### Notes $\def\Z{\mathbb Z} \def\Q{\mathbb Q} \def\C{\mathbb C} \def\Torelli{\mathcal I}$

These are some miscellaneous notes I have written. They are not intended for publication and for the most part do not contain original material.

 [pdf] [abstract] Knot complements and Spanier-Whitehead duality [pdf] [abstract] Periodic billiard paths on smooth tables [pdf] [abstract] My favorite proof of the Pythagorean Theorem [pdf] [abstract] Smith theory and Bredon homology [pdf] [abstract] Hopf's theorem via geometry [pdf] [abstract] Spines of manifolds and the freeness of fundamental groups of noncompact surfaces [pdf] [abstract] The word problem for surface groups and hyperbolic geometry [pdf] [abstract] Lie groups and automatic continuity [pdf] [abstract] Two nonstandard constructions of free groups [pdf] [abstract] Classifying spaces and Brown representability [pdf] [abstract] Representation theory without character theory [pdf] [abstract] The symplectic representation of the mapping class group is surjective [pdf] [abstract] Realizing homology classes by simple closed curves [pdf] [abstract] The action of the deck group on the homology of finite covers of surfaces [pdf] [abstract] The Noetherianity of group rings [pdf] [abstract] One-relator groups [pdf] [abstract] A quick proof of the Seifert-Van Kampen theorem [pdf] [abstract] The isoperimetric inequality in the plane [pdf] [abstract] The generalized Schoenflies theorem [pdf] [abstract] Homotopy groups of spheres and low-dimensional topology [pdf] [abstract] The complex of cycles on a surface (after Bestvina-Bux-Margalit) [pdf] [abstract] The Torelli group and congruence subgroups of the mapping class group [pdf] [abstract] Rochlin's theorem on signatures of spin 4-manifolds via algebraic topology [pdf] [abstract] The congruence subgroup problem for SLn($\Z$) [pdf] [abstract] The fundamental theorem of projective geometry [pdf] [abstract] The Borel density theorem [pdf] [abstract] The action on homology of finite groups of automorphisms of surfaces and graphs [pdf] [abstract] A quick proof of the classification of surfaces [pdf] [abstract] The abelianization of the level L mapping class group