Archive for the ‘seasons’ Category

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.

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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.

Season: Crescendo.

September 12, 2011

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.

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.

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.

Season: Pulling together and writing up.

May 20, 2011

1. What mathematical activities? What level of rigor?

After a period of creative research (of days, weeks, months,…), it’s necessary to consolidate and pull together results. This involves very carefully retracing steps, chronologically, and lining up ideas and proofs. Initially, results are scattered throughout my notes; or proofs haven’t been written down; or some statements are wrong, or outdated, or improved upon.

Because the original path to the result is almost always not the most direct, everything must be restructured. The goal here is to present ideas and proofs in the conventional form, an explanation to a particular audience. So there is a pure logic component, of lining up arguments correctly, but also a conversational component, as I decide how much detail to include, how much exposition, how much rigor.

This task is relatively easy and straightforward. It can feel administrative at times, for example when compiling a list of references. Virtually all math is type-set in Tex, so writing up involves hours and hours of typing Tex code, which is not very intellectually gripping.

I try to keep a running list of random ideas or questions that pop into my head as I’m writing something up. But I won’t pursue these until I’ve finished, since switching back and forth seems to make the writing up process less efficient.

2. What relevant interactions with other mathematicians?

This is maybe the most independent extended math experience I know. I might need to check some work with someone else, but presumably at this stage I’ve already solidified the results. It’s helpful to ask for tips on Tex syntax. I might have someone check that I’ve included the right amount of justification and exposition for my target audience. When submitting a paper, there is a well-established process of refereeing, which involves recursive feedback and reworking, and this can drag out past any self-contained “writing up” experience.

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

Pulling things together can be affirming, and satisfying. It feels good to solidify knowledge. Writing up can be relaxing, or mildly frustrating. It’s unnerving when I find a mistake I made a long time ago, and have to fix it.

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

Pulling together feels easy, methodical, and uniquely compartmentalized. I only need to worry about one proof or handful of ideas at a time, and can safely ignore the periphery. Of course, I try to stay open to the occasional random new idea or question, but I intentionally stop my mind from wandering too much from the concrete task at hand. Writing up results is conversational and performative – my mind traces through the ideas as though I were explaining them out loud, in real time.

This kind of math is also relatively easy to turn on and off. Sometimes while doing it my mind wanders from math, and I get lost in a daydream. It’s maybe the closest that math comes to being a “day job.”

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

As mentioned above, I try to keep a balance between capturing any possibly-valuable peripheral thoughts, and not getting too distracted from finishing the write-up. So I allow myself to use the restructuring and reviewing as an opportunity to gain perspective, but this perspective only comes if I keep some distance and don’t get wrapped up in following new leads. Maintaining this balance requires some self-reflection. In fact, it seems that the better I’m attuned to this balance, the closer I can get to simultaneously maximizing perspective and efficiency.

6. An everyday metaphor for the experience?

Pulling together and writing up is just like washing dishes. The goal is to sanitize all the mess of discovery, and to dry off any trace of the restructuring. We present a stack of dry, clean, glistening ideas, full of order and necessity, untouched by humans. These spotless ideas are complete in themselves, but sit ready to be used and rearranged as vessels and tools for someone else’s new mess.

The dish-washing process is narrative (to me), relaxing, and mechanical. I can let my mind wander, to some extent. There are definitely more efficient and less efficient ways to do it.

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

A good day ends with a few new pages of nice, clean Texed math. On a bad day, I find a gap or hole, and can’t fix it.

8. What did you do when you were stuck?

Getting stuck might mean finding a gap in some argument; this needs to be fixed. Or it might mean that I lost or can’t find some proof, so I have to reprove it. Or it might be that I don’t know the Tex syntax for the symbol I want, which means I have to hunt through Tex documentation.

9. When and why did it end?

It ends when the results typed up and pretty.


Connections:

Terry Tao has advice on writing mathematics.

Season: Entering a Field.

April 27, 2011

1. What mathematical activities? What level of rigor?

This experience lasted from the time I began reading courses with my soon-to-be PhD advisor, until right before my General Exam. My activities were divided into two main types.

First, I was loosely trying to map out the new terrain. It was like someone dropped me in a new city, with a bicycle, and said, “You have a week to map out the whole city.” In a non-rigorous, and sometimes cursory way, I was rapidly traversing whatever conceptual avenues presented themselves to me. Over time, the same words and references kept popping up, and I built a rough map of what ideas were important nodes, what papers or people were central. It was helpful to build a historical/chronological map of the ideas, in addition to my content-based map. This exploration was done largely online, but also through talking with professors and other grad students.

Second, I was endeavoring to slowly, methodically build a solid foundation of understanding. In an extremely rigorous, pedantic way I inched my way through book after book, paper after paper. I tried to do every exercise, tried to understand every clause of every proof. Each field has its own collection of common proof techniques and tricks, and it’s necessary to get a functional grasp of these tools, add them to your toolbox.

2. What relevant interactions with other mathematicians?

Frequent advisor meetings were important, for pointed questions on proofs (“How does he get from here to here?”), as well as vague intuition-building ponderings (“How important is it to localize at p?”). Other professors and grad students were great at filling in a picture of what matters, what people care about, what’s really going on.

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

This was a fun and exciting time, since I enjoyed the field (algebraic topology) that I was entering (see #5 below). In the first case, it felt like I was an explorer in a new country, trying to understand the history, landscape, important people and places. In the second case, I was starting the foundation for a new mathematical castle, and was finding the material understandable and aesthetically beautiful. Having a solid, comprehensive understanding of any subject is an intellectual’s dream.

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

In the first case, my mind was very chaotic, mercurial, dispersed, and outwardly oriented. My goal was breadth and big-picture understanding, which is like flying above the forest with nowhere to land. Nothing really made sense, I was just exposing myself to what was out there and trying to get used to it, trying to accept whatever pathways and features were laid out below me. This was unsettling and in some ways very shallow and unsatisfying.

But I complemented this with the second approach, of depth and rigor. My mind was like a frog or snail, slowly coming to understand a few trees in the forest.

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

I recognized these two very different approaches, and how they were complementing each other. This allowed me to intentionally play them off of each other. Reading history and mapping the social network of algebraic topologists would unnerve me as being shallow and “un-mathematical”, and I would switch to crawling through some proof or exercise. If this began to feel hopelessly specialized, I would switch back to a broader perspective – and I would find that studying one tree had helped me to speak the language of the whole forest, and now my broader study was yielding more fruit. Back and forth, between breadth and depth, using self-reflection to decide when it was time to switch. (As we said on the Appalachian Trail, “walk until you feel like stopping, then stop until you feel like walking.”) If math is too broad and shallow, or math is too deep and specific, it leaves a bad taste in your awareness.

I think the shallower, more holistic type of understanding is often under-appreciated by mathematicians, or at least no one taught me that I should intentionally pursue a big-picture map. (On the contrary, a professor once told me, “The thing about you, Luke, is that you’re not really a mathematician, you’re more like a spectator of mathematics. You’re really enthusiastic about knowing all the stats of all the players, but you never get out on field.”)  Self-reflection showed me this prejudice about what it means to “understand,” and helped me catch myself when I was getting unnecessarily unnerved.

6. An everyday metaphor for the experience?

This season was very much like the first weeks and months of a new relationship. There’s lots of getting to know each other, lots of cans of worms that need to be  opened one-by-one and dealt with, lots of snippets and intuitions that need to be gradually pulled together to form a holistic map. There are both chronological and as-you-are-today maps to be sketched. Shallow, yet broad, shared experiences complement those that are deeper and more focused.

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

On a good day I would make progress on some paper or book I was reading, and then sit down and have a Wikipedia wandering session, and find that I was recognizing more and more of the words that showed up. On a bad day, I would get stuck on a proof and be too stubborn to give up, or would procrastinate facing the rolling-up-of-the-sleeves-anxiety that follows me everywhere.

8. What did you do when you were stuck?

I would switch between the two approaches (breadth and depth), and wait until my next advisor meeting to move forward.

9. When and why did it end?

In a sense, you never really stop learning about your field. But the initial culture-shock phase of mapping out a foreign land has ended, as has the original foundation building. I do occasionally visit neighboring cities and try to map them out, or methodically add a new story or wing to my rendition of the algebraic topology castle.