Weird and Wacky Manifolds, or Learning How to Appreciate Geometry and Topology

By Michael Zhao, The University of Sydney

It’s hard talking to my friends and family about higher-level maths in general, where even though I try and impress on them the fact that (in the areas I’m interested in) there’s really not all that many equations or numbers. Trying to explain (relatively basic) concepts in abstract algebra like group theory or results in analysis is already hard enough, but having to explain other areas related to projects I’ve done – algebraic topology, representation theory, to name two typical honours-level classes – trying to explain these in any meaningful sense without throwing away all the detail (category theorists don’t have this particular hangup) is virtually impossible.

So, you can imagine my excitement upon starting my project in low-dimensional topology, because, well, it’s easy to visualise and explain the basics of what’s going on. Topological invariants like genus, or the number of holes, and its close cousin the Euler characteristic, are all very well-motivated in 2 dimensions, and even going up to 3 dimensions it is more or less alright. But once we go up to 4 dimensions, it is very easy to hand-wave a lot of things and just say

“Well, imagine what goes on in 3 dimensions, and now do that one dimension higher.”

To tell the truth, when I first started working with this I too was also complaining, so you can imagine what’s going through the minds of my audience (who are, at this point if I’m lucky, just his cat). Unlike in other areas of, say, abstract algebra, where (despite the name) there are quite rigid definitions that can be worked with, in this field of maths it does seem to the naive eye incredibly loose.

So with all of this semi-complaining, where does the appreciation part come in?

Well, for me personally anyway figuring out the attractiveness of this area came rather slowly. Just like in our differential geometry course, and arguably for many fields of maths, things don’t really make sense until you work on it for a while. The pieces fit together quite nicely (there should be something related to geometry and topology here, but I can’t figure out what), and it turns out that the visualisation of these low-dimensional manifolds, despite the occasional hand-wave, does feel quite natural. Chasing down tighter bounds of complexity using different topological perspectives requires a great deal of creativity that I personally believe this area is more conducive to prompting – again, the visualisation aspect gives a certain ‘pliable’ feeling to applying various techniques, which doesn’t come as easily in other areas of maths.

Despite the occasional bouts of brain sickness that jump me when I’m reading through the material, there does seem to be an inescapable feeling of true appreciation for all of this geometric and topological stuff. And if I ever get asked as a follow-up to the visualisation, an application of this, I can always hand-wave harder into the wild land of theoretical physics.

 

Michael Zhao was a recipient of a 2018/19 AMSI Vacation Research Scholarship.

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