Flavor: Finishing the PhD.

1. What’s going on mathematically?

I graduated with a mathematics PhD.

2. What is the emotional and logistical context?

Many years (for me, six) had gone by since I signed up for this. The last few involved working for hundreds, probably thousands, of hours on writing a dissertation that a handful (really!) of people will ever read.

3. What thoughts are there?

For me, “I am invincible” and “It’s over.”

4. What quality of awareness?

Sublime, peaceful, and quiet. My mind was light and free, as a huge burden was gone. For the first time in six years, there was no work to do, and I could let go.

There was also an extreme intimacy with my own mind and being. Remember the scene in Star Wars, when Luke Skywalker is the last hope for destroying the Death Star, and at the last minute he disconnects himself from his radio and homing equipment, to just use the force? During that last act, he is alone with himself. I thought of this scene often, and it captures my mindset from the time right before my dissertation defense, up until the graduation receptions I attended.

5. What emotions?

My operational metaphor during years 3 – 6 of graduate school were that of a pine tree in the Cascades during winter, burdened with snow but beautiful. Finishing the PhD, the snow fell to the ground and the bough rebounded and oscillated with uncertainty. I felt expansive elation and lightness, and spent hours crying. There was a lot of deep sleep. Eventually, there was several weeks of stratified relaxation, during which I realized just how tense and focused the lead-up to graduation was.

Handing in the dissertation, there was a feeling of invincibility but also a sort of creative ecstasy. Condensing so much effort into a creative act and object brings a deep joy and feeling of meaning.

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

There’s more work, eventually, and next steps. After emailing out my thesis to folks I thought might care, I took a few months off.

7. How frequent is this flavor?

Once, unless you really want to do it again…

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

I think you’re entitled to do whatever you want, no questions asked, for a little while.

Flavor: Preparing a (research) talk.

1. What’s going on mathematically?

Often I have to give a talk about my research, or a survey talk about relatively advanced mathematics. This post is not about giving the talk, but preparing it.

2. What is the emotional and logistical context?

Before starting to prep, there’s always some degree of nervousness and anticipation, depending on how important the talk is and how comfortable I am with the material.

I’ve experimented with a wide range of logistical contexts for prepping talks. This is one experience for which the math culture permits a significant amount of flexibility and alchemy. Among mathematicians, “He/She has to prepare for a conference talk next week” translates to “he/she is going to be acting a little strange; give him/her some extra space.” I’ve tried locking myself away until it’s done, or prepping during a long hike without writing anything down, or sitting in front of a waterfall and practicing my words for hours. There isn’t much common context.

3. What thoughts are there?

The goal is to develop a human connection to the content of the talk, in order to figure out how to communicate the content so it can be best understood by the range of people in the audience. There is usually some learning and relearning of material, piece by piece. But the biggest challenge is in reorganizing the ideas into a story that is linear enough to flow as a narrative, but nonlinear enough to convey the robust intuitive interconnections. The second-biggest challenge is aiming for your audience – including the right balance of detail and metaphor, rigor and exposition.

A good starting point is immersion in the subject – getting comfortable with all the important ideas, the different perspectives on those ideas, the history of the ideas. I recently spent 15 hours preparing for a one-hour talk. Then it’s necessary to zoom out, try to see the big picture, and from a good vantage point to construct your story. Towards the end of the preparation, the talk becomes more gestural in my mind – it has organized itself into parts, with transitions, and an engaging structure and flow. The talk is ready when I can see it all in my head – all the information batched into articulate clumps, which are batched into sections, each section occupying a place in a conceptual outline whose shape I see clearly in my head.

4. What quality of awareness?

The hardest, but perhaps most important, task is to constantly and consciously shift between the different levels of detail. Zooming in, zooming out – the talk needs to work on multiple levels, so that the audience can comfortably engage the talk and follow it, from a range of backgrounds and focus. It’s easy to passively let the math lead you, while prepping a talk, but the result usually lacks perspective and is boring.

I try to see the content for the first time, to be my own audience. I try to forget that I already know how the story ends, so I can perfect the story-telling. It feels like trying to hear your own voice, from outside your head.

Each expository challenge starts with a tension, which is broken by a moment of freshness and newness, which then rearranges itself into an enjoyable obviousness.

When the talk is ready, I feel like I’ve transcended the math/non-math boundary. I’m holding, in my mind, an object (the talk) that is mathematically rigorous and true, but crafted to interface with the multi-dimensional, non-rational, colorful and robust real world.

5. What emotions?

There are the emotions associated with the impending performace – fear, adrenaline, excitement, nausea. The preparation itself is, while usually an anxious experience, also very pleasant. It is a gradual shift from confusion to certainty, confidence, familiarity, and friendship. They say that you don’t really understand something until you’ve tried to explain it to someone else.

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

It might take 30 minutes or 15 hours to prep the talk, but it’s clear to me when it’s done. I stop thinking about it, move on to something else.

7. How frequent is this flavor?

About four times a year. There are lots of lectures to prepare when I’m teaching a course; the experience is similar to prepping a research talk but much more mellow, since I’ve only taught 300-level courses.

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

I have a strict post-talk-prep routine, which consists of actively doing nothing. I’ve found it’s important to rest, not worry, get sleep, do enjoyable things like listen to music or eat, maybe meditate or get some exercise. The talk is prepped; there’s nothing to worry about. It’s out of my hands, and now my job is to be the best performer I can be. If the talk was prepared well, the performance of the talk is really fun. The worst thing to do is to indulge in nervousness and anxiety.

Flavor: Mathache.

1. What’s going on mathematically?

I’ve been doing lots of math, maybe too much: research, writing up a paper, teaching, applying for jobs, other projects. There’s a lot more to do. I stop to self-reflect.

2. What is the emotional and logistical context?

Maybe after a long day of conference talks; or a frustrating afternoon of getting nowhere with research; or late at night when my brain has stopped working and I should be done by now, but I have more to do.

3. What thoughts are there?

My head feels saturated with information, strained from too many self-imposed cognitive tasks. I keep thinking of how many more things I have to do. Often there’s a frustrating problem that I’m stuck on but can’t seem to give up. Usually my body is struggling as well as my mind, and I start to think about my physical aches and discomforts.

4. What quality of awareness?

I’m existing very shallowly. Even if I do non-math things, there’s such a loud noise in my head, of math ideas chasing each other, that I can’t focus on or process life much. The noise has a wide spectrum of frequencies – some of it conscious math thoughts, some of it low-frequency “percolation” (my word for when a math idea hijacks mental bandwidth for an indeterminate amount of time, for mostly-subconscious learning and processing). I’m very aware of physical discomfort and fatigue.

It reminds me of the drained and saturated feeling after a long day of socializing and talking to people, when you want to lock yourself away and listen to silence. But it’s hard to lock the math out of your head.

5. What emotions?

I feel pain, tightness in between my shoulders, often a headache or stinging eyes. Tired, drained. Disconnected from real life emotions and experiences. I feel behind, and sometimes like I’m drowning. It is a particularly physical math experience, and an unpleasant one.

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

Eventually things get done, or I sleep, or take a break. somehow my cup gets emptied a little. But it might take a while, and it might get worse first. There have been points of graduate school that really tested me, with burnout or breakdown a nebulous possibility.

7. How frequent is this flavor?

Sometimes for days or weeks on end. As a flavor, it usually comes in the late afternoon, maybe five times a month.

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

One thing that helps is physical rest – laying down, consciously fixing my awareness on my breath and on relaxing my body. Meditation is one of the best things to do, although depending on the state of my practice and the degree of mathache, sometimes I fail at dissolving the anxiety. Sometimes a good cry seems necessary, and makes me feel better.

The worst thing is to let it take over, and to wallow in mathache. My cutoff point is societal: if I start to become a mean person, then it’s gone too far.

Flavor: Discussing with a colleague.

1. What’s going on mathematically?

A live conversation with a colleague, about research. For example, with my PhD advisor, or someone at a conference.

2. What is the emotional and logistical context?

The context is pleasant. We’re sitting together, with paper and pen or at a chalk/whiteboard. I’m mildly prepped, with comments or questions. There might be coffee. There is always time, and patience.

3. What thoughts are there?

A dynamic back and forth, sharing and building, going beyond either individual. We express old ideas, new ideas, shared ideas. Sometimes there is a bit of translation involved (e.g. topologists say “finite” and “smash”; algebraists say “compact” and “tensor”), or effort in communication. But overall the thoughts themselves seem to move unhindered in our shared collective mind, via common mental imagery. There is a lot of mathematical “body language” – conceptual shorthand, written scribbles and diagrams, and physical gestures that convey so much.

4. What quality of awareness?

My awareness is always sharp, like lightning. I feel like I’m reading someone else’s mind, seeing inside their head. Often I’m simply absorbed in the moment, without self-reflection; there’s nothing else around, no time, no bodies. When I talk with my advisor at our weekly meetings, I have enough perspective to reflect on the experience while it’s happening, to witness the mind-meld from outside as well as inside.

5. What emotions?

These discussions are usually exhilarating; I’m on the edge of my seat. There’s a deep, deep pleasure in connecting and speaking the same esoteric language, especially with someone who is a stranger in so many other ways. The math is a strong bond, of a common research philosophy (of how to think about things), common research modes (of how to go about doing research), and common upbringing (of learned content). The precision of our language allows us to go very deep very quickly, in spite of other cultural differences.

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

There are usually some of the following: new answers, new questions, new directions, and/or new perspectives.

7. How frequent is this flavor?

Usually once a week, with my advisor. Possibly twice a day, at a conference. During my recent Solo Math Intensive, I only video-chatted once a month.

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

It’s helpful to quickly review any notes from the discussion, to document the new ideas, add things to a To Do list, or hunt down new references. Getting frustrated, intimidated, or discouraged is very unproductive.

Season: Crescendo.

1. What mathematical activities? What level of rigor?

Everything is building. I’m writing papers, the research is coming together. I’m seeing new vistas, with new leads to follow. And there are other projects – writing for blogs, applying for jobs, teaching, etc. There are moments of pure rigor and moments of rigor-less scheming.

2. What relevant interactions with other mathematicians?

As many as possible – with my advisor, with other grad students, with others in my field.

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

I’m unnervingly busy. There’s too much to do. It’s exhilarating but tiring. I’m anxious, and slightly worried about where this is heading, worried that I’ll burn out. I want to be available to allow the math research to grow and spread as it wants to, but it’s growing faster and faster and demanding more and more.

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

My mind is constantly on, and needs to be constantly on. There’s no chance to be dull or to drop the ball on anything; I feel like I’m performing 24/7. I’m asking for an on-demand delivery of focus or dispersion, stability or chaos, and my mind is indulging me. For now.

There’s a constant noise in the background, of unattended-to things, of lists.

Occasionally I find myself staring at stable geometry – staircases, buildings, trees – and finding comfort in the solidity of their existence.

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

I notice a growing demand for personal time, to process my experiences. I need to find new time-management schemes. I need to consciously leave behind certain things – research directions or interesting projects. I’m aware of the threat of burnout, and am able to mindfully shift to new modes, to rest or actively refresh the different parts of my mind.

I’m amazed at just how much I can push my mind – how much new and old information I can juggle at once. It helps to appreciate this, to be grateful and positive.

6. An everyday metaphor for the experience?

It reminds me of building a sand castle. If you want to go higher and higher, your castle needs to get a bigger and bigger base. But the relationship is not linear; consecutive increases in height require larger investments in the growing base. I also have the image of pinching a sheet of fabric and pulling it up. As my research progresses, I need to call on a larger and larger circles of knowledge. As my various projects continue, their demand on resources seems to increase quadratically or exponentially.

In terms of state of mind, I think of a cup that is full of water, that threatens to overflow. I’m constantly intentionally emptying the cup, and more water is constantly pouring in. The situation can be sustainable and reach an equilibrium, as long as the emptying and filling rates are balanced.

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

On a good day, I get a lot done and have the time to appreciate it. Sometimes the math itself fuels me and no effort is required.

On a bad day I push myself too hard, or feel like the math is in control of me and I can’t say no.

8. What did you do when you were stuck?

Rest is always good, or giving myself a pep talk, or positive affirmation and gentleness.

9. When and why did it end?

This season is going on currently. I hope that after I get these few papers out, write my thesis, and get a PhD and a job, then there might be a decrescendo season.

Season: Doing computations.

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.

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.

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

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.

Flavor: Breakthrough I: Getting stuck.

(The four posts in this series describe the steps in a breakthrough.)

1. What’s going on mathematically?

Sometimes it seems that, in order to have a big breakthrough, I need to have something big and thick to break through. These bursts of insight are always worth the effort, but, like walking a sawtooth function, playing this game means spending most of my time enduring the growing confusion and stuckness.

The spiral into stuckness is subtle. I’m working on some train of thought, but keep hitting dead ends or gaps in my understanding. I shift my perspective, try new examples, zoom out or zoom in, try to fill some gaps. At first this works, and I proceed a few more steps. But over time the dead ends and dark confusion surround me, and I’m totally stuck.

2. What is the emotional and logistical context?

This happens so often, in so many different contexts. I’m just sitting somewhere doing math. I could be happy or sad or relaxed or stressed. It could be morning or night, in a coffeeshop or in the back of my jeep or in a park.

3. What thoughts are there?

Over the course of several hours, I follow many lines of thought, sometimes striding through familiar territory, sometimes treading carefully around or through dark spots. Gradually the land becomes stranger and the visibility decreases. The dead ends and darkness become more and more frequent.

4. What quality of awareness?

Again, a wide range of possibilities here; it starts out as the generic research experience – I could be dull or sharp, open or muddy. There is an observer, the Teacher-in-my-mind, that watches as I venture into the new territory, keeps track of the growing confusion, where it is, what the fringes taste like. There is a mild awareness of where this may lead (see Breakthrough II-IV posts), but the Teacher allows the Student to innocently follow its curiosity.

5. What emotions?

As an explorer, I relish the new territory. When it becomes clear that I’m getting more and more stuck, I start to get a little frustrated. I start to notice things like hunger or body ache.

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

It’s possible that I’ll figure something out – I’ll successfully cross into the unknown and come back unscathed. But this amounts to avoiding getting stuck.

Sometimes when I’m stuck, I’ll give up immediately, perhaps deferring further progress until after I’ve talked with my advisor. But then I miss a potentially big breakthrough…

Sometimes I stick with it.

7. How frequent is this flavor?

Most research sessions end with me getting stuck to a greater or lesser extent, so this happens between 2 and 10 times a week.

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

See Breakthrough II post.

 

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Connections:

Thomas Edison is famous for saying, “Genius is 1% inspiration and 99% perspiration.” But he didn’t say where the inspiration shows up among the perspiration. For an inventor I imagine it’s close to the beginning – you have an insight, and then work hard to make it a reality. With math, the inspired idea seems to come more often at the end of a long period of hard work and stuckness.