# Mathematics

The undergraduate degree in mathematics emphasizes knowledge and awareness of:

- basic real analysis of one variable;
- calculus of several variables and vector analysis;
- basic linear algebra and theory of vector spaces;
- the structure of mathematical proofs and definitions; and
- at least one additional specialized area of mathematics.

In addition, students completing a degree in mathematics are expected to acquire the ability and skills to:

- use techniques of differentiation and integration of one and several variables;
- solve problems using differentiation and integration;
- solve systems of linear equations;
- give direct proofs, proofs by contradiction, and proofs by induction;
- formulate definitions;
- read mathematics without supervision; and
- utilize mathematics.

Course code for this program is MATH.

### Bachelor's Degree Program(s)

## Bachelor’s Degree in Mathematics

The Department of Mathematics offers three tracks leading to a BA in mathematics. To earn a BA in mathematics, a student must complete the general requirements of the College of Arts and Sciences as well as the five mathematics core courses and the six additional courses for one of the three tracks (described below). Students must earn a grade of *C-* or better in each of the mathematics core courses and the six additional courses, and have at least a *C* average for all attempted work in mathematics.

**Mathematics Core** (required of all mathematics majors)

*Required Courses and Semester Credit Hours*

- Calculus 1 (e.g., MATH 1300 or APPM 1350)—5
- Calculus 2 (e.g., MATH 2300 or APPM 1360)—5
- Calculus 3 (e.g., MATH 2400 or APPM 2350)—5
- MATH 2001 An Introduction to Discrete Mathematics—3
- MATH 3001 Analysis I—3
- MATH 3130 Linear Algebra—3

### Comprehensive Track

Aimed at students seeking a general background in mathematics or intending to pursue graduate work in mathematics.

**Additional Courses Required**

*Required Courses and Semester Credit Hours*

- MATH 3140 Abstract Algebra I—3
- One of MATH 4140 Abstract Algebra II or MATH 4001 Analysis II—3
- Plus 3 upper-division MATH or approved APPM courses (at least one at the 4000-level)—9

### Applicable Track

Aimed at students seeking a background in applied or applicable mathematics.

**Additional Courses Required**

*Required Courses and Semester Credit Hours*

- MATH 4510 Introduction to Probability—3
- MATH 4430 Introduction to Ordinary Differential Equations—3
- One of MATH 4520 Mathematical Statistics or MATH 4470 Partial Differential Equations 1—3
- Plus 2 upper-division MATH or approved APPM courses—6

### Secondary Education Track

Aimed at students intending to teach mathematics at the secondary level.

**Additional Courses Required**

*Required Courses and Semester Credit Hours*

- MATH 3110 Introduction to the Theory of Numbers or MATH 3140 Abstract Algebra I—3
- MATH 3120 Functions and Modeling—3
- MATH 3210 Euclidean and Non-Euclidean Geometry I—3
- MATH 3510 Introduction to Probability and Statistics—3
- MATH 4820 History of Mathematical Ideas—3

*NOTE: Completion of the Secondary Education Track does not provide the student a teaching license. For information about the secondary education teaching licensure program, see the School of Education section.*

Computational Track

Aimed at students interested in both mathematics and computation.

**Additional Courses Required**

- CSCI 1300 Computer Science 1: Programming---3
- CSCI 2270 Computer Science 2: Programming—3
- CSCI 3104 Algorithms—3
- MATH/APPM 4650Intermediate Numerical Analysis I—3
- Plus three MATH or approved APPM or approved CSCI courses, at least one of which must be at the 4000-level. At most one of these courses may be a computer science course—9

Graduating in Four Years

Consult the Four-Year Guarantee Requirements for information on eligibility. The concept of “adequate progress,” as it is used here, refers to maintaining eligibility for the four-year guarantee; it is not a requirement for the major.

**Comprehensive Track **

- Declare major by the beginning of the second semester.
- Complete Calculus 1, Calculus 2, Calculus 3, MATH 2001, 3001, and 3130 by the end of the fourth semester.
- Complete MATH 3140, 4140, or 4001, and one additional approved MATH or APPM course by the end of the sixth semester.
- Complete the major by the end of the eighth semester.

**Applicable Track**

- Declare major by the beginning of the second semester.
- Complete Calculus 1, Calculus 2, Calculus 3, MATH 2001, 3001, and 3130 by the end of the fourth semester.
- Complete MATH 4510 and 4430, and either MATH 4520 or 4470 by the end of the sixth semester.
- Complete the major by the end of the eighth semester.

**Secondary Education Track**

- Declare major by the beginning of the second semester.
- Complete Calculus 1, Calculus 2, MATH 2001, 3001, 3120, and 3130 by the end of the fourth semester.
- Complete MATH 3510 and 3210, and either MATH 3110 or 3140 by the end of the sixth semester.
- Complete the major by taking MATH 4210 and the remaining of MATH 3110 or 3140 the end of the eighth semester.

**Computational Track **

- Declare major by beginning of the second semester.
- Complete Computer Science 1, Computer Science 2, Calculus 1, Calculus 2, Calculus 3, MATH 2001, 3001, and 3130 by the end of the fourth semester.
- Complete CSCI 3104 Algorithms, MATH 4650, or APPM 4650, and one of the additionally required upper-division MATH classes by the end of the sixth semester.
- Complete the major by the end of the eighth semester.

### Residency Requirement

For the BA degree in mathematics, all students must have completed at least 12 credit hours of upper-division mathematics courses, with grades of *C* (2.00) or better, taken in the College of Arts and Sciences on the Boulder campus. Additional courses transferred from other universities or from other campuses of the University of Colorado that are used to meet the minimum 24-hour upper-division requirement must be approved by the Department of Mathematics. Courses accepted as mathematics credit but excluded from the minimum 24-hour upper-division requirement still count in the 45 maximum hours allowed in mathematics.

Undergraduate students planning to do graduate work in mathematics should follow the Comprehensive Track, as well as fulfill the arts and sciences language requirement with German, French, or Russian.

### Minor Program

A minor is offered in mathematics. Declaration of a minor is open to any student enrolled at CU-Boulder, regardless of college or school.

**Required Courses and Semester Credit Hours**

- Calculus 1 (e.g., MATH 1300 or APPM 1350)—5
- Calculus 2 (e.g.,MATH 2300 or APPM 1360)—5
- Calculus 3 (e.g., MATH 2400 or APPM 2350)—4
- MATH 2001 Discrete Mathematics—3
- MATH 3130 Linear Algebra—3
- Plus two upper-division MATH courses (at least one at the 4000-level)—6

### Graduate Degree Program(s)

## Graduate Study in Mathematics

The Department of Mathematics offers programs leading to the degrees MA or PhD in mathematics and MS in applied mathematics. Students interested in any of these programs should read carefully the materials describing the university requirements in the Graduate School section. The student is responsible for satisfying these requirements at the proper time.

Applicants must have demonstrated mathematical maturity and accomplishment roughly at the level of a successful mathematics major at CU-Boulder. Applicants must also demonstrate mathematical potential: success in courses in advanced calculus and abstract algebra help demonstrate this potential. General and Mathematics GRE subject scores are required.

The basic requirements for the various degrees are summarized here, and full details are available in the department office or online at www.colorado.edu/math. For fulfillment of all course requirements, mathematics courses must be numbered 5000 or higher excluding MATH 5800.

## Master's Degree in Mathematics

- All master’s degree students must complete 30 hours of approved graduate credit. No more than 6 credit hours of seminars or independent study may be included for this requirement.
- All master’s degree students must take two 2-semester sequences. No language is required of master’s degree students.
- To earn an MA or MS degree, a student must pass a master’s examination based on the particular program of the student.
- For the MA degree in mathematics, students can pursue a “thesis option,” which requires 4–6 credit hours of thesis work, and a thesis defense.
- For the MS degree in applied mathematics, 6–12 credit hours must be in an approved minor program outside the mathematics department.

## Doctoral Degree in Mathematics

Before being admitted to candidacy for the PhD degree in mathematics, a student must pass two out of the three preliminary exams in algebra, real analysis, and geometry/topology; take courses in topology and complex analysis; and pass a comprehensive exam. The basic requirements for a PhD degree in mathematics are as follows:

- Demonstrate reading knowledge of French, German, Italian, or Russian.
- Complete at least 30 credit hours of graduate course work and 30 credit hours of thesis work
- Prepare a written thesis that contains substantial original contributions to mathematics
- Successfully complete a final examination.

For more current and precise details, contact the mathematics office or go to the department website at www.colorado.edu/math.