M > Recent Status Updates

Finished Chapter 4. Not sure if everything made sense to me, especially the last two were rough. But connectivity makes sense at least, now I know what topologists mean by it. - 15:27, 05.03.2026

Finished Chapter 3. Nice to finally see a proof of the Ham-Sandwich theorem and really be familiar with the Schrijver graph (ironic given my work). - 01:27, 22.02.2026

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.

finished "The finite-field Nikodym problem"

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.

Skipped the remaining exercises and read about (abstract and geometric) simplicial complexes. That was pretty straightforward. No idea how to do the Exercise in Subsection 1.5 though.

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

finished Chapter 1, kinda. I don't get the last few propositions and lemmas, but I am cool with that for now. I feel like I am slowly getting it, but man, it hurts my brain. - 08.01.2026, 23:48

Finished till Induced Homomorphisms. At least the proof that pi_1(S^1) = Z clicked with me now, though I skipped the proof of statement (c) on page 30. Managed to get to the 2-d Borsuk-Ulam theorem, which is good. My head hurts now, so I will stop here at "Induced Homeomorphisms".

In the middle of reading about differential entropy. - 22:22, 04.01.2026.

Stopped in the middle of proof of Theorem 1.7. (That S^1 has fundamental group Z.) I skipped parts of the introduction that were already too advanced for me and got stuck here. That was before 2026. - 18:00, 02.01.2026

Skipped the problems because reading on a bus is maybe not the best environment for deep thought. So instead started the next chapter on data compression and got till Kraft inequality. I remember learning about it as a side remark in my introductory TCS class in undergrad. Didn't remember the proof being so clean.

Finished Chapter on the entropy rates of stochastic processes. Very fun and light intro, though I am kinda struggling what I am missing with the first exercise.

Pretty good. The AEP chapter was nice and short. I like the book motivated the entropy rate of a stationary stochastic process too.

Finished Chapter 1 till summary. Was good so far, though I don't have a good intuition for some equalities, such as the chain rule for mutual information. - 23:16, 18.08.2025

Got stuck in the proof of Theorem 2.10.1 (Fano's inequality). I don't have a good feeling how the probability Pr(X \neq \tilde{X}) is random, and the inequalities in 2.135 don't make sense to me yet.

Read up till Section 2.7.

Finally got a feeling for what mutual information and the KL divergence is. I especially liked the interpretation for encoding a random variable X. The KL divergence is the inefficiency, the additional needed when one uses an encoding assuming X follows q(x) but actually the true probability mass function is p(x).

The Venn-Diagram with H(X | Y), H(Y | X), I(X;Y) was helpful too.