305 – Homework I

January 23, 2012

This homework set is due Wednesday, February 1st, at the beginning of lecture, but feel free to turn it in earlier if possible.

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David Hilbert

January 23, 2012

Today is the 150th anniversary of Hilbert’s birthday.

Hilbert is one of the mathematicians I admire the most. I believe I first learned about him while in High School, through “Higher Geometry” by N.V. Efimov. My copy of Reid’s “Hilbert” is one of the first books I bought when I arrived in the States (in 1997).

Here is a link to the reprint in the Bulletin of the AMS of Mathematical Problems.

I would love to have a poster sized copy of the picture above. Springer sold them years ago, but now they seem impossible to find.


515 – The Dehn-Sydler theorem

January 23, 2012

As mentioned in lecture, Hilbert’s third problem was an attempt to understand whether the Bolyai-Gerwien theorem could generalize to {\mathbb R}^3:

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305 – Solving cubic and quartic equations

January 23, 2012

Ars Magna, “The Great Art”, by Gerolamo Cardano.

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414/514 – The Schoenberg functions

January 23, 2012

Here is Jeremy Ryder’s project from last term, on the Schoenberg functions. Here we have a space-filling continuous map f:x\mapsto(\phi_s(x),\psi_s(x)) whose coordinate functions \phi_s and \psi_s are nowhere differentiable.

The proof that \phi_s,\psi_s are continuous uses the usual strategy, as the functions are given by a series to which Weierstrass M-test applies.

The proof that f is space filling is nice and short. The original argument can be downloaded here. A nice graph of the first few stages of the infinite fractal-like process that leads to the graph of f can be seen in page 49 of Thim’s master thesis.