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:
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Vector spaces and linear transformations.
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Systems of equations, determinants.
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Spectral theorem.
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Ordinary, linear, and nonlinear differential equations.
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Existence and uniqueness theorems.
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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:
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Algebraic Fundamentals
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Sets and Numbers
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Groups, Rings and Fields
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Vector Spaces
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Topological Fundamentals
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Norms
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Metrics
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Topological Spaces
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Compact Sets
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Mappings
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Continuous Mappings
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Differentiable Mappings
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Series
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Linear Mappings
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Endomorphisms
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Inner Product Spaces
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Geometry of Mappings
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The Inverse Mapping Theorem
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The Implicit Function Theorem
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The Rank Theorem
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Integration
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Riemann Integral
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Calculation of Integrals
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Differential Forms
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Line Integrals
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Differential Forms
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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:
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Getting started: vectors and linear combinations, lengths and dot products, matrices, systems of linear equations
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Solving linear systems by Gaussian elimination
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Rules for matrix operations
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Inverse matrices
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A matrix perspective on Gaussian elimination via LU factorization
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Transposes and permutations
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Vector spaces and subspaces
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The null space of a matrix, and the complete solution of Ax = b
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Independence, basis, and dimension
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Dimensions of the four basic subspaces
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An introduction to linear programming
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Orthogonal complements and projections
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Introduction to Fourier series
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Least square approximation
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The Gram-Schmidt procedure for finding an orthogonal basis
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Determinants
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Eigenvalues and eigenvectors
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Diagonalization, when it is possible
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The exponential of a matrix, and use of linear algebra to solve ODE’s
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Symmetric matrices and positive definiteness
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The Perron-Frobenius theorem with applications to Markov processes and economics
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Singular value decomposition
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Application of SVD to principal component analysis
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Linear transformations
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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:
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Determinants
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Volume
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Change of variables formula
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Parallelograms and their volumes
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Parameterizations
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Computing volumes of manifolds
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Forms on Rn
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Form fields and parametrized domains
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Orientation of manifolds
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Integrating forms over oriented manifolds
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Forms in the language of vector calculus
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Boundary orientation
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The exterior derivative
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Grad, curl, div, and all that
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The generalized Stokes's theorem
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The integral theorems of vector calculus
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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:
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Introduction
This course will provide a rigorous introduction to abstract algebra, including group theory and linear algebra. Topics include:
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Set theory. Formalization of Z, Q, R, C.
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Linear algebra. Vector spaces and transformations over R and C. Other ground fields. Eigenvectors. Jordan form.
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Multilinear algebra. Inner products, quadratic forms, alternating forms, tensor products, determinants.
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Abstract groups.
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Groups, symmetry, and representations.
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Set Theory
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Vector spaces
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Polynomials
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Linear Operators
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Inner product spaces
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Bilinear forms
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Trace and determinant
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Introduction to Group Theory
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Symmetry
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Finite group theory
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Representation theory
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Group presentations
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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:
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Chapter 1. Preliminaries to Complex Analysis
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Complex numbers and the complex plane
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Basic properties
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Convergence
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Sets in the complex plane
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Functions on the complex plane
2.1 Continuous functions
2.2 Holomorphic functions
2.3 Power series
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Integration along curves
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Exercises
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Chapter 2. Cauchy’s Theorem and Its Applications
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Goursat’s theorem
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Local existence of primitives and Cauchy’s theorem in a disc
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Evaluation of some integrals
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Cauchy’s integral formulas
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Further applications
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Morera’s theorem
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Sequences of holomorphic functions
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Holomorphic functions defined in terms of integrals
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Schwarz reflection principle
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Runge’s approximation theorem
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Exercises
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Problems
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Chapter 3. Meromorphic Functions and the Logarithm
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Zeros and poles
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The residue formula
2.1 Examples
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Singularities and meromorphic functions
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The argument principle and applications
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Homotopies and simply connected domains
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The complex logarithm
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Fourier series and harmonic functions
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Exercises
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Problems
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Chapter 4. The Fourier Transform
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The class F
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Action of the Fourier transform on F
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Paley-Wiener theorem
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Exercises
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Problems
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Chapter 5. Entire Functions
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Jensen’s formula
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Functions of finite order
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Infinite products
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Generalities
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Example: the product formula for the sine function
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Weierstrass infinite products
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Hadamard’s factorization theorem
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Exercises
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Problems
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Chapter 6. The Gamma and Zeta Functions
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The gamma function
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Analytic continuation
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Further properties of Γ
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The zeta function
2.1 Functional equation and analytic continuation
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Exercises
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Problems