Math Honors
College
Course Outline
Honors Linear Algebra and Differential Equations:
Course Description:
Linear transformations and matrices, spectral theorem, ordinary differential equations, existence and uniqueness theorems, phase space, stability, oscillations.
Course Outline:

Vector spaces and linear transformations.

Systems of equations, determinants.

Spectral theorem.

Ordinary, linear, and nonlinear differential equations.

Existence and uniqueness theorems.

Phase space, stability, and periodic points.
Honors Introduction to Analysis:
Course Description:
Rigorous theoretical introduction to the foundations of analysis in one and several variables: covers basic set theory, vector spaces, metric and topological spaces, continuous and differentiable mappings between ndimensional real vector spaces.
Course Outline:

Algebraic Fundamentals

Sets and Numbers

Groups, Rings and Fields

Vector Spaces

Topological Fundamentals

Norms

Metrics

Topological Spaces

Compact Sets

Mappings

Continuous Mappings

Differentiable Mappings

Series

Linear Mappings

Endomorphisms

Inner Product Spaces

Geometry of Mappings

The Inverse Mapping Theorem

The Implicit Function Theorem

The Rank Theorem

Integration

Riemann Integral

Calculation of Integrals

Differential Forms

Line Integrals

Differential Forms

Integrals of Differential Forms
Honors Linear Algebra:
Course Description:
This honors section of Linear Algebra is intended for wellprepared students who have already developed some mathematical maturity. Its scope will include the usual Linear Algebra syllabus, however this class will move faster, covering additional topics and going deeper. Vector spaces, linear dependence, basis and dimension, matrices, determinants, solving linear equations, eigenvalues and eigenvectors, quadratic forms, applications such as optimization or linear regression.
Course Outline:

Getting started: vectors and linear combinations, lengths and dot products, matrices, systems of linear equations

Solving linear systems by Gaussian elimination

Rules for matrix operations

Inverse matrices

A matrix perspective on Gaussian elimination via LU factorization

Transposes and permutations

Vector spaces and subspaces

The null space of a matrix, and the complete solution of Ax = b

Independence, basis, and dimension

Dimensions of the four basic subspaces

An introduction to linear programming

Orthogonal complements and projections

Introduction to Fourier series

Least square approximation

The GramSchmidt procedure for finding an orthogonal basis

Determinants

Eigenvalues and eigenvectors

Diagonalization, when it is possible

The exponential of a matrix, and use of linear algebra to solve ODE’s

Symmetric matrices and positive definiteness

The PerronFrobenius theorem with applications to Markov processes and economics

Singular value decomposition

Application of SVD to principal component analysis

Linear transformations

More on linear transformations
Honors Multivariable Calculus:
Course Description:
This course is the third in the Mathematics Department's Honors sequence. It covers vector calculus in Rn: the change of variables formula, integration of differential forms, exterior derivative, generalized Stokes' theorem, conservative vector fields, potentials.
Course Outline:

Determinants

Volume

Change of variables formula

Parallelograms and their volumes

Parameterizations

Computing volumes of manifolds

Forms on Rn

Form fields and parametrized domains

Orientation of manifolds

Integrating forms over oriented manifolds

Forms in the language of vector calculus

Boundary orientation

The exterior derivative

Grad, curl, div, and all that

The generalized Stokes's theorem

The integral theorems of vector calculus

Potentials
Honors Abstract Algebra:
Course Description:
We will spend most of the semester studying bilinear algebra and lay the groundwork for further topics in abstract algebra, culminating with the main results of either Galois theory or the representation theory of finite groups, if not both. These are fundamental tools for much of modern mathematics.
Course Outline:

Introduction
This course will provide a rigorous introduction to abstract algebra, including group theory and linear algebra. Topics include:

Set theory. Formalization of Z, Q, R, C.

Linear algebra. Vector spaces and transformations over R and C. Other ground fields. Eigenvectors. Jordan form.

Multilinear algebra. Inner products, quadratic forms, alternating forms, tensor products, determinants.

Abstract groups.

Groups, symmetry, and representations.

Set Theory

Vector spaces

Polynomials

Linear Operators

Inner product spaces

Bilinear forms

Trace and determinant

Introduction to Group Theory

Symmetry

Finite group theory

Representation theory

Group presentations

Knots and the fundamental group
Honors Complex Analysis:
Course Description:
This is a rigorous introduction to complex analysis and is considered an Introduction to Proofs (IP) course. Topics covered in the course will include, review of complex numbers, Cauchy's theorem, holomorphic and meromorphic functions and some applications of complex analysis.
Course Outline:

Chapter 1. Preliminaries to Complex Analysis

Complex numbers and the complex plane

Basic properties

Convergence

Sets in the complex plane


Functions on the complex plane
2.1 Continuous functions
2.2 Holomorphic functions
2.3 Power series

Integration along curves

Exercises

Chapter 2. Cauchy’s Theorem and Its Applications

Goursat’s theorem

Local existence of primitives and Cauchy’s theorem in a disc

Evaluation of some integrals

Cauchy’s integral formulas

Further applications

Morera’s theorem

Sequences of holomorphic functions

Holomorphic functions defined in terms of integrals

Schwarz reflection principle

Runge’s approximation theorem


Exercises

Problems

Chapter 3. Meromorphic Functions and the Logarithm

Zeros and poles

The residue formula
2.1 Examples

Singularities and meromorphic functions

The argument principle and applications

Homotopies and simply connected domains

The complex logarithm

Fourier series and harmonic functions

Exercises

Problems

Chapter 4. The Fourier Transform

The class F

Action of the Fourier transform on F

PaleyWiener theorem

Exercises

Problems

Chapter 5. Entire Functions

Jensen’s formula

Functions of finite order

Infinite products

Generalities

Example: the product formula for the sine function


Weierstrass infinite products

Hadamard’s factorization theorem

Exercises

Problems

Chapter 6. The Gamma and Zeta Functions

The gamma function

Analytic continuation

Further properties of Γ


The zeta function
2.1 Functional equation and analytic continuation

Exercises

Problems