29. My PhD Thesis: Part 0 - Setup and Contents
(Epistemic status: A creased, stained map to what were once my favorite hunting grounds. Accessible to anyone who can support substantial abstraction; prior math knowledge is not necessary. In particular, ignorance of calculus is not an obstruction here, but total ignorance of geometry or like, arithmetic or logic, will be. Extremely dense and probably won’t get you there, but at least you’ll ask better questions. Partially dedicated to DG.)
It’s been said that you don’t truly understand what you can’t explain to a lay audience. By that logic, I don’t truly understand my doctoral thesis now, and I probably never have. Five years later, I’ll remedy that wrong. I’ll avoid jargon as much as it’s helpful that I avoid it, and avoid equations almost completely.
Here’s the main result of my thesis, stated as accessibly as I can manage: Take any group G that’s finitely generated and whose elements we can separate using appropriate maps to other finite groups, and which has only finitely many meaningfully different ways to write down what it does in terms of linear transformations on complex-entried vectors. Also, take any shape S where no matter where inside it you go, the nearby volume looks like flat 3-d space and clockwise is a meaningful concept, but which globally has hyperbolic geometry, finite volume, and exactly one direction you can walk along forever - the long thin part where this is true is called a cusp. Now, chop off that cusp - the exposed slice will look like a hollow doughnut, because the cusp looked like a line’s worth of shrinking hollow doughnuts - and glue on a solid doughnut that’s been internally twisted around some number of times, both around and through the doughnut hole; construct the group H from the resulting glued-together shape which tells you what kinds of loops can be pulled tight inside the shape. Then we can tell G apart from H just by looking at their maps to finite groups - even if G came from a very similar gluing procedure on S - almost all of the time.
Don’t worry if you can’t understand much of that right now! By the time we’re done here, you hopefully will; I encourage you to go back and look at this definition after you’ve read through everything. It’s fine if you don’t recognize any of the terms I use - they’re just names for things, and you shouldn’t feel bad for not knowing names for things you’ve never seen before. I use the terms I do because they’re how I learned about them, and so that you can go looking them up on your own if you want.
For a little initial orientation, my thesis topic was geometric group theory, itself something like a branch of algebraic topology, itself the main active branch of topology as a whole. Accordingly, in the first post of this series, I’ll start with some basics of topology and algebra, and some algebraic topology. Then in the second post, I’ll discuss model theory and representation theory. Finally, I’ll get into geometric group theory results, my own model theory technology, and my main thesis results. Until then!
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It’s been said that you don’t truly understand what you can’t explain to a lay audience. By that logic, I don’t truly understand my doctoral thesis now, and I probably never have. Five years later, I’ll remedy that wrong. I’ll avoid jargon as much as it’s helpful that I avoid it, and avoid equations almost completely.
Here’s the main result of my thesis, stated as accessibly as I can manage: Take any group G that’s finitely generated and whose elements we can separate using appropriate maps to other finite groups, and which has only finitely many meaningfully different ways to write down what it does in terms of linear transformations on complex-entried vectors. Also, take any shape S where no matter where inside it you go, the nearby volume looks like flat 3-d space and clockwise is a meaningful concept, but which globally has hyperbolic geometry, finite volume, and exactly one direction you can walk along forever - the long thin part where this is true is called a cusp. Now, chop off that cusp - the exposed slice will look like a hollow doughnut, because the cusp looked like a line’s worth of shrinking hollow doughnuts - and glue on a solid doughnut that’s been internally twisted around some number of times, both around and through the doughnut hole; construct the group H from the resulting glued-together shape which tells you what kinds of loops can be pulled tight inside the shape. Then we can tell G apart from H just by looking at their maps to finite groups - even if G came from a very similar gluing procedure on S - almost all of the time.
Don’t worry if you can’t understand much of that right now! By the time we’re done here, you hopefully will; I encourage you to go back and look at this definition after you’ve read through everything. It’s fine if you don’t recognize any of the terms I use - they’re just names for things, and you shouldn’t feel bad for not knowing names for things you’ve never seen before. I use the terms I do because they’re how I learned about them, and so that you can go looking them up on your own if you want.
For a little initial orientation, my thesis topic was geometric group theory, itself something like a branch of algebraic topology, itself the main active branch of topology as a whole. Accordingly, in the first post of this series, I’ll start with some basics of topology and algebra, and some algebraic topology. Then in the second post, I’ll discuss model theory and representation theory. Finally, I’ll get into geometric group theory results, my own model theory technology, and my main thesis results. Until then!
32768/32768
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