[pdf] [abstract] 

Onerelator groups
Abstract:
We give a classically flavored introduction to the theory of onerelator 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
onerelator groups with torsion, and Lyndon's Identity Theorem together with the fact that the
presentation 2complex for a torsionfree onerelator group is aspherical.


[pdf] [abstract] 

A quick proof of the SeifertVan Kampen theorem
Abstract:
This note contains a very short and elegant proof of the SeifertVan Kampen theorem that is due to Grothendieck.


[pdf] [abstract] 

The isoperimetric inequality in the plane
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.


[pdf] [abstract] 

The generalized Schoenflies theorem
Abstract:
The generalized Schoenflies theorem asserts that if
$\phi$:S^{n1} $\rightarrow$ S^{n}
is a topological embedding and A is the closure of a component of
S^{n} $\setminus$ $\phi$(S^{n1}), 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.


[pdf] [abstract] 

Homotopy groups of spheres and lowdimensional topology
Abstract:
We give a modern account of Pontryagin's approach to calculating
$\pi$_{n+1}(S^{n}) and
$\pi$_{n+2}(S^{n})
using techniques from lowdimensional topology.


[pdf] [abstract] 

The complex of cycles on a surface (after BestvinaBuxMargalit)
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 BestvinaBuxMargalit 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 BestvinaBuxMargalit.


[pdf] [abstract] 

The Torelli group and congruence subgroups of the mapping class group
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".


[pdf] [abstract] 

Rochlin's theorem on signatures of spin 4manifolds via algebraic topology
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.


[pdf] [abstract] 

The congruence subgroup problem for SL_{n}($\Z$)
Abstract:
Following BassMilnorSerre, we prove that SL_{n}($\Z$)
has the congruence subgroup property for n $\geq$ 3. This
was originally proved by Mennicke and BassLazardSerre.


[pdf] [abstract] 

The fundamental theorem of projective geometry
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.


[pdf] [abstract] 

The Borel density theorem
Abstract:
We discuss the Borel density theorem and prove it for
SL_{n}($\Z$).


[pdf] [abstract] 

The action on homology of finite groups of automorphisms of surfaces and graphs
Abstract:
We prove that aside from trivial cases, finiteorder 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.


[pdf] [abstract] 

A categorical construction of free groups
Abstract:
This note contains a proof (from Lang's book on algebra, though it does not seem to be widely known) that
free groups exist. In contrast to the usual proof, words in the generators do not appear. Even worse,
uncountable groups appear even when constructing the free group on two generators. This proof is
essentially a specialization to the situation at hand of the usual proof of the adjoint functor theorem.


[pdf] [abstract] 

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


[pdf] [abstract] 

The abelianization of the level L mapping class group
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.
