# Mathematics (BA)

**Chair: Byung-Jay Kahng, PhD**

## Introduction

The Department of Mathematics and Statistics strives to transmit an understanding and appreciation of mathematics: its substance, its applicability, its literature, its current directions and problems, its historical development, its human worth and values; and to promote the development of skill in the practice of mathematics. For a more detailed description of the program, faculty, facilities, academic and co-curricular opportunities please go to the Mathematics and Statistics website.

## Qualifications

Students must maintain a 2.0 GPA in their major and a 2.0 overall average to graduate with a degree in Mathematics and Statistics. Students must have a minimum grade of C- in all courses in the major.

## Advisement

All students should have an advisor in the major and should contact the department directly to have an advisor assigned if they do not already have one. Meetings with academic advisors are required prior to students receiving their PIN for course registration each semester. All majors should work closely with their advisor in discussing career expectations, choosing their major electives, developing their entire academic program and planning their co-curricular or supplemental academic experiences.

## Double Majors

Students who wish to expand their educational opportunities may decide to declare a double major. The decision may be based on career goals or planned graduate studies. Before a student declares a double major, it is important to meet with the appropriate academic departments for advisement. Some double major combinations can be completed within the minimum 120 credit hour degree requirement, but in some cases additional course work may be required. In order to declare a double major, the student must complete the appropriate double major request form and get the signature of each department chairperson and the appropriate associate dean. Please note that students will receive only one degree, regardless of the number of majors they complete.

Popular double majors with mathematics are economics, computer science, education, and physics, but we have also had students double major in math and music, English, political science, philosophy, Spanish, psychology, pre-medicine, and other fields.

## Minors in Other Disciplines

Minors provide students the opportunity to pursue additional interests but generally do not require as many courses as a major. Minors generally range from five to eight required courses. The minors page provides a complete list of minors and provides links to each minor. Some majors and minors can be completed within the minimum 120 credit hour degree requirement, but in some cases additional coursework may be required. Students must complete the appropriate minor request form.

## Curriculum

### An Ignatian Foundation

All undergraduate students must complete either the Canisius Core Curriculum or the All-College Honors Curriculum. Many schools refer to their college-wide undergraduate requirements as "general education" requirements. We believe that the core curriculum and the honors curriculum are more than a series of required classes; they provide the basis for a Jesuit education both with content and with required knowledge and skills attributes that are central to our mission.

### Free Electives

Students may graduate with more but not less than 120 credit hours. Free electives are courses in addition to the Canisius Core Curriculum or All-College Honors Curriculum and major requirements sufficient to reach the minimum of 120 credit hours required for graduation.

### Major Requirements

#### Option A (Mathematics Major)

Code | Title | Credits |
---|---|---|

MAT 111 | Calculus I | 4 |

MAT 112 | Calculus II | 4 |

MAT 211 | Calculus III | 4 |

MAT 219 | Linear Algebra | 4 |

MAT 222 | Differential Equations | 3 |

or MAT 351 | Probability & Statistics I | |

MAT 230 | Logic, Set Theory, and Proofs | 4 |

MAT 311 | Abstract Algebra | 4 |

MAT 321 | Real Analysis | 4 |

MAT 380 | Mathematics Seminar | 1 |

MAT 381 | Mathematics Seminar | 1 |

MAT 480 | Mathematics Seminar | 1 |

Select one of the following: | 3 | |

Topics in Algebra | ||

Topics in Analysis | ||

Probability & Statistics II | ||

Complex Analysis | ||

Select four courses from the following: | 12-16 | |

Any additional 300 or 400-level mathematics courses | ||

Mathematical Analysis for Physicists | ||

Quantum Mechanics I | ||

Choose one of the following two sequences: | 6-8 | |

Any two Economics (ECO) classes | ||

Total Credits | 55-61 |

#### Option B (Mathematics with a Concentration in Another Area)

Code | Title | Credits |
---|---|---|

MAT 111 | Calculus I | 4 |

MAT 112 | Calculus II | 4 |

MAT 211 | Calculus III | 4 |

MAT 219 | Linear Algebra | 4 |

MAT 222 | Differential Equations | 3 |

or MAT 351 | Probability & Statistics I | |

MAT 230 | Logic, Set Theory, and Proofs | 4 |

MAT 311 | Abstract Algebra | 4 |

MAT 321 | Real Analysis | 4 |

MAT 380 | Mathematics Seminar | 1 |

MAT 381 | Mathematics Seminar | 1 |

MAT 480 | Mathematics Seminar | 1 |

Select one of the following: | 3 | |

Topics in Algebra | ||

Topics in Analysis | ||

Probability & Statistics II | ||

Complex Analysis | ||

Select two of the following: | 6-8 | |

any additional 300 or 400-level mathematics courses | ||

Mathematical Analysis for Physicists | ||

Quantum Mechanics I | ||

Choose one of the following two sequences: | 6-8 | |

Any two Economics (ECO) classes | ||

Four courses in another area approved by the Department of Mathematics and Statistics | 12 | |

Total Credits | 61-65 |

#### Option C (Statistics)

Code | Title | Credits |
---|---|---|

MAT 111 | Calculus I | 4 |

MAT 112 | Calculus II | 4 |

MAT 211 | Calculus III | 4 |

MAT 219 | Linear Algebra | 4 |

MAT 222 | Differential Equations (or 300 or 400-level mathematics course) | 3 |

MAT 230 | Logic, Set Theory, and Proofs | 4 |

MAT 321 | Real Analysis | 4 |

Select one of the following | 3-4 | |

Abstract Algebra | ||

Topics in Analysis | ||

Complex Analysis | ||

MAT 351 | Probability & Statistics I | 3 |

MAT 352 | Probability & Statistics II | 3 |

MAT 353 | Regression Analysis | 3 |

MAT 354 | Experimental Design and Statistical Computing | 3 |

MAT 380 | Mathematics Seminar | 1 |

MAT 381 | Mathematics Seminar | 1 |

MAT 480 | Mathematics Seminar | 1 |

Select one additional upper-level MAT elective | 3-4 | |

Choose one of the following two sequences: | 6-8 | |

Any two Economics (ECO) classes | ||

Total Credits | 54-58 |

### Additional Course Considerations

**Option A** is designed to prepare the student for positions in which mathematical competence is sought, as well as for advanced study in graduate school.

**Option B** allows the student to develop expertise in another area besides mathematics. Many students combine this option with a major or minor in another department. Students interested in education can get a dual major to give them a strong background in mathematics while fulfilling the requirements of the School of Education.

**Option C** prepares the student for advanced work or graduate study and the many careers which utilize statistical thinking. This option, together with appropriate courses in business, constitutes a good preparation for a career in actuarial science.

## Roadmap

### Recommended Semester Schedule for Major Courses

#### Option A

Freshman | |
---|---|

Fall | Spring |

MAT 111 | MAT 112 |

PHY 223 (or ECO course) | PHY 224 (or ECO course) |

Sophomore | |

Fall | Spring |

MAT 211 | MAT 219 |

MAT 230 | MAT 222 or 351 |

Junior | |

Fall | Spring |

MAT 311 | MAT 381 |

MAT 321 | MAT elective |

MAT 380 | |

Senior | |

Fall | Spring |

MAT 480 | Select one of the following ("restricted elective"): |

MAT elective | |

MAT elective | |

MAT 352 (MAT352 is typically offered in the fall semester.) | |

MAT elective |

#### Option B

Freshman | |
---|---|

Fall | Spring |

MAT 111 | MAT 112 |

PHY 223 (or ECO course) | PHY 224 (or ECO course) |

Sophomore | |

Fall | Spring |

MAT 211 | MAT 219 |

MAT 230 | MAT 222 or 351 |

Junior | |

Fall | Spring |

MAT 311 | MAT 381 |

MAT 321 | MAT elective |

MAT 380 | |

Senior | |

Fall | Spring |

MAT 480 | Select one of the following ("restricted elective"): |

MAT elective | |

MAT 352 (MAT352 is typically offered in the fall semester.) | |

#### Option C

Freshman | |
---|---|

Fall | Spring |

MAT 111 | MAT 112 |

PHY 223 (or ECO course) | PHY 224 (or ECO course) |

Sophomore | |

Fall | Spring |

MAT 211 | MAT 219 |

MAT 230 | MAT 351 |

Junior | |

Fall | Spring |

MAT 321 | MAT 381 |

MAT 352 | Select one of the following statistics electives: |

MAT 380 | |

MAT 222 (or MAT elective) | |

Senior | |

Fall | Spring |

MAT 480 | Select one of the following statistics electives: |

MAT elective | |

Select one of the following ("restricted elective"): | |

MAT 311 (MAT311 is typically offered in the fall semester.) | |

## Learning Goals & Objectives

### Student Learning Goal 1:

#### Mathematics majors will perform tasks requiring logical reasoning.

##### Students will:

**Objective A**: Understand the different types of mathematical statements and how they are used, including definitions, axioms, hypotheses, conclusions, theorems, corollaries, lemmata, and conjectures;**Objective B:**Understand methods of proof.

### Student Learning Goal 2

#### Mathematics majors will know the content of the fundamental fields of mathematics and can perform tasks requiring complex reasoning.

##### Students will:

**Objective A**: Use basic skills to manipulate expressions;**Objective B**: Know the basic definitions and theorems of mathematics;**Objective C**: Be able to perform tasks requiring complex reasoning.

### Student Learning Goal 3

#### Mathematics majors will advance their understanding and knowledge of mathematics and their ability to convey mathematical concepts through currently available technology.

##### Students will:

**Objective A**(Information Literacy): Use the internet and/or library resources to obtain relevant information concerning historical information or mathematical content in regards to current course or project. Students will be expected to both look up sources and learn to search for their own sources;**Objective B**(Computation): Use computers or graphing calculators to perform labor-intensive calculations and/or create graphical displays. Programs include, but are not limited to,*Excel, Minitab*(or other statistical software)*, and Mathematica*(or other software);**Objective C**(Presentation): Use technology for the purpose of elegantly presenting mathematical ideas, theories or results. Technologies include PowerPoint, Prezzi, Jing, Beamer, Latex, Word, and graphical tools.

### Student Learning Goal 4

#### Mathematics majors will communicate mathematical ideas with precision and clarity.

##### Students will:

**Objective A:**Present mathematical material in writing;**Objective B:**Present mathematical material orally.

## Minor

### Mathematics Minor

(7 courses)

Code | Title | Credits |
---|---|---|

Required Courses | ||

MAT 111 | Calculus I | 4 |

or MAT 115 | Calculus for Business | |

MAT 112 | Calculus II | 4 |

MAT 211 | Calculus III | 4 |

MAT 219 | Linear Algebra | 4 |

MAT 230 | Logic, Set Theory, and Proofs | 4 |

Restricted Electives | ||

Select one of the following tracks and take two of the courses in that track: | 6-8 | |

Probability and Statistics: | ||

Probability & Statistics I | ||

Probability & Statistics II | ||

Applied Mathematics: | ||

Differential Equations | ||

Numerical Analysis | ||

Graph Theory | ||

Climate and Sustainability | ||

Mathematical Analysis for Physicists | ||

Theoretical Mathematics: | ||

Abstract Algebra | ||

Topics in Algebra | ||

Real Analysis | ||

Topics in Analysis | ||

Complex Analysis | ||

Mathematics and Culture: | ||

History of Mathematics | ||

Geometry | ||

Total Credits | 26-28 |

## Courses

**MAT 105 Finite Mathematics 3 Credits**

Introduction to finite (non-calculus) mathematics and its applications: linear, quadratic, exponential and logarithmic functions and equations; systems of linear equations and linear programming; compound interest problems and annuities.

**Fulfills College Core:** Field 7 (Mathematical Sciences)

**Offered:** every fall & spring.

**MAT 106 Calculus for the Non-Sciences 3 Credits**

Fundamentals of calculus for students in business, or social or behavioral sciences. Credit not allowed if student already has credit for MAT 109 & MAT 110; or MAT 111 or MAT 115.

**Prerequisite:** MAT 105

**Fulfills College Core:** Field 7 (Mathematical Sciences)

**Offered:** fall & spring.

**MAT 108 Precalculus for Sciences 3 Credits**

Polynomials and rational, exponential, logarithmic, and trigonometric functions. Solving equations. Graphing functions. Trigonometric identities. Credit not allowed if student already has credit for MAT 109, MAT 111, or MAT 115.

**Fulfills College Core:** Field 7 (Mathematical Sciences)

**Offered:** summer.

**MAT 109 Calculus with Review I 4 Credits**

For science and mathematics majors. Calculus, with topics from pre-calculus. Logarithmic and exponential functions, trigonometric functions, limits, differentiation. Credit not allowed if student already has credit for MAT 111 or MAT 115.

**Fulfills College Core:** Field 7 (Mathematical Sciences)

**Offered:** fall.

**MAT 110 Calculus with Review II 4 Credits**

Continuation of MAT 109. Differentiation, related rates, optimization problems, anti-differentiation, definite integral. Credit not allowed if student already has credit for MAT 111 or MAT 115.

**Prerequisite:** MAT 109

**Offered:** spring.

**MAT 111 Calculus I 4 Credits**

For science and mathematics majors. Calculus of functions of single variable. Functions, limits, differentiation, continuity, graphing, logarithm, exponential and inverse trigonometric functions, related rates, optimization problems, mean value theorem, l'Hospital's rule, anti-differentiation, definite integral. Credit not allowed if student already has credit for MAT 109 and MAT 110 or for MAT 115.

**Fulfills College Core:** Field 7 (Mathematical Sciences)

**Offered:** fall & spring.

**MAT 112 Calculus II 4 Credits**

Applications of integration, integration techniques, improper integrals, sequences, series, convergence tests, Taylor's series, applications; parametric and polar curves.

**Prerequisite:** minimum grade of C- in one of the following MAT 109 & MAT 110, MAT 111 or MAT 115.

**Offered:** fall & spring.

**MAT 115 Calculus for Business 4 Credits**

Calculus for business students. Differentiation and integration of functions of one variable. Applications, concepts, examples and problems in economics and business. Credit not allowed if student already has credit for either MAT 109 and MAT 110 or for MAT 111.

**Prerequisite:** 3 1/2 years of high school mathematics.

**Fulfills College Core:** Field 7 (Mathematical Sciences)

**Offered:** fall.

**MAT 121 Mathematics through History 3 Credits**

Liberal arts mathematics course. Tracing the development of mathematical ideas globally and through history, with emphasis on problem solving techniques, quantitative thinking, and deductive reasoning.

**Prerequisite:** 3 years of high school math or equivalent; sophomore standing or higher.

**Fulfills College Core:** Field 7 (Mathematical Sciences), Global Awareness

**Offered:** once a year.

**MAT 131 Statistics for Social Sciences 3 Credits**

A first course for majors in social or health sciences. Descriptive statistics, calculators, computer programs and introduction to inferential statistics. Credit not allowed if student already has credit for MAT 141 or MAT 351.

**Prerequisite:** 3 years of high school mathematics or equivalent.

**Fulfills College Core:** Field 7 (Mathematical Sciences)

**Offered:** fall & spring.

**MAT 141 Inferential Statistics and Computers for Science 4 Credits**

Elementary probability theory, descriptive statistics, hypothesis testing, estimation, correlation and regression. The computer will be used with one of the standard statistical packages. Credit not allowed if student already has credit for MAT 131 or MAT 351.

**Prerequisite:** 3 1/2 years of high school mathematics.

**Fulfills College Core:** Field 7 (Mathematical Sciences)

**Offered:** fall & spring.

**MAT 150 Mathematics and Politics 3 Credits**

Liberal arts course emphasizing applications of math in the social sciences. Covers topics such as voting theory, decisions made by groups, measurement of political power.

**Prerequisite:** 3 years of high school mathematics or equivalent.

**Fulfills College Core:** Field 7 (Mathematical Sciences), Justice

**Offered:** once a year.

**MAT 161 Mathematics for Elementary Teachers 3 Credits**

Designed to provide a solid foundation for the mathematical topics encountered in elementary schools. The primary goal is to develop a deep understanding of mathematical concepts so future teachers can teach with knowledge and confidence. The main topics are: problem solving processes and strategies, elementary set theory, the theory behind basic arithmetic, number systems, basic probability and statistics, elementary geometry.

**Prerequisite:** 3 years of high school mathematics or equivalent.

**Fulfills College Core:** Field 7 (Mathematical Sciences)

**Offered:** occasionally.

**MAT 191 Introduction to Discrete Mathematics 4 Credits**

Fundamental topics with computer science applications. Sets and logic, propositional and predicate calculus, elements of combinatorics and counting, elementary discrete probability, functions and relations, and graphs.

**Prerequisite:** sophomore standing or instructor's permission.

**Offered:** fall.

**MAT 211 Calculus III 4 Credits**

Continuation of MAT 111 and MAT 112. Analytic geometry of 3-dimensional space and calculus of functions of several variables.

**Prerequisite:** minimum grade of C- in MAT 112.

**Offered:** fall & spring.

**MAT 219 Linear Algebra 4 Credits**

Vector spaces and inner product spaces. Linear transformations and matrices. Eigenvectors, eigenvalues, and applications. Orthogonal transformations. Quadratic forms and quadric surfaces.

**Prerequisite:** MAT 112 or permission of instructor.

**Offered:** spring.

**MAT 222 Differential Equations 3 Credits**

Introduction to the theory and applications of ordinary differential equations.

**Prerequisite:** minimum grade of C- in MAT 211.

**Offered:** spring.

**MAT 230 Logic, Set Theory, and Proofs 4 Credits**

Transition from calculus to upper division courses: logic and methods of proof; set theory; relations, orders and functions; number systems; cardinality.

**Prerequisite:** minimum grade of C- in MAT 112 or permission of instructor.

**Offered:** fall.

**MAT 301 History of Mathematics 3 Credits**

Development and interrelations of major areas of mathematics, from ancient to modern times. Emphasis on both historical context and mathematical content. Mathematics as cultural heritage.

**Prerequisite:** MAT 111 & MAT 112.

**Offered:** occasionally.

**MAT 311 Abstract Algebra 4 Credits**

Introduction to the basic structures of abstract algebra; groups, rings, fields.

**Prerequisite:** minimum grade of C- in both MAT 219 and MAT 230.

**Offered:** fall.

**MAT 312 Topics in Algebra 3 Credits**

Continuation of MAT 311. Selected topics from groups, rings, integral domains, field extensions and Galois theory.

**Prerequisite:** MAT 311.

**Offered:** spring of odd-numbered years.

**MAT 313 Number Theory 3 Credits**

Divisibility, prime numbers, numerical functions, congruencies, quadratic reciprocity, Diophantine equations.

**Prerequisite:** MAT 311 or permission of instructor.

**Offered:** occasionally.

**MAT 321 Real Analysis 4 Credits**

Advanced study of limits, continuity, differentiation and integration of functions.

**Prerequisite:** minimum grade of C- in MAT 211, MAT 219 & MAT 230.

**Offered:** fall.

**MAT 322 Topics in Analysis 3 Credits**

Continuation of MAT 321. Selected topics from functions of one or several variables.

**Prerequisite:** minimum grade of C- in MAT 321.

**Offered:** occasionally.

**MAT 331 Geometry 3 Credits**

Axioms for geometry and their consequences: independence of the parallel postulate. Rudiments of hyperbolic geometry. Geometric transformations.

**Prerequisite:** MAT 230.

**Offered:** fall of even-numbered years.

**MAT 341 Numerical Analysis 3 Credits**

The methods used to obtain numerical solutions of functional and differential equations. Polynomial interpolation. Various approximation algorithms.

**Prerequisite:** MAT 219 & an elementary knowledge of computer programming.

**Offered:** occasionally.

**MAT 342 Graph Theory 3 Credits**

This course covers basic constructions on graphs. Complete graphs, bipartite graphs, trees, cycles, and more general graphs are studied along with their combinatorial properties. Euler circuits, Hamilton circuits, spanning trees. Applications of graphs to optimization problems such as the Traveling Salesperson Problem and Construction of the Minimal Spanning Trees.

**Prerequisite:** MAT 111 or MAT 115.

**Offered:** spring of even-numbered years.

**MAT 345 Climate and Sustainability 3 Credits**

This course, for majors from all the sciences, develops the use of conceptual models in understanding complicated situations, while illustrating the role of scientific arguments in societal debates about climate change and sustainability. Topics include: the Earth's energy balance, historical data for ocean and atmosphere and temperature and their circulation and oscillation patterns, the carbon cycle and biological systems, and handling large data. The increasingly sophisticated examples explored in the course will introduce the use of various areas of mathematics: linear models and networks, calculus-based dynamical systems, statistics and data assessment, and analysis of periodic phenomena. Students will learn about these topics by seeing simple examples of their use. The last part of the course will center on sustainability issues, and connections to global awareness, diversity, ethics, and justice.

**Prerequisite:** MAT 111 or equivalent. **Restriction:** seniors only.

**Fulfills College Core:** Core Capstone, (part of the Sustainability Core Pathway)

**Offered:** occasionally.

**MAT 351 Probability & Statistics I 3 Credits**

Introduction to the mathematical aspects of modern probability theory and the theory of statistics.

**Prerequisite:** MAT 211.

**Offered:** spring.

**MAT 352 Probability & Statistics II 3 Credits**

Continuation of MAT 351. Introduction to the mathematical aspects of modern probability theory and the theory of statistics.

**Prerequisite:** MAT351.

**Offered:** fall.

**MAT 353 Regression Analysis 3 Credits**

Linear regression and correlation. Covariance, residual sum of squares, residual variance, correlation coefficient, tests of significance for correlation coefficient and for regression coefficients. Non-linear regression.

**Prerequisite:** MAT 351 & MAT 352, or permission of instructor.

**Offered:** spring of odd-numbered years.

**MAT 354 Experimental Design and Statistical Computing 3 Credits**

Analyzing data; one-way/two-way blocking; chi-square, goodness of fit. Statistical computing package; Monte-Carlo simulation-subset selection, central limit theorem; residual plots.

**Prerequisite:** MAT 351 & MAT 352, or permission of instructor.

**Offered:** spring of even-numbered years.

**MAT 361 Probability for Actuaries 1 Credit**

This course prepares students to take Exam P of the Society of Actuaries or Exam 1 from the Casualty Actuary Society. Key probability concepts are reviewed and students learn to apply these tools to problems encountered by actuaries. The course will develop a thorough command of calculus and probability topics. Additionally, a very basic knowledge of insurance and risk management is introduced.

**Prerequisite:** MAT 211 & MAT 351. **Corequisite:** MAT 352 or permission of instructor.

**Offered:** fall.

**MAT 362 Financial Mathematics for Actuaries 1 Credit**

This course prepares students to take Exam FM of the Society of Actuaries or Exam 2 from the Casualty Actuary Society. It covers interest theory (discrete and continuous) and an introduction to derivative securities.

**Prerequisite:** MAT 111 & MAT 112 or permission of instructor.

**Offered:** spring.

**MAT 380 Mathematics Seminar 1 Credit**

Mathematics seminar for majors. To be taken for three semesters.

**Prerequisite:** junior standing.

**Offered:** fall & spring.

**MAT 381 Mathematics Seminar 1 Credit**

Mathematics seminar for majors. To be taken for three semesters.

**Prerequisite:** junior standing.

**Offered:** fall & spring.

**MAT 411 Topology 3 Credits**

An introduction to topology, stressing concrete examples including surfaces. Point-set, geometric and algebraic topology with interconnections and applications.

**Prerequisite:** MAT 230.

**Offered:** occasionally.

**MAT 421 Complex Analysis 3 Credits**

Complex numbers, analytic functions and mappings. Cauchy-Riemann equations, harmonic functions, Cauchy's theorem, integral formula and inequalities. Power series, residues, singularities and zeros, Rouche's Theorem.

**Prerequisites:** MAT 321 or PHY 335.

**Offered:** occasionally.

**MAT 480 Mathematics Seminar 1 Credit**

Mathematics seminar for majors. To be taken for three semesters.

**Prerequisite:** junior standing.

**Fulfills College Core:** Oral Communication

**Offered:** fall & spring.

**MAT 498 Internship in Mathematics 1-3 Credits**

Internship involving non-routine tasks linking academic concepts to practical experience. May be used for free elective credit only. Internships require an application and approval by the associate dean.

**Prerequisite:** permission of the chair & associate dean.

**Offered:** occasionally.

**MAT 499 Independent Study 1-4 Credits**

Study and work with a faculty supervisor. Project to be determined by faculty agreement. Independent studies require an application and approval by the associate dean.

**Prerequisite:** permission of the instructor, department chair, & associate dean.

**Offered:** occasionally.