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.