Paper about this blog published in the Journal of Humanistic Mathematics

January 31, 2013

I wrote a paper half about this blog, and half about contemplative education, and it appeared in January’s issue of the Journal of Humanistic Mathematics.  You can download it here.

If you’re new to this blog, the About the project page (to the right) explains what it’s about, and there’s also a handy Table of Contents page.

(Final?) Reflections on the blog

March 15, 2013

I’ve decided to take a (permanent?) hiatus from writing this blog, and wanted to conclude with some reflections and analysis on the posts to date.

A look at the Table of Contents shows that there are 25 flavors and 7 seasons, and a few posts on background and analysis.  I think that these posts capture the main recurring mathematical experiences that I’ve had as a graduate student and now a postdoc.  In a strange way, the blog feels complete, not to say done.  When I reflect on my day-to-day and month-to-month experience doing research math, I mostly find that I’m revisiting experiences, not undertaking new ones.  Am I approaching a comprehensive list?  I’m sure it can’t be so, and look forward to the unknown.

The process of writing the blog has, I think, accomplished the goals I set out with.  Writing posts has forced me to be more self-reflective about how I do math.  This self-reflection has helped me to tweak my process, to be more efficient, productive, and enjoyable.  It has also brought meaning and depth to my academic research life.  Writing posts has also helped me articulate the recurring aspects of my experience, and confirmed for me the stability and intersubjectivity of these experiences.  Although no one else’s voice appears on this blog, in conversations and other mathematicians’ writings I’ve seen parallel experiences described.  I think that other mathematicians would find a lot (of course not 100%!) to agree with in my posts.

If I were to do it all again, I would probably use the same procedure and the same questions for flavors and seasons.  Except I’d change flavor question #6 to “what experience does it resolve to, after how much time?”, and maybe rephrase the questions as actual grammatical sentences.

Where to go from here?  

This blog aimed to articulate recurring math research experiences.  I left out quite a few singular experiences, and a careful description of these might also be interesting.  For example, the night I submitted my first paper to a journal, then spent hours walking the streets in an intense rain and wind storm.  Or when I found out my first paper got published.  But maybe this becomes too subjective and less relevant.

This blog avoided discussing the experience of specific mathematical ideas (what does it feel like to think about a Bousfield lattice? what do you see in your mind? how?).  Doing so could be interesting.

My next step, however, will be learning more about first-person methods for understanding experience.  I’m obviously not the first one to try to carefully describe his or her lived experience, and lately I’ve been finding out about a body of research that establishes specific techniques for doing so, uses these techniques to examine certain experiences and draw conclusions, and analyzes the limitations and pitfalls of these techniques.  I wrote about these first-person methods in a paper I wrote for the Journal of Humanistic Mathematics about contemplation in mathematics.  So I’ve been reading, and will continue reading, these books and related material: On Becoming Aware, The View From Within, and Ten Years of Viewing From Within.  With more theory under my belt, I’m excited to think about how I could integrate this sort of intentioned self-reflection into a classroom setting (as a kind of metacognition), to get students thinking about (and consequently improving) their learning process.

Flavor: Giving a (research) talk.

February 15, 2013

1. What’s going on mathematically?

I’ve been asked to give a talk about my research. After preparing (see Preparing a research talk) for a day or two, eventually I’m standing in front of a quiet room and have just been introduced.

2. What is the emotional and logistical context?

The room is quiet and everyone is looking at me. I may be very nervous, mildly nervous, or not nervous at all. Then I start talking.

3. What thoughts are there?

The mathematical story I start telling is always very integrated with the slides I’m showing, or the material I’m writing on the board. The visuals guide my thoughts, which are translated into explanations and narrations. The translation of my mathematical thoughts to words requires some thinking about the audience, what they might or might not know, what I’ve already said to them, what they’re probably thinking about now. There are also occasional flashes of thoughts that snag on random and inconsequential details during the experience, like I find myself thinking about the color of the board, or where my right hand is, or the sound of a certain word.

4. What quality of awareness?

This is a very, very absorbed state, with little self-awareness. Time doesn’t seem to flow, although I occasionally glance at the clock to notice that it is later than before. I often feel like just a conduit for my thoughts to be spoken, and end up speaking every single thought that enters my awareness (for better or for worse). I feel comfortable but not free, not in control, not myself. My awareness is heightened to an unsustainable level, perhaps like I’m being attacked or about to crash a motorcycle.

The self-awareness I do have is usually about my body movement and tone of language. I wish I was aware of and able to control my eye movement, but I haven’t been able to train myself to do this; I don’t even know where I look, let alone know if I should change where I look.

My level of self-awareness is correlated in an interesting way with my distance from the board/slides versus the audience. The closer I move to the audience, the more I’m aware of being a human who is communicating using body language and eye contact in addition to speech. As I move towards the board/slides, it’s like I get submerged in my own internal monologue.

5. What emotions?

Personally, I enjoy giving talks, and performing generally. I enjoy the adrenaline. During the talk, I’m not very aware of my feelings, except to notice that I feel high on adrenaline. After the talk, I’m always relieved.

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

The talk ends and there are questions. During the questions I’m only slightly more aware of my body language and eye contact, and maybe start to feel more mixed emotions (happiness if the talk went well, or frustration if it didn’t). But the heightened awareness and absorption continues. After the questions, I just want to go be by myself and not think about math.

7. How frequent is this flavor?

Every few months.

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

Since the experience itself is so singularly focused, I think it’s a good idea to sort of debrief myself afterwards. How did it go? I always tell myself that I should’ve been more mindful of eye contact, but of course there are other ways that the talk could’ve been improved. Giving a talk is an art form, that only improves with practice and reflection (if it improves at all). It’s healthy to acknowledge some successful aspects and some unsuccessful aspects. And there are usually mathematical follow-ups to pursue, from the questions and comments.

There are always imperfections in the performance, and it’s useless to dwell on them. Gandhi said something like: “Freedom is not worth having if it does not connote the freedom to make mistakes.”

Flavor: Wandering around (math) for fun.

December 23, 2012

1. What’s going on mathematically?

I’m surveying some new mathematical terrain, mostly out of curiosity and for fun.

As a child and young student, I remember falling in love with math by exploring, and following my curiosity. Once math becomes a vocation, there’s more at stake and one is less free to just flitter about dabbling and seeking math pleasure. But occasionally, I do it anyways. This flavor of math is the grown-up analog of the childhood experience.

2. What is the emotional and logistical context?

There’s time. Perhaps I’m going to a conference, and want to know more about the talks, the speakers, and the content. Maybe there’s a seminar I’m about to attend, and the title and abstract are intriguing but slightly out of my area. Or maybe I don’t have an excuse, and am just curious about something.

3. What thoughts are there?

I use the internet, and my resources are usually: Wikipedia, papers on the arXiv, personal websites of mathematicians, and the Mathematics Genealogy site.

Like a detective, I start with a few terms on Wikipedia. I follow leads, to read about the people behind the ideas (when and where they studied, from whom; where they are now; who else they work with; what else they’ve done). I try to reconstruct a history of the development of the ideas, while piecing together the theory itself. (I find chronology a very useful tool for unpacking knowledge.)

Certain words, terms, and theorems keep appearing, and I try to pin down precise definitions, as well as get a gist of main properties and consequences. Gradually, a rough map of ideas comes together. It is superficial, to a larger or smaller extent, depending on how familiar I am already with the ideas.

The exploration is not completely aimless, unlike the times I start reading the news about Syria, then end up scouring the internet for information about the millions of pounds of chemical weapons the US has dumped in the ocean, then reading about underwater landslides and the currents they produce, and giant squids, etc, etc… I stick to math, and stick to the rough topic I started with, and if necessary I abandon an intriguing thread and return to the starting point.

4. What quality of awareness?

A productive session has a good dose of self-awareness, to keep pulling myself back to the central idea(s). There are periods of absorption, when I try to let my curiosity lead me, as unfettered as possible. These alternate with stepping back, checking in, closing some browser tabs, and returning to somewhere familiar, to head off in a different direction.

There are also good periods of spacing out. After skimming one too many papers, my eyes glaze over and my mind detaches from the reading, to go out on its own adventure. I catch myself staring off into space, like I’m listening to the new ideas resonate within the context of everything else that I know.

5. What emotions?

Getting lost in curiosity is delightful. There’s a bittersweet tug every time I stop myself and return to the central idea(s). The space-out periods are also pleasant, the pure joy of absorbed thought. I think there is value in this type of unconscious percolation, although it is hard to quantify or describe.

There’s a feeling of irresponsibility, that is enjoyable. As I piece together an understanding like a detective, a narrative emerges. I can enjoy the characters and what they’ve gone through, without feeling accountable. It’s like watching a movie of mathematics. I can relate. It is meaningful to hear the human story, and understand the conceptual inter-relations. Even if the content is different in substance, analysis instead of algebra, say, the architecture of the content is still logical and rigorous, and the humans responsible for the development are still members of the same community. I relish the taste of this mathematical essence.

Interestingly, there’s always a gradual trend within me towards anxiety, that I watch and try to keep in check. Inevitably, I’m confronted with the vastness of Math That I Will Never Know, or with things I used to know but have forgotten, or with a long list of articles that I really should read and understand. Traveling quickly and superficially around the mathematical terrain is frightening, just like traveling quickly through any unfamiliar terrain for too long. What’s tragic and somewhat fascinating is watching this anxiety (which I never had during the childhood periods of mathematical exploration) sneak up on me, every time.

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

There’s really no concrete end, since one can always go farther or deeper. So I just decide to stop at some point. Usually after an hour or a few hours. Most times, I will have found some papers that I genuinely want or ought to revisit in more depth, and these get saved as PDFs and put in a certain spot.

7. How frequent is this flavor?

Maybe once or twice a month. I ought to do it more often.

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

In the best cases, I even write out a quick outline or sketch of the ideas that I’ve mapped out; this helps me internalize the exploration, and remember some of what I’ve learned. This could be an outline, a chronology, or an idea map.

This flavor is a great sort of intermission, to working more seriously on some project. So afterwards, hopefully, my mind is relaxed and I feel happy, and I can dive back in to “work.”

In the worst cases, the anxiety catches me and I can’t get perspective on it, so I’m left in a sort of panic, facing the overwhelming deluge of information.

Flavor: Pondering.

December 23, 2012

1. What’s going on mathematically?

I decide to try to see (mathematical) things differently. It’s a relaxed, playful attempt to play with math. Similar to an “intentioned immersion” but looser and less aggressive.

2. What is the emotional and logistical context?

Usually I’m riding on public transit, commuting, in a car, waiting for something, or taking a long walk where I won’t be interrupted. I’m alone. I won’t be able to work very rigorously, so I have low expectations of what will happen; still, there’s an intention to engage math.

3. What thoughts are there?

There’s a range of pondering modes, or games you can play. Here are a few.

Sometimes I take a specific object or idea and try to visualize it. I try to describe its essence. I imagine seeing this image or description for the first time, and see what questions arise.

Sometimes I juxtapose two things via a nonsensical premise, and see what happens.

Sometimes I imagine that I can turn any statement I want into a true statement, like I had a magic truth wand. What would be the most beautiful situation in a given context? What do I really want to be true? And then what are the relevant hypotheses?

Sometimes I construct an internal dialog with an imaginary colleague, or pretend I’m teaching a class.

Again, the basic idea is to play. Rearrange, experiment, start again… but in a light way.

Usually I don’t write, or don’t write very much, so my lines of reasoning can’t get very far. If the game I was playing was like a chess game, then I don’t try to strategize out four moves ahead, but rather just imagine different possible vague developments in the layout of the board.

4. What quality of awareness?

It takes effort, to stay focused on my pondering, and to think in ways I’m not used to. There is a strange juxtaposition of the mathematical headspace and the real world. My perception and awareness jumps back and forth, usually in a disjoint way, between these two realities. So my hearing of the ambient surroundings cuts in and out, perhaps alternating with the sound of an internal mathematical narrative. Or my spacial sense of where I am cuts in and out. Or my visual perception of what’s in front of me gets interrupted by mathematical imagery.

There have been a few times when the two realities sort of coalesced. Rather than a juxtaposition, or cutting back and forth between the two, there was a synthesis. I don’t quite know how to describe these experiences, except as a kind of integration of mind and body.

5. What emotions?

It’s usually fun, and sometimes funny. Sometimes it’s too effortful though, because I’m too distracted. It never feels like I’m getting anywhere, and so it often feels pointless and dumb. But there’s a pleasant feeling that comes with bringing some cognitive depth into a somewhat numbing situation like riding the metro or waiting on line.

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

Often I do get some new idea (of questionable worth). Or a new perspective, or a new question. I think I need about 15 minutes of stable pondering, if I’m going to settle into it and get anywhere.

7. How frequent is this flavor?

Mathematical pondering doesn’t come naturally to me, but I feel like it should. (I ponder other things much more effortlessly.) Maybe once a week, unless I intentionally practice it.

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

I always think that pondering will become easier, or more fruitful, eventually. It seems like the transitions from being in the real world to being in the math headspace, and vice versa, should become easier and easier to traverse. It seems like there’s a lot of insight in being able to enter the math headspace in different ways, at different times, and so pondering is a way of practicing this transition.

I try to write down any insights I gain from pondering, no matter how vague. But other times I give up pondering because it’s hard and doesn’t feel productive, or even fun.

Flavor: Natural inspiration.

November 28, 2012

1. What’s going on mathematically?

Mathematics is overlaid onto a nature experience. Nature seems to present a vague metaphor, that inspires some mathematical insight.

2. What is the emotional and logistical context?

I’m out in nature. Maybe walking, sitting and looking, or traveling. I’m relaxed, and not particularly thinking about math at first, except for some pondering perhaps.

3. What thoughts are there?

It starts with an acute focus on the natural experience. Maybe watching and listening to wind blowing in trees, or looking at clouds, or looking at interference patterns of raindrops in puddles. I’m absorbed in the sights, sounds, smell; it’s a full experience.

As the scope of my awareness expands (see #4), I start to overlap the nature experience with some math idea. This is usually not an explicit direct metaphor (maybe it would be for a physicist, or a mathematician studying geometry or dynamical systems; the math I do is too abstract, too structural to ever really correspond to anything in physical reality). Instead it’s a loose structural metaphor, a vague feeling like nature is presenting me with the secret I’ve been looking for. Often I have the thought that I’m looking at, or experiencing, the Answer, I just don’t quite see how to interpret it.

Mathematicians are only as good as their imagery and metaphors, which allow them to fasten abstract ideas to commonplace intuition. In these moments of natural inspiration, it feels like I’m getting hints of new images, metaphors, and structures on which to hang my math ideas.

4. What quality of awareness?

First there’s a strong focus on physical sensations, and then absorption into those sensations. This brings an expansion of awareness to a totality, engaging all the senses in one singular experience of the moment. Bringing in the math requires a small conscious intention to do so, to bring some math ideas into my awareness. And then I just hold the two flavors simultaneously: the natural experience and the math ideas. Sometimes this juxtaposition takes effort, sometimes it seems effortless.

5. What emotions?

At the start I’m usually relaxed and happy, then become more happy and slightly less relaxed, and towards the end I just feel peaceful, and a feeling like everything is going to be okay.

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

There’s usually no concrete insight, just a sort of spreading out and a new perspective. I hold the juxtaposition for five minutes to 45 minutes (if I’m on a hike, for example), then let go of it. I usually leave it and do non-math things afterwards.

7. How frequent is this flavor?

About once every 2 months.

8. What are good/bad ways to change or follow it up?
As I said, there’s usually no specific insight or answers that arise. Just a vague feeling of fresh perspective and harmony. I think it’s a mistake to try to squeeze some insight out of these vague, almost subconscious, mental interactions. It’s also a mistake to try to hold onto the delicious peace that accompanies them. Better to not try to understand everything, and enjoy the moment as it passes. I’ve always trusted, blindly perhaps, that these experiences make me a better mathematician.

Flavor: Filling out results.

November 28, 2012

1. What’s going on mathematically?

This is a mathematical flavor that happens during a “Pulling together and writing up” season. I have some collection of results that I’ve pulled together, and I’m trying to turn it into a coherent paper or talk. To turn it into a whole, instead of a bunch of pieces. In a good paper, the results are “complete” in some sense. But most of the time, math sprawls continuously off to the horizon. Choosing a paper-size chunk of terrain, and trying to develop it into a coherent whole, is what I call filling out the results.

2. What is the emotional and logistical context?

There’s usually some excitement that things are coming together. For me there’s always some confusion about what’s interesting, or rather what will be considered interesting to others. There’s usually time pressure from a deadline.

3. What thoughts are there?

There’s an element of logic, maybe even necessity. On the one hand, I need to know where to draw the line. To make up an example, imagine that I proved something is true for every positive value of some parameter n. I could stop there, or I could work for a few months longer to try to prove it for every negative value as well, or every real-valued n, or complex-valued n, etc. At a certain point, I draw the line somewhere. This decision takes into consideration the background required, and the proof methods used, and is usually a straightforward decision. On the other hand, maybe I know n can only take values 0, 1, or 2. If I’ve only addressed the n=0 and n=1 case, there’s a feeling of necessity that I should consider the n=2 case. If it is significantly harder, or different, or uninteresting, I should at least mention that this is the case. I think every mathematician would agree that omitting any mention of the n=2 case would be a shortsight.

But mostly the thoughts are centered on aesthetic considerations. The paper needs to “flow”, to be “complete”, to go “far enough” but not “too far” (not to mention the proofs must go “deep enough”, but not “too deep”). These are all culturally-defined aesthetic qualities, that nevertheless most mathematicians would, for the most part, agree on. You know it when you see it. When filling out results, the challenge is that at first you don’t see it. The results are incomplete, and your job is to complete them.

4. What quality of awareness?

Very fluid and open. I’m trying to step back and see the collection of results as a whole, possibly for the first time. It takes an open, flexible mind to see the best way to organize the ideas. Or rather, it’s more like the ideas self-organize into a natural flow, if I can only hold them all in my mind at once, in a big open awareness. Filling out the results means, while holding that big awareness, also noticing all the dark areas that need to be explored, or at least addressed, to complete the whole.

5. What emotions?

Manipulating my own results is always emotional. Often some hard-won proofs are subsumed, or irrelevant, or improved upon. Filling out results means identifying holes and small-minded reasoning, in other words, flaws. This is the phase when the ideas and proofs are depersonalized and objectified as much as possible, and it can be heart-wrenching.

Also, probing the aesthetics engages my emotions, as I try to decide when enough is enough, when results are interesting or complete rather than irrelevant or partial. In general, mathematical exposition is really an art form.

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

Personally I haven’t had many hugely climactic results. So after I’ve filled out the results I do have, there’s a feeling of having drawn an arbitrary line somewhere, and there are always copious new directions to pursue. See also the post on “Pulling together and writing up.

7. How frequent is this flavor?

This happens in the lead-up to every paper, write-up, or research talk.

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

I’m not very good at drawing a line and deciding enough is enough; I’m more inclined to keep proving and proving, until I really get a big tangled mess. Recognizing some necessary arbitrariness is healthy, then.

The aesthetic judgement part is not easy either, and I can get confused and frustrated. It’s helpful to step back and appreciate whatever nice flow of results is already there, and recognize that the sense for mathematical aesthetics is only grown slowly, through practice.

Flavor: Making a big mistake.

October 23, 2012

1. What’s going on mathematically?

Someone points out that I’ve made a very big mathematical mistake. Perhaps in a paper that I’ve submitted, or during a research talk. Mathematicians are devout truth-seekers, and mathematical truth can be harsh.

2. What is the emotional and logistical context?

The context is social. I’m presenting what I believe to be correct, with confidence. Someone points out a serious flaw.

3. What thoughts are there?

The flaw is usually presented with a counterexample. So first there is cold certainty (of the mistake), and then along with an unpleasant emotional response there are non-mathematical thoughts about the non-mathematical implications. Does this mean the paper is junk, or the theorem fails? Maybe I won’t get a good letter of recommendation then? Does this mean I’m actually an idiot and should quit mathematics? How can I acknowledge the mistake and recover in a way that saves some face?

4. What quality of awareness?

There is a shock and a vivid immediacy. I feel acutely aware, but with an instinctual fight-or-flight quality. My thoughts actually move very slowly, and I find myself focusing acutely on a single symbol on the blackboard, the texture of my seat, or my breathing.

5. What emotions?

There is usually an unpleasant mix of embarrassment, terror, panic, and disgust. Once there was a feeling of being betrayed. There is also a strong feeling of surrender and letting go, which is maybe the silver lining of this mathematical experience. I’m forced to surrender my pride and a bit of ego, in the face of incontrovertible mathematical truth.

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

It’s necessary to immediately admit that a mistake was made. There may be an effort to fix or learn from the mistake, but it might be clear that there is no fix. Some face-saving gestures, some wound-licking.

7. How frequent is this flavor?

Maybe once or twice a year for me. Of course, there are smaller mistakes all the time. I wonder if there are mathematicians that have never made such a big mistake.

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

There can be something very freeing, about surrendering pride and ego and accepting reality. Mathematical truth is uncompromising and absolute, and mathematicians are harsh truth-seekers. When I can step back from the frustration and embarrassment, I can watch my ego dissolve a little, and find new freedom in the experience. When my mathematical world has just broken so dramatically, it seems like the real world is going to break — and yet it doesn’t.

The worst thing to do is get caught up in the emotions, to the detriment of others. Sometimes I find myself getting angry and trying to place blame on someone else. In the end, however, it’s absolutely clear who is responsible for the mistake: me.

I’ve had mistakes pointed out graciously and ungraciously, and there’s something to be said for compassion and understanding in these moments. Doing mathematics means being stuck, confused, and wrong most of the time. It is our job to clarify and get to the bottom of things, even if that means pointing out a fatal mistake to a colleague. By going through the process of making a big mistake and having it pointed out, I’ve developed a little more empathy for others in the same situation.

Season: Applying for jobs.

October 23, 2012

1. What mathematical activities? What level of rigor?

The math job application process is described in more pragmatic detail in many places, for example here or here. This post will focus on the interior experience of applying. You must write a research statement, which is a sort of demonstration of ability with rigorous logic. You must write a teaching statement, which is less rigorous and more personal. There is a lot of non-rigorous soul-searching, about what job would be best, what the future might hold, is mathematics really worth doing?

In a way, you apply your mathematical problem-solving skills to the problem of getting a job. This requires researching different mathematical lifestyles, asking yourself good questions, methodically clarifying your wants and abilities, and articulating these in the application materials. It’s a sort of math problem, but with your life.

2. What relevant interactions with other mathematicians?

As a grad student, it’s not clear what life is like as a postdoc, or a professor, or someone working in industry. So there’s lots of question-asking. Getting career advice is also fun. Maybe editing and proof-reading help.

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

For me there were lots of ups and downs. It’s necessary to balance an artificial, constructed hubris with humility and longing. I could work myself up into a high mood of idealism, dreaming about the future, being excited for change. I could also exhaust myself with low moods of fear, competition, existential crisis, and administrative fatigue. It’s important to be passionate and indifferent at the same time.

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

In fact, my mind was focused on the applications, a determined focus to make the best applications possible. There was not much clarity about my desires and hopes for the future, more of a two-month tunnel vision centered on application paperwork. I figured I could apply first, and then think about what I wanted; I don’t know if this was the best strategy. Gradually, this determination and focus left my mind confused and drained.

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

It was extremely helpful to recognize that I would have to balance artificial, constructed hubris with my more genuine and self-reflective longing and confusion. Holding this contradiction in my awareness, I could present myself as passionate, confident, and all around really awesome, and could get excited for a range of possible futures, while also recognizing how humbled and confused I also felt. In a sense, I could self-reflect on the self-reflection that the job application process was stirring up, and give it a space to unfold. Gradually, on its own schedule, some clarity did emerge about what I wanted for my future.

6. An everyday metaphor for the experience?

Everyone (hopefully) knows what it’s like to apply for a job. However, applying for an academic math job is a little unique. I applied to over a hundred positions, and received rejections that explained they had over 600 applicants for a single opening. So chances there were worse than flipping a coin and getting 9 heads in a row.

Furthermore, the job cycle is almost a full year. So applying for an academic math job is like writing someone else’s application for a job that starts a year later. Who will you be and what will you have actually accomplished by the time that job starts? That’s who you must present in the application.

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

On a good day the application material looks good and well-written. On a bad day, it feels like I’m spending hours applying for jobs I don’t want.

8. What did you do when you were stuck?

It was always nice to return to doing actual math, research or teaching, rather than just writing about it. I found it really helpful to discuss plans, and vent, with friends and family who knew me well.

9. When and why did it end?

No matter how exhausting and nerve-wracking the application process is, at least it has specific deadlines. You know it will end at some point, hopefully with an accepted job offer.

Season: Rest.

September 30, 2012

1. What mathematical activities? What level of rigor?

These blog posts attempt to describe the common experiences of a research mathematician… all of them. Most mathematical seasons seem to involve hard work, even overwork. This post is about the necessary seasons of rest and gentleness. They occur during vacations, or after significant accomplishments (e.g. passing qualifying exams, or finishing a PhD). It’s a time to recharge the batteries, but there may be some light math — reading blogs or news, checking out what others have been working on, maybe attending a conference as a participant. However, of utmost importance is allowing oneself to “do nothing,” which is hard! Low rigor, high relaxation (hopefully).

2. What relevant interactions with other mathematicians?

Almost none, except perhaps social interaction. If I’m interacting mathematically with someone, I’m not resting.

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

Relaxing… great!

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

Any state of mind that is a release, or distancing, from mathematical states of mind. This is a chance to get some fresh air, and in my experience this means not just thinking about non-math things, but actively experiencing fresh and different states of mind. So I often go traveling, which often induces a chaotic, dispersed being in the world. Or I focus on friends and family, which is a chance to stabilize and concentrate on meaningful non-mathematical things.

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

Mathematicians need to develop not just mathematically, but also emotionally, morally, politically, culturally, etc. For me, a period of rest is an important time to ask, “Who else am I?” and “How else can I engage the world?” So there is self-reflection about the world and my place in it.

There is also much reflection about my mathematical lifestyle. When I rest, I reflect that my life in math mode seems unsustainable, in one way or another. How, besides taking periods of rest, could I change to make it more sustainable? Should I do less (one fewer math-art collaboration?), or do things differently (more sleep, less coffee?), or shift priorities, or change my short- or medium- or long-term mathematical goals?

Furthermore, there is usually some good self-reflection about the threads of mathematical content in my life. Was that last proof really the best one I could find? What do I think is really going on with that circle of ideas? I have some of my best mathematical clarity during vacations, or in the first few weeks after I return to working. I’ve learned to gently introduce this pondering, towards the end of a rest period.

In a sense, I rest precisely because I need to self-reflect in these ways. If you’re not self-reflecting, you’re not resting properly.

6. An everyday metaphor for the experience?

Of course, everyone knows what rest is. Imagine how you feel after staying up late, or through the night, or staying up for two nights in a row — that closed, confused, muddy state of mind and rigid, almost non-sensical, way of thinking. Imagine then the surrender to deep sleep, and then the wonderfully free and light mind of waking up, really rested. So important! Don’t let anyone convince you otherwise!

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

On good days I am gentle, and allow myself to rest. Maybe it’s even a “productive” rest, in which I can contemplate the bigger picture of my mathematical doings. On bad days, I don’t allow myself to do this.

8. What did you do when you were stuck?

If I catch myself working when I should be resting, or not allowing myself to rest, I remind myself that gentleness is a virtue, too. Although it might not be easy, it’s something worth striving for.

9. When and why did it end?

Vacation is over.