# Department of Mathematical Sciences

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. in Mathematical Sciences degree 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, 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. Note that 21-269 will be offered pending Fall 2011 approval by the MCS College Council.

##### Mathematical Studies

Following the 21-242/21-269 sequence, we intend to offer Mathematical Studies Analysis I/II and Mathematical Studies Algebra I/II, pending Fall 2011 approval by the MCS College Council. Mathematical Studies provides an 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 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.

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.

These programs are described in the catalog section on interdisciplinary programs.

### Curricula

For each concentration, we provide a list of the requirements and a suggested schedule that takes prerequisites into account. A Mathematical Sciences, Statistics, or Computer Science Elective refers to a course from any of the Departments of Mathematical Sciences, Statistics or Computer Science. The only restrictions on these electives are that a mathematical sciences course must be beyond the calculus sequence, a statistics course must have at least 36-225 as a prerequisite, and a computer science course must be at the 15-200 level or above.

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 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.

In addition to the courses in the suggested schedules below, a student majoring in mathematical sciences also takes the one unit course 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

Units | ||

21-120 | Differential and Integral Calculus | 10 |

21-122 | Integration, Differential Equations and Approximation | 10 |

21-127 | Concepts of Mathematics | 9 |

21-201 | Undergrad Colloquium | 1 |

21-228 | Discrete Mathematics | 9 |

or | 21-301 Discrete Mathematics (9 units) | |

or | 21-484 Discrete Mathematics (9 units) | |

21-241 | Matrices and Linear Transformations | 10 |

or | 21-242 Matrices and Linear Transformations (10 units) | |

21-341 | Matrices and Linear Transformations | 9 |

21-259 | Calculus in Three Dimensions | 9 |

or | 21-268 Calculus in Three Dimensions (9 units) | |

or | 21-269 Calculus in Three Dimensions (9 units) | |

21-260 | Differential Equations | 9 |

or | 21-261 Differential Equations (10 units) | |

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

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

21-373 | Algebraic Structures | 9 |

103 |

five Mathematical Sciences electives

##### Other courses

Units | ||

15-110 | Principles of Computing | 10 |

36-225 | Probability | 9 |

or | 21-325 Probability (9 units) | |

19 |

three Mathematical Sciences, Statistics, or Computer Science electives

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

seven free electives

### Suggested Schedule

##### Freshman Year

Fall | Units | |

21-120 | Differential and Integral Calculus | 10 |

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

15-110 | Principles of Computing | 10 |

03-121 | Modern Biology | 9 |

76-101 | Interpretation and Argument | 9 |

99-101 | Computing @ Carnegie Mellon | 3 |

53 |

Spring | Units | |

21-122 | Integration, Differential Equations and Approximation | 10 |

21-127 | Concepts of Mathematics | 9 |

21-241 | Matrices and Linear Transformations | 10 |

or | 21-242 Matrices and Linear Transformations (10 units) | |

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

xx-xxx | H&SS Elective | 9 |

50 |

##### Sophomore Year

Fall | Units | |

21-228 | Discrete Mathematics | 9 |

or | 21-301 Discrete Mathematics (9 units) | |

or | 21-484 Discrete Mathematics (9 units) | |

21-268 | Calculus in Three Dimensions | 10 |

or | 21-269 Calculus in Three Dimensions (9 units) | |

21-201 | Undergrad Colloquium | 1 |

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

xx-xxx | H&SS Elective | 9 |

xx-xxx | Elective | 9 |

48 |

Spring | Units | |

21-261 | Differential Equations | 10 |

21-373 | Algebraic Structures | 9 |

21-201 | Undergrad Colloquium | 1 |

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

xx-xxx | H&SS Elective | 9 |

xx-xxx | Elective | 9 |

47 |

##### Junior Year

Fall | Units | |

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

36-225 | Probability | 9 |

or | 21-325 Probability (9 units) | |

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

xx-xxx | H&SS Elective | 9 |

xx-xxx | Elective | 9 |

45 |

Spring | Units | |

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

21-341 | Matrices and Linear Transformations | 9 |

21-xxx | Mathematical Sciences Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | 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 | Elective | 9 |

xx-xxx | 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 | Elective | 9 |

45 |

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

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 | Discrete Mathematics | 9 |

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

21-372 | Partial Differential Equations | 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 | Ordinary Differential Equations | 9 |

21-484 | Discrete Mathematics | 9 |

Note that courses 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 an independent work. The Department offers 21-499 Undergraduate Research Topic and 21-599 Undergraduate Reading and Research for this purpose.

### 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 Mathematical Sciences 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

Units | ||

21-120 | Differential and Integral Calculus | 10 |

21-122 | Integration, Differential Equations and Approximation | 10 |

21-127 | Concepts of Mathematics | 9 |

21-201 | Undergrad Colloquium | 1 |

21-228 | Discrete Mathematics | 9 |

or | 21-301 Discrete Mathematics (9 units) | |

or | 21-484 Discrete Mathematics (9 units) | |

21-241 | Matrices and Linear Transformations | 10 |

or | 21-242 Matrices and Linear Transformations (10 units) | |

21-259 | Calculus in Three Dimensions | 9 |

or | 21-268 Calculus in Three Dimensions (9 units) | |

or | 21-269 Calculus in Three Dimensions (9 units) | |

21-260 | Differential Equations | 9 |

or | 21-261 Differential Equations (10 units) | |

21-292 | Operations Research I | 9 |

21-369 | Numerical Methods | 9 |

21-393 | Operations Research II | 9 |

94 |

##### Statistics

Units | ||

36-225 | Probability | 9 |

or | 21-325 Probability (9 units) | |

36-226 | Introduction to Statistical Inference | 9 |

36-401 | Modern Regression | 9 |

36-402 | Advanced Data Analysis | 9 |

36-410 | Introduction to Probability Modeling | 9 |

45 |

##### Depth Electives

The detailed curriculum below includes five depth electives. These are to be chosen from among the following (21-261 , 21-268 , and 21-355 are 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 |

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

36-461 | Topics in Statistics: Statistical Methods in Epidemiology | 9 |

36-462 | Topics in Statistics: Statistical Learning | 9 |

36-463 | Topics in Statistics: Multilevel Hierarchical Models | 9 |

36-464 | Topics in Statistics: Applied Multivariate Methods | 9 |

70-371 | Production/Operations Management | 9 |

70-460 | Mathematical Models for Consulting | 9 |

70-471 | Logistics and Supply Chain Management | 9 |

##### Other Courses

Units | ||

15-110 | Principles of Computing | 10 |

70-122 | Introduction to Accounting | 9 |

73-100 | Principles of Economics | 9 |

73-150 | Intermediate Microeconomics | 9 |

73-200 | Intermediate Macroeconomics | 9 |

46 |

##### MCS humanities, social sciences, and science core (120

units, including , 73-150 and 73-200 )

Five free electives

### Suggested Schedule

##### Freshman Year

Fall | Units | |

21-120 | Differential and Integral Calculus | 10 |

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

15-110 | Principles of Computing | 10 |

03-121 | Modern Biology | 9 |

76-101 | Interpretation and Argument | 9 |

99-101 | Computing @ Carnegie Mellon | 3 |

53 |

Spring | Units | |

21-122 | Integration, Differential Equations and Approximation | 10 |

21-127 | Concepts of Mathematics | 9 |

21-241 | Matrices and Linear Transformations | 10 |

or | 21-242 Matrices and Linear Transformations (10 units) | |

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

xx-xxx | H&SS Elective | 9 |

50 |

##### Sophomore Year

Fall | Units | |

21-228 | Discrete Mathematics | 9 |

or | 21-301 Discrete Mathematics (9 units) | |

or | 21-484 Discrete Mathematics (9 units) | |

21-259 | Calculus in Three Dimensions | 9 |

or | 21-268 Calculus in Three Dimensions (9 units) | |

or | 21-269 Calculus in Three Dimensions (9 units) | |

21-201 | Undergrad Colloquium | 1 |

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

73-100 | Principles of Economics | 9 |

38 |

Spring | Units | |

21-260 | Differential Equations | 9 |

or | 21-261 Differential Equations (10 units) | |

21-292 | Operations Research I | 9 |

21-201 | Undergrad Colloquium | 1 |

70-122 | Introduction to Accounting | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Elective | 9 |

46 |

##### Junior Year

Fall | Units | |

21-369 | Numerical Methods | 9 |

xx-xxx | Depth Elective | 9 |

36-225 | Probability | 9 |

or | 21-325 Probability (9 units) | |

73-150 | Intermediate Microeconomics | 9 |

xx-xxx | Elective | 9 |

45 |

Spring | Units | |

xx-xxx | Depth Elective | 9 |

36-226 | Introduction to Statistical Inference | 9 |

36-410 | Introduction to Probability Modeling | 9 |

xx-xxx | H&SS Elective | 9 |

73-200 | Intermediate Macroeconomics | 9 |

45 |

##### Senior Year

Fall | Units | |

21-393 | Operations Research II | 9 |

xx-xxx | Depth Elective | 9 |

36-401 | Modern Regression | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Elective | 9 |

45 |

Spring | Units | |

36-402 | Advanced Data Analysis | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | 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 /36-226 Introduction to Probability and Statistics I/II 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

Units | ||

21-120 | Differential and Integral Calculus | 10 |

21-122 | Integration, Differential Equations and Approximation | 10 |

21-127 | Concepts of Mathematics | 9 |

21-201 | Undergrad Colloquium | 1 |

21-228 | Discrete Mathematics | 9 |

or | 21-301 Discrete Mathematics (9 units) | |

or | 21-484 Discrete Mathematics (9 units) | |

21-241 | Matrices and Linear Transformations | 10 |

or | 21-242 Matrices and Linear Transformations (10 units) | |

21-259 | Calculus in Three Dimensions | 9 |

or | 21-268 Calculus in Three Dimensions (9 units) | |

or | 21-269 Calculus in Three Dimensions (9 units) | |

21-260 | Differential Equations | 9 |

or | 21-261 Differential Equations (10 units) | |

21-292 | Operations Research I | 9 |

21-369 | Numerical Methods | 9 |

21-393 | Operations Research II | 9 |

94 |

##### Statistics

Units | ||

36-225 | Probability | 9 |

or | 21-325 Probability (9 units) | |

36-226 | Introduction to Statistical Inference | 9 |

36-401 | Modern Regression | 9 |

36-402 | Advanced Data Analysis | 9 |

36-410 | Introduction to Probability Modeling | 9 |

45 |

##### Depth Electives

The detailed curriculum below includes six depth electives. These are to be chosen from among the following including at least one statistics course (21-268 and 21-355 are 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 |

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

36-461 | Topics in Statistics: Statistical Methods in Epidemiology | 9 |

36-462 | Topics in Statistics: Statistical Learning | 9 |

36-463 | Topics in Statistics: Multilevel Hierarchical Models | 9 |

36-464 | Topics in Statistics: Applied Multivariate Methods | 9 |

##### Other Courses

Units | ||

15-110 | Principles of Computing | 10 |

15-122 | Principles of Imperative Computation | 10 |

73-100 | Principles of Economics | 9 |

29 |

##### (114 units, including 73-100 )MCS humanities, social sciences, and science core

four free electives

### Suggested Schedule

##### Freshman Year

Fall | Units | |

21-120 | Differential and Integral Calculus | 10 |

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

15-110 | Principles of Computing | 10 |

03-121 | Modern Biology | 9 |

76-101 | Interpretation and Argument | 9 |

99-101 | Computing @ Carnegie Mellon | 3 |

53 |

Spring | Units | |

21-122 | Integration, Differential Equations and Approximation | 10 |

21-127 | Concepts of Mathematics | 9 |

21-241 | Matrices and Linear Transformations | 10 |

or | 21-242 Matrices and Linear Transformations (10 units) | |

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

41 |

##### Sophomore Year

Fall | Units | |

21-228 | Discrete Mathematics (21-301) | 9 |

21-259 | Calculus in Three Dimensions | 9 |

or | 21-268 Calculus in Three Dimensions (9 units) | |

or | 21-269 Calculus in Three Dimensions (9 units) | |

21-201 | Undergrad Colloquium | 1 |

73-100 | Principles of Economics | 9 |

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

xx-xxx | H&SS Elective | 9 |

47 |

Spring | Units | |

15-122 | Principles of Imperative Computation | 10 |

21-260 | Differential Equations | 9 |

or | 21-261 Differential Equations (10 units) | |

21-292 | Operations Research I | 9 |

21-201 | Undergrad Colloquium | 1 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | Elective | 9 |

47 |

##### Junior Year

Fall | Units | |

21-369 | Numerical Methods | 9 |

36-225 | Probability | 9 |

or | 21-325 Probability (9 units) | |

xx-xxx | Depth Elective | 9 |

xx-xxx | Depth Elective | 9 |

xx-xxx | H&SS Elective | 9 |

45 |

Spring | Units | |

xx-xxx | Depth Elective | 9 |

36-226 | Introduction to Statistical Inference | 9 |

36-410 | Introduction to Probability Modeling | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | 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 | Elective 4 | 9 |

45 |

Spring | Units | |

36-402 | Advanced Data Analysis | 9 |

xx-xxx | Depth Analysis | 9 |

xx-xxx | Depth Analysis | 9 |

xx-xxx | H&SS Elective | 9 |

xx-xxx | 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 (122 units)

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, Differential Equations and Approximation | 10 |

21-127 | Concepts of Mathematics | 9 |

21-201 | Undergrad Colloquium | 1 |

21-300 | Basic Logic | 9 |

21-301 | Discrete Mathematics | 9 |

21-341 | Matrices and Linear Transformations | 9 |

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

21-373 | Algebraic Structures | 9 |

21-484 | Discrete Mathematics | 9 |

116 |

##### Discrete Mathematics and Logic

Three of the following: (27 to 36 units) | ||

21-329 | Set Theory | 9 |

21-374 | Field Theory | 9 |

80-405 | Game Theory | 9 |

80-411 | Proof Theory | 9 |

80-413 | Category Theory | 9 |

21-441 | Number Theory | 9 |

##### Computer Science electives: (18 units)

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.

##### Technical Electives: (36 units)

Any four Mathematical Sciences courses at the 300 level or above, or from the following list: | ||

21-259 | Calculus in Three Dimensions | 9 |

or | 21-268 Calculus in Three Dimensions (9 units) | |

or | 21-269 Calculus in Three Dimensions (9 units) | |

21-260 | Differential Equations | 9 |

or | 21-261 Differential Equations (10 units) | |

21-292 | Operations Research I | 9 |

36-217 | Probability Theory and Random Processes | 9 |

80-405 | Game Theory | 9 |

80-411 | Proof Theory | 9 |

80-413 | Category Theory | 9 |

##### Other Courses:

MCS Humanities, Science and Computer Skills Core: (114 units) Free Electives: (Sufficient to meet the minimum requirement of 360 units.)

### Suggested Schedule

##### Freshman Year

Fall | Units | |

15-122 | Principles of Imperative Computation | 10 |

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 |

44 |

Spring | Units | |

21-122 | Integration, Differential Equations and Approximation | 10 |

21-127 | Concepts of Mathematics | 9 |

21-241 | Matrices and Linear Transformations | 10 |

or | 21-242 Matrices and Linear Transformations (10 units) | |

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

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

51 |

##### Sophomore Year

Fall | Units | |

15-150 | Principles of Functional Programming | 10 |

21-268 | Calculus in Three Dimensions | 10 |

or | 21-269 Calculus in Three Dimensions (9 units) | |

21-301 | Discrete Mathematics | 9 |

21-373 | Algebraic Structures | 9 |

21-201 | Undergrad Colloquium | 1 |

xx-xxx | Humanities Elective | 9 |

48 |

Spring | Units | |

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

xx-xxx | Discrete Math/Logic | 9 |

21-201 | Undergrad Colloquium | 1 |

xx-xxx | Technical Elective | 9 |

03-121 | Modern Biology | 9 |

xx-xxx | Humanities Elective | 9 |

xx-xxx | Humanities Elective | 9 |

58 |

##### Junior Year

Fall | Units | |

15-xxx | Computer Science Elective | 9 |

21-300 | Basic Logic | 9 |

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

xx-xxx | Humanities Elective | 9 |

xx-xxx | Elective | 9 |

45 |

Spring | Units | |

15-xxx | Computer Science Elective | 9 |

21-484 | Discrete Mathematics | 9 |

21-341 | Matrices and Linear Transformations | 9 |

xx-xxx | Humanities Elective | 9 |

xx-xxx | Elective | 9 |

45 |

##### Senior Year

Fall | Units | |

xx-xxx | Discrete Math/Logic | 9 |

xx-xxx | Technical Elective | 9 |

xx-xxx | Humanities Elective | 9 |

xx-xxx | Elective | 9 |

xx-xxx | Elective | 9 |

45 |

Spring | Units | |

xx-xxx | Discrete Math/Logic | 9 |

xx-xxx | Technical Elective | 18 |

xx-xxx | Humanities Elective | 9 |

xx-xxx | 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 requiring significant skills in computation and problem solving. Beginning at the level of quantifying or modeling a problem, students will develop skills in 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: (101 Units)

Units | ||

21-120 | Differential and Integral Calculus | 10 |

21-122 | Integration, Differential Equations and Approximation | 10 |

21-127 | Concepts of Mathematics | 9 |

21-201 | Undergrad Colloquium | 1 |

21-228 | Discrete Mathematics | 9 |

or | 21-301 Discrete Mathematics (9 units) | |

or | 21-484 Discrete Mathematics (9 units) | |

21-241 | Matrices and Linear Transformations | 10 |

or | 21-242 Matrices and Linear Transformations (10 units) | |

21-259 | Calculus in Three Dimensions | 9 |

or | 21-268 Calculus in Three Dimensions (9 units) | |

or | 21-269 Calculus in Three Dimensions (9 units) | |

21-260 | Differential Equations | 9 |

or | 21-261 Differential Equations (10 units) | |

21-320 | Symbolic Programming Methods | 9 |

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

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

21-369 | Numerical Methods | 9 |

103 |

##### Five of the following distribution courses:

(A minimum of 45 units) | ||

15-211 | Fundamental Data Structures and Algorithms | 12 |

21-292 | Operations Research I | 9 |

21-370 | Discrete Time Finance | 9 |

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

21-372 | Partial Differential Equations | 9 |

21-393 | Operations Research II | 9 |

21-476 | Ordinary Differential Equations | 9 |

21-470 | Selected Topics in Analysis ^{*} | 9 |

21-xxx | Mathematical Sciences Elective | |

36-410 | Introduction to Probability Modeling | 9 |

^{* Topics have included the following (student may take more than one):}

^{Calculus of Variations}

^{Finite Difference Equations}

##### Other Courses: (19 units)

Units | ||

15-122 | Principles of Imperative Computation | 10 |

36-225 | Probability | 9 |

or | 21-325 Probability (9 units) | |

19 |

##### MCS humanities, science and computer skills course (114 units)

##### Free electives (sufficient to meet minimum of 360 units)

### Suggested Schedule

##### Freshman Year

Fall | Units | |

21-120 | Differential and Integral Calculus | 10 |

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

15-121 | Principles of Computing | 10 |

76-101 | Interpretation and Argument | 9 |

99-101 | Computing @ Carnegie Mellon | 3 |

44 |

Spring | Units | |

21-122 | Integration, Differential Equations and Approximation | 10 |

21-127 | Concepts of Mathematics | 9 |

21-241 | Matrices and Linear Transformations | 10 |

or | 21-242 Matrices and Linear Transformations (10 units) | |

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

xx-xxx | Humanities Elective | 9 |

50 |

##### Sophomore Year

Fall | Units | |

03-121 | Modern Biology | 9 |

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

21-268 | Calculus in Three Dimensions | 10 |

or | 21-269 Calculus in Three Dimensions (9 units) | |

21-201 | Undergrad Colloquium | 1 |

xx-xxx | Humanities Elective | 9 |

xx-xxx | Elective | 9 |

48 |

Spring | Units | |

21-228 | Discrete Mathematics | 9 |

or | 21-301 Discrete Mathematics (9 units) | |

or | 21-484 Discrete Mathematics (9 units) | |

21-261 | Differential Equations | 10 |

21-201 | Undergrad Colloquium | 1 |

xx-xxx | Distribution Course | 9 |

xx-xxx | Humanities Elective | 9 |

xx-xxx | Elective | 9 |

47 |

##### Junior Year

Fall | Units | |

21-320 | Symbolic Programming Methods | 9 |

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

36-225 | Probability | 9 |

or | 21-325 Probability (9 units) | |

xx-xxx | Humanities Elective | 9 |

xx-xxx | Elective | 9 |

45 |

Spring | Units | |

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

21-369 | Numerical Methods | 9 |

xx-xxx | Distribution Course | 9 |

xx-xxx | Humanities Elective | 9 |

xx-xxx | Elective | 9 |

45 |

##### Senior Year

Fall | Units | |

xx-xxx | Distribution Course | 9 |

xx-xxx | Distribution Course | 9 |

xx-xxx | Humanities Elective | 9 |

xx-xxx | Elective | 9 |

xx-xxx | Elective | 9 |

45 |

Spring | Units | |

xx-xxx | Distribution Course | 9 |

xx-xxx | Humanities Elective | 9 |

xx-xxx | Elective | 9 |

xx-xxx | Elective | 9 |

xx-xxx | 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 Undergrad 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 21-120 /21-122 or equivalent courses. Students planning to include 21-373 Algebraic Structures as a Mathematical Sciences Elective should choose 21-341 Matrices and Linear Transformations. 21-241 or 21-242, and 21-341 cannot both count toward the minor.

21-127 | Concepts of Mathematics | 9 |

21-228 | Discrete Mathematics | 9 |

or | 21-301 Discrete Mathematics (9 units) | |

or | 21-484 Discrete Mathematics (9 units) | |

21-241 | Matrices and Linear Transformations | 10 |

or | 21-242 Matrices and Linear Transformations (10 units) | |

or | 21-341 Matrices and Linear Transformations (9 units) | |

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.

A student who completes the Mathematical Studies sequence plus two recommended electives (typically 21-470 Selected Topics in Analysis and 21-374 Field Theory ) will receive a Minor in Mathematical Sciences. Excluded as acceptable electives are the following: 21-105 , 21-111 , 21-112 , 21-120 , 21-122 , 21-259 or 21-268 or 21-269, and 21-260 or 21-261, and courses intended for H&SS or undergraduate business students, such as 21-110 , 21-256 and 21-257 .

### 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 |

21-301 | Discrete Mathematics | 9 |

21-341 | Matrices and Linear Transformations | 9 |

21-484 | Discrete Mathematics | 9 |

Two of the following: | ||

21-329 | Set Theory | 9 |

21-374 | Field Theory | 9 |

21-441 | Number Theory | 9 |

### 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. In the application process the Department will hold to the same high standards which apply to admission to any graduate program.

##### Honors Program Requirements:

###### 60 unitsFive graduate mathematics courses

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 , 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 .

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

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