Course Descriptions

**MATH 1000, Precalculus**

A preparatory course to assist students at FC who intend to take calculus. A placement test is required before students can register. Topics include inequalities; linear, polynomial, rational, exponential, logarithm and inverse functions and their graphs; distance, lengths and area of simple regions; and trigonometric functions. This course does not satisfy the mathematical reasoning core area requirement.

**MATH 1001, Math for Business: Precalculus**

A preparatory course to assist students at the GSB to take MATH 1109, Math for Business: Calculus. Topics include inequalities; linear, polynomial, rational, exponential, logarithm and inverse functions and their graphs; distance, lengths and area of simple regions. This course does not satisfy the mathematical reasoning core area requirement.

**MATH 1100, Finite Mathematics**

Topics include solutions to systems of linear equations, counting techniques including Venn diagrams, permutations, combinations, probability, Bayes theorem, Markov chains.

**MATH 1108, Math for Business: Finite** (*open only to GSB students*)

Topics include solutions to systems of linear equations, elementary matrix theory, linear programming, elementary counting techniques, probability, mathematics of finance.

**MATH 1109, Math for Business: Calculus** (*open only to GSB students*)

Topics include derivatives of polynomial, rational, exponential, and logarithmic functions, curve sketching, optimization problems, definite integrals, applications to business and economics.

**MATH 1203, Applied Calculus I**

*This is an elementary course in calculus intended primarily for nonscience majors.*

Derivatives of polynomial, rational, exponential, and logarithmic functions, curve sketching, optimization problems, definite integrals.

**MATH 1204, Applied Calculus II**

Topics include derivatives of trigonometric functions, methods of integration and applications, calculus of functions of several variables, Lagrange multipliers.

*Prerequisite: MTRU/LU 1203 or equivalent.*

**MATH 1205, Applied Statistics**

Course designed for students in fields that emphasize quantitative methods. It includes calculus based preliminary probability material followed by introduction to the basic statistical methods such as estimation, hypothesis testing, correlation and regression analysis. Illustrations are taken from a variety of fields. The course provides practical experience with statistical software. Prerequisite: MATH 1203 or equivalent.

*Prerequisite: MTRU/LU 1203 or equivalent.*

**MATH 1206, Calculus I (4 credits)**

This calculus course is for science and math majors and math minors. Topics include: functions, limits, continuity, Intermediate Value Theorem, the derivative, its interpretations, and rules for computation, differentiation of trigonometric functions, applications to curve sketching and optimization problems, antiderivatives and initial value problems, Riemann sums, definite integrals, and the Fundamental Theorem of Calculus.

**MATH 1207 - CALCULUS II (4 credits)**

This course is a continuation of MATH 1206. Topics include: definite integrals, arc length, area, volume, work, logarithmic and exponential functions, inverse functions, techniques of integration, including substitutions and integration by parts. parametric curves, polar coordinates, sequences and series, Taylor polynomials with remainder, and Taylor series.

*Prerequisite: MATH 1206 or equivalent.*

**MATH 1700, Mathematical**** Modeling**

Building and using discrete and continuous mathematical models. Applications to mathematics, physical sciences, b**i**ology, economics. Computer explorations with software packages like Maple.

*Prerequisite**:** MATH 1206 **or equivalent**.*

**MATH 2001, Discrete Mathematics**

Topics include elementary logic, set theory, basic counting techniques including generating functions, induction, recursion, recurrence.

*Prerequisite: **MATH 1206** *

**MATH 2004, Multivariable Calculus (4 credits)**

This course is a continuation of MATH 1207. Topics include three-dimensional Cartesian coordinates, vector methods of solid geometry, vector-valued function, functions of several variables, partial derivatives, gradients, optimization and Lagrange multipliers, Implicit Function Theorem, double and triple integrals in Cartesian coordinates.

*Prerequisite: **MATH 1207 or equivalent*.

MATH 2005, Multivariable Calculus II (4 credits)

This course is a continuation of MATH 2004. Topics include vector fields and their derivatives, multiple integrals in curvilinear coordinates, line and surface integrals, the theorems of Gauss, Green, and Stokes. Additional topics, as time permits, may cover one or more of the following: differential forms, functions of a complex variable, equations of fluid mechanics, Mean and Gauss curvature. *Prerequisite: MATH 2004 or equivalent.*

**MATH 2006, Linear Algebra**

Topics include systems of linear equations, real and complex vector spaces, linear independence, dimension, linear transformations, matrix representations, kernel and range, determinants and eigenvalues.

*Prerequisites:** MATH 1206 .*

MATH 3001, Linear Algebra II

Topics include vector spaces over arbitrary fields, Jordan canonical form, inner product spaces, coding theory.

*Prerequisite: **MATH 2001 and** **MATH 2006**.*

**MATH 3002, Differential Equations**

Topics include existence and uniqueness Theorems for ordinary differential equations, linear differential equations, power series solutions, and numerical methods.

*Prerequisite:** MATH 2004.*

**MATH 3003, Real Analysis**

Topics include cardinality of sets, limits, continuity, uniform continuity, sequences and series of numbers and functions, modes of convergence, compact sets and associated Theorems.

*Prerequisite:** **MATH 2005.*

**MATH 3004, Complex Analysis**

Topics include complex numbers and mappings, analytic functions, Cauchy-Riemann equations, Cauchy integral Theorem, Taylor and Laurent series expansions, residue theory.

*Prerequisite:** MATH 2005.*

**MATH 3005, Abstract Algebra**

Topics include well-ordering and induction, unique factorization, modular arithmetic, groups, subgroups, Lagrange's Theorem, normality, homomorphisms of groups, permutation groups, simple groups.

*Prerequisite: **MATH 2001 and 2006.*

**MATH 3006, Probability**

Topics include discrete and continuous probability models in one and several variables, expectation and variance, limit Theorems, applications.

*Prerequisite: **MATH 2004.*

**MATH 3007, Statistics**

Topics include sampling distributions, estimation, testing hypotheses, analysis of variance, regression and correlation, nonparametric methods, time series.

*Prerequisite: **MATH 3006.*

**MATH 3008, Number Theory**

Topics include divisibility and related concepts, congruencies, quadratic residues, number theoretic functions, addtive number theory, some Diophantine equations.

*Prerequisite:** MATH 2001 and 2006.*

MATH 3009, Mathematics of Finance

The market for options, a type of contract in finance, has grown quickly in the past 50 years. In this course we will explore the Nobel Prize=winning Black-Scholes-Merton model for valuing these contracts. We will introduce basic notions of probability (such as Brownian motion) as well as basic notions from finance (such as the No Arbitrage Principle) and use these to derive and solve the Black-Scholes equation.

*Prerequisites: MATH 20014 and either one of the following two courses: MATH 3006 or ECON 2140*.

MATH 3021, Graph Theory

Elements of graph theory and digraphs, matrix representations of graphs, shortest paths, applications of graph theory to transport networks, graph colorings, matching theory, and graphical algorithms.

Prerequisite: *MATH 2001 or CISC 1400, and any programming experience* *.*

**MATH 4000, Mathematical Ethics Practicum, 4 credits**

In this class, which fulfills the Senior Values seminar requirement of the Core Curriculum and serves as a capstone to both the pure and applied tracks of the Mathematics major, students will learn the ethical responsibilities of mathematicians, both as interpreters and as creators of mathematics. The course will combine historical and contemporary case studies with practical training in the skills and disciplines students must master to assume full ownership of their mathematics.

**MATH 4001, Operations Research**

Topics include introduction to linear programming, statistical inference, decision theory, queuing theory, inventory theory, Markov chains.

*Prerequisites: MATH 2006 and MATH 3002.*

**MATH 4003, Abstract Algebra II**

Topics include Sylow Theorems, solvable groups, field extensions, Galois theory, polynomial equations.

*Prerequisite: MATH 3005.*

**MATH 4004, Topology**

Topics include open sets and continuity in metric and topological spaces, subspaces and quotient topologies, compact sets, connected sets.

*Prerequisites: MATH 2001 and 2006.*

**MATH 4006, Numerical Analysis**

Topics include approximation of functions, interpolation, solutions to systems of equations, numerical integration, solutions to differential equations, error analysis.

*Prerequisites: Math 2004 and 2006.*

**MATH 4009, Geometry**

This course focuses on the study of both Euclidean and non-Euclidean Geometry using axiomatic and discovery based approaches. We review some of the basics in logic and study some of the proofs presented in Euclid's Elements before focusing on more advanced geometric properties and proofs. We will use Geometer's Sketchpad in making discoveries and conjectures. We ill study the history of the parallel postulate, the discovery of non=Euclidean geometry, and it's philosophical implications. We will build models and focus on some interesting properties in hyperbolic geometry.

Prerequisite: *MATH 2001 or MATH 2004 and permission of instructor**.*

MATH 4010, Topics in Topology

The course consists of set theory. Topics include sets and functions, Hilbert's Infinite Hotel, infinite sets. Included are countable sets, uncountable sets, cardinalities of sets, ordinals and cardinals, the collection of all sets, and the logic of statements that are neither true nor false.

Prerequisite: *MATH 4004**.*

**MATH 4020, Differential Geometry**

This course introduces the geometry of curved spaces in many dimensions, which are the basis of subjects such as Einstein's theory of gravitation. Topics include manifolds, tangent spaces, the Gauss map, the shape operator, curvature, and geodesics.

*Prerequisite: **MATH 2004 and** **MATH 2006**.*

**MATH 4022, Partial Differential Equation (4 credits)**

This course is an introduction to the theory of partial differential equations. It covers first order linear, wave, diffusion, and Laplace's equations. Topics include the method of characteristics, maximum principle, reflection and sources, separation of variables, Fourier series, completeness, Poisson's intergral formula and the mean value formula.

The department of mathematics also offers the following advanced courses upon request:

**MATH 4007, Topics in Algebra**

**MATH 4008, Topics in Analysis**

**MATH 4011, Topics in Applied Mathematics**

**MATH 4013, Problem Seminar**

**MATH 4999, Independent Study**

**MATH 3012, Mathematics of Infinity**

Elementary set and function theory, notion of counting infinite sets, including Hilbert's infinite hotel. Cardinality and infinite cardinals. Cantor's work on infinite sets. Additional topics may include: well-ordered sets and math induction; prime number generators; the Riemann zeta function; logic and metamathematics.

Prerequisite: *MATH 2001 and MATH 2006.*