A treatment of the theory of differential equations. The objective is to provide the student with an ability to solve problems in this field. Administered and staffed by the Mechanical Engineering Department (MAE).

# All Current Undergraduate courses

#### Showing courses for option:Upper division - all 300, 400, & 500 level courses

All 300, 400, and 500 level courses - more advanced courses for people who already know their major.

##### MAT301 Mathematics in Engineering I

##### MAT302 Mathematics in Engineering II

This course provides an introduction to partial differential equations, covering PDEs of relevant interest in engineering and science problems.

##### MAT320 Introduction to Real Analysis

Introduction to real analysis, including the theory of Lebesgue measure and integration on the line and n-dimensional space, and the theory of Fourier series. Prerequisite: MAT201 and MAT202 or equivalent.
(Replaces MAT314 beginning AY2012-13)

##### MAT321 Numerical Methods

Introduction to numerical methods with emphasis on algorithms, applications and computer implementation issues. Topics covered include solution of nonlinear equations; numerical differentiation, integration and interpolation; direct and iterative methods for solving linear systems; numerical solutions of differential equations; two-point boundary value problems; and approximation theory. Lectures are supplemented with numerical examples using MATLAB. Prerequisites: MAT201 and MAT202; or MAT203 and MAT204; or equivalent.

##### MAT322 Introduction to Differential Equations

An introduction to differential equations, covering both applications and fundamental theory.

##### MAT323 Topics in Mathematical Modeling - Mathematical Neuroscience

Draws problems from the sciences & engineering for which mathematical models have been developed and analyzed to describe, understand and predict natural and man-made phenomena. Emphasizes model building strategies, analytical and computational methods, and how scientific problems motivate new mathematics.
This interdisciplinary course in collaboration with Molecular Biology, Psychology and the Program in Neuroscience is directed toward upperclass undergraduate students and first-year graduate students with knowledge of linear algebra and differential equations.

##### MAT325 Analysis I: Fourier Series and Partial Differential Equations

Fourier series, Fourier transforms, and applications to the classical partial differential equations.
Prerequisites: MAT215 or MAT218 or consent of instructor.
(Replaces MAT330 beginning AY 2012-13)

##### MAT330 Complex Analysis with Applications

Calculus of functions of one complex variable, power series expansions, residues, and conformal mapping. (Replaces MAT317 beginning AY 2012-13)

##### MAT335 Analysis II: Complex Analysis

Study of functions of a complex variable, with emphasis on interrelations with other parts of mathematics. Cauchy's theorems, singularities, contour integration, power series, infinite products. The gamma and zeta functions and the prime number theorem. Elliptic functions, theta functions, Jacobi's triple product and combinatorics. An overall view of Special Functions via the hypergeometric series. The second course in a four-semester sequence, but may be taken independently.
(Replaces MAT331 beginning AY 2012-13)

##### MAT340 Applied Algebra

An applied algebra course that integrates the basics of theory and modern applications for students in mathematics, applied mathematics, physics, chemistry, computer science and electrical engineering. Intended for students who have taken a semester of linear algebra and are interested in the structures, properties and applications of groups, rings and fields. Applications and algorithmic aspects of algebra will be emphasized throughout. New in Fall 2014.

##### MAT345 Algebra I

Covers the basics of symmetry and group theory, with applications. Topics include the fundamental theorem of finitely generated abelian groups, Sylow theorems, group actions and the representation theory of finite groups. Prerequisites: MAT202 or MAT204 or MAT217.
(Replaces MAT322 beginning AY 2012-13)

##### MAT350 Intoduction to Differentiable Manifolds

Begins with the calculus of differential forms, Stokes's theorem, Poincare Lemma and the deRham complex, and then continues with the theorem of Frobenius and its applications.
New in Fall 2014. Recommended sophomore course for students who took MAT215/217 in the freshman year.

##### MAT355 Introduction to Differential Geometry

Riemannian geometry of surfaces. Surfaces in Euclidan space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Bonnet Theorem, uniformization of surfaces. Prerequisites: MAT218 or equivalent.
(Replaces MAT327 beginning AY 2012-13)

##### MAT365 Topology

An introduction to point set topology, the fundamental group, covering spaces, methods of calculation and applications. Prerequisites: MAT202 or MAT204 or MAT218 or equivalent.
(Replaces MAT325 beginning AY 2012-13)

##### MAT375 Introduction to Graph Theory

This course will cover the fundamental theorems and algorithms of graph theory. Topics include: connectivity, mathchings, graph coloring, planarity, the four-color theorem, extremal problems, network flows, and related algorithms. Prerequisite: MAT202 or MAT204 or MAT217, or equivalent.
(Replaces MAT306 beginning AY 2012-13)

##### MAT377 Combinatorial Mathematics

Combinatorics is the study of enumeration and structure of discrete objects. These structures are widespread throughout mathematics, including geometry, topology and algebra, as well as computer science, physics, and optimization. This course will give an introduction to modern techniques in the field, and how they relate to objects such as polytopes, permutations, and hyperplane arrangements. Current work and open problems will also be discussed.

##### MAT378 Theory of Games

The mathematical concept of a game is an abstraction which encompasses conflict-cooperation situations in which strategy (not just chance) plays a role. Games in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications. Prerequisites: MAT202 or 204 or 217 or equivalent. MAT215 or equivalent is recommended.
(Replaces MAT308 beginning AY 2012-13)

##### MAT385 Probability Theory

Sequences of independent trials, applications to number theory and analysis, Monte Carlo method. Markov chains, ergodic theorem for Markov chains, Entropy and McMillan theorem. Random walks, recurrence and non-recurrence; connection with linear difference equations. Strong laws of large numbers, random series and products. Weak convergence of probability measures, weak Helly theorems, Fourier transforms of distributions. Limit theorems of probability theory. Prerequisites: MAT203 or MAT218 or equivalent.
(Replaces MAT390 beginning AY 2012-13)

##### MAT415 Analytic Number Theory

An introduction to classical results in analytic number theory, presenting fundamental theorems with detailed proofs and highlighting the tight connections between them. Topics covered might include: the prime number theorem, Dirichlet L-functions, zero-free regions, sieve methods, representation by quadratic forms, and Gauss sums. Prerequisites: MAT335 (Complex Analysis) and MAT345 (Algebra I).

##### MAT419 Topics in Number Theory: Algebraic Number Theory

Course on algebraic number theory. Topics covered include number fields and their integer rings, class
groups, zeta and L-functions.

##### MAT425 Analysis III: Integration Theory and Hilbert Spaces

The theory of Lebesgue integration in n-dimensional space. Differentiation theory. Hilbert space theory and applications to Fourier transforms, and partial differential equations. Introduction to fractals. The third semester of a four-semester sequence, but may be taken independently. Prerequisites: MAT215 or MAT218 or equivalent.
(Replaces MAT332 beginning AY 2012-13)

##### MAT427 Ordinary Differential Equations

The course concerns explicit solution of simple differential equations. Methods of proving that one has found all the solutions are discussed. For this purpose, a brief review of foundational concepts in real analysis is provided. The second part concerns explicit solutions of simultaneous linear differential equations with constant coefficients, a topic closely connected with linear algebra (assumed prerequisite knowledge). The third part concerns the proof of the basic existence and uniqueness theorem for ordinary differential equations. Students will do simple proofs.
(Replaces MAT303 beginning AY 2012-13)

##### MAT443 Cryptography

An introduction to modern cryptography with an emphasis on the fundamental ideas.

##### MAT447 Commutative Algebra

This course will cover the standard material in a first course on commutative algebra. Topics include: ideals in and modules over commutative rings, localization, primary decomposition, integral dependence, Noetherian rings and chain conditions, discrete valuation rings and Dedekind domains, completion; dimension theory. Prerequisites: Algebra I & II, MAT345-6.

##### MAT449 Topics in Algebra: Representation Theory

An introduction to representation theory of Lie groups and Lie algebras. The goal is to cover roughly the
first half of Knapp's book.

##### MAT455 Advanced Topics in Geometry - Lie Theory

Lie algebras and Lie groups are important in many areas of mathematics as well as theoretical physics. The course gives an introduction to the topic.

##### MAT486 Random Processes

Wiener measure, Stochastic differential equations, Markov diffusion processes, Linear theory of stationary processes, Ergodicity, mixing, central limit theorem for stationary processes, Gibbs random field. If time permits, the theory of products of random matrices and PDE's with random coefficients will be discussed. Prerequisite: MAT390 in the old numbering system or MAT385 in the new system.
(Replaces MAT391 beginning AY 2012-13)

##### MAT522 Introduction to Partial Differential Equations

Introduction to the techniques necessary for the formulation and solution of problems involving partial differential equations in the natural sciences and engineering, with detailed study of the heat and wave equations. Topics include method of eigenfunction expansions, Fourier series, the Fourier transform, inhomogeneous problems, the method of variation of parameters. Prerequisite MAT202 or MAT204 or MAT218.

##### MAT523 Advanced Analysis

The course covers the essentials of the first eleven chapters of the textbook, "Analysis" by Lieb and Loss. Topics include Lebesque integrals, Measure Theory, L^p Spaces, Fourier Transforms, Distributions, Potential Theory, and some illustrative examples of applications of these topics.