# Department of Mathematical Sciences

Thomas Bohman, Department Head

William J. Hrusa, Associate Head

John F. Mackey, Associate Head

Office: Wean Hall 6113

http://www.math.cmu.edu

Mathematics provides much of the language and quantitative underpinnings of the natural and social sciences, and mathematical scientists have been responsible for the development of many of the most commonly used tools in business management as well as for laying the foundation for computational and computer science. The name of the Department of Mathematical Sciences reflects its tradition of outstanding research and teaching of applicable mathematics relating to these areas. Indeed, the Department contains highly ranked research groups in Applied Mathematics, Discrete Mathematics, Logic, and Mathematical Finance. These research strengths are reflected in the variety of options that the Department provides for its undergraduate majors.

The Department offers a B.S. degree in Mathematical Sciences with concentrations in Mathematics, Operations Research and Statistics, Statistics, Discrete Mathematics and Logic, and Computational and Applied Mathematics.

The B.S. in Mathematics Curriculum is the least structured of our programs, in recognition of the wide variety of interests that can be productively coupled with the study of mathematical sciences. It can be an appropriate choice for students planning for graduate study in mathematics or seeking to design their curriculum to take advantage of the many opportunities for a second major from another department in the University.

The Operations Research and Statistics Concentration prepares students to enter the area of operations research, which is expected to be among the growth occupations over the next decade. Mathematicians with a background in operations research are especially valuable in such diverse activities as project planning, production scheduling, market forecasting and finance. Such applications are found in virtually all industrial and governmental settings.

The Statistics Concentration prepares students to contribute to a wide variety of research areas. Applications range from experimental design and data analysis in the physical and social sciences, medicine and engineering, to modeling and forecasting in business and government, to actuarial applications in the financial and insurance industries. This is also a useful second major for students planning for graduate study and research in subject areas requiring a strong statistical background.

The Discrete Mathematics and Logic Concentration provides a background in discrete mathematics, mathematical logic, and theoretical computer science. This concentration prepares the student to do research in these and related fields, or to apply their ideas elsewhere.

Finally, the Computational and Applied Mathematics Concentration provides the background needed to support the computational and mathematical analysis needs of a wide variety of businesses and industries and is well suited to students with an interest in the physical sciences and engineering.

The Department places great emphasis on the advising of students. This is critical if students are to make the most of their years at the University. Students are urged to work carefully with their advisor and other faculty to formulate their degree programs. Study abroad is encouraged, and an interested student should investigate the opportunities available in the Undergraduate Options section of the catalog.

## Special options within the Department

The Department offers special opportunities for the exceptionally well-prepared and intellectually ambitious student. These options are available to students from any department in the University.

##### Matrix Theory and Vector Analysis

For selected Freshmen entering the University, we offer the Fall/Spring sequence of 21-242 Matrix Theory and 21-269 Vector Analysis, which include a rigorous introduction to proofs and abstract mathematics. Typically, a student choosing this sequence has mastered the operational aspects of high school mathematics and now seeks a deeper conceptual understanding.

##### Mathematical Studies

Following the 21-242/21-269 sequence, we offer Mathematical Studies Analysis I/II and Mathematical Studies Algebra I/II. These courses provide excellent preparation for graduate study, with many of the participants taking graduate courses as early as their Junior year. The typical enrollment of about 15 students allows for close contact with faculty. Admission to Mathematical Studies is by invitation, and interested students should apply during the Spring of their Freshman year.

##### Honors Degree Program

This demanding program qualifies the student for two degrees: The Bachelor of Science and the Master of Science in Mathematical Sciences. This program typically includes the Mathematical Studies option. For students who complete the Mathematical Studies sequence, the Master of Science degree in Mathematical Sciences may be earned together with a Bachelor of Science from another department.

##### Interdisciplinary Programs

Several interdisciplinary options enable a student to combine mathematics with other disciplines.

- The Bachelor of Science and Arts program allows a student to combine mathematics with study in any of the five schools in the College of Fine Arts.
- The Science and Humanities Scholars program includes an option shared with the Statistics Department in the Humanities and Social Sciences College that leads to a BS in Mathematics and Statistics.
- The Bachelor of Science in Mathematics and Economics is a flexible program which allows students to develop depth in both fields of study. Note: for students whose home college is Dietrich College, this major is known as the Bachelor of Science in Economics and Mathematical Sciences.
- Finally, a joint program with the Heinz College of Public Policy and Management and the Tepper School of Business leads to the degree Bachelor of Science in Computational Finance.

## Curricula

For each concentration, we provide a list of the requirements and a suggested schedule that takes prerequisites into account. A Mathematical Sciences, Computer Science, Physics, Statistics Elective refers to any course from the Departments of Mathematical Sciences, Computer Science, Physics, Statistics, respectively, satisfying the following restrictions: a mathematical sciences course must be at the 21-300 level or above or 21-270 or 21-292, a computer science course must be at the 15-200 level or above, a physics course must be at the 33-300 level or above, and a statistics course must have at least 36-225 as a prerequisite.

Mathematical Sciences majors are required to complete an introductory computer science course, either 15-110 or 15-112. Students who plan to take further computer science courses must complete 15-112.

An H&SS Elective refers to a course in the Dietrich College of Humanities and Social Sciences requirements as described in the catalog section for the Mellon College of Science. A course listed as an Elective is a free elective with the only restriction that the maximum total of ROTC, STUCO, and Physical Education units that will be accepted for graduation is nine.

For a list of courses required for all MCS students see "First Year for Science Students."

In addition to the courses in the suggested schedules below, a student majoring in mathematical sciences also takes the one unit course 21-201 Undergraduate Colloquium each semester of the Sophomore year. This course plays an important role in introducing students to career opportunities, graduate school preparation, and student and faculty research in the Department.

### Mathematics Degree

This program is the most flexible available to our majors. The flexibility to choose eight electives within the major plus seven humanities courses and seven free electives allows the student to design a program to suit his or her individual needs and interests. The requirements for the Mathematics Degree are:

##### Mathematical Sciences Courses (required)

The alternative courses 21-242, 21-261, and 21-268 (or 21-269) are particularly recommended for a student planning to pursue graduate work.

Courses | Units | |

21-120 | Differential and Integral Calculus | 10 |

21-122 | Integration and Approximation | 10 |

21-127 | Concepts of Mathematics | 10 |

21-201 | Undergraduate Colloquium (Taken twice in Sophomore year.) | 1 |

21-228 | Discrete Mathematics | 9-12 |

or 15-251 | Great Theoretical Ideas in Computer Science | |

21-241 | Matrices and Linear Transformations | 10 |

or 21-242 | Matrix Theory | |

36-225 | Introduction to Probability Theory | 9 |

or 21-325 | Probability | |

21-259 | Calculus in Three Dimensions | 9-10 |

or 21-268 | Multidimensional Calculus | |

or 21-269 | Vector Analysis | |

21-260 | Differential Equations | 9-10 |

or 21-261 | Introduction to Ordinary Differential Equations | |

or 33-231 | Physical Analysis | |

21-341 | Linear Algebra | 9 |

21-355 | Principles of Real Analysis I | 9 |

21-356 | Principles of Real Analysis II | 9 |

21-373 | Algebraic Structures | 9 |

113-118 |

Forty-five units of (required) Mathematical Sciences electives (at the 21-300 level or above or 21-270 or 21-292).

Three (required) Mathematical Sciences (at the 21-300 level or above or 21-270 or 21-292), or Computer Science (at the 15-200 level or above), or Physics (at the 33-300 level or above), or Statistics (must have at least 36-225 as a prerequisite) electives.

##### MCS General Education (required)

MCS or SHS humanities, social sciences, and science core (114 units)

##### Mathematical Sciences Electives for Students Intending Graduate Studies

Students preparing for graduate study in mathematics should consider the following courses as Mathematical Sciences electives, choosing among them according to the desired area of graduate study:

21-301 | Combinatorics | 9 |

21-371 | Functions of a Complex Variable | 9 |

21-372 | Partial Differential Equations and Fourier Analysis | 9 |

21-374 | Field Theory | 9 |

21-441 | Number Theory | 9 |

21-465 | Topology | 9 |

21-467 | Differential Geometry | 9 |

21-470 | Selected Topics in Analysis | 9 |

21-476 | Introduction to Dynamical Systems | 9 |

21-484 | Graph Theory | 9 |

21-600 | Mathematical Logic I | 12 |

21-602 | Introduction to Set Theory I | 12 |

21-603 | Model Theory I | 12 |

21-610 | Algebra I | 12 |

21-620 | Real Analysis | 6 |

21-621 | Introduction to Lebesgue Integration | 6 |

21-630 | Ordinary Differential Equations | 12 |

21-640 | Introduction to Functional Analysis | 12 |

21-651 | General Topology | 12 |

21-660 | Introduction to Numerical Analysis I | 12 |

21-701 | Discrete Mathematics | 12 |

21-720 | Measure and Integration | 12 |

21-721 | Probability | 12 |

21-737 | Probabilistic Combinatorics | 12 |

21-738 | Extremal Combinatorics | 12 |

Note that courses 21-600 and above carry graduate credit. Courses at the 600 level are designed as transitional courses to graduate study. A student preparing for graduate study should also consider undertaking independent work. The Department offers 21-499 Undergraduate Research Topic and 21-599 Undergraduate Reading and Research for this purpose.

Courses 21-700 and above can be used with the permission of both the instructor and the department.

#### Suggested Schedule

##### Freshman Year

Fall | Units | |

21-120 | Differential and Integral Calculus | 10 |

21-127 | Concepts of Mathematics | 10 |

33-111 | Physics I for Science Students | 12 |

76-101 | Interpretation and Argument | 9 |

99-101 | Computing @ Carnegie Mellon | 3 |

44 |

Spring | Units | |

15-110 | Principles of Computing | 10 |

21-122 | Integration and Approximation | 10 |

21-241 | Matrices and Linear Transformations | 10 |

or 21-242 | Matrix Theory | |

33-112 | Physics II for Science Students | 12 |

xx-xxx | H&SS Elective | 9 |

51 |

##### Sophomore Year

Fall | Units | |

03-121 | Modern Biology | 9 |

09-105 | Introduction to Modern Chemistry I | 10 |

21-201 | Undergraduate Colloquium | 1 |

21-228 | Discrete Mathematics | 9-12 |

or 15-251 | Great Theoretical Ideas in Computer Science | |

21-268 | Multidimensional Calculus | 10 |

or 21-269 | Vector Analysis | |

xx-xxx | H&SS Elective | 9 |

48-51 |

Spring | Units | |

21-201 | Undergraduate Colloquium | 1 |

21-261 | Introduction to Ordinary Differential Equations | 10 |

21-373 | Algebraic Structures | 9 |

xx-xxx | Mathematical Sci, Statistics, or Computer Sci Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

47 |

##### Junior Year

Fall | Units | |

21-355 | Principles of Real Analysis I | 9 |

36-225 | Introduction to Probability Theory | 9 |

or 21-325 | Probability | |

xx-xxx | Mathematical Sci, Statistics, or Computer Sci Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

Spring | Units | |

21-341 | Linear Algebra | 9 |

21-356 | Principles of Real Analysis II | 9 |

21-xxx | Mathematical Sciences Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

##### Senior Year

Fall | Units | |

21-xxx | Mathematical Sciences Elective | 9 |

21-xxx | Mathematical Sciences Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

Spring | Units | |

21-xxx | Mathematical Sciences Elective | 9 |

21-xxx | Mathematical Sciences Elective | 9 |

xx-xxx | Mathematical Sci, Statistics, or Computer Sci Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

###### 360Minimum number of units for the degree:

### Operations Research and Statistics Concentration

An operations research professional employs quantitative and computational skills toward enhancing the function of an organization or process. Students choosing this concentration will develop problem-solving abilities in mathematical and statistical modeling and computer-based simulation in areas such as network design, transportation scheduling, allocation of resources and optimization. In addition to courses in mathematics and statistics, a basic background in economics and accounting is included. Since problems in business and industry are often solved by teams, the program also includes a group project to be undertaken in the Senior year. Students choosing this concentration may not pursue an additional major or minor in Statistics in the Humanities and Social Sciences College.

The requirements for the concentration in Operations Research and Statistics are:

##### Mathematical Sciences Courses (required)

The alternative courses 21-242, 21-261, and 21-268 (or 21-269) are particularly recommended for a student planning to pursue graduate work.

Courses | Units | |

21-120 | Differential and Integral Calculus | 10 |

21-122 | Integration and Approximation | 10 |

21-127 | Concepts of Mathematics | 10 |

21-201 | Undergraduate Colloquium (Taken twice in Sophomore year.) | 1 |

21-228 | Discrete Mathematics | 9-12 |

or 15-251 | Great Theoretical Ideas in Computer Science | |

21-241 | Matrices and Linear Transformations | 10 |

or 21-242 | Matrix Theory | |

21-259 | Calculus in Three Dimensions | 9-10 |

or 21-268 | Multidimensional Calculus | |

or 21-269 | Vector Analysis | |

21-260 | Differential Equations | 9-10 |

or 21-261 | Introduction to Ordinary Differential Equations | |

or 33-231 | Physical Analysis | |

21-292 | Operations Research I | 9 |

21-369 | Numerical Methods | 9 |

21-393 | Operations Research II | 9 |

95-100 |

##### Statistics Courses (required)

Courses | Units | |

36-225 | Introduction to Probability Theory | 9 |

or 21-325 | Probability | |

36-226 | Introduction to Statistical Inference | 9 |

36-401 | Modern Regression | 9 |

36-402 | Advanced Methods for Data Analysis | 9 |

36-410 | Introduction to Probability Modeling | 9 |

45 |

##### Economics, Business, and Computer Science Courses (required)

Courses | Units | |

15-110 | Principles of Computing | 10 |

70-122 | Introduction to Accounting | 9 |

73-100 | Principles of Economics | 9 |

73-230 | Intermediate Microeconomics | 9 |

73-240 | Intermediate Macroeconomics | 9 |

46 |

##### Depth Electives (required)

Five depth electives (required), to be chosen from the list below. The course 21-355 is particularly recommended for a student planning to pursue graduate work.

15-122 | Principles of Imperative Computation | 10 |

15-150 | Principles of Functional Programming | 10 |

15-210 | Parallel and Sequential Data Structures and Algorithms | 12 |

21-270 | Introduction to Mathematical Finance | 9 |

21-301 | Combinatorics | 9 |

21-341 | Linear Algebra | 9 |

21-355 | Principles of Real Analysis I | 9 |

21-356 | Principles of Real Analysis II | 9 |

21-365 | Projects in Applied Mathematics | 9 |

21-366 | Topics in Applied Mathematics | 9 |

21-370 | Discrete Time Finance | 9 |

21-373 | Algebraic Structures | 9 |

21-420 | Continuous-Time Finance | 9 |

21-484 | Graph Theory | 9 |

36-461 | Special Topics | 9 |

36-462 | Special Topics: Data Mining | 9 |

36-463 | Special Topics: Multilevel and Hierarchical Models | 9 |

36-464 | Special Topics: Applied Multivariate Methods | 9 |

70-371 | Operations Management | 9 |

70-460 | Mathematical Models for Consulting | 9 |

70-471 | Supply Chain Management | 9 |

##### MCS General Education (required)

MCS or SHS humanities, social sciences, and science core (114 units)

Note that 73-100, 73-230, and 73-240 satisfy requirements from the MCS general education core.

#### Suggested Schedule

##### Freshman Year

Fall | Units | |

03-121 | Modern Biology | 9 |

21-120 | Differential and Integral Calculus | 10 |

21-127 | Concepts of Mathematics | 10 |

33-111 | Physics I for Science Students | 12 |

76-101 | Interpretation and Argument | 9 |

99-101 | Computing @ Carnegie Mellon | 3 |

53 |

Spring | Units | |

15-110 | Principles of Computing | 10 |

21-122 | Integration and Approximation | 10 |

21-241 | Matrices and Linear Transformations | 10 |

or 21-242 | Matrix Theory | |

33-112 | Physics II for Science Students | 12 |

xx-xxx | H&SS Elective | 9 |

51 |

##### Sophomore Year

Fall | Units | |

09-105 | Introduction to Modern Chemistry I | 10 |

21-201 | Undergraduate Colloquium | 1 |

21-228 | Discrete Mathematics | 9-12 |

or 15-251 | Great Theoretical Ideas in Computer Science | |

21-259 | Calculus in Three Dimensions | 9-10 |

or 21-268 | Multidimensional Calculus | |

or 21-269 | Vector Analysis | |

73-100 | Principles of Economics | 9 |

38-42 |

Spring | Units | |

21-260 | Differential Equations | 9-10 |

or 21-261 | Introduction to Ordinary Differential Equations | |

or 33-231 | Physical Analysis | |

21-292 | Operations Research I | 9 |

21-201 | Undergraduate Colloquium | 1 |

70-122 | Introduction to Accounting | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

46-47 |

##### Junior Year

Fall | Units | |

21-369 | Numerical Methods | 9 |

36-225 | Introduction to Probability Theory | 9 |

or 21-325 | Probability | |

73-230 | Intermediate Microeconomics | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

Spring | Units | |

36-226 | Introduction to Statistical Inference | 9 |

36-410 | Introduction to Probability Modeling | 9 |

73-240 | Intermediate Macroeconomics | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

45 |

##### Senior Year

Fall | Units | |

21-393 | Operations Research II | 9 |

36-401 | Modern Regression | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

Spring | Units | |

36-402 | Advanced Methods for Data Analysis | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

### Statistics Concentration

Statistics is concerned with the process by which inferences are made from data. Statistical methods are essential to research in a wide variety of scientific disciplines. For example, principles of experimental design that assist chemists in improving their yields also help poultry farmers grow bigger chickens. Similarly, time series analysis is used to better understand radio waves from distant galaxies, hormone levels in the blood, and concentrations of pollutants in the atmosphere. This diversity of application is an exciting aspect of the field, and it is one reason for the current demand for well-trained statisticians.

The courses 36-225 Introduction to Probability Theory and 36-226 Introduction to Statistical Inference taken in the Junior year serve as the basis for all further statistics courses. The course 21-325 is a more mathematical alternative to 36-225.

The Statistics Concentration is jointly administered by the Department of Mathematical Sciences and the Department of Statistics. The Department of Statistics considers applications for the master's program from undergraduates in the Junior year. Students who are accepted are expected to finish their undergraduate studies, using some electives in the Senior year to take courses recommended by the Department of Statistics. This will ensure a strong background to permit completion of the master's program in one year beyond the baccalaureate. The requirements for the Statistics Concentration are:

##### Mathematical Sciences Courses (required)

The alternative courses 21-242, 21-261, and 21-268 (or 21-269) are particularly recommended for a student planning to pursue graduate work.

Courses | Units | |

21-120 | Differential and Integral Calculus | 10 |

21-122 | Integration and Approximation | 10 |

21-127 | Concepts of Mathematics | 10 |

21-201 | Undergraduate Colloquium (Taken twice in Sophomore year.) | 1 |

21-228 | Discrete Mathematics | 9-12 |

or 15-251 | Great Theoretical Ideas in Computer Science | |

21-241 | Matrices and Linear Transformations | 10 |

or 21-242 | Matrix Theory | |

21-259 | Calculus in Three Dimensions | 9-10 |

or 21-268 | Multidimensional Calculus | |

or 21-269 | Vector Analysis | |

21-260 | Differential Equations | 9-10 |

or 21-261 | Introduction to Ordinary Differential Equations | |

or 33-231 | Physical Analysis | |

21-292 | Operations Research I | 9 |

21-369 | Numerical Methods | 9 |

21-393 | Operations Research II | 9 |

95-100 |

##### Statistics Courses (required)

Courses | Units | |

36-225 | Introduction to Probability Theory | 9 |

or 21-325 | Probability | |

36-226 | Introduction to Statistical Inference | 9 |

36-401 | Modern Regression | 9 |

36-402 | Advanced Methods for Data Analysis | 9 |

36-410 | Introduction to Probability Modeling | 9 |

45 |

##### Economics and Computer Science Courses (required)

Courses | Units | |

15-112 | Fundamentals of Programming and Computer Science | 12 |

15-122 | Principles of Imperative Computation | 10 |

73-100 | Principles of Economics | 9 |

31 |

##### Depth Electives (required)

Four depth electives, including at least one statistics course, to be chosen from the list below. The course 21-355 Principles of Real Analysis I is particularly recommended for a student planning to pursue graduate work.

15-150 | Principles of Functional Programming | 10 |

15-210 | Parallel and Sequential Data Structures and Algorithms | 12 |

21-270 | Introduction to Mathematical Finance | 9 |

21-341 | Linear Algebra | 9 |

21-355 | Principles of Real Analysis I | 9 |

21-356 | Principles of Real Analysis II | 9 |

21-365 | Projects in Applied Mathematics | 9 |

21-366 | Topics in Applied Mathematics | 9 |

21-370 | Discrete Time Finance | 9 |

21-373 | Algebraic Structures | 9 |

21-420 | Continuous-Time Finance | 9 |

21-484 | Graph Theory | 9 |

36-461 | Special Topics | 9 |

36-462 | Special Topics: Data Mining | 9 |

36-463 | Special Topics: Multilevel and Hierarchical Models | 9 |

36-464 | Special Topics: Applied Multivariate Methods | 9 |

##### MCS General Education (required)

MCS or SHS humanities, social sciences, and science core (114 units)

Note that 73-100 satisfies the requirement from the MCS core.

#### Suggested Schedule

##### Freshman Year

Fall | Units | |

03-121 | Modern Biology | 9 |

21-120 | Differential and Integral Calculus | 10 |

21-127 | Concepts of Mathematics | 10 |

33-111 | Physics I for Science Students | 12 |

76-101 | Interpretation and Argument | 9 |

99-101 | Computing @ Carnegie Mellon | 3 |

53 |

Spring | Units | |

15-112 | Fundamentals of Programming and Computer Science | 12 |

21-122 | Integration and Approximation | 10 |

21-241 | Matrices and Linear Transformations | 10 |

or 21-242 | Matrix Theory | |

33-112 | Physics II for Science Students | 12 |

44 |

##### Sophomore Year

Fall | Units | |

09-105 | Introduction to Modern Chemistry I | 10 |

21-201 | Undergraduate Colloquium | 1 |

21-228 | Discrete Mathematics | 9-12 |

or 15-251 | Great Theoretical Ideas in Computer Science | |

21-259 | Calculus in Three Dimensions | 9-10 |

or 21-268 | Multidimensional Calculus | |

or 21-269 | Vector Analysis | |

73-100 | Principles of Economics | 9 |

xx-xxx | H&SS Elective | 9 |

47-51 |

Spring | Units | |

15-122 | Principles of Imperative Computation | 10 |

21-260 | Differential Equations | 9-10 |

or 21-261 | Introduction to Ordinary Differential Equations | |

or 33-231 | Physical Analysis | |

21-292 | Operations Research I | 9 |

21-201 | Undergraduate Colloquium | 1 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

47-48 |

##### Junior Year

Fall | Units | |

21-369 | Numerical Methods | 9 |

36-225 | Introduction to Probability Theory | 9 |

or 21-325 | Probability | |

xx-xxx | Depth Elective | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

45 |

Spring | Units | |

36-226 | Introduction to Statistical Inference | 9 |

36-410 | Introduction to Probability Modeling | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

##### Senior Year

Fall | Units | |

21-393 | Operations Research II | 9 |

36-401 | Modern Regression | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

Spring | Units | |

36-402 | Advanced Methods for Data Analysis | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

###### 360Minimum number of units required for the degree:

### Discrete Mathematics and Logic Concentration

Discrete mathematics is the study of finite and countable structures and algorithms for the manipulation and analysis of such structures, while mathematical logic is the study of axiomatic systems and their mathematical applications. Both are flourishing research areas and have close ties with computer science.

The Discrete Mathematics and Logic Concentration provides a firm background in discrete mathematics and mathematical logic, together with the elements of theoretical computer science. It prepares the student to pursue research in these fields, or to apply their ideas in the many disciplines (ranging from philosophy to hardware verification) where such ideas have proved relevant.

The requirements for the Discrete Mathematics and Logic Concentration are:

##### Mathematical Sciences and Computer Science Courses (required)

The alternative course 21-242 is particularly recommended for a student planning to pursue graduate work. Students who plan to pursue graduate study in mathematical logic are strongly advised to take 21-300.

Units | ||

15-122 | Principles of Imperative Computation | 10 |

15-150 | Principles of Functional Programming | 10 |

15-210 | Parallel and Sequential Data Structures and Algorithms | 12 |

21-120 | Differential and Integral Calculus | 10 |

21-122 | Integration and Approximation | 10 |

21-127 | Concepts of Mathematics | 10 |

21-201 | Undergraduate Colloquium (Taken twice in Sophomore year.) | 1 |

21-241 | Matrices and Linear Transformations | 10 |

or 21-242 | Matrix Theory | |

21-228 | Discrete Mathematics | 9-12 |

or 15-251 | Great Theoretical Ideas in Computer Science | |

21-300 | Basic Logic | 9 |

or 15-317 | Constructive Logic | |

21-301 | Combinatorics | 9 |

21-341 | Linear Algebra | 9 |

21-355 | Principles of Real Analysis I | 9 |

21-373 | Algebraic Structures | 9 |

127-130 |

##### Computer Science electives (required)

Any two courses at the 300 level or above. The following are specifically suggested: | ||

15-312 | Foundations of Programming Languages | 12 |

15-451 | Algorithm Design and Analysis | 12 |

15-453 | Formal Languages, Automata, and Computability | 9 |

Students pursuing this concentration who minor in Computer Science must take two additional Computer Science courses at the 300 level or above to avoid excessive double counting.

##### Mathematical Sciences Electives (required)

Seven courses from lists 1 and 2 below, including at least three chosen from list 1.

##### List 1 (Discrete Mathematics and Logic Electives)

21-325 | Probability | 9 |

21-329 | Set Theory | 9 |

21-374 | Field Theory | 9 |

21-441 | Number Theory | 9 |

21-484 | Graph Theory | 9 |

21-602 | Introduction to Set Theory I | 12 |

21-603 | Model Theory I | 12 |

21-610 | Algebra I | 12 |

21-700 | Mathematical Logic II | 12 |

80-405 | Game Theory | 9 |

80-411 | Proof Theory | 9 |

80-413 | Category Theory | 9 |

##### List 2 (General Mathematics Electives)

21-259 | Calculus in Three Dimensions | 9-10 |

or 21-268 | Multidimensional Calculus | |

or 21-269 | Vector Analysis | |

21-260 | Differential Equations | 9-10 |

or 21-261 | Introduction to Ordinary Differential Equations | |

or 33-231 | Physical Analysis | |

21-292 | Operations Research I | 9 |

21-356 | Principles of Real Analysis II | 9 |

21-371 | Functions of a Complex Variable | 9 |

21-372 | Partial Differential Equations and Fourier Analysis | 9 |

21-393 | Operations Research II | 9 |

21-465 | Topology | 9 |

21-467 | Differential Geometry | 9 |

21-476 | Introduction to Dynamical Systems | 9 |

##### MCS General Education (required)

MCS or SHS humanities, social sciences, and science core (114 units)

#### Suggested Schedule

##### Freshman Year

Fall | Units | |

15-112 | Fundamentals of Programming and Computer Science | 12 |

21-120 | Differential and Integral Calculus | 10 |

33-111 | Physics I for Science Students | 12 |

76-101 | Interpretation and Argument | 9 |

99-101 | Computing @ Carnegie Mellon | 3 |

46 |

Spring | Units | |

15-122 | Principles of Imperative Computation | 10 |

21-122 | Integration and Approximation | 10 |

21-127 | Concepts of Mathematics | 10 |

21-241 | Matrices and Linear Transformations | 10 |

or 21-242 | Matrix Theory | |

33-112 | Physics II for Science Students | 12 |

52 |

##### Sophomore Year

Fall | Units | |

15-150 | Principles of Functional Programming | 10 |

21-268 | Multidimensional Calculus | 10 |

or 21-269 | Vector Analysis | |

21-301 | Combinatorics | 9 |

21-373 | Algebraic Structures | 9 |

21-201 | Undergraduate Colloquium | 1 |

xx-xxx | H&SS Elective | 9 |

48 |

Spring | Units | |

03-121 | Modern Biology | 9 |

09-105 | Introduction to Modern Chemistry I | 10 |

15-210 | Parallel and Sequential Data Structures and Algorithms | 12 |

xx-xxx | Discrete Math/Logic Elective | 9 |

21-201 | Undergraduate Colloquium | 1 |

xx-xxx | Mathematics Elective | 9 |

xx-xxx | H&SS Elective | 9 |

59 |

##### Junior Year

Fall | Units | |

15-xxx | Computer Science Elective | 9 |

21-300 | Basic Logic | 9 |

or 15-317 | Constructive Logic | |

21-355 | Principles of Real Analysis I | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

Spring | Units | |

15-xxx | Computer Science Elective | 9 |

21-341 | Linear Algebra | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

##### Senior Year

Fall | Units | |

xx-xxx | Discrete Math/Logic Elective | 9 |

xx-xxx | Mathematics Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

Spring | Units | |

xx-xxx | Discrete Math/Logic Elective | 9 |

xx-xxx | Mathematics Elective | 9 |

xx-xxx | Mathematics Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

###### 360Minimum number of units required for degree:

### Computational and Applied Mathematics Concentration

This concentration is designed to prepare students for careers in business or industry which require significant analytical, computational and problem solving skills. It also prepares students with interest in computational and applied mathematics for graduate school.

The students in this concentration develop skills to choose the right framework to quantify or model a problem, analyze it, simulate and in general use appropriate techniques for carrying the effort through to an effective solution. The free electives allow the student to develop an interest in a related area by completing a minor in another department, such as Engineering Studies, Economics, Information Systems or Business Administration.

The requirements for the Computational and Applied Mathematics Concentration are:

##### Mathematical Sciences Courses (required)

Courses | Units | |

21-120 | Differential and Integral Calculus | 10 |

21-122 | Integration and Approximation | 10 |

21-127 | Concepts of Mathematics | 10 |

21-201 | Undergraduate Colloquium (Taken twice in Sophomore year.) | 1 |

21-228 | Discrete Mathematics | 9-12 |

or 15-251 | Great Theoretical Ideas in Computer Science | |

21-241 | Matrices and Linear Transformations | 10 |

or 21-242 | Matrix Theory | |

21-259 | Calculus in Three Dimensions | 9-10 |

or 21-268 | Multidimensional Calculus | |

or 21-269 | Vector Analysis | |

21-260 | Differential Equations | 9-10 |

or 21-261 | Introduction to Ordinary Differential Equations | |

or 33-231 | Physical Analysis | |

21-320 | Symbolic Programming Methods | 9 |

36-225 | Introduction to Probability Theory | 9 |

or 21-325 | Probability | |

21-355 | Principles of Real Analysis I | 9 |

21-356 | Principles of Real Analysis II | 9 |

21-369 | Numerical Methods | 9 |

113-118 |

##### Computer Science Courses (required)

Courses | Units | |

15-122 | Principles of Imperative Computation | 10 |

##### Depth Electives (required)

Students must take 45 units from the list below:

15-210 | Parallel and Sequential Data Structures and Algorithms | 12 |

21-292 | Operations Research I | 9 |

21-341 | Linear Algebra | 9 |

21-365 | Projects in Applied Mathematics | 9 |

21-366 | Topics in Applied Mathematics | 9 |

21-370 | Discrete Time Finance | 9 |

21-371 | Functions of a Complex Variable | 9 |

21-372 | Partial Differential Equations and Fourier Analysis | 9 |

21-393 | Operations Research II | 9 |

21-420 | Continuous-Time Finance | 9 |

21-467 | Differential Geometry | 9 |

21-470 | Selected Topics in Analysis | 9 |

21-476 | Introduction to Dynamical Systems | 9 |

21-499 | Undergraduate Research Topic | 9 |

21-599 | Undergraduate Reading and Research | Var. |

21-620 | Real Analysis | 6 |

21-621 | Introduction to Lebesgue Integration | 6 |

21-630 | Ordinary Differential Equations | 12 |

21-640 | Introduction to Functional Analysis | 12 |

21-651 | General Topology | 12 |

21-660 | Introduction to Numerical Analysis I | 12 |

21-690 | Methods of Optimization | 12 |

21-720 | Measure and Integration | 12 |

21-721 | Probability | 12 |

21-732 | Partial Differential Equations I | 12 |

36-410 | Introduction to Probability Modeling | 9 |

21-366 Topics in Applied Mathematics and 21-470 Selected Topics in Analysis have content that varies from year to year. These courses can be taken more than once (with permission).

Note that courses 21-600 and above carry graduate credit. 600 level courses are designed as transitional courses to graduate study.

A student preparing for graduate study should also consider undertaking independent work. The Department offers 21-499 Undergraduate Research Topic and 21-599 Undergraduate Reading and Research for this purpose. These courses can be taken as part of satisfying the Depth Elective requirement, but require permission of both the instructor and the department.

Courses 21-700 and above can be used with the permission of both the instructor and the department.

##### MCS General Education (required)

MCS or SHS humanities, social sciences, and science core (114 units).

Students not in MCS are required to take 15-110 Principles of Computing (10 units).

#### Suggested Schedule

##### Freshman Year

Fall | Units | |

09-101 | Introduction to Experimental Chemistry | 3 |

21-120 | Differential and Integral Calculus | 10 |

21-126 | Introduction to Mathematical Software | 3 |

21-127 | Concepts of Mathematics | 10 |

33-111 | Physics I for Science Students | 12 |

76-101 | Interpretation and Argument | 9 |

47 |

Spring | Units | |

21-122 | Integration and Approximation | 10 |

21-228 | Discrete Mathematics | 9 |

21-241 | Matrices and Linear Transformations | 10 |

33-112 | Physics II for Science Students | 12 |

xx-xxx | H&SS Elective | 9 |

50 |

##### Sophomore Year

Fall | Units | |

09-105 | Introduction to Modern Chemistry I | 10 |

15-112 | Fundamentals of Programming and Computer Science | 12 |

21-268 | Multidimensional Calculus | 10 |

or 21-269 | Vector Analysis | |

21-201 | Undergraduate Colloquium | 1 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

51 |

Spring | Units | |

03-121 | Modern Biology | 9 |

15-122 | Principles of Imperative Computation | 10 |

21-261 | Introduction to Ordinary Differential Equations | 10 |

21-201 | Undergraduate Colloquium | 1 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

48 |

##### Junior Year

Fall | Units | |

21-320 | Symbolic Programming Methods | 9 |

21-325 | Probability | 9 |

21-355 | Principles of Real Analysis I | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

Spring | Units | |

21-356 | Principles of Real Analysis II | 9 |

21-369 | Numerical Methods | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

##### Senior Year

Fall | Units | |

xx-xxx | Depth Elective | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

Spring | Units | |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Free Elective | 9 |

xx-xxx | Free Elective | 9 |

xx-xxx | Free Elective | 9 |

45 |

###### 360Minimum units required for degree:

## Double Major Requirements

All degrees offered by the Department are available as a second major to students majoring in other departments. Interested students should contact the Department for further information and guidance. In general the requirements for a second major include all the required courses except the MCS core, free electives and 21-201 Undergraduate Colloquium.

## The Minor in Mathematical Sciences

The Minor includes six courses. 21-127 Concepts of Mathematics is a prerequisite for 21-228 and recommended for 21-241 . The minimum preparation required for 21-355 Principles of Real Analysis I is 21-122 and 21-127 or equivalent courses.

21-127 | Concepts of Mathematics | 10 |

21-228 | Discrete Mathematics | 9-12 |

or 15-251 | Great Theoretical Ideas in Computer Science | |

21-241 | Matrices and Linear Transformations | 10 |

or 21-242 | Matrix Theory | |

21-355 | Principles of Real Analysis I | 9 |

21-3xx | Mathematical Sciences Elective | |

21-3xx | Mathematical Sciences Elective |

To avoid excessive double counting, the two Mathematical Sciences Electives may not also count toward the student's major.

## The Minor in Discrete Mathematics and Logic

This minor develops the fundamentals of discrete mathematics and logic necessary to understand the mathematical foundations of many computer related disciplines. Required courses are:

21-300 | Basic Logic | 9 |

or 15-317 | Constructive Logic | |

21-301 | Combinatorics | 9 |

Four of the following: | ||

21-329 | Set Theory | 9 |

21-341 | Linear Algebra | 9 |

21-373 | Algebraic Structures | 9 |

21-374 | Field Theory | 9 |

21-441 | Number Theory | 9 |

21-484 | Graph Theory | 9 |

21-602 | Introduction to Set Theory I | 12 |

21-603 | Model Theory I | 12 |

21-610 | Algebra I | 12 |

21-700 | Mathematical Logic II | 12 |

## The Honors Degree Program

This demanding program leads to an M.S. in Mathematical Sciences, normally in four years, in addition to the student's B.S. degree. The key element in the program is usually the Mathematical Studies sequence. Admission to the Honors Program, in the Junior year, requires an application while admission to the Math Studies sequences is by invitation only. In the application process the Department will hold to the same high standards which apply to admission to any graduate program.

The core undergraduate honors courses are:

##### Freshman Year

Fall | Units | |

21-242 | Matrix Theory (Honors version of 21-241 Matrices and Linear Transformations) | 10 |

Spring | ||

21-269 | Vector Analysis (Honors version of 21-268 Multidimensional Calculus) | 10 |

##### Sophomore Year

Fall | ||

21-235 | Mathematical Studies Analysis I (Honors version of 21-355 Principles of Real Analysis I) | 10 |

21-237 | Mathematical Studies Algebra I (Honors version of 21-373 Algebraic Structures) | 10 |

Spring | ||

21-236 | Mathematical Studies Analysis II (Honors version of 21-356 Principles of Real Analysis II) | 10 |

21-238 | Mathematical Studies Algebra II (Honors version of 21-341 Linear Algebra) | 10 |

##### Honors Program Requirements:

21-901 | Masters Degree Research | Var. |

###### Five graduate mathematics courses: 60 units

Each student in the honors degree program will have a thesis advisor in addition to his or her academic advisor. In practice, the student must start thinking about the thesis as early as possible. For this reason we include some thesis work, 3 units of 21-901 Masters Degree Research, in the Fall semester of the Senior year to allow for exploratory work under supervision. The actual thesis work is then planned for the final semester with 15 units of 21-901 Masters Degree Research. The student must give a public presentation and will be examined on the thesis and related mathematics.

The five graduate courses must include at least one course from each of the following areas:

- Analysis: for example, Measure and Integration, Complex Analysis, Functional Analysis
- Algebra, Logic, Geometry and Topology: for example, Mathematical Logic I, Algebra I, General Topology, Discrete Mathematics, Commutative Algebra, Differential Geometry
- Applied Mathematics: for example, Introduction to Continuum Mechanics, Probability Measures, Probability Theory, Graphs and Network Flows, Ordinary Differential Equations, Methods of Optimization, Introduction to Numerical Analysis I, Partial Differential Equations I, Sobolev Spaces.