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Existence and uniqueness for integral curves
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.


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


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


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Algebraicity of matrix entries of representations
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.


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Representations of products
Abstract:
We give a charactertheory 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.


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The Jacobson density theorem
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 finitedimensional
irreducible complex representation of G, then the natural map
$\C$[G] $\rightarrow$ End_{$\C$}(V) is surjective.


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Identifying primes in polynomial time: the AKS algorithm
Abstract:
We prove a remarkable theorem AgrawalKayalSaxena 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.


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The representation theory of SL_{n}($\Z$)
Abstract:
We give a fairly complete description of the finitedimensional characteristic 0 representation theory of SL_{n}($\Z$) for
n $\geq$ 3, following work of Margulis and Lubotzky.


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Knot complements and SpanierWhitehead duality
Abstract:
We explain SpanierWhitehead 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).


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Periodic billiard paths on smooth tables
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.


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


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Smith theory and Bredon homology
Abstract:
We discuss theorems of P. Smith and Floyd
connecting the cohomology of a simplicial complex equipped with
an action of a finite pgroup 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.


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


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Spines of manifolds and the freeness of fundamental groups of noncompact surfaces
Abstract:
We prove a theorem of Whitehead that says that a smooth noncompact nmanifold deformation
retracts onto an (n1)dimensional spine. As a consequence, we deduce a theorem of
Johansson that says that the fundamental group of a noncompact surface is free.


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


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


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


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Classifying spaces and Brown representability
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.


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Representation theory without character theory
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.


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


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


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The action of the deck group on the homology of finite covers of surfaces
Abstract:
We give two proofs of a theorem of ChevalleyWeil that describes the homology of a cover of a surface as a
representation of the deck group.


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The Noetherianity of group rings
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 nonfinitelygenerated 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.


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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.


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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.


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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.


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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.


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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.


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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.


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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".


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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.


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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.


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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.


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


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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.


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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.


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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.
