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This is an archived copy of the 2011-2012 Catalog. To access the most recent version of the catalog, please visit http://coursecatalog.web.cmu.edu.

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
or21-301 Discrete Mathematics (9 units)
or21-484 Discrete Mathematics (9 units)
21-241 Matrices and Linear Transformations 10
or21-242 Matrices and Linear Transformations (10 units)
21-341 Matrices and Linear Transformations 9
21-259 Calculus in Three Dimensions 9
or21-268 Calculus in Three Dimensions (9 units)
or21-269 Calculus in Three Dimensions (9 units)
21-260 Differential Equations 9
or21-261 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
or21-325 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
FallUnits
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


SpringUnits
21-122 Integration, Differential Equations and Approximation 10
21-127 Concepts of Mathematics 9
21-241 Matrices and Linear Transformations 10
or21-242 Matrices and Linear Transformations (10 units)
33-112 Physics II for Science Students 12
xx-xxx H&SS Elective 9
  50
Sophomore Year
FallUnits
21-228 Discrete Mathematics 9
or21-301 Discrete Mathematics (9 units)
or21-484 Discrete Mathematics (9 units)
21-268 Calculus in Three Dimensions 10
or21-269 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


SpringUnits
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
FallUnits
21-355 Principles of Real Analysis I 9
36-225 Probability 9
or21-325 Probability (9 units)
xx-xxx Mathematical Sci, Statistics, or Computer Sci Elective 9
xx-xxx H&SS Elective 9
xx-xxx Elective 9
  45


SpringUnits
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
FallUnits
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


SpringUnits
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
or21-301 Discrete Mathematics (9 units)
or21-484 Discrete Mathematics (9 units)
21-241 Matrices and Linear Transformations 10
or21-242 Matrices and Linear Transformations (10 units)
21-259 Calculus in Three Dimensions 9
or21-268 Calculus in Three Dimensions (9 units)
or21-269 Calculus in Three Dimensions (9 units)
21-260 Differential Equations 9
or21-261 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
or21-325 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
FallUnits
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


SpringUnits
21-122 Integration, Differential Equations and Approximation 10
21-127 Concepts of Mathematics 9
21-241 Matrices and Linear Transformations 10
or21-242 Matrices and Linear Transformations (10 units)
33-112 Physics II for Science Students 12
xx-xxx H&SS Elective 9
  50
Sophomore Year
FallUnits
21-228 Discrete Mathematics 9
or21-301 Discrete Mathematics (9 units)
or21-484 Discrete Mathematics (9 units)
21-259 Calculus in Three Dimensions 9
or21-268 Calculus in Three Dimensions (9 units)
or21-269 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


SpringUnits
21-260 Differential Equations 9
or21-261 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
FallUnits
21-369 Numerical Methods 9
xx-xxx Depth Elective 9
36-225 Probability 9
or21-325 Probability (9 units)
73-150 Intermediate Microeconomics 9
xx-xxx Elective 9
  45


SpringUnits
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
FallUnits
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


SpringUnits
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
or21-301 Discrete Mathematics (9 units)
or21-484 Discrete Mathematics (9 units)
21-241 Matrices and Linear Transformations 10
or21-242 Matrices and Linear Transformations (10 units)
21-259 Calculus in Three Dimensions 9
or21-268 Calculus in Three Dimensions (9 units)
or21-269 Calculus in Three Dimensions (9 units)
21-260 Differential Equations 9
or21-261 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
or21-325 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
FallUnits
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


SpringUnits
21-122 Integration, Differential Equations and Approximation 10
21-127 Concepts of Mathematics 9
21-241 Matrices and Linear Transformations 10
or21-242 Matrices and Linear Transformations (10 units)
33-112 Physics II for Science Students 12
  41
Sophomore Year
FallUnits
21-228 Discrete Mathematics (21-301) 9
21-259 Calculus in Three Dimensions 9
or21-268 Calculus in Three Dimensions (9 units)
or21-269 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


SpringUnits
15-122 Principles of Imperative Computation 10
21-260 Differential Equations 9
or21-261 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
FallUnits
21-369 Numerical Methods 9
36-225 Probability 9
or21-325 Probability (9 units)
xx-xxx Depth Elective 9
xx-xxx Depth Elective 9
xx-xxx H&SS Elective 9
  45


SpringUnits
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
FallUnits
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


SpringUnits
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
or21-268 Calculus in Three Dimensions (9 units)
or21-269 Calculus in Three Dimensions (9 units)
21-260 Differential Equations 9
or21-261 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
FallUnits
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


SpringUnits
21-122 Integration, Differential Equations and Approximation 10
21-127 Concepts of Mathematics 9
21-241 Matrices and Linear Transformations 10
or21-242 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
FallUnits
15-150 Principles of Functional Programming 10
21-268 Calculus in Three Dimensions 10
or21-269 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


SpringUnits
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
FallUnits
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


SpringUnits
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
FallUnits
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


SpringUnits
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
or21-301 Discrete Mathematics (9 units)
or21-484 Discrete Mathematics (9 units)
21-241 Matrices and Linear Transformations 10
or21-242 Matrices and Linear Transformations (10 units)
21-259 Calculus in Three Dimensions 9
or21-268 Calculus in Three Dimensions (9 units)
or21-269 Calculus in Three Dimensions (9 units)
21-260 Differential Equations 9
or21-261 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
or21-325 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
FallUnits
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


SpringUnits
21-122 Integration, Differential Equations and Approximation 10
21-127 Concepts of Mathematics 9
21-241 Matrices and Linear Transformations 10
or21-242 Matrices and Linear Transformations (10 units)
33-112 Physics II for Science Students 12
xx-xxx Humanities Elective 9
  50
Sophomore Year
FallUnits
03-121 Modern Biology 9
09-105 Introduction to Modern Chemistry I 10
21-268 Calculus in Three Dimensions 10
or21-269 Calculus in Three Dimensions (9 units)
21-201 Undergrad Colloquium 1
xx-xxx Humanities Elective 9
xx-xxx Elective 9
  48


SpringUnits
21-228 Discrete Mathematics 9
or21-301 Discrete Mathematics (9 units)
or21-484 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
FallUnits
21-320 Symbolic Programming Methods 9
21-355 Principles of Real Analysis I 9
36-225 Probability 9
or21-325 Probability (9 units)
xx-xxx Humanities Elective 9
xx-xxx Elective 9
  45


SpringUnits
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
FallUnits
xx-xxx Distribution Course 9
xx-xxx Distribution Course 9
xx-xxx Humanities Elective 9
xx-xxx Elective 9
xx-xxx Elective 9
  45


SpringUnits
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
or21-301 Discrete Mathematics (9 units)
or21-484 Discrete Mathematics (9 units)
21-241 Matrices and Linear Transformations 10
or21-242 Matrices and Linear Transformations (10 units)
or21-341 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.

 

Faculty

PETER B. ANDREWS, Professor – Ph.D., Princeton University; Carnegie Mellon, 1963–.JEREMY AVIGAD, Professor – Ph.D., University of California, Berkeley; Carnegie Mellon, 1996–.EGON BALAS, University Professor – Ph.D., University of Brussels; Carnegie Mellon, 1968–.ALBERT A. BLANK, Emeritus – Ph.D., New York University; Carnegie Mellon, 1969–.MANUEL BLUM, University Professor – Ph.D., Massachusetts Institute of Technology; Carnegie Mellon, 1999–.TOM BOHMAN, Professor – Ph.D., Rutgers University; Carnegie Mellon, 1998–.DEBORAH BRANDON, Associate Teaching Professor – Ph.D., Carnegie Mellon University; Carnegie Mellon, 1991–.CHARLES V. COFFMAN, Emeritus – Ph.D., Johns Hopkins University; Carnegie Mellon, 1962–.GERARD CORNUEJOLS, University Professor – Ph.D., Cornell University; Carnegie Mellon, 1978–.JAMES CUMMINGS, Professor – Ph.D., Cambridge University; Carnegie Mellon, 1996–.HASAN DEMIRKOPARAN, Assistant Teaching Professor – Ph.D., Michigan State University; Carnegie Mellon, 2005–.TIMOTHY FLAHERTY, Assistant Teaching Professor – Ph.D., University of Pittsburgh,; Carnegie Mellon, 1999–.IRENE M. FONSECA, Professor – Ph.D., University of Minnesota; Carnegie Mellon, 1987–.ALAN M. FRIEZE, Professor – Ph.D., University of London; Carnegie Mellon, 1987–.IRINA GHEORGHICIUC, Assistant Teaching Professor – Ph.D., University of Pennsylvania; Carnegie Mellon, 2007–.JAMES M. GREENBERG, Emeritus – Ph.D., Brown University; Carnegie Mellon, 1995–.RAMI GROSSBERG, Professor – Ph.D., Hebrew University of Jerusalem; Carnegie Mellon, 1988–.MORTON E. GURTIN, Emeritus – Ph.D., Brown University; Carnegie Mellon, 1966–.SIMI HABER, Nehari Visiting Assisant Professor – Ph.D., Tel Aviv University; Carnegie Mellon, 2011–.DAVID HANDRON, Associate Teaching Professor – Ph.D., Rice University; Carnegie Mellon, 1999–.WILLIAM J. HRUSA, Professor – Ph.D., Brown University; Carnegie Mellon, 1982–.GAUTAM IYER, Assistant Professor – Ph.D., University of Chicago; Carnegie Mellon, 2009–.GREGORY JOHNSON, Visiting Assistant Professor – Ph.D., University of Maryland; Carnegie Mellon, 2009–.DAVID KINDERLEHRER, Professor – Ph.D., University of California at Berkeley; Carnegie Mellon, 1990–.DMITRY KRAMKOV, Professor – Ph.D., Steklov Mathematical Institute; Carnegie Mellon, 2000–.KASPER LARSEN, Assistant Professor – Ph.D., University of Southern Denmark; Carnegie Mellon, 2007–.JOHN P. LEHOCZKY, Professor – Ph.D., Stanford University; Carnegie Mellon, 1969–.GIOVANNI LEONI, Professor – Ph.D., University of Minnestota; Carnegie Mellon, 2002–.PO-SHEN LOH, Assistant Professor – Ph.D., Princeton University; Carnegie Mellon, 2009–.JOHN MACKEY, Teaching Professor – Ph.D., University of Hawaii; Carnegie Mellon, 2003–.DANIELA MIHAI, Assistant Teaching Professor – Ph.D., University of Pittsburgh; Carnegie Mellon, 2007–.RICHARD A. MOORE, Emeritus – Ph.D., Washington University; Carnegie Mellon, 1956–.ROY A. NICOLAIDES, Professor – Ph.D., University of London; Carnegie Mellon, 1984–.WALTER NOLL, Emeritus – Ph.D., Indiana University; Carnegie Mellon, 1956–.MARION L. OLIVER, Teaching Professor – Ph.D., Carnegie Mellon University; Carnegie Mellon, 2004–.DAVID R. OWEN, Emeritus – Ph.D., Brown University; Carnegie Mellon, 1967–.ROBERT L. PEGO, Professor – Ph.D., University of California at Berkeley; Carnegie Mellon, 2004–.AGOSTON PISZTORA, Associate Professor – Ph.D., ETH Zurich,; Carnegie Mellon, 1996–.SCOTT ROBERTSON, Assistant Professor – Ph.D., Boston University; Carnegie Mellon, 2011–.JOHN W. SCHAEFFER, Professor – Ph.D., Indiana University; Carnegie Mellon, 1983–.JUAN J. SCHäFFER, Professor – Ph.D., Universitt Zrich; Carnegie Mellon, 1968–.ERNEST SCHIMMERLING, Professor – Ph.D., University of California at Los Angeles; Carnegie Mellon, 1998–.DANA SCOTT, Emeritus – Ph.D., Princeton University; Carnegie Mellon, 1981–.ROBERT F. SEKERKA, University Professor – Ph.D., Harvard University; Carnegie Mellon, 1969–.STEVEN E. SHREVE, Professor – Ph.D., University of Illinois; Carnegie Mellon, 1980–.DEJAN SLEPCEV, Associate Professor – Ph.D., University of Texas at Austin; Carnegie Mellon, 2006–.RICHARD STATMAN, Professor – Ph.D., Stanford University; Carnegie Mellon, 1984–.SHLOMO TA'ASAN, Professor – Ph.D., Weizmann Institute; Carnegie Mellon, 1994–.LUC TARTAR, University Professor – Ph.D., University of Paris; Carnegie Mellon, 1987–.RUSSELL C. WALKER, Teaching Professor – D.A., Carnegie Mellon University ; Carnegie Mellon, 1984–.NOEL S. WALKINGTON, Professor – Ph.D., University of Texas at Austin; Carnegie Mellon, 1989–.WILLIAM O. WILLIAMS, Emeritus – Ph.D., Brown University; Carnegie Mellon, 1966–.

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Faculty

PETER B. ANDREWS, Professor – Ph.D., Princeton University; Carnegie Mellon, 1963–.JEREMY AVIGAD, Professor – Ph.D., University of California, Berkeley; Carnegie Mellon, 1996–.EGON BALAS, University Professor – Ph.D., University of Brussels; Carnegie Mellon, 1968–.ALBERT A. BLANK, Emeritus – Ph.D., New York University; Carnegie Mellon, 1969–.MANUEL BLUM, University Professor – Ph.D., Massachusetts Institute of Technology; Carnegie Mellon, 1999–.TOM BOHMAN, Professor – Ph.D., Rutgers University; Carnegie Mellon, 1998–.DEBORAH BRANDON, Associate Teaching Professor – Ph.D., Carnegie Mellon University; Carnegie Mellon, 1991–.CHARLES V. COFFMAN, Emeritus – Ph.D., Johns Hopkins University; Carnegie Mellon, 1962–.GERARD CORNUEJOLS, University Professor – Ph.D., Cornell University; Carnegie Mellon, 1978–.JAMES CUMMINGS, Professor – Ph.D., Cambridge University; Carnegie Mellon, 1996–.HASAN DEMIRKOPARAN, Assistant Teaching Professor – Ph.D., Michigan State University; Carnegie Mellon, 2005–.TIMOTHY FLAHERTY, Assistant Teaching Professor – Ph.D., University of Pittsburgh,; Carnegie Mellon, 1999–.IRENE M. FONSECA, Professor – Ph.D., University of Minnesota; Carnegie Mellon, 1987–.ALAN M. FRIEZE, Professor – Ph.D., University of London; Carnegie Mellon, 1987–.IRINA GHEORGHICIUC, Assistant Teaching Professor – Ph.D., University of Pennsylvania; Carnegie Mellon, 2007–.JAMES M. GREENBERG, Emeritus – Ph.D., Brown University; Carnegie Mellon, 1995–.RAMI GROSSBERG, Professor – Ph.D., Hebrew University of Jerusalem; Carnegie Mellon, 1988–.MORTON E. GURTIN, Emeritus – Ph.D., Brown University; Carnegie Mellon, 1966–.SIMI HABER, Nehari Visiting Assisant Professor – Ph.D., Tel Aviv University; Carnegie Mellon, 2011–.DAVID HANDRON, Associate Teaching Professor – Ph.D., Rice University; Carnegie Mellon, 1999–.WILLIAM J. HRUSA, Professor – Ph.D., Brown University; Carnegie Mellon, 1982–.GAUTAM IYER, Assistant Professor – Ph.D., University of Chicago; Carnegie Mellon, 2009–.GREGORY JOHNSON, Visiting Assistant Professor – Ph.D., University of Maryland; Carnegie Mellon, 2009–.DAVID KINDERLEHRER, Professor – Ph.D., University of California at Berkeley; Carnegie Mellon, 1990–.DMITRY KRAMKOV, Professor – Ph.D., Steklov Mathematical Institute; Carnegie Mellon, 2000–.KASPER LARSEN, Assistant Professor – Ph.D., University of Southern Denmark; Carnegie Mellon, 2007–.JOHN P. LEHOCZKY, Professor – Ph.D., Stanford University; Carnegie Mellon, 1969–.GIOVANNI LEONI, Professor – Ph.D., University of Minnestota; Carnegie Mellon, 2002–.PO-SHEN LOH, Assistant Professor – Ph.D., Princeton University; Carnegie Mellon, 2009–.JOHN MACKEY, Teaching Professor – Ph.D., University of Hawaii; Carnegie Mellon, 2003–.DANIELA MIHAI, Assistant Teaching Professor – Ph.D., University of Pittsburgh; Carnegie Mellon, 2007–.RICHARD A. MOORE, Emeritus – Ph.D., Washington University; Carnegie Mellon, 1956–.ROY A. NICOLAIDES, Professor – Ph.D., University of London; Carnegie Mellon, 1984–.WALTER NOLL, Emeritus – Ph.D., Indiana University; Carnegie Mellon, 1956–.MARION L. OLIVER, Teaching Professor – Ph.D., Carnegie Mellon University; Carnegie Mellon, 2004–.DAVID R. OWEN, Emeritus – Ph.D., Brown University; Carnegie Mellon, 1967–.ROBERT L. PEGO, Professor – Ph.D., University of California at Berkeley; Carnegie Mellon, 2004–.AGOSTON PISZTORA, Associate Professor – Ph.D., ETH Zurich,; Carnegie Mellon, 1996–.SCOTT ROBERTSON, Assistant Professor – Ph.D., Boston University; Carnegie Mellon, 2011–.JOHN W. SCHAEFFER, Professor – Ph.D., Indiana University; Carnegie Mellon, 1983–.JUAN J. SCHäFFER, Professor – Ph.D., Universitt Zrich; Carnegie Mellon, 1968–.ERNEST SCHIMMERLING, Professor – Ph.D., University of California at Los Angeles; Carnegie Mellon, 1998–.DANA SCOTT, Emeritus – Ph.D., Princeton University; Carnegie Mellon, 1981–.ROBERT F. SEKERKA, University Professor – Ph.D., Harvard University; Carnegie Mellon, 1969–.STEVEN E. SHREVE, Professor – Ph.D., University of Illinois; Carnegie Mellon, 1980–.DEJAN SLEPCEV, Associate Professor – Ph.D., University of Texas at Austin; Carnegie Mellon, 2006–.RICHARD STATMAN, Professor – Ph.D., Stanford University; Carnegie Mellon, 1984–.SHLOMO TA'ASAN, Professor – Ph.D., Weizmann Institute; Carnegie Mellon, 1994–.LUC TARTAR, University Professor – Ph.D., University of Paris; Carnegie Mellon, 1987–.RUSSELL C. WALKER, Teaching Professor – D.A., Carnegie Mellon University ; Carnegie Mellon, 1984–.NOEL S. WALKINGTON, Professor – Ph.D., University of Texas at Austin; Carnegie Mellon, 1989–.WILLIAM O. WILLIAMS, Emeritus – Ph.D., Brown University; Carnegie Mellon, 1966–.