vague. How do you induce an open, relaxed mind? I do it by going out into nature, emptying my mind, and then intentionally and effortfully pondering.

I don’t think I would’ve very productive, building this theory, if it were, say, halfway through a school term. The clarity and openness of my mind slowly gets worn down as the term goes on. The breaks refresh me. I’m not saying that there’s an inherent conflict between expansive awareness and responsibility, but I personally struggle to balance them.

5. What type of self-reflection during the experience, and did it help?

I felt very sensitive to the quality of my awareness, throughout the day, from week to week. How does a writer know when to sit down and write? How does a musician cultivate the sparks? I felt free to indulge in the most imaginative and hopeful math – even sitting with the question, “What should math be?” This was because of the huge grains of salt I’d swallowed; there was a decent chance that all of this was going to crumble, so why not make it as beautiful as possible right now?

6. An everyday metaphor for the experience?

This doesn’t quite happen every day, but imagine that you and your friend are on opposite sides of a chasm – how do you build a bridge? One way: try to start with a connection – any connection. You could shoot an arrow across, with a small, lightweight thread attached. Once your friend has the thread, you can tie a thicker string to the thread, and he or she could use the thread to pull across the string, which could pull a rope, which could pull across some sort of rope bridge, which you could use to build a more solid bridge.

One challenge: getting the arrow with the thread to make it across. Another: procuring the string, rope, etc, and figuring out how to connect them.

7. An example of a good day and a bad day?

On a good day, I might decorate the interior of one of the rooms in my imaginary castle of sand – maybe find an interesting function between the homological and cohomological Bousfield classes, one that seemed to have nice properties.

On bad days, I might stall on the meaninglessness inherent in assuming so much. Or my constructions, whose existence was at the time only justified by their aesthetics, would seem ugly and trivial.

8. What did you do when you were stuck?

Go outside into the sun and the wonderful fresh Seattle air.

9. When and why did it end?

After two or three months, I reached a point of diminishing returns, and so stopped and switched to projects that had a better chance of being true or false. To date, we haven’t figured out whether there is a set or class of cohomological Bousfield classes, so all the work from those months is tucked away in a notebook, waiting.

Contemplative Education

August 7, 2011

I recently attended a regional meeting of the Association for Contemplative Mind in Higher Education.  Simply put, it’s an organization of college professors that incorporate contemplative practice in their classes.  The conference was very diverse – with people from the arts, social sciences, activist programs, and even a few scientists.  Some talks were detailing success stories, others were participatory and demonstrated specific contemplative practices to use, still others were theoretical and visionary.

Let me back up and explain what I’ve learned about this thing: contemplative education.  First, a quote:

“The faculty of voluntarily bringing back a wandering attention, over and over again, is the very root of judgment, character, and will. . . An education which should improve this faculty would be the education par excellence” 

-William James, 1890

If you believe this, then maybe you’ll believe a second one:

“Universities have forgotten their larger educational role for college students. They succeed, better than ever, as creators and repositories of knowledge. But they have forgotten that the fundamental job of undergraduate education is… to help them grow up, to learn who they are, to search for a larger purpose for their lives, and to leave college as better human beings. So totally has the goal of scholarly excellence overshadowed universities’ educational role that they have forgotten that the two need not be in conflict.”

-Harry R. Lewis, former dean of Harvard College

If you believe the first and second quotes, and you care about teaching, then you might start to wonder if there’s a way to develop “the faculty of voluntarily bringing back a wandering attention,” for example.  There’s a word for this.  Focus.   And there’s a way to become more focused.  Meditate.

If a technique for becoming more “focused” seems like a good idea to you, something worth looking into, but maybe needing some scientific grounding to be more appealing, well, guess what?  You’re living in the right decade.  In recent years, there have been dozens of scientific studies – neuroscience, psychology, health care, education – more or less confirming the claims of meditators.  These (secular and non-secular) meditators base their techniques in millenia-old wisdom traditions from around the world.  Here is one link to a list of research articles and books.  Here is a link to a review of research that pertains specifically to the benefits that meditation brings to a college classroom.

This is not sketchy science.  This is, for example, recurring 5-day workshops hosted in India by the Dalai Lama, bringing together Buddhist scholars and Nobel-prize-winning scientists like (current US Secretary of Energy) Steven Chu.  This is, for example, panel discussions at MIT and Stanford, with thousands of academics attending.  If you believe in global warming, you should believe in the benefits of a meditation practice.

Increased focus is just one of these “proven” benefits of meditation.  Prior to the act of refocusing attention is the act of noticing when it wanders – this is called mindfulness.  There are meditation techniques that improve mindfulness, and then there are meditation techniques that use that mindfulness to improve focus and concentration.  The same techniques will help you to be aware of and in control of emotions (like stress).  And then there are techniques to go deeper into the objects of attention, and for example cultivate curiosity, creativity, open-mindedness.  And there are techniques to foster concern and compassion for those around you.

The teachers at the conference had been using meditation in their classes, with some subset of these “goals” in mind.  As I said, there were many success stories told – coming from chemists, physicists, law teachers, as well as art and social science teachers.  (Here are some syllabi used, in a range of disciplines.)  But many of them have also experimented with other “contemplative practices” – things like contemplative movement or deep listening.  Here is a page with a helpful diagram of the diversity of contemplative practices, and some info about many of them

The post up until now has been discussing a teaching pedagogy, one that I think is fascinating and holds a lot of potential, and I’ll be experimenting with in the near future.  But this blog is supposed to be about research, not teaching.

All the benefits of meditation – greater awareness, focus, balance of mind, insight, creativity, interpersonal communication (and more!) – are yours for the taking, IF you’re willing to establish a personal contemplative practice.  I say this from personal experience, and with the above research articles as empirical evidence.  If you don’t like sitting meditation, then look into one of the other contemplative practices.

I’ve heard two nice analogies for the role meditation might play in a balanced life.

One is hygienic.  You keep your body clean, so you should keep your mind clean.  You nourish and exercise your body, so you should nourish and exercise your mind.  Meditation is a way of clearing out the clutter, of giving wholesome food to your mind and letting it go for a quiet walk outside.

The second is more scientific.  In order to perform experiments, a chemist needs a lab with the right tools.  The untrained mind is unwieldy – easily distracted, prone to dullness, never still but always jittery and burdened.  Meditation cultivates your mind as a tool – steadies it, sharpens it, gives you practice in controlling it.  Of course, traditionally the purpose of this was to allow meditators to go deeper into the nature of reality, in order to find the most universal truths and embrace the world with the most expansive compassion.  But you can use it to do better math, too.