At the moment I have about six big projects going.
1. Teaching
Various courses at the University of Southern California.
2. Conventional research
My conventional research is in homotopy theory, algebraic topology, homological algebra, derived categories, and tensor triangulated category theory. Lately I've been interested in localized categories of spectra, such as the harmonic category, looking at localizations and Bousfield classes there. One dream is to understand the Bousfield lattices of categories of module spectra.
I've also been getting into topological data analysis, and the ways that category theory can be applied to study "the shape of data". Read more in my year-old research statement. My GitHub page has several sample analyses, and interactive visualization software I wrote for playing with persistence barcodes (see a screencast and screenshots).
3. User's Guides
A user's guide is an expository paper accompanying a published research paper, providing additional metadata on the results and their experiential context. Five mathematicians work together to write user's guides for their papers, collaboratively editing and refereeing a compilation that comes out each fall. The first issue came out October 2015, the second in September 2016, and the third is in progress. I'm the creator and head editor.
The website for the project is mathusersguides.com.
4. Mathematical phenomenology of groups
I'm working with a philosopher of mathematics and ex-physicist, Alexandra Van-Quynh, on a project in mathematical phenomenology. Alexandra's recent work has used phenomenological methodology to investigate what it is like when a mathematical intuition arises in a working mathematician, and is fascinating (see her recent paper).
Our project looks at at the mathematician's lived experience of groups (the kind from abstract algebra). We conducted elicitation interviews, in the tradition of Vermersch. Our first paper, recently accepted, looks at the lived experience of having an advanced understanding of a group, and elucidates the essential structure of advanced understanding.
5. Contemplative pedagogy
At Lawrence I'm applying contemplative pedagogy in my classes with positive results, and wrote a paper about contemplation in mathematics. When I have free time I read about contemplative education and metacognition. I created and spend time managing a collaborative wiki site for contemplative mathematics pedagogy.
6. Himalayan measurement and math-as-walking
My most recent math-art project began in August 2015, when Elizabeth McTernan and I conduct various math-related actions in the Indian Himalaya. Some artifacts (sketches, math scratchwork, writing) were the focus of an exhibition at the HORSEANDPONY gallery in Berlin in October 2016. We are planning a follow-up project above the Arctic Circle in Finland for summer of 2017, involving a smartphone app that models the competing forces of post-glacial land uplift and climate change-induced sea-level rise.
