A brief look at topology

The following is a re-upload of a post in my previous blog (which was destroyed by mistake).

Topology was something that had always fascinated me. When looking for introductory materials to read up on, I came across the AIMS 2014 Lecture Series on Topology and Geometry by Dr Tadashi Tokieda. It's a 15 day lecture, across 3 weeks. A concluding overview of the entire lecture series is on the last day (here).

I quite enjoyed it. It wasn't really rigorous with all the notations, but I think the purpose of the lectures was to excite and gently introduce the subject to the class. Tadashi strongly believed in pictorial thinking and strong usage of examples to illustrate the meaning of definitions and usage of theorems. Apart from technical knowledge, I've also picked other "softer" learning/thinking points from the lecture series.

  • Problem transformation
  • Sources of knowledge are often related
  • Get going (Life is short)
  • Draw and visualise concepts
  • Generic/Common cases vs. Degenerate/Corner cases

Problem transformation

While in NUS, I picked up a useful way of thinking about problems from Prof Leong Hon Wai. It can be visualised as such:

The idea is as follows: If it is hard to directly solve a problem, it may be useful to transform/map it to a model and use well-studied methods to obtain a solution in the model world. We can then map it back to the actual context after that. This framework helps in both solving new problems and showing that certain problems are hard (see below).

Now, let's see this screenshot from Day 2 of the lecture series:

The diagram looks awfully similar, doesn't it? The strategy mentioned by Tadashi is to deform away degeneracies in topological problems, solve them, then map the solution to the actual problem. He has a very nice puzzle highlighting how problem transformation works in the last part of Day 14.

Sources of knowledge are often related

Tadashi drew links between the topological concepts he covered and things that are seemingly unrelated. For example, he emphasised the link between dynamical systems and topology via the Poincare-Hopf theorem.

This resonated well with me as I always liked to see cross-fertilisation of ideas across different domains, such as biologically inspired algorithms like Ant Colony Optimisation, etc. We should all learn broadly and be curious about how things work :)

Get going (Life is short)

Throughout the lectures, Tadashi is slightly annoyed by the attendees' slow responses. One of the more prominent examples is on Day 11 (from the 5:00 mark onwards). In other parts of the lecture, students would give wrong answers to his questions until they are asked to come up to the board and write it down - They already knew the answers but did not think it through properly before replying.

The takeaway here is simple: Life is short, we should get going and stop dilly-dallying. Things like "being shy" is a waste of everyone's time.

Draw and visualise concepts

Throughout the lecture series, Tadashi drew several diagrams to illustrate the topological concepts. He's also a strong believer of using examples to motivate the intuition of his students on the newly taught concepts. I couldn't agree more.

Generic/Common cases vs. Degenerate/Corner cases

Often, we are warned to guard against corner cases1 and how such special cases can (and will) entirely invalidate mathematical statements. However, throughout this lecture series, I was introduced to a different point of view - Always handle the normal case first, then see the corner case works out. Often, it can be deformed to something we can handle.

The idea of "generic vs. degenerate" was brought up very early on2 and Tadashi made it clear that we should focus on the former as the latter. He showed that in topology, we can often analyse the latter by applying continuous deformation / taking limits from generic cases to degenerate ones.

  1. There's a well known Computer Science joke: There are two hard things in Computer Science: cache invalidation, naming things, and off-by-one errors.

  2. According to Tadashi: Generic cases are those that can be drawn on a moving train. He then acted a scene of someone trying to draw parallel lines on a bumpy train ride.