Sample Projects
Algebra
Classification of Rings
Possible texts: Dummit and Foote, Abstract Algebra
Read through the various types of rings and how they inter-relate.
Prerequisites: abstract algebra
Vector Spaces and Modules
Possible texts: Dummit and Foote, Abstract Algebra
Expand on linear algebra with such generalizations as infinite dimensions, basis-free notation, and
modules (where you can't divide by the constants).
Prerequisites: abstract algebra
Representation theory of finite groups
Possible texts: James & Liebeck, Representations and Characters of Groups;
Fulton & Harris, Representation Theory: A First Course, Ch 11
This is an introduction to representation theory, shedding light on how linear algebra and group theory,
together, can yield interesting results. We shall develop this understanding by working with modules. Examples
of the representation theory of finite groups will abound, along with excursions to other examples
like SL2(Z).
Prerequisites: linear algebra, groups
Analysis
Gamma function
Possible text: Artin, The Gamma Function
The familiar factorial function n! is only defined for n an integer. The Gamma function
is essentially a function that deserves to be called x! where x can be any real number.
Artin's delightful little note introduces many notions in analysis while discussing all of the amazing
properties of the Gamma function.
Prerequisites: [none]
Fourier Series and Transforms
Possible text: Pinkus and Zafrany, Fourier Series and Integral Transforms
We will begin with introducing the machinery of inner product spaces and orthonormal systems. We will
explore Fourier series in a very example-oriented way. We will discuss pointwise and uniform convergence,
explaining Gibbs phenomena. The goal of the project is to work through the main source, with special
attention given to the applications at the end of each chapter.
Prerequisites: some analysis
Hilbert Spaces
Possible text: Lax Functional Analysis;
Folland, Real Analysis
We explore the inner product and motivation for the concept of a Hilbert space. Specifically, we will
discuss the Schwarz inequality, the parallelogram law, the closed subspace decomposition theorem, the Riesz
representation theorem, orthonormal bases, completeness, Parseval's identity, separability of Hilbert spaces
with a countable orthonormal basis, and the Stone-Weierstrass theorem.
Prerequisites: some analysis
Combinatorics
Introductory Enumerative Combinatorics
Possible text: Martin, Counting: The Art of Enumerative Combinatorics
We cover the basic counting techniques of enumerative combinatorics, including the principle of
inclusion-exclusion, generating functions, and recurrence relations.
Prerequisites: [none]
Generating functions
Text: Wilf, Generatingfunctionology
The basic theory of generating functions uses little more than formal Taylor series and manipulation of
polynomials, yet is an extremely powerful tool. We will use generating functions to find closed formulas for
recursively defined sequences that are important in combinatorics and computer science. If time permits, we
will also discuss a different sort of generating function: the zeta function.
Prerequisites: [none]
Geometry / Topology
Knot theory
Possible texts: Adams, The Knot Book;
Livingstone, Knot Theory
Knot theory is the study of ways the circle can be embedded in three-dimensional space. The primary
questions in the subject are when two knots are equivalent, when can a knot be untangled, and how many
different types of knots are possible. These questions are addressed in part through the introduction of
algebraic invariants. For instance, the Jones polynomial can be used to distinguish between knots of eight
or fewer crossings. The subject involves drawing lots of pictures, and has surprising applications to
chemistry and biology.
Prerequisites: [none]
Geometry of Surfaces
Possible text: Stillwell, Geometry of Surfaces
This project treats the geometry of the Euclidean plane, the sphere, and the hyperbolic plane, and the
surfaces that arise from these as quotients by groups of isometries. We will study the groups of isometries
of each of these three spaces in depth and how quotient surfaces inherit the geometry of their covering spaces.
We will see that all complete constant-curvature surfaces arise in this way.
Prerequisites: group theory, multivariable analysis, basic topology
Bezout's Theorem in algebraic geometry
Possible texts: Kirwan, Complex Algebraic Curves;
Shafarevich, Basic Algebraic Geometry
After an introduction to algebraic curves and varieties in projective space, we prove Bezout's theorem,
which states that the number of points of intersection of two plane curves, counted right, equals the product
of their degrees.
Prerequisites: abstract algebra, complex analysis, topology
Hurwitz's formula for Riemann surfaces
Possible text: Miranda, Algebraic Curves and Riemann surfaces
We introduce Riemann Surfaces and functions between them, and study examples. Hurwitz's formula
relates arithmetic information of two surfaces and a map between them.
Prerequisites: complex analysis, algebra, topology
Morse theory
Possible texts: Milnor, Morse Theory;
Matsumoto, An Introduction to Morse Theory
Morse theory is a collection of results that allow you to study the topology of manifolds based on
the critical points of a function on the manifold. It is a very geometric subject.
Prerequisites: topology
Number theory
Pseudoprimes and RSA
Possible texts: Koblitz, A course in Number Theory and Cryptography;
Ribenboim, The Little Book of Big Primes
An investigation of computational methods for finding "industrial primes" and their application to
RSA (a public key cryptosystem). We will study pseudoprimes and Carmichael numbers along the way.
Prerequisites: [none]
Geometry of numbers
Possible text: Sharlau, Opolka, From Fermat to Minkowski
We introduce lattices in Euclidean space and Minkowski's classic pigeonhole-type results: regions,
if big enough and regular enough, are forced to contain lattice points. From geometric properties like
these, we deduce number theoretic consequences, for example proving that every natural number is the sum
of four squares.
Prerequisites: [none]
Prime number theorem
We begin by studying basic properties of modular arithmetic and elementary number-theoretic results
(e.g. Fermat's Little Theorem), and getting an account of the prime number theorem which says that the
number of primes up to n is asymptotic to n/log n. Other topics from analytic number
theory could then be pursued.
Prerequisites: [none]
p-adic numbers
Possible texts:
Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-functions;
Gouvea, p-adic Numbers: An Introduction
From scratch, we develop the theory of p-adics by exploring alternatives to absolute value for
distance functions on the rational numbers. Possible directions include: p-adic interpolation,
p-adic exponential and logarithm, the structure of p-adic fields.
Prerequisites: basic analysis
Elliptic curves
Possible texts: Silverman and Tate, Rational Points on Elliptic Curves;
Koblitz, A Course in Cryptography;
Elliptic curves appear algebraically as the solutions of cubic equations (called Weierstrass equations)
or complex analytically as the quotient of the complex plane by a lattice. Elliptic curves have fascinating
properties with applications to many areas of math, most notably number theory. The reason they are special
is that they are groups, and can be defined over arbitrary fields. The symmetry arising from the group
structure and the geometry arising from the complex analytic description can be used to shed light on
arithmetic properties. Possible focuses of such a project could be an introduction to algebraic geometry,
cryptosystems based on elliptic curves, or a remarkable algorithm discovered by Lenstra which uses elliptic
curves to factor large integers.
Prerequisites: algebra
Set theory
Fractals
Possible text: Crilly et al, Fractals and Chaos
This project begins with the idea of shapes generated by iterations. Main examples are
Cantor-type sets, and line bendings. Hausdorff dimension is introduced and it is shown that these
shapes can have fractional dimension. This ties in with notions of self-similarity on all scales.
The idea of chaos can be understood by random behavior of orbits under iterated application of
functions. A simple example is given by the iteration of a complex polynomial like
f(z) = z2 + c. Through studying the dynamics of repeatedly
applying this map in the complex plane, we arrive at the famous Julia sets and, parameterizing those,
the Mandelbrot set.
Prerequisites: [none]
Introduction to set theory
Possible text: Kaplansky, Set Theory and Metric Spaces
We cover basic set theory, cardinal numbers, ordinal numbers, the axiom of choice, basic properties
of metric spaces, continuity, separability, compactness.
Prerequisites: [none]
Banach-Tarski paradox
Possible text: Wagon, The Banach-Tarski Paradox
There is a family of spatial results which seem paradoxical -- one of the most famous formulations
says that you can begin with a solid ball in R3, cut it into a finite number of
disjoint pieces, and rearrange those pieces by rigid motions to reassemble two solid balls, each of
the same size as the original. The main paradox and its proof will be covered in the first half of
this project. The main ingredients are free groups, isometries, countable sets, and the axiom of choice.
There are many directions where this can lead, including hyperbolic paradoxes, measure theory and
invariant means, amenable groups, and representation theory.
Prerequisites: set theory, basic group theory
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