A Sad Day for Mathematics

It is with tremendous regret to the long-term effects to mathematics that I bring you this news…

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Reading, and reading, and teaching, and reading, and reading, and…

So, to summarize the direction of my most recent mathematical endeavors: I woke up and decided that part of my aspiration was to become a geometric topologist, and I did that despite the fact that topology is (far and away) my worst subject.

That sounds precisely as terrible as it probably is.

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Study Plan, tentatively, + Algebraic Geometry Exercises

So I think it’s probably best to have a rotating study plan schedule that allows me to do certain topics on certain days. So far, I’m thinking of having a rotation that looks something like:

Differential Geometry -> Algebra -> Clifford Stuff -> Algebraic Topology (optional),

and since yesterday was (unofficially) differential geometry day, I’m going to spend today doing algebra.

First order of business: Eisenbud and Harris. And, since I’ve been meaning to write down some of the solutions to exercises I’ve passed, I guess I’ll do that here.

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Continuous, Nowhere Differentiable Functions

A few days ago, I posted about a conversation I had with my friend L. We spent some time catching up and, in so doing, spent a little time talking about this particular plot of space on the grand ol’ internet. He mentioned a couple blog topics for me to consider and also asked if I was contemplating research in algebra/topology; looking back, the fact that L’s an analyst, the fact that I have very few analysis posts here, and the fact that the topics he suggested were analysis topics made me realize I really do need to do a better job representing my enjoyment for analysis. Consider this entry step one of that, perhaps.

Rather than spending a bunch of time researching stuff I’d never seen before, I decided to type up a little summary thing of an interesting article I found online when I was a master’s student. For a little perspective as to why this particular article is important, we’ll have to take a trip into so-called higher education and examine the topic that generally serves as most people’s introduction to grown up mathematics, i.e., calculus. A (really really over-simplified, primitive, simplistic) synopsis of calculus can be summed up in this way: Calculus is a class that abstracts the unknown variable quantities thrown at you in Algebra I/II into unknown variable quantities that themselves can vary by way of limiting arguments.

And that’s basically it: In America, calculus is really just taught as the algebra of limits. As such, some basic limit-intrinsic notions such as continuity, differentiability, and integrability are touched on / hinted at, and at the end of fourteen weeks of being fooled into thinking you’re finally understanding what math is, you’re sent on your way. For most, that’s the end of the story, but for a self-selecting few, the journey through mathematics continues, and new techniques / ideas get thrown at you in hopes that they’ll stick and that you’ll be able to use them for something special….

…and at the same time, for that self-selecting few, it’s not uncommon at all for certain somewhat obvious questions to go unasked through the years. For example: It’s invariably shown in Calculus I that f(x)=|x| fails to be differentiable at x=0 because of the sharp edge there. It stands to reason, then, that combinations and scalings of the absolute value function with two, three, four, etc. sharp edges would fail to be differentiable at two, three, four, etc. values of x. This idea isn’t a hard one to grasp for a calculus student. But then the next question: How many points of non-differentiability can a function have? Or how about, Construct a function that fails to be differentiable at infinitely many points. Most students would be quick to adapt previous examples and notice that a saw-blade function with sharp points at each value x=n, n\in\mathbb{Z}, proves the existence of functions with infinitely many points of discontinuity. Again, no big deal.

So what, then? Can we have functions that are non-differentiable at uncountably many points? How about functions that are differentiable nowhere? By and large, these are ideas that escape lots of students – even students nearing the end of a traditional math major curriculum at an average American institution. I know this because I was once one of those students, and have since taught several myself: I see how students fail to comprehend non-differentiability and even the everywhere-discontinuity of functions like g(x)=\chi_{\mathbb{Q}}(x). It’s simply something that fails to register for the average student.

Coincidentally, it doesn’t always stop there. L was actually telling me a story once about a statistics professor we both knew who claimed, absent-mindedly, that most continuous functions are differentiable. That, of course, is a big statement, and for the inquisitive audience-member, the natural response is: Prove it. Hence the aforementioned paper…. Continue reading