Season: Entering a Field.

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.


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