M’s Reviews > Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry > Status Update

Read the subsection on the Ham-Sandwich theorem.

For Exercise 1: By the intermediate value theorem, there is a line such that on either half-planes we have half of the mass. Define two new mass distributions by restricting on the half-planes. Apply Ham-Sandwich theorem.

For Exercise 2, consider 4 points in a line L, 3 on another line K such that the "middle" point among the three is the intersection of L and K.

Finished the Chapter on Tucker's lemma. I skipped the algorithmic proof of Tucker's lemma as it seemed long and tedious. The topological proof using the degree of the function is beautiful, quite ingenious. Probably the first time I got how homology could be useful. - 22:48, 09.02.2026.

Read till Section 2.2. Most stuff made sense, though I find the exercises not that simple. - 12:40, 29.01.2026

Read 1.6 now too, pretty slick! I remember that the moment curve comes up when proving that any graph can be drawn in R^3 using only straight segments. I guess it then makes sense that this strategy works in general.

Completed everything till Subsection 1.2. It takes me a bit to really understand it, currently stuck at Exercise 4 of that subsection. But enjoyed it so far, I like it being more to the point and being upfront about nontrivial (not proven) facts that are nonetheless important. I feel like it will help me with Hatcher. - 22:21, 15.01.2026