Possible text: Atiyah & MacDonald, Introduction to Commutative Algebra
Commutative algebra studies the properties of commutative rings and has geometric realizations in algebraic geometry. Commutative algebra is a necessary prerequisite for studying algebraic geometry and is used in combinatorics.
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
Wedderburn's theorem and central simple algebras
Possible texts: Dennis & Farb, Noncommutative Algebra; Lam, Noncommutative Rings
We study the structure theory of certain noncommutative rings and algebras. Wedderburn's theorem characterizes the algebras of matrices.
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.
Introduction to probability
Possible text: Feller, Introduction to Probability
We start with the basic notions (discrete sample spaces, random variables, expectation, etc), and move on to study Bernoulli trials and the binomial distribution. We end by proving the weak law of large numbers, Kolmogorov's inequality and finally the strong law of large numbers. Time permitting, other topics include the central limit theorem and the study of random walks.
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
Possible text: 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
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.
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.
Geometry / Topology
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.
Introduction to Topology
Possible text: Munkres, Topology: a first course
Topology is the study of continuous functions and the properties that they preserve. For example, it has been said that a topologist cannot tell the difference between a coffee cup and a donut. Topologies are a generalization of the class of all open sets in a metric space and the theory of topologies is useful in analysis and topology, as well as deep and interesting on its own.
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
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.
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.
Elementary number theory
Possible texts: Dudley, Elementary Number Theory; Ireland & Rosen, A Classical Introduction to Modern Number Theory
We begin with elementary number theory (modular arithmetic, Fermat's little theorem, Wilson's theorem, Euler's function) and work up to the beautiful law of quadratic reciprocity.
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.
Prime number theorem
Possible text: Ribenboim, The little book of big primes
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.
Possible text: Cuoco, Visualizing the p-adic integers
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: the action of p-adics on trees, representations of the p-adics, and classification of local fields.
Prerequisites: basic analysis
Possible texts: Silverman, Elliptic Curves; Koblitz, A Course in Cryptography; Lenstra, Factoring integers with elliptic curves (Ann. of Math. 1987)
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.
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.
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.
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