Over the next four months or so*, I will be writing a series of blogposts which outline in broad, descriptive terms the area/s of mathematics I am most interested in—a large swathe of low-dimensional topology and its environs, for the most part—, some sets of tools which are useful to these areas, and how they relate to one another and to the rest of mathematics and to other bits of human inquiry.

The purpose of (writing) these posts is mostly to seek an at least mildly coherent answer to the question, “so what is it you do?” and the often-unasked but—at least to me—entirely natural follow-up, “but why?”

In order to do this I will attempt to describe just what each of these bits of mathematics is about, how it came about, why its originators found it interesting, and why I find it interesting. Along the way, as our dear doctoral chair would advise, there should be plenty of concrete illustrative examples. In some cases, I may also describe current directions of exploration and active research.

*A rough deadline … or more of a broad scheduling guideline. To some extent, I’m inclined to treat [large parts of] this as an ongoing project, with posts—particularly ones covering areas I am particularly focused on / interested in—subject to active and ongoing revision.

**An outline of the ground to be covered**, to be updated with links and edits as I go along:

- Focus areas
- Deformation spaces and Teichmüller theory
- Higher Teichmüller theory; Anosov representations
- Projective geometry
- Weil-Petersson geometry (I) (II)

- Geometric group theory (I) (II)
- Around hyperbolicity and NPC

- Deformation spaces and Teichmüller theory
- More peripheral areas
- Polygonal billiards, translation surfaces and the magic wand
- Knot theory

- Connections and applications
- Low-dimensional topology in physics
- Topology as squinting
- Algorithms from geometry
- Groups, graphs, and computations

- Tools
- Ergodic theory and smooth dynamics
- 2- and 3-manifold theory (including bits of differential topology, Riemannian geometry, etc.)
- Morse theory
- Complex geometry and cousins
- Algebraic topology
- Functional analysis