In this course, (taught joint with Uli Wagner), we offered a number of attractive topics based mostly on Matoušek's book. Namely, we went over the Borsuk-Ulam theorem, and its multiple applications including the proof of Kneser's conjecture, Helly's theorem and its relatives and even the more esoteric evasiveness conjecture.
Don't worry, despite the very similar name, this course, taught with Zuzka Patáková is instead a tour of basic topology and convex geometry culminating in the introduction to Topological Data analysis, including varous simplicial complexes (Čech, Vietoris-Rips, alpha), persistence, barcodes, persistent diagrams and introduction to useful pieces of software.