Short expository notes on various mathematical topics

• 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.

• 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.

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 thesis² 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.

• 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.