HeyJaredThe Deranged Mathematician
The personal Substack of mathematician Senia Sheydvasser, all about learning math and its applications.
- Indexed issues, last 90 days
- 13
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- Sep 30, 2026
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- Aug 8, 2026
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An Overview of Differential Geometry (opens the original)
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Differential geometry has a reputation as something thatmany non-mathematicians want to learn (e.g., it is very relevant to physics, particularly general relativity), butis so expansive and in parts arcane that it is difficult to know where to start.To give you an idea, the only reasonably well-known textbook that I could find treating differential geometry at an undergraduate level (and, mind you, this is meant for the last couple of years of a mathematics undergraduate major) is Pressley’s <a
The Power of Abstraction (opens the original)
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Last time, we went over how to attack a problem by making things concrete: how to search for examples and counterexamples and extract information out of them.This time, we’re looking at something that might seem diametrically opposite.Make things general. Try to see if a generalization of what you are investigating is true.As I noted when I originally gave this principle, this idea is counterintuitive to most people: they feel like if you make a problem more general and more abstract, then it be
Heuristics and Numerics (opens the original)
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Last time, we talked about how, once you have a feeling for what is true or false, you can try to search for counterexamples or contradictions.One thing that I did not talk about is how one should obtain such a feeling, and that is precisely what I want to talk about today—we’ll be going over the next proof-generating principle.Make things concrete. Try to find an example (or, preferably, multiple examples) of the general principle that you are investigating. Understanding that example very well
Counterexamples and Contradictions (opens the original)
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Last time, I laid out eight big-picture principles that I see as fundamental to how we approach proofs. We then went through the first two, which together amount to what we must do initially to properly understand a problem.This time, we’ll go through the next two principles.Search for a counter-example. Once you have a sense of whether something is true or not, try to find an example to contradict the given statement.Search for contradictions. Does what we are trying to prove or disprove contra
The Importance of Understanding (opens the original)
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Last time, we introduced this entire series and discussed why proof-writing is important at all.Now, let’s talk about how one actually proves things, anyway. Here are the big-picture steps that I suggest to my students.Rephrase the statement in your own language. You cannot solve a problem that you can’t understand. First and foremost, make sure that you know precisely what is being asked. Restating the question is a good way to do this.Fit the statement into your conceptual web<spa
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