Description: Systems of equations, matrix algebra, vector spaces, linear transformations, geometry of R^n, eigenvalues, eigenvectors, diagonalization, inner products. Emphasis on mathematical reasoning, proofs, and concepts.

Description: Separable and exact equations, linear equations and variation of parameters, higher order linear equations, Laplace Transforms, first-order linear systems, classification of singularities, nonlinear systems, partial differential equations and Fourier Series, existence and uniqueness theorems. Emphasis on theory.

Description: Emphasis on foundations and metric topology. Convergence of sequence and series, continuity of functions. Differentiation and integration in one dimension.

Description: Theory of groups, Sylow theory, the structure of finite Abelian groups, ring theory, ideals, homomorphisms, and polynomial rings.

Description: Probability, conditional probability and independence. Random variables. Discrete, continuous, univariate, and multivariate distributions. Expectation and its properties, moment generating functions. Law of large numbers, central limit theorem.

Description: Continuation of MTH 327H. Convergence of sequences and series of functions, differentiation and integration in higher dimensional settings. Inverse and implicit function theorems.

Description: Algebraic field extensions, Galois theory. Classification of finite fields. Fundamental Theorem of Algebra.

Description: The two primary topics of the course will be Fourier Series and the Fourier Transform. Fourier series are used to express a periodic function in terms of infinite sums of sines and cosines. We will investigate the convergence of Fourier series, which turns out to be a subtle matter. The Fourier transform represents a given function in terms of a continuum of “frequencies”, and has various applications in areas such as Partial Differential Equations, Mathematical Physics, Signal Processing and Medical Imaging. We will develop the theory of the Fourier transform and illustrate its use in the aforementioned areas.

Additional Topics we might cover include (time permitting): Discrete Fourier Transforms, the Fast Fourier Transform (FFT) algorithm, Orthogonal Functions, Abstract Inner Product Spaces, Distributions, and Time-Frequency Analysis.

Description: Binomial and multinomial theorems. Graphs and digraphs, graph coloring. Generating functions, asymptotic analysis, trees. Representing graphs in computers.

Description: Standard methods and tools for computational modeling and data analysis using simulation techniques, data mining, and machine learning.

Description: Study of limits, continuous functions, derivatives, integrals and their applications.