## 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.

## 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 ${\mathbb R}^3:$

## 305 – Solving cubic and quartic equations

January 23, 2012

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

## 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.