Thinking about waves

I’ve had fun the last week preparing for an upcoming talk to the senior citizen group OASIS. This time around I’m talking about Fourier series, or more picturesquely, how our lives are influenced greatly by the technique of thinking about everything as sums of simple waves. Besides the many, many applications of that idea in disparate areas, from heat flow, to telecommunications, to astronomy, to quantum mechanics, I’ll be talking about the revolution in mathematics that was largely spurred on by Fourier series, starting in 1807.

Soon I’ll post video of the talk itself, but the short version is that Fourier series worked so well, yet seemed to go so against what was known or assumed about functions and calculus, that they required a revamping of both ideas. That then led to a revamping of the ideas of numbers, sets, mathematical rigor, logic, and the whole foundational structure and culture of mathematics. (It’s of course an oversimplification to say that this was all just due to Fourier’s ideas, but it’s not really so inaccurate to say so.) It links up with my previous talk about infinity, since Cantor’s work came directly out of a question about Fourier series. Interestingly enough, there’s a very natural endpoint to the story, exactly 100 years after Fourier’s initial bombshell of 1807, when the Riesz-Fischer Theorem gave what is still the nicest way to explain how and why Fourier’s waves work so well. It’s a great story, and I’m looking forward to talking about it. If I have a lot more time (unlikely) I may go further on YouTube and present the ideas at a somewhat higher (but still accessible) level.

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