Mathematics-Algebraic Stru (MAS)

Introduction to the basic concepts and theorems of linear algebra. Topics include matrices and determinants, matrix operations and canonical factorizations, Gauss-Jordan elimination, vector spaces, linear transformations, basis, independence, orthogonality, eigenvalues and eigenvectors, similarity, diagonalization, and an introduction to Markov chains.

Topics in Linear Algebra drawn from a broad range of applications such as large-matrix computation techniques, pseudo-inverses and linear regression, non-negative matrices and the PageRank algorithm, low rank approximation and compressed sensing, computer graphics, multivariate Gaussian and weighted least squares, Markov chains, optimization, neural networks and machine learning.

Basic principles of algebraic number theory. Topics include divisibility and residues, congruences, primes, pseudoprimes and primality tests, systems of congruences and Chinese remaindering, multiplicative functions, continued fractions.

Introduction to fundamental concepts of modern algebra. Topics include group axioms, subgroups, Lagrange's Theorem, homomorphism, quotient groups, permutation and symmetry groups, rings, integral domains and fields, rings of polynomials, field of quotients.

Continuation of MAS 4301. Topics include subgroups and Sylow theorems, homomorphisms and quotient groups, ideals in rings, principal ideal domains and Euclidean domains, quotient rings, fields and extension fields. Emphasis is on skills and topics needed for graduate study in mathematics.

Training in formulating and solving problems with the aid of a computer algebra system. A wide-ranging selection of exercises will be addressed with Maple 10 or a similar system.

Numerical methods underlying a variety of modern machine learning and data analysis techniques are discussed. Topics include block-matrix multiplication and its role in high-performance computing, Identification of fundamental low-dimensional structure or parameters in high-dimensional problems, and model reduction through Principal Component Analysis, Data fitting and classification through multi-objective and constrained least squares, Introduction to data fitting and classification through nonlinear least squares, Bias-Variance tradeoff in Lasso (L1) and Ridge (L2) regression in data fitting problems, and ill conditioning.

Advanced topics in linear algebra, to include abstract vector spaces, subspaces, linear transformations and matrices, canonical forms and decompositions, eigenvalues and singular values, quadratic form, and positive definite matrices.

Course will feature an accelerated review of elementary concepts in sufficient detail to support a more intensive study of the classical problems in algebraic number theory. Permission of instructor required.

This course is an intensive review of basic structures of abstract and linear algebra. Topics may include groups, rings and ideals, polynomials and factorization, vector spaces and modules, linear transformations, and the classical Galois theory of fields. It is intended as a refresher course for students in the Mathematics Certificate program prior to enrolling in MAS 5311. It is not available for credit to students enrolled in the M.S. in Mathematics program.

Graduate level introduction to basic theory of groups and rings. Topics include groups, subgroups, homomorphism, quotient groups, rings, fields, and Sylow theory.