The art of turning shapes into algebra. Fundamental groups, homology & cohomology theories, covering spaces, and the eternal question: is this coffee mug actually a donut?
Finding the shape of data. Persistent homology, Vietoris–Rips complexes, the Mapper algorithm, and barcodes that tell you which holes in your data actually matter.
Where abstract algebra meets the machine. Gröbner bases, primary decomposition, syzygies, and making Macaulay2 do the heavy lifting.
"A topologist is someone who can't tell the difference between a coffee cup and a donut, but can spot a homology class from across the room." — The mathematical community, probably