# 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:

1. Vector spaces and linear transformations.

2. Systems of equations, determinants.

3. Spectral theorem.

4. Ordinary, linear, and nonlinear differential equations.

5. Existence and uniqueness theorems.

6. 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 n-dimensional real vector spaces.

Course Outline:

1. Algebraic Fundamentals

1. Sets and Numbers

2. Groups, Rings and Fields

3. Vector Spaces

1. Topological Fundamentals

1. Norms

2. Metrics

3. Topological Spaces

4. Compact Sets

1. Mappings

1. Continuous Mappings

2. Differentiable Mappings

3. Series

1. Linear Mappings

1. Endomorphisms

2. Inner Product Spaces

1. Geometry of Mappings

1. The Inverse Mapping Theorem

2. The Implicit Function Theorem

3. The Rank Theorem

1. Integration

1. Riemann Integral

2. Calculation of Integrals

1. Differential Forms

1. Line Integrals

2. Differential Forms

3. Integrals of Differential Forms

Honors Linear Algebra:

Course Description:

This honors section of Linear Algebra is intended for well-prepared 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:

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

2. Solving linear systems by Gaussian elimination

3. Rules for matrix operations

4. Inverse matrices

5. A matrix perspective on Gaussian elimination via LU factorization

6. Transposes and permutations

7. Vector spaces and subspaces

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

9. Independence, basis, and dimension

10. Dimensions of the four basic subspaces

11. An introduction to linear programming

12. Orthogonal complements and projections

13. Introduction to Fourier series

14. Least square approximation

15. The Gram-Schmidt procedure for finding an orthogonal basis

16. Determinants

17. Eigenvalues and eigenvectors

18. Diagonalization, when it is possible

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

20. Symmetric matrices and positive definiteness

21. The Perron-Frobenius theorem with applications to Markov processes and economics

22. Singular value decomposition

23. Application of SVD to principal component analysis

24. Linear transformations

25. 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:

1. Determinants

2. Volume

3. Change of variables formula

4. Parallelograms and their volumes

5. Parameterizations

6. Computing volumes of manifolds

7. Forms on Rn

8. Form fields and parametrized domains

9. Orientation of manifolds

10. Integrating forms over oriented manifolds

11. Forms in the language of vector calculus

12. Boundary orientation

13. The exterior derivative

14. Grad, curl, div, and all that

15. The generalized Stokes's theorem

16. The integral theorems of vector calculus

17. 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:

1. Introduction

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

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

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

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

4. Abstract groups.

5. Groups, symmetry, and representations.

1. Set Theory

2. Vector spaces

3. Polynomials

4. Linear Operators

5. Inner product spaces

6. Bilinear forms

7. Trace and determinant

8. Introduction to Group Theory

9. Symmetry

10. Finite group theory

11. Representation theory

12. Group presentations

13. 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:

1. Chapter 1. Preliminaries to Complex Analysis

1. Complex numbers and the complex plane

1. Basic properties

2. Convergence

3. Sets in the complex plane

2. Functions on the complex plane

2.1 Continuous functions

2.2 Holomorphic functions

2.3 Power series

1. Integration along curves

2. Exercises

1. Chapter 2. Cauchy’s Theorem and Its Applications

1. Goursat’s theorem

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

3. Evaluation of some integrals

4. Cauchy’s integral formulas

5. Further applications

1. Morera’s theorem

2. Sequences of holomorphic functions

3. Holomorphic functions defined in terms of integrals

4. Schwarz reflection principle

5. Runge’s approximation theorem

6. Exercises

7. Problems

1. Chapter 3. Meromorphic Functions and the Logarithm

1. Zeros and poles

2. The residue formula

2.1 Examples

1. Singularities and meromorphic functions

2. The argument principle and applications

3. Homotopies and simply connected domains

4. The complex logarithm

5. Fourier series and harmonic functions

6. Exercises

7. Problems

1. Chapter 4. The Fourier Transform

1. The class F

2. Action of the Fourier transform on F

3. Paley-Wiener theorem

4. Exercises

5. Problems

1. Chapter 5. Entire Functions

1. Jensen’s formula

2. Functions of finite order

3. Infinite products

1. Generalities

2. Example: the product formula for the sine function

4. Weierstrass infinite products

6. Exercises

7. Problems

1. Chapter 6. The Gamma and Zeta Functions

1. The gamma function

1. Analytic continuation

2. Further properties of Γ

2. The zeta function

2.1 Functional equation and analytic continuation

1. Exercises

2. Problems