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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, 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 at the 21-300 level or above or 21-270 or 21-292, 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 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
21-120Differential and Integral Calculus10
21-122Integration, Differential Equations and Approximation10
21-127Concepts of Mathematics10
21-201Undergraduate Colloquium
(Taken twice in Sophomore year.)
1
21-228Discrete Mathematics9-12
or 15-251 Great Theoretical Ideas in Computer Science
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
36-225Introduction to Probability Theory9
or 21-325 Probability
21-259Calculus in Three Dimensions9-10
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
21-260Differential Equations9-10
or 21-261 Introduction to Ordinary Differential Equations
21-341Linear Algebra9
21-355Principles of Real Analysis I9
21-356Principles of Real Analysis II9
21-373Algebraic Structures9
 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 Statistics (must have at least 36-225 as a prerequisite), or Computer Science (at the 15-200 level or above) 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:

21-301Combinatorics9
21-371Functions of a Complex Variable9
21-372Partial Differential Equations and Fourier Analysis9
21-374Field Theory9
21-441Number Theory9
21-465Topology9
21-467Differential Geometry9
21-470Selected Topics in Analysis9
21-476Introduction to Dynamical Systems9
21-484Graph Theory9
21-600Mathematical Logic I12
21-602Introduction to Set Theory I12
21-603Model Theory I12
21-610Algebra I12
21-620Real Analysis6
21-621Introduction to Lebesgue Integration6
21-630Ordinary Differential Equations12
21-640Introduction to Functional Analysis12
21-651General Topology12
21-660Introduction to Numerical Analysis I12
21-701Discrete Mathematics12
21-720Measure and Integration12
21-721Probability12
21-737Probabilistic Combinatorics12
21-738Extremal Combinatorics12

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
21-120Differential and Integral Calculus10
21-127Concepts of Mathematics10
33-111Physics I for Science Students12
76-101Interpretation and Argument9
99-101Computing @ Carnegie Mellon3
 44


Spring Units
15-110Principles of Computing10
21-122Integration, Differential Equations and Approximation10
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
33-112Physics II for Science Students12
xx-xxxH&SS Elective9
 51
Sophomore Year
Fall Units
03-121Modern Biology9
09-105Introduction to Modern Chemistry I10
21-201Undergraduate Colloquium1
21-228Discrete Mathematics9-12
or 15-251 Great Theoretical Ideas in Computer Science
21-268Multidimensional Calculus10
or 21-269 Vector Analysis
xx-xxxH&SS Elective9
 48-51


Spring Units
21-201Undergraduate Colloquium1
21-261Introduction to Ordinary Differential Equations10
21-373Algebraic Structures9
xx-xxxMathematical Sci, Statistics, or Computer Sci Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 47
Junior Year
Fall Units
21-355Principles of Real Analysis I9
36-225Introduction to Probability Theory9
or 21-325 Probability
xx-xxxMathematical Sci, Statistics, or Computer Sci Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 45


Spring Units
21-341Linear Algebra9
21-356Principles of Real Analysis II9
21-xxxMathematical Sciences Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 45
Senior Year
Fall Units
21-xxxMathematical Sciences Elective9
21-xxxMathematical Sciences Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
 45


Spring Units
21-xxxMathematical Sciences Elective9
21-xxxMathematical Sciences Elective9
xx-xxxMathematical Sci, Statistics, or Computer Sci Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 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
21-120Differential and Integral Calculus10
21-122Integration, Differential Equations and Approximation10
21-127Concepts of Mathematics10
21-201Undergraduate Colloquium
(Taken twice in Sophomore year.)
1
21-228Discrete Mathematics9-12
or 15-251 Great Theoretical Ideas in Computer Science
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
21-259Calculus in Three Dimensions9-10
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
21-260Differential Equations9-10
or 21-261 Introduction to Ordinary Differential Equations
21-292Operations Research I9
21-369Numerical Methods9
21-393Operations Research II9
 95-100
Statistics Courses (required)
Courses Units
36-225Introduction to Probability Theory9
or 21-325 Probability
36-226Introduction to Statistical Inference9
36-401Modern Regression9
36-402Advanced Data Analysis9
36-410Introduction to Probability Modeling9
 45
Economics, Business, and Computer Science Courses (required)
Courses Units
15-110Principles of Computing10
70-122Introduction to Accounting9
73-100Principles of Economics9
73-230Intermediate Microeconomics9
73-240Intermediate Macroeconomics9
 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.

15-122Principles of Imperative Computation10
15-150Principles of Functional Programming10
15-210Parallel and Sequential Data Structures and Algorithms12
21-270Introduction to Mathematical Finance9
21-301Combinatorics9
21-341Linear Algebra9
21-355Principles of Real Analysis I9
21-356Principles of Real Analysis II9
21-365Projects in Applied Mathematics9
21-366Topics in Applied Mathematics9
21-370Discrete Time Finance9
21-373Algebraic Structures9
21-420Continuous-Time Finance9
21-484Graph Theory9
36-461Special Topics9
36-462Topics in Statistics: Data Mining9
36-463Multilevel and Hierarchical Models9
36-464Topics in Statistics: Applied Multivariate Methods9
70-371Operations Management9
70-460Mathematical Models for Consulting9
70-471Supply Chain Management9

 

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
03-121Modern Biology9
21-120Differential and Integral Calculus10
21-127Concepts of Mathematics10
33-111Physics I for Science Students12
76-101Interpretation and Argument9
99-101Computing @ Carnegie Mellon3
 53


Spring Units
15-110Principles of Computing10
21-122Integration, Differential Equations and Approximation10
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
33-112Physics II for Science Students12
xx-xxxH&SS Elective9
 51
Sophomore Year
Fall Units
09-105Introduction to Modern Chemistry I10
21-201Undergraduate Colloquium1
21-228Discrete Mathematics9-12
or 15-251 Great Theoretical Ideas in Computer Science
21-259Calculus in Three Dimensions9-10
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
73-100Principles of Economics9
 38-42


Spring Units
21-260Differential Equations9-10
or 21-261 Introduction to Ordinary Differential Equations
21-292Operations Research I9
21-201Undergraduate Colloquium1
70-122Introduction to Accounting9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 46-47
Junior Year
Fall Units
21-369Numerical Methods9
36-225Introduction to Probability Theory9
or 21-325 Probability
73-230Intermediate Microeconomics9
xx-xxxDepth Elective9
xx-xxxFree Elective9
 45


Spring Units
36-226Introduction to Statistical Inference9
36-410Introduction to Probability Modeling9
73-240Intermediate Macroeconomics9
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
 45
Senior Year
Fall Units
21-393Operations Research II9
36-401Modern Regression9
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 45


Spring Units
36-402Advanced Data Analysis9
xx-xxxDepth Elective9
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 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
21-120Differential and Integral Calculus10
21-122Integration, Differential Equations and Approximation10
21-127Concepts of Mathematics10
21-201Undergraduate Colloquium
(Taken twice in Sophomore year.)
1
21-228Discrete Mathematics9-12
or 15-251 Great Theoretical Ideas in Computer Science
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
21-259Calculus in Three Dimensions9-10
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
21-260Differential Equations9-10
or 21-261 Introduction to Ordinary Differential Equations
21-292Operations Research I9
21-369Numerical Methods9
21-393Operations Research II9
 95-100
Statistics Courses (required)
Courses Units
36-225Introduction to Probability Theory9
or 21-325 Probability
36-226Introduction to Statistical Inference9
36-401Modern Regression9
36-402Advanced Data Analysis9
36-410Introduction to Probability Modeling9
 45
 Economics and Computer Science Courses (required)
Courses Units
15-112Fundamentals of Programming and Computer Science12
15-122Principles of Imperative Computation10
73-100Principles of Economics9
 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.

15-150Principles of Functional Programming10
15-210Parallel and Sequential Data Structures and Algorithms12
21-270Introduction to Mathematical Finance9
21-341Linear Algebra9
21-355Principles of Real Analysis I9
21-356Principles of Real Analysis II9
21-365Projects in Applied Mathematics9
21-366Topics in Applied Mathematics9
21-370Discrete Time Finance9
21-373Algebraic Structures9
21-420Continuous-Time Finance9
21-484Graph Theory9
36-461Special Topics9
36-462Topics in Statistics: Data Mining9
36-463Multilevel and Hierarchical Models9
36-464Topics in Statistics: Applied Multivariate Methods9

 

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
03-121Modern Biology9
21-120Differential and Integral Calculus10
21-127Concepts of Mathematics10
33-111Physics I for Science Students12
76-101Interpretation and Argument9
99-101Computing @ Carnegie Mellon3
 53


Spring Units
15-112Fundamentals of Programming and Computer Science12
21-122Integration, Differential Equations and Approximation10
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
33-112Physics II for Science Students12
 44
Sophomore Year
Fall Units
09-105Introduction to Modern Chemistry I10
21-201Undergraduate Colloquium1
21-228Discrete Mathematics9-12
or 15-251 Great Theoretical Ideas in Computer Science
21-259Calculus in Three Dimensions9-10
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
73-100Principles of Economics9
xx-xxxH&SS Elective9
 47-51


Spring Units
15-122Principles of Imperative Computation10
21-260Differential Equations9-10
or 21-261 Introduction to Ordinary Differential Equations
21-292Operations Research I9
21-201Undergraduate Colloquium1
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 47-48
Junior Year
Fall Units
21-369Numerical Methods9
36-225Introduction to Probability Theory9
or 21-325 Probability
xx-xxxDepth Elective9
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
 45


Spring Units
36-226Introduction to Statistical Inference9
36-410Introduction to Probability Modeling9
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 45
Senior Year
Fall Units
21-393Operations Research II9
36-401Modern Regression9
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective 9
 45


Spring Units
36-402Advanced Data Analysis9
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
 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.

Units
15-122Principles of Imperative Computation10
15-150Principles of Functional Programming10
15-210Parallel and Sequential Data Structures and Algorithms12
21-120Differential and Integral Calculus10
21-122Integration, Differential Equations and Approximation10
21-127Concepts of Mathematics10
21-201Undergraduate Colloquium
(Taken twice in Sophomore year.)
1
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
21-228Discrete Mathematics9-12
or 15-251 Great Theoretical Ideas in Computer Science
21-300Basic Logic9
21-301Combinatorics9
21-341Linear Algebra9
21-355Principles of Real Analysis I9
21-373Algebraic Structures9
 127-130
 Computer Science electives (required)
Any two courses at the 300 level or above. The following are specifically suggested:
15-312Foundations of Programming Languages12
15-451Algorithm Design and Analysis12
15-453Formal Languages, Automata, and Computability9

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)
21-325Probability9
21-329Set Theory9
21-374Field Theory9
21-441Number Theory9
21-484Graph Theory9
21-602Introduction to Set Theory I12
21-603Model Theory I12
21-610Algebra I12
21-700Mathematical Logic II12
80-405Game Theory9
80-411Proof Theory9
80-413Category Theory9

 

List 2 (General Mathematics Electives)
21-259Calculus in Three Dimensions9-10
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
21-260Differential Equations9-10
or 21-261 Introduction to Ordinary Differential Equations
21-292Operations Research I9
21-356Principles of Real Analysis II9
21-371Functions of a Complex Variable9
21-372Partial Differential Equations and Fourier Analysis9
21-393Operations Research II9
21-465Topology9
21-467Differential Geometry9
21-476Introduction to Dynamical Systems9

 

MCS General Education (required)

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

 

Suggested Schedule

Freshman Year
Fall Units
15-112Fundamentals of Programming and Computer Science12
21-120Differential and Integral Calculus10
33-111Physics I for Science Students12
76-101Interpretation and Argument9
99-101Computing @ Carnegie Mellon3
 46


Spring Units
15-122Principles of Imperative Computation10
21-122Integration, Differential Equations and Approximation10
21-127Concepts of Mathematics10
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
33-112Physics II for Science Students12
 52
Sophomore Year
Fall Units
15-150Principles of Functional Programming10
21-268Multidimensional Calculus10
or 21-269 Vector Analysis
21-301Combinatorics9
21-373Algebraic Structures9
21-201Undergraduate Colloquium1
xx-xxxH&SS Elective9
 48


Spring Units
03-121Modern Biology9
09-105Introduction to Modern Chemistry I10
15-210Parallel and Sequential Data Structures and Algorithms12
xx-xxxDiscrete Math/Logic Elective9
21-201Undergraduate Colloquium1
xx-xxxMathematics Elective9
xx-xxxH&SS Elective9
 59
Junior Year
Fall Units
15-xxxComputer Science Elective9
21-300Basic Logic9
21-355Principles of Real Analysis I9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 45


Spring Units
15-xxxComputer Science Elective9
21-341Linear Algebra9
xx-xxxH&SS Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 45
Senior Year
Fall Units
xx-xxxDiscrete Math/Logic Elective9
xx-xxxMathematics Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
 45


Spring Units
xx-xxxDiscrete Math/Logic Elective9
xx-xxxMathematics Elective9
xx-xxxMathematics Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 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)

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
21-120Differential and Integral Calculus10
21-122Integration, Differential Equations and Approximation10
21-127Concepts of Mathematics10
21-201Undergraduate Colloquium
(Taken twice in Sophomore year.)
1
21-228Discrete Mathematics9-12
or 15-251 Great Theoretical Ideas in Computer Science
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
21-259Calculus in Three Dimensions9-10
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
21-260Differential Equations9-10
or 21-261 Introduction to Ordinary Differential Equations
21-320Symbolic Programming Methods9
36-225Introduction to Probability Theory9
or 21-325 Probability
21-355Principles of Real Analysis I9
21-356Principles of Real Analysis II9
21-369Numerical Methods9
 113-118
 Computer Science Courses (required)
Courses Units
15-122Principles of Imperative Computation10

 

Depth Electives (required)

Students must take 45 units from the list below:

15-214Principles of Software Construction: Objects, Design, and Concurrency12
21-292Operations Research I9
21-365Projects in Applied Mathematics9
21-366Topics in Applied Mathematics9
21-370Discrete Time Finance9
21-371Functions of a Complex Variable9
21-372Partial Differential Equations and Fourier Analysis9
21-393Operations Research II9
21-420Continuous-Time Finance9
21-467Differential Geometry9
21-470Selected Topics in Analysis9
21-476Introduction to Dynamical Systems9
21-499Undergraduate Research Topic9
21-599Undergraduate Reading and ResearchVar.
21-620Real Analysis6
21-621Introduction to Lebesgue Integration6
21-630Ordinary Differential Equations12
21-640Introduction to Functional Analysis12
21-651General Topology12
21-660Introduction to Numerical Analysis I12
21-690Methods of Optimization12
21-720Measure and Integration12
21-721Probability12
21-732Partial Differential Equations I12
36-410Introduction to Probability Modeling9

 

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
09-101Introduction to Experimental Chemistry3
21-120Differential and Integral Calculus10
21-126Introduction to Mathematical Software3
21-127Concepts of Mathematics10
33-111Physics I for Science Students12
76-101Interpretation and Argument9
 47

 

Spring Units
21-122Integration, Differential Equations and Approximation10
21-228Discrete Mathematics9
21-241Matrices and Linear Transformations10
33-112Physics II for Science Students12
xx-xxxH&SS Elective9
 50
Sophomore Year
Fall Units
09-105Introduction to Modern Chemistry I10
15-112Fundamentals of Programming and Computer Science12
21-268Multidimensional Calculus10
or 21-269 Vector Analysis
21-201Undergraduate Colloquium1
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 51


Spring Units
03-121Modern Biology9
15-122Principles of Imperative Computation10
21-261Introduction to Ordinary Differential Equations10
21-201Undergraduate Colloquium1
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
 48
Junior Year
Fall Units
21-320Symbolic Programming Methods9
21-325Probability9
21-355Principles of Real Analysis I9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 45


Spring Units
21-356Principles of Real Analysis II9
21-369Numerical Methods9
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 45
Senior Year
Fall Units
xx-xxxDepth Elective9
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
 45


Spring Units
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
 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 21-120 /21-122 and 21-127 or equivalent courses. 

 

21-127Concepts of Mathematics10
21-228Discrete Mathematics9-12
or 15-251 Great Theoretical Ideas in Computer Science
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
21-355Principles of Real Analysis I9
21-3xxMathematical Sciences Elective
21-3xxMathematical 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:

21-300Basic Logic9
21-301Combinatorics9


Four of the following:
21-329Set Theory9
21-341Linear Algebra9
21-373Algebraic Structures9
21-374Field Theory9
21-441Number Theory9
21-484Graph Theory9
21-602Introduction to Set Theory I12
21-603Model Theory I12
21-610Algebra I12
21-700Mathematical Logic II12

 

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.

The core undergraduate honors courses are:

Freshman Year
Fall Units
21-242Matrix Theory
(Honors version of 21-241 Matrices and Linear Transformations)
10

 

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

 

Sophomore Year
Fall
21-235Mathematical Studies Analysis I
(Honors version of 21-355 Principles of Real Analysis I)
10
21-237Mathematical Studies Algebra I
(Honors version of 21-373 Algebraic Structures)
10

 

Spring
21-236Mathematical Studies Analysis II
(Honors version of 21-356 Principles of Real Analysis II)
10
21-238Mathematical Studies Algebra II
(Honors version of 21-341 Linear Algebra)
10

 

Honors Program Requirements:
21-901Masters Degree ResearchVar.
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.

Faculty

PETER B. ANDREWS, Emeritus – 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–.THOMAS BOHMAN, Professor – Ph.D., Rutgers University; Carnegie Mellon, 1998–.DEBORAH BRANDON, Associate Teaching Professor – Ph.D., Carnegie Mellon University; Carnegie Mellon, 1991–.BORIS BUKH, Assistant Professor – Ph.D., Princeton University; Carnegie Mellon, 2012–.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–.DAVID HANDRON, Associate Teaching Professor – Ph.D., Rice University; Carnegie Mellon, 1999–.WILLIAM J. HRUSA, Professor – Ph.D., Brown University; Carnegie Mellon, 1982–.JOSE IOVINO, Visiting Associate Professor – Ph.D., University of Illinois; Carnegie Mellon, 2012–.GAUTAM IYER, Assistant Professor – Ph.D., University of Chicago; Carnegie Mellon, 2009–.GREGORY JOHNSON, Assistant Teaching Professor – Ph.D., University of Maryland; Carnegie Mellon, 2009–.DAVID KINDERLEHRER, Professor – Ph.D., University of California at Berkeley; Carnegie Mellon, 1990–.ALEXEI KOLESNIKOV, Shelly Visiting Professor – Ph.D., Carnegie Mellon University; Carnegie Mellon, 2013–.NATASHA KOMAROV, Visiting Assistant Professor – Ph.D., Dartmouth University; Carnegie Mellon, 2013–.DMITRY KRAMKOV, Professor – Ph.D., Steklov Mathematical Institute; Carnegie Mellon, 2000–.KASPER LARSEN, Associate 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–.WESLEY PEGDEN, Assistant Professor – Ph.D., Rutgers University; Carnegie Mellon, 2013–.ROBERT L. PEGO, Professor – Ph.D., University of California at Berkeley; Carnegie Mellon, 2004–.MICHAEL PICOLLELLI, Shelly Visiting Assistant Professor – Ph.D., Carnegie Mellon ; Carnegie Mellon, 2012–.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–.MATTHEW SZUDZIK, Assistant Teaching Professor – Ph.D., Carnegie Mellon; Carnegie Mellon, 2011–.SHLOMO TA'ASAN, Professor – Ph.D., Weizmann Institute; Carnegie Mellon, 1994–.LUC TARTAR, Emeritus – Ph.D., University of Paris; Carnegie Mellon, 1987–.IAN TICE, Assistant Professor – Ph.D., New York University; Carnegie Mellon, 2012–.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, Emeritus – 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–.THOMAS BOHMAN, Professor – Ph.D., Rutgers University; Carnegie Mellon, 1998–.DEBORAH BRANDON, Associate Teaching Professor – Ph.D., Carnegie Mellon University; Carnegie Mellon, 1991–.BORIS BUKH, Assistant Professor – Ph.D., Princeton University; Carnegie Mellon, 2012–.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–.DAVID HANDRON, Associate Teaching Professor – Ph.D., Rice University; Carnegie Mellon, 1999–.WILLIAM J. HRUSA, Professor – Ph.D., Brown University; Carnegie Mellon, 1982–.JOSE IOVINO, Visiting Associate Professor – Ph.D., University of Illinois; Carnegie Mellon, 2012–.GAUTAM IYER, Assistant Professor – Ph.D., University of Chicago; Carnegie Mellon, 2009–.GREGORY JOHNSON, Assistant Teaching Professor – Ph.D., University of Maryland; Carnegie Mellon, 2009–.DAVID KINDERLEHRER, Professor – Ph.D., University of California at Berkeley; Carnegie Mellon, 1990–.ALEXEI KOLESNIKOV, Shelly Visiting Professor – Ph.D., Carnegie Mellon University; Carnegie Mellon, 2013–.NATASHA KOMAROV, Visiting Assistant Professor – Ph.D., Dartmouth University; Carnegie Mellon, 2013–.DMITRY KRAMKOV, Professor – Ph.D., Steklov Mathematical Institute; Carnegie Mellon, 2000–.KASPER LARSEN, Associate 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–.WESLEY PEGDEN, Assistant Professor – Ph.D., Rutgers University; Carnegie Mellon, 2013–.ROBERT L. PEGO, Professor – Ph.D., University of California at Berkeley; Carnegie Mellon, 2004–.MICHAEL PICOLLELLI, Shelly Visiting Assistant Professor – Ph.D., Carnegie Mellon ; Carnegie Mellon, 2012–.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–.MATTHEW SZUDZIK, Assistant Teaching Professor – Ph.D., Carnegie Mellon; Carnegie Mellon, 2011–.SHLOMO TA'ASAN, Professor – Ph.D., Weizmann Institute; Carnegie Mellon, 1994–.LUC TARTAR, Emeritus – Ph.D., University of Paris; Carnegie Mellon, 1987–.IAN TICE, Assistant Professor – Ph.D., New York University; Carnegie Mellon, 2012–.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–.