# Short expository notes on various mathematical topics

## Table of Contents

## Short stories about category or homotopy theory

- Turning simplicial complexes into simplicial sets.
- The homotopy fiber of the map on classifying spaces.
- Equivalent ways of computing total homotopy fibers.
- Homotopy orbits of representation spheres are Thom spaces.
- The universal way to invert a morphism.
- If you think of a \(G\)-set \(X\) as a functor from the one object
category \(G\) to the category of sets, then its Grothendieck
construction is just the usual action groupoid.
^{1} - Lots of categories are contractible.
- Taking orbits of a finite group action commutes with inverse limits.
- The nine model category structures on the category of sets.

## Miscellaneous mathematical short stories

- Tiling polygons by similar polygons.
- Some things I was surprised to learn are false.
- The uniqueness of reduced row-echelon form. This was written for beginning linear algebra students and uses as little background as I could manage.
- A short proof of the invariance of Lebesgue measure under isometries.
- The Petr-Neumann-Douglas theorem in plane geometry. I don’t want to ruin the problem for anyone: if you want to read the statement of the theorem before you look at the proof, look here.
- A proper local homeomorphism is a covering map. (Well, at least if the spaces satisfy some mild assumptions.)
- The reflection properties of conics through calculus.
- Linear algebra around a configuration of lines in the plane, a lovely problem of Hossein Movasati’s.

## Longer expository articles

A Whirlwind Tour of the World of (∞,1)-categories (PDF).

I am (still planning to finish) writing rough lecture notes for a course I taught on the fundamental groupoid. The section on the van Kampen theorem is done (PDF).

I wrote my *minor thesis*^{2} on the iteration of rational maps on the
Riemann sphere. It includes fairly complete proofs of Sullivan’s No
Wandering Domains theorem and the classification of the periodic Fatou
components and is, I hope, fairly light reading —and is also
available in PDF.

## Artículos expositorios en español

- Un intento de motivar la definición de espacio topológico.
- La jerarquía de conmutatividad en la revista electrónica Motivos Matemáticos.
- Un producto infinito de \(\mathbb{Z}\)s no es libre.
*Todo lleno de círculos*(PDF), sobre varios problemas acerca de llenar espacios euclídeanos o esferas con esferas o bolas que nos planteamos Juan José Alba González y yo.- La obra matemática de Jacob Lurie en la revista electrónica
`universo.math`

. - El teorema de Barbier.
- La historia de Grunwald y Wang.
- π²/6 > φ.

^{1}

No link needed because the whole story fit into the title!

^{2}

This an excellent program requirement: a chance to make a fool of yourself in writing by producing an exposition of a topic outside your field of expertise in a limited amount of time. I loved it! All math departments should have this.