# Mathematics Immersion

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## Overview

• This immersion is closed to students majoring in applied statistics and actuarial science, applied mathematics, and computational mathematics.
• Students are required to complete at least one course at the 300-level or above as part of the immersion.

## Curriculum

Course
Prerequisites
MATH-181
Project-based Calculus I (or equivalent)
This is the first in a two-course sequence intended for students majoring in mathematics, science, or engineering. It emphasizes the understanding of concepts, and using them to solve physical problems. The course covers functions, limits, continuity, the derivative, rules of differentiation, applications of the derivative, Riemann sums, definite integrals, and indefinite integrals.
Plus one of the following:
MATH-182
Project-based Calculus II
This is the second in a two-course sequence intended for students majoring in mathematics, science, or engineering. It emphasizes the understanding of concepts, and using them to solve physical problems. The course covers techniques of integration including integration by parts, partial fractions, improper integrals, applications of integration, representing functions by infinite series, convergence and divergence of series, parametric curves, and polar coordinates.
MATH-190
Discrete Mathematics for Computing
This course introduces students to ideas and techniques from discrete mathematics that are widely used in Computer Science. Students will learn about the fundamentals of propositional and predicate calculus, set theory, relations, recursive structures and counting. This course will help increase students’ mathematical sophistication and their ability to handle abstract problems.
MATH-200
Discrete Mathematics and Introduction to Proof
This course prepares students for professions that use mathematics in daily practice, and for mathematics courses beyond the introductory level where it is essential to communicate effectively in the language of mathematics. It covers various methods of mathematical proof, starting with basic techniques in propositional and predicate calculus and set theory, and then moving to applications in advanced mathematics.
Electives*
Choose three of the following:
MATH-219
Multivariable Calculus †
This course is principally a study of the calculus of functions of two or more variables, but also includes the study of vectors, vector-valued functions and their derivatives. The course covers limits, partial derivatives, multiple integrals, and includes applications in physics. Credit cannot be granted for both this course and MATH-221.
MATH-221
Multivariable and Vector Calculus † §
This course is principally a study of the calculus of functions of two or more variables, but also includes a study of vectors, vector-valued functions and their derivatives. The course covers limits, partial derivatives, multiple integrals, Stokes' Theorem, Green's Theorem, the Divergence Theorem, and applications in physics. Credit cannot be granted for both this course and MATH-219.
MATH-231
Differential Equations ‡
This course is an introduction to the study of ordinary differential equations and their applications. Topics include solutions to first order equations and linear second order equations, method of undetermined coefficients, variation of parameters, linear independence and the Wronskian, vibrating systems, and Laplace transforms.
MATH-233
Linear Systems and Differential Equations ‡
This is an introductory course in linear algebra and ordinary differential equations in which a scientific computing package is used to clarify mathematical concepts, visualize problems, and work with large systems. The course covers matrix algebra, the basic notions and techniques of ordinary differential equations with constant coefficients, and the physical situation in which they arise.
MATH-241
Linear Algebra §
This course is an introduction to the basic concepts of linear algebra, and techniques of matrix manipulation. Topics include linear transformations, Gaussian elimination, matrix arithmetic, determinants, vector spaces, linear independence, basis, null space, row space, and column space of a matrix, eigenvalues, eigenvectors, change of basis, similarity and diagonalization. Various applications are studied throughout the course.
MATH-251
Probability and Statistics I
This course introduces sample spaces and events, axioms of probability, counting techniques, conditional probability and independence, distributions of discrete and continuous random variables, joint distributions (discrete and continuous), the central limit theorem, descriptive statistics, interval estimation, and applications of probability and statistics to real-world problems. A statistical package such as Minitab or R is used for data analysis and statistical applications.
MATH-311
Linear Optimization
This course presents the general linear programming problem. Topics include a review of pertinent matrix theory, convex sets and systems of linear inequalities, the simplex method of solution, artificial bases, duality, parametric programming, and applications.
MATH-312
Non-linear Optimization
This course provides a study of the theory of optimization of non-linear functions of several variables with or without constraints. Applications of this theory in business, management, engineering and the sciences are considered. Algorithms for practical applications will be analyzed and implemented. The course may require the use of specialized software to analyze problems. Students taking this course will be expected to complete applied projects and/or case studies.
MATH-321
Game Theory
This course introduces solution techniques and applications of Game Theory. Topics include game trees, matrix games, linear inequalities, convex sets, the minimax theorem, and n-person games.
MATH-326
Boundary Value Problems
This course provides an introduction to boundary value problems. Topics include Fourier series, separation of variables, Laplace's equation, the heat equation, and the wave equation in Cartesian and polar coordinate systems.
MATH-331
Dynamical Systems
The course revisits the equations of spring-mass system, RLC circuits, and pendulum systems in order to view and interpret the phase space representations of these dynamical systems. The course begins with linear systems followed by a study of the stability analysis of nonlinear systems. Matrix techniques are introduced to study higher order systems. The Lorentz equation will be studied to introduce the concept of chaotic solutions.
MATH-361
Combinatorics
This course introduces the mathematical theory of enumeration of discrete structures. Topics include enumeration, combinatorial proofs, recursion, inclusion-exclusion, and generating functions.
MATH-367
Codes and Ciphers
This course will introduce, explain and employ the basic techniques of cryptography, both classical and modern. Topics will include the Vignere cipher, affine ciphers, Hill ciphers, one-time pad encryption, Enigma, cryptosystems such as DES (Data Encryption Standard) and AES (Advanced Encryption Standard), public key encryption scheme (RSA), and hash functions. The course will include an introduction to number theoretic tools used in cryptography.
MATH-381
Complex Variables
This course covers the algebra of complex numbers, analytic functions, Cauchy-Riemann equations, complex integration, Cauchy's integral theorem and integral formulas, Taylor and Laurent series, residues, and the calculation of real-valued integrals by complex-variable methods.
MATH-341
This is a second course in linear algebra that provides an in-depth study of fundamental concepts of the subject. It focuses largely on the effect that a choice of basis has on our understanding of and ability to solve problems with linear operators. Topics include linear transformations, similarity, inner products and orthogonality, QR factorization, singular value decomposition, and the Spectral Theorem. The course includes both computational techniques and the further development of mathematical reasoning skills.
MATH-351
Graph Theory
This course covers the theory of graphs and networks for both directed and undirected graphs. Topics include graph isomorphism, Eulerian and Hamiltonian graphs, matching, covers, connectivity, coloring, and planarity. There is an emphasis on applications to real world problems and on graph algorithms such as those for spanning trees, shortest paths, and network flows.
MATH-371
Number Theory
This course provides an introduction to the study of the set of integers and their algebraic properties. Topics include prime factorization and divisibility, linear Diophantine equations, congruences, arithmetic functions, primitive roots, and quadratic residues.
MATH-411
Numerical Analysis
This course covers numerical techniques for the solution of nonlinear equations, interpolation, differentiation, integration, and the solution of initial value problems.
MATH-412
Numerical Linear Algebra
This course covers numerical techniques for the solution of systems of linear equations, eigenvalue problems, singular values and other decompositions, applications to least squares, boundary value problems, and additional topics at the discretion of the instructor.
MATH-431
Real Variables I
This course is an investigation and extension of the theoretical aspects of elementary calculus. Topics include mathematical induction, real numbers, sequences, functions, limits, and continuity. The workshop will focus on helping students develop skill in writing proofs.
MATH-432
Real Variables II
This course is a continuation of MATH-431. It concentrates on differentiation, integration (Riemann and Riemann-Stieltjes integrals), power series, and sequences and series of functions.
MATH-441
Abstract Algebra I
This course covers basic set theory, number theory, groups, subgroups, cyclic and permutation groups, Lagrange and Sylow theorems, quotient groups, and isomorphism theorems. Group Theory finds applications in other scientific disciplines like physics and chemistry.
MATH-442
Abstract Algebra II
This course covers the basic theory of rings, integral domains, ideals, modules, and abstract vector spaces. It also covers the key constructions including direct sums, direct products, and field extensions. These topics serve as the foundation of mathematics behind advanced topics such as algebraic geometry and various applications like cryptography and coding theory.
MATH-461
Topology
This course defines metric spaces and topological spaces. For metric spaces it examines continuity spaces of continuous functions and completeness in Euclidean spaces. For topological spaces it examines compactness, continuous functions, and separation axioms.
MATH-505
Stochastic Processes
This course explores Poisson processes and Markov chains with an emphasis on applications. Extensive use is made of conditional probability and conditional expectation. Further topics, such as renewal processes, Brownian motion, queuing models and reliability are discussed as time allows.

* At least one course must be taken at the 300-level or above.

† Students may choose one of these courses, but not both

‡ Students may choose one of these courses, but not both

§ This course has honors-designated sections taught occasionally.