Abstract: We give a simple proof of Schur-Weyl duality and discuss some of its consequences.

Abstract: We try to give a geometrically motivated account of the homotopy excision theorem (aka the Blakers-Massey theorem), using smooth manifold techniques instead of the usual simplicial ones.

Abstract: We prove the well-known "half lives, half dies" theorem, and as an application prove that the signatures of boundaries are 0.

Abstract: We give the most efficient proofs we know of a number of basic results in calculus: Taylor’s theorem with remainder, L'Hopital’s rule, the change of variables theorem (both in single-variable calculus and in multi-variable calculus), and the two fundamental theorems of calculus.

Abstract: We give two proofs of a theorem of Dehn that says that if a rectangle can be tiled by squares, then its side lengths must be commensurable.

Abstract: We give three proofs that functions on subgraphs of finite graphs can be uniquely extended to functions that are harmonic on the remainder of the graph.

Abstract: We give a proof of the Cauchy-Binet formula for the determinant of the product of two matrices that mostly avoids explicit matrix manipulations.

Abstract: We explain why integral curves to vector fields exist if the vector fields are continuous and are unique if the vector fields are locally Lipshitz.

Abstract: We give a simple and direct proof of Tychonoff's Theorem.

Abstract: We prove Burnside's theorem saying that a group of order p^aq^b for primes p and q is solvable.

Abstract: We prove that with respect to an appropriate basis, the matrices associated to complex representations of finite groups have entries lying in algebraic number rings.

Abstract: We give a character-theory free proof that for finite groups G and H, the irreducible representations of G$\times$H over an algebraically closed field of characteristic 0 consist of tensor products of irreducible representations of G and H. Because our proof avoids character theory, it generalizes to many other similar situations.

Abstract: We prove the Jacobson density theorem concerning simple modules over rings. This implies, for instance, that if G is a finite group and V is a finite-dimensional irreducible complex representation of G, then the natural map $\C$[G] $\rightarrow$ End_$\C$(V) is surjective.

Abstract: We prove a remarkable theorem Agrawal-Kayal-Saxena saying that there is an algorithm to determine if an integer is prime whose running time is a polynomial in the number of digits of the integer.

Abstract: We give a fairly complete description of the finite-dimensional characteristic 0 representation theory of SL_n($\Z$) for n $\geq$ 3, following work of Margulis and Lubotzky.

Abstract: We explain Spanier-Whitehead duality, which gives a simple geometric explanation for the fact that the homology of a knot complement is independent of the knot (a fact usually derived from Alexander duality).

Abstract: We explain a theorem of Birkhoff that says that a smooth convex billiard table always has periodic billiard paths of any given prime period.

Abstract: I explain my favorite proof of the Pythagorean Theorem, which uses scaling and similar triangles.

Abstract: We discuss theorems of P. Smith and Floyd connecting the cohomology of a simplicial complex equipped with an action of a finite p-group to the cohomology of its fixed points. The proof we give is the original one, but phrased using the modern language of Bredon homology (to which we give a concise introduction). An older version of these notes that doesn't use Bredon homology (and thus is a bit shorter and more direct, though less conceptual) can be found here.

Abstract: We show that elementary ideas about bordism allow a simple and natural proof of Hopf's theorem in group homology.

Abstract: We prove a theorem of Whitehead that says that a smooth noncompact n-manifold deformation retracts onto an (n-1)-dimensional spine. As a consequence, we deduce a theorem of Johansson that says that the fundamental group of a noncompact surface is free.

Abstract: We explain Dehn's solution to the word problem for fundamental groups of surfaces using hyperbolic geometry.

Abstract: We prove a theorem of Pettis that says that all measurable homomorphisms between Lie groups are continuous.

Abstract: We give two nonstandard constructions of free groups, one using geometric topology and the other inspired by category theory.

Abstract: We sketch the proof of the Brown representability theorem and give a few applications of it, the most important being the construction of the classifying space for principal $G$-bundles.

Abstract: We give short, direct proofs that if G is a finite group, then the group ring $\C$[G] decomposes as a direct sum of dim(V) copies of every irreducible representation V of G and that the number of irreducible representations of G is the same as the number of conjugacy classes of G.

Abstract: We give an efficient proof that the symplectic representation of the mapping class group is surjective.

Abstract: We give an efficient proof that primitive classes in the first homology group of a surface can be realized by simple closed curves.

Abstract: We give two proofs of a theorem of Chevalley-Weil that describes the homology of a cover of a surface as a representation of the deck group.

Abstract: We discuss conditions under with the group ring of a group is and isn't Noetherian. There are two main results. The first is a folklore theorem that says that if a group contains a non-finitely-generated subgroup, then its group ring is not Noetherian. The second is a theorem of Phillip Hall that says that group rings of virtually polycyclic groups are Noetherian.

Abstract: We give a classically flavored introduction to the theory of one-relator groups. Topics include Magnus's Freiheitsatz, the solution of the word problem, the classification of torsion, Newman's Spelling Theorem together with the hyperbolicity (and thus solution to the conjugacy problem) for one-relator groups with torsion, and Lyndon's Identity Theorem together with the fact that the presentation 2-complex for a torsion-free one-relator group is aspherical.

Abstract: This note contains a very short and elegant proof of the Seifert--Van Kampen theorem that is due to Grothendieck.

Abstract: The classical isoperimetric inequality says the circle encloses the most area among simple closed curves in $\mathbb{R}$² of a fixed length. We give a short and fairly geometric proof of this.

Abstract: The generalized Schoenflies theorem asserts that if $\phi$:S^n-1 $\rightarrow$ Sⁿ is a topological embedding and A is the closure of a component of Sⁿ $\setminus$ $\phi$(S^n-1), then A $\cong$ $\mathbb{D}$ⁿ as long as A is a manifold. This was originally proved by Barry Mazur and Morton Brown using rather different techniques. We give both of these proofs.

Abstract: We give a modern account of Pontryagin's approach to calculating $\pi$_n+1(Sⁿ) and $\pi$_n+2(Sⁿ) using techniques from low-dimensional topology.

Abstract: The complex of cycles on a surface is a cell complex that encodes all the ways that an element of first homology can be written as an embedded cycle. It was introduced by Bestvina-Bux-Margalit and plays an important role in their calculation of the cohomological dimension of the Torelli group. We give a detailed proof that this complex is contractible, expanding upon one of the proofs given by Bestvina-Bux-Margalit.

Abstract: These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class group, following my papers "The second rational homology group of the moduli space of curves with level structures" and "The Picard group of the moduli space of curves with level structures".

Abstract: We give the original proof of Rochlin's famous theorem on signatures of smooth spin $4$-manifolds, which uses techniques from algebraic topology. We have attempted to include enough background and details to make this proof understandable to a geometrically minded topologist. We also include a fairly complete discussion of spin structures on manifolds.

Abstract: Following Bass-Milnor-Serre, we prove that SL_n($\Z$) has the congruence subgroup property for n $\geq$ 3. This was originally proved by Mennicke and Bass-Lazard-Serre.

Abstract: We prove the fundamental theorem of projective geometry. In addition to the usual statement, we also prove a variant in the presence of a symplectic form.

Abstract: We discuss the Borel density theorem and prove it for SL_n($\Z$).

Abstract: We prove that aside from trivial cases, finite-order homeomorphisms of surfaces and graphs must act nontrivially on homology. For surfaces, this classical theorem is usually deduced from the Lefschetz fixed point theorem, while for graphs it is usually deduced via combinatorial manipulations. Our proof is different and is in the same spirit as the original proof (due to Hurwitz) of this theorem for surfaces.

Abstract: This note contains a very short and elegant proof of the classification of surfaces that is due to Zeeman.

Abstract: We calculate the abelianizations of the level L subgroup of the genus g mapping class group and the level L congruence subgroup of the 2g $\times$ 2g symplectic group for L odd and g$\geq$3.