Thomas Bohman, Department Head
William J. Hrusa, Associate Head
Jason Howell, Director of Undergraduate Studies
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.  Concentrations within the degree include Mathematical Sciences, Operations Research and Statistics, Statistics, Discrete Mathematics and Logic, and Computational and Applied Mathematics.

The Mathematical Sciences concentration 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 21-235 Mathematical Studies Analysis I/21-236 Mathematical Studies Analysis II and 21-237 Mathematical Studies Algebra I/21-237 Mathematical Studies Algebra I.  These courses provide excellent preparation for graduate study, with many of the participants taking graduate courses as early as their Junior year.  The typical enrollment of about 15 students allows for close contact with faculty.  Admission to Mathematical Studies is by invitation, and interested students should apply during the Spring of their Freshman year.

Honors Degree Program

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

Interdisciplinary Programs

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

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

Curricula

For each concentration, we provide a list of the requirements and a suggested schedule that takes prerequisites into account. A Mathematical Sciences, Computer Science, Physics, Statistics Elective refers to any course from the Departments of Mathematical Sciences, Computer Science, Physics, or Statistics and Data Science, respectively, satisfying the following restrictions: a mathematical sciences course must be at the 21-300 level or above or 21-270 or 21-272 or 21-292, a  computer science course must be at the 15-200 level or above, a physics course must be at the 33-300 level or above, and a statistics course must be at the 36-300 level or above and have at least 36-225 as a prerequisite.

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

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

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

Mathematical Sciences Concentration

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 and Approximation10
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
21-228Discrete Mathematics9
or 15-251 Great Ideas in Theoretical 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
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
21-260Differential Equations9
or 21-261 Introduction to Ordinary Differential Equations
or 33-231 Physical Analysis
21-341Linear Algebra9
21-355Principles of Real Analysis I9
21-356Principles of Real Analysis II9
21-373Algebraic Structures9
 112

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

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

MCS General Education (required)

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

Mathematical Sciences Electives for Students Intending Graduate Studies

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

Courses Units
21-272Introduction to Partial Differential Equations9
21-301Combinatorics9
21-360Differential Geometry of Curves and Surfaces9
21-371Functions of a Complex Variable9
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
21-632Introduction to 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
21-723Advanced Real Analysis12
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 for students without AP credit

Freshman Year
Fall Units
21-120Differential and Integral Calculus10
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
38-101First-year Seminar6
76-101Interpretation and Argument9
99-101Computing @ Carnegie Mellon3
xx-xxxLife/Physical Sciences Course9
 47
Spring Units
15-110Principles of Computing10
or 15-112 Fundamentals of Programming and Computer Science
21-122Integration and Approximation10
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
xx-xxxPhysical/Life Sciences Course9
xx-xxxH&SS Elective9
 48
Sophomore Year
Fall Units
21-228Discrete Mathematics9
or 15-251 Great Ideas in Theoretical Computer Science
21-268Multidimensional Calculus10
xx-xxxSTEM Course9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 46
Spring Units
21-261Introduction to Ordinary Differential Equations10
21-373Algebraic Structures9
xx-xxxMathematical Sci, Statistics, or Computer Sci Elective9
xx-xxxSTEM Course9
xx-xxxH&SS Elective9
 46
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-xxxCultural/Global Understanding Elective9
xx-xxxFree Elective9
 45
Senior Year
Fall Units
21-xxxMathematical Sciences Elective9
21-xxxMathematical Sciences Elective9
xx-xxxFree 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-xxxFree Elective9
xx-xxxFree Elective9
 45

Minimum number of units required for degree:360

Suggested Schedule for Students with AP Credit

Freshman Year
Fall Units
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
38-101First-year Seminar6
76-101Interpretation and Argument9
99-101Computing @ Carnegie Mellon3
xx-xxxLife/Physical Sciences Course9
 47
Spring Units
15-110Principles of Computing10
or 15-112 Fundamentals of Programming and Computer Science
21-228Discrete Mathematics9
or 15-251 Great Ideas in Theoretical Computer Science
21-268Multidimensional Calculus10
or 21-269 Vector Analysis
xx-xxxH&SS Elective9
 38
Sophomore Year
Fall Units
36-225Introduction to Probability Theory9
or 21-325 Probability
xx-xxxMathematical Sci, Statistics, or Computer Sci Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 36
Spring Units
21-261Introduction to Ordinary Differential Equations10
21-355Principles of Real Analysis I9
xx-xxxSTEM Course9
xx-xxxH&SS Elective9
 37
Junior Year
Fall Units
21-356Principles of Real Analysis II9
21-xxxMathematical Sciences Elective9
xx-xxxCultural/Global Understanding Course9
xx-xxxFree Elective9
 36
Spring Units
21-341Linear Algebra9
xx-xxxMathematical Sci, Statistics, or Computer Sci Elective9
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-xxxFree 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-xxxFree Elective9
xx-xxxFree Elective9
 45

Minimum number of units required for degree:360

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 and Approximation10
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
21-228Discrete Mathematics9
or 15-251 Great Ideas in Theoretical Computer Science
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
21-259Calculus in Three Dimensions9
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
21-260Differential Equations9
or 21-261 Introduction to Ordinary Differential Equations
or 33-231 Physical Analysis
21-292Operations Research I9
21-369Numerical Methods9
Starting Fall 2018, 21-369 will be 12 units with recitation.
21-393Operations Research II9
 94

Statistics Courses (required)

Courses Units
36-225Introduction to Probability Theory9
or 21-325 Probability
36-226Introduction to Statistical Inference9
36-401Modern Regression9
36-402Advanced Methods for 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
73-102Principles of Microeconomics9
73-103Principles of Macroeconomics9
73-230Intermediate Microeconomics9
or 73-240 Intermediate Macroeconomics
 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.

Courses Units
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
21-377Monte Carlo Simulation for Finance9
21-378Mathematics of Fixed Income Markets9
21-420Continuous-Time Finance9
21-484Graph Theory9
36-461Special Topics: Statistical Methods in Epidemiology9
36-462Special Topics: Data Mining9
36-463Special Topics: Multilevel and Hierarchical Models9
36-464Special Topics: 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 73-102, 73-230, and 73-240 satisfy requirements from the MCS general education core.
 

Suggested Schedule

Freshman Year
Fall Units
21-120Differential and Integral Calculus10
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
38-101First-year Seminar6
76-101Interpretation and Argument9
99-101Computing @ Carnegie Mellon3
xx-xxxLife/Physical Sciences Course9
 47
Spring Units
15-110Principles of Computing10
or 15-112 Fundamentals of Programming and Computer Science
21-122Integration and Approximation10
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
xx-xxxPhysical/Life Sciences Course9
xx-xxxH&SS Elective9
 48
Sophomore Year
Fall Units
21-228Discrete Mathematics9
or 15-251 Great Ideas in Theoretical Computer Science
21-259Calculus in Three Dimensions9
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
73-102Principles of Microeconomics9
xx-xxxSTEM Course9
 36
Spring Units
21-260Differential Equations9
or 21-261 Introduction to Ordinary Differential Equations
or 33-231 Physical Analysis
21-292Operations Research I9
70-122Introduction to Accounting9
xx-xxxH&SS Elective9
xx-xxxSTEM Elective9
 45
Junior Year
Fall Units
21-369Numerical Methods9
36-225Introduction to Probability Theory9
or 21-325 Probability
73-103Principles of Macroeconomics9
xx-xxxDepth Elective9
xx-xxxFree Elective9
 45
Spring Units
36-226Introduction to Statistical Inference9
36-410Introduction to Probability Modeling9
xx-xxxDepth Elective9
73-230Intermediate Microeconomics9
or 73-240 Intermediate Macroeconomics
xx-xxxCultural/Global Understanding Course9
 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 Methods for Data Analysis9
xx-xxxDepth Elective9
xx-xxxDepth Elective9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 45

Minimum number of units required for degree:360

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 and Data Science. The Department of Statistics and Data Science 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 and Data Science. 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 and Approximation10
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
21-228Discrete Mathematics9
or 15-251 Great Ideas in Theoretical Computer Science
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
21-259Calculus in Three Dimensions9
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
21-260Differential Equations9
or 21-261 Introduction to Ordinary Differential Equations
or 33-231 Physical Analysis
21-292Operations Research I9
21-369Numerical Methods9
Starting Fall 2018, 21-369 will be 12 units with recitation.
21-393Operations Research II9
 94

Statistics Courses (required)

Courses Units
36-225Introduction to Probability Theory9
or 21-325 Probability
36-226Introduction to Statistical Inference9
36-401Modern Regression9
36-402Advanced Methods for 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
73-102Principles of Microeconomics9
 31

Depth Electives (required)

Five 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.

Courses Units
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
21-377Monte Carlo Simulation for Finance9
21-378Mathematics of Fixed Income Markets9
21-420Continuous-Time Finance9
21-484Graph Theory9
36-461Special Topics: Statistical Methods in Epidemiology9
36-462Special Topics: Data Mining9
36-463Special Topics: Multilevel and Hierarchical Models9
36-464Special Topics: Applied Multivariate Methods9

MCS General Education (required)

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

Note that 73-102 satisfies the requirement from the MCS core.

Suggested Schedule

Freshman Year
Fall Units
21-120Differential and Integral Calculus10
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
38-101First-year Seminar6
76-101Interpretation and Argument9
99-101Computing @ Carnegie Mellon3
xx-xxxLife/Physical Sciences Course9
 47
Spring Units
15-112Fundamentals of Programming and Computer Science12
21-122Integration and Approximation10
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
xx-xxxSTEM Course9
xx-xxxPhysical/Life Sciences Course9
 50
Sophomore Year
Fall Units
21-228Discrete Mathematics9
or 15-251 Great Ideas in Theoretical Computer Science
21-259Calculus in Three Dimensions9
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
73-102Principles of Microeconomics9
xx-xxxSTEM Course9
xx-xxxH&SS Elective9
 45
Spring Units
15-122Principles of Imperative Computation10
21-260Differential Equations9
or 21-261 Introduction to Ordinary Differential Equations
or 33-231 Physical Analysis
21-292Operations Research I9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 46
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-xxxCultural/Global Understanding Course9
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 Methods for Data Analysis9
xx-xxxDepth Elective9
xx-xxxDepth Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
 45

Minimum number of units required for degree:360

Discrete Mathematics and Logic Concentration

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

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

The requirements for the Discrete Mathematics and Logic Concentration are:

Mathematical Sciences and Computer Science Courses (required)

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

Courses 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 and Approximation10
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
21-300Basic Logic9
or 15-317 Constructive Logic
21-228Discrete Mathematics9
or 15-251 Great Ideas in Theoretical Computer Science
21-301Combinatorics9
21-341Linear Algebra9
21-355Principles of Real Analysis I9
21-373Algebraic Structures9
 126

 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)
Courses Units
21-325Probability9
21-329Set Theory9
21-374Field Theory9
21-400Intermediate Logic9
21-441Number Theory9
21-484Graph Theory9
21-602Introduction to Set Theory I12
21-603Model Theory I12
21-610Algebra I12
21-701Discrete Mathematics12
80-405Game Theory9
80-411Proof Theory9
80-413Category Theory9
List 2 (General Mathematics Electives)
Courses Units
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
or 33-231 Physical Analysis
21-272Introduction to Partial Differential Equations9
21-292Operations Research I9
21-356Principles of Real Analysis II9
21-366Topics in Applied Mathematics9
21-369Numerical Methods9
Starting Fall 2018, 21-369 will be 12 units with recitation.
21-370Discrete Time Finance9
21-371Functions of a Complex Variable9
21-393Operations Research II9
21-420Continuous-Time Finance9
21-470Selected Topics in Analysis9
21-476Introduction to Dynamical Systems9
21-499Undergraduate Research Topic9
Any graduate course in mathematics at the 600 and 700 level not included in List 1.

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
38-101First-year Seminar6
76-101Interpretation and Argument9
99-101Computing @ Carnegie Mellon3
xx-xxxLife/Physical Sciences Course9
 49
Spring Units
15-122Principles of Imperative Computation10
21-122Integration and Approximation10
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
xx-xxxPhysical/Life Sciences Course9
 49
Sophomore Year
Fall Units
15-150Principles of Functional Programming10
21-268Multidimensional Calculus10
or 21-269 Vector Analysis
21-301Combinatorics9
21-373Algebraic Structures9
xx-xxxH&SS Elective9
 47
Spring Units
15-210Parallel and Sequential Data Structures and Algorithms12
xx-xxxDiscrete Math/Logic Elective9
xx-xxxMathematics Elective9
xx-xxxSTEM Course9
xx-xxxH&SS Elective9
 48
Junior Year
Fall Units
15-xxxComputer Science Elective9
21-300Basic Logic9
or 15-317 Constructive Logic
21-355Principles of Real Analysis I9
xx-xxxH&SS Elective9
xx-xxxSTEM Course9
 45
Spring Units
15-xxxComputer Science Elective9
21-341Linear Algebra9
xx-xxxH&SS Elective9
xx-xxxCultural/Global Understanding Course9
xx-xxxFree Elective9
 45
Senior Year
Fall Units
xx-xxxDiscrete Math/Logic Elective9
xx-xxxMathematics Elective9
xx-xxxMathematics Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
 45
Spring Units
xx-xxxDiscrete Math/Logic Elective9
xx-xxxMathematics Elective9
xx-xxxMathematics Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
 45

Minimum number of units required for degree:360

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 and Approximation10
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
21-228Discrete Mathematics9
or 15-251 Great Ideas in Theoretical Computer Science
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
21-259Calculus in Three Dimensions9
or 21-268 Multidimensional Calculus
or 21-269 Vector Analysis
21-260Differential Equations9
or 21-261 Introduction to Ordinary Differential Equations
or 33-231 Physical Analysis
36-225Introduction to Probability Theory9
or 21-325 Probability
21-272Introduction to Partial Differential Equations9
21-355Principles of Real Analysis I9
21-369Numerical Methods9
Starting Fall 2018, 21-369 will be 12 units with recitation.
 103

Two courses from:

Courses Units
21-341Linear Algebra9
21-356Principles of Real Analysis II9
21-380Introduction to Mathematical Modeling9
21-435Applied Harmonic Analysis9
21-469Numerical Methods II: Scientific Computing9

 Computer Science Courses (required)

Courses Units
15-122Principles of Imperative Computation10

Mathematics Electives (required)

Students must take 36 units either from the three remaining courses in List 1 or from the list below:

Courses Units
21-292Operations Research I9
21-365Projects in Applied Mathematics9
21-366Topics in Applied Mathematics9
21-370Discrete Time Finance9
21-371Functions of a Complex Variable9
21-373Algebraic Structures9
21-377Monte Carlo Simulation for Finance9
21-378Mathematics of Fixed Income Markets9
21-393Operations Research II9
21-420Continuous-Time Finance9
21-467Differential Geometry9
21-470Selected Topics in Analysis9
21-476Introduction to Dynamical Systems9
21-484Graph Theory9
21-620Real Analysis6
21-621Introduction to Lebesgue Integration6
21-630Ordinary Differential Equations12
21-632Introduction to 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
21-723Advanced Real Analysis12
21-724Sobolev Spaces12
21-732Partial Differential Equations I12
21-832Partial Differential Equations II12

Students must take 9 additional units of Mathematical Sciences (at the 21-300 level or above or 21-270 or 21-272 or 21-292), or Computer Science (at the 15-200 level or above), or Physics (at the 33-300 level or above), or Statistics (must have at least 36-225 as a prerequisite) electives.

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
21-120Differential and Integral Calculus10
21-126Introduction to Mathematical Software3
21-127Concepts of Mathematics10
or 21-128 Mathematical Concepts and Proofs
38-101First-year Seminar6
76-101Interpretation and Argument9
xx-xxxLife/Physical Sciences Course9
 47
Spring Units
21-122Integration and Approximation10
21-228Discrete Mathematics9
21-241Matrices and Linear Transformations10
or 21-242 Matrix Theory
xx-xxxPhysical/Life Sciences Course9
xx-xxxH&SS Elective9
 47
Sophomore Year
Fall Units
15-112Fundamentals of Programming and Computer Science12
21-268Multidimensional Calculus10
or 21-269 Vector Analysis
xx-xxxSTEM Course9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 49
Spring Units
15-122Principles of Imperative Computation10
21-261Introduction to Ordinary Differential Equations10
21-355Principles of Real Analysis I9
xx-xxxSTEM Course9
xx-xxxH&SS Elective9
 47
Junior Year
Fall Units
21-320Symbolic Programming Methods9
21-325Probability9
21-356Principles of Real Analysis II9
xx-xxxH&SS Elective9
xx-xxxFree Elective9
 45
Spring Units
xx-xxxMathematics Elective9
21-369Numerical Methods9
xx-xxxDepth Elective9
xx-xxxCultural/Global Understanding Elective9
xx-xxxFree Elective9
 45
Senior Year
Fall Units
xx-xxxMathematics Elective9
xx-xxxMathematics Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
 45
Spring Units
xx-xxxMathematics Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
xx-xxxFree Elective9
 45

Minimum number of units required for degree:360

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 and free electives.

The Minor in Mathematical Sciences

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

21-127Concepts of Mathematics10
21-228Discrete Mathematics9-12
or 15-251 Great Ideas in Theoretical 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:

21-228Discrete Mathematics 19-12
or 15-251 Great Ideas in Theoretical Computer Science
21-300Basic Logic9
or 15-317 Constructive Logic
21-301Combinatorics9

121-127 Concepts of Mathematics is a prerequisite for 21-228.

Three of the following (at least one from each group):

Logic
21-329Set Theory9
21-400Intermediate Logic9
21-602Introduction to Set Theory I12
21-603Model Theory I12
21-700Mathematical Logic II12
Algebra and Discrete Mathematics
21-341Linear Algebra9
21-373Algebraic Structures9
21-374Field Theory9
21-441Number Theory9
21-484Graph Theory9
21-610Algebra I12
21-701Discrete Mathematics12

The Honors Degree Program 

This demanding program leads to an M.S. in Mathematical Sciences, normally in four years, in addition to the student's B.S. degree. The key element in the program is usually the Mathematical Studies sequence. Admission to the Honors Program, in the Junior year, requires an application while admission to the Math Studies sequences is by invitation only.  Students with a grade of B or higher in 21-235 Mathematical Studies Analysis I will be allowed to register for 21-236 Mathematical Studies Analysis II.  Students with a grade of B or higher in 21-237 Mathematical Studies Algebra I will be allowed to register for 21-238 Mathematical Studies Algebra II . 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 I12
21-237Mathematical Studies Algebra I
(Honors version of 21-373 Algebraic Structures)
12
Spring
21-236Mathematical Studies Analysis II
(Honors version of 21-356 Principles of Real Analysis II)
12
21-238Mathematical Studies Algebra II
(Honors version of 21-341 Linear Algebra)
12
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.

Course Descriptions

Note on Course Numbers

Each Carnegie Mellon course number begins with a two-digit prefix which designates the department offering the course (76-xxx courses are offered by the Department of English, etc.). Although each department maintains its own course numbering practices, typically the first digit after the prefix indicates the class level: xx-1xx courses are freshmen-level, xx-2xx courses are sophomore level, etc. xx-6xx courses may be either undergraduate senior-level or graduate-level, depending on the department. xx-7xx courses and higher are graduate-level. Please consult the Schedule of Classes each semester for course offerings and for any necessary pre-requisites or co-requisites.

21-101 Freshman Mathematics Seminar
Fall: 3 units
This course is offered in the Fall semester for first semester Freshmen interested in majoring in mathematics. Topics vary from year to year. Recent topics have included Fermat's last theorem, finite difference equations, convexity, and fractals. 3 hrs. lec.
21-105 Pre-Calculus
Summer: 9 units
Review of basic concepts, logarithms, functions and graphs, inequalities, polynomial functions, complex numbers, and trigonometric functions and identities. Special summer program only. 3 hrs lec., 1 hr.rec.
21-111 Differential Calculus
Fall and Spring: 10 units
Review of basic algebra, functions, limits, derivatives of algebraic, exponential and logarithmic functions, curve sketching, maximum-minimum problems. Successful completion of 21-111 and 21-112 entitles a student to enroll in any mathematics course for which 21-120 is a prerequisite. 3 hrs. lec., 2 hrs. rec.
21-112 Integral Calculus
Fall and Spring: 10 units
Definite and indefinite integrals, and hyperbolic functions; applications of integration, integration by substitution and by parts. Successful completion of 21-111 and 21-112 entitles a student to enroll in any mathematics course for which 21-120 is a prerequisite. 3 hrs. lec., 2 hrs. rec.
Prerequisite: 21-111
21-115 Basic Differential Calculus
Summer: 5 units
Functions, limits, derivatives, curve sketching, Mean Value Theorem, trigonometric functions, related rates, linear and quadratic approximations, maximum-minimum problems. Special summer program only.
21-120 Differential and Integral Calculus
All Semesters: 10 units
Functions, limits, derivatives, logarithmic, exponential, and trigonometric functions, inverse functions; L'Hospital's Rule, curve sketching, Mean Value Theorem, related rates, linear and quadratic approximations, maximum-minimum problems, inverse functions, definite and indefinite integrals, and hyperbolic functions; applications of integration, integration by substitution and by parts. 3 hrs lec., 2 hrs. rec.
21-122 Integration and Approximation
Fall and Spring: 10 units
Integration by trigonometric substitution and partial fractions; arclength; improper integrals; Simpson's and Trapezoidal Rules for numerical integration; separable differential equations, Newton's method, Euler's method, Taylor's Theorem including a discussion of the remainder, sequences, series, power series. Parametric curves, polar coordinates, vectors, dot product. 3 hrs lec., 2 hrs. rec.
Prerequisites: 21-112 or 21-120
21-124 Calculus II for Biologists and Chemists
Spring: 10 units
This is intended as a second calculus course for biology and chemistry majors. It uses a variety of computational techniques based around the use of MATLAB or a similar system. Topics to be covered include: Integration: techniques and numerical integration. Ordinary differential equations: techniques for solving ODEs and numerical methods. Modeling with ODEs (e.g., infection, population models). Linear algebra: matrices, complex numbers, eigenvalues, eigenvectors. Systems of ordinary differential equations (if time allows: stability of differential systems). Probability: discrete and continuum probability, conditional probability and independence, limit theorems, important distributions, probabilistic models. 3 hrs. lec., 2 hrs. rec. Prerequisite: 21-112 or 21-120.
Prerequisites: 21-112 or 21-120
21-126 Introduction to Mathematical Software
Spring: 3 units
This course provides an introduction to the use of several software packages, which are useful to mathematics students. Among the packages are Maple and Mathematica for symbolic computing, TeX and LaTeX for mathematical documents, and Matlab for numerical computing. The course will also introduce the mathematical facilities built into spreadsheets such as Excel. The aim of the course is to provide the student with some basic skills in the use of this software without attempting complete coverage. A deeper knowledge of the software will be easy to obtain after completing this course. There are no prerequisites for the course, other than basic computer literacy and a knowledge of elementary mathematics. It is suggested that the course should be taken during the first two years of undergraduate studies.
21-127 Concepts of Mathematics
Fall and Spring: 10 units
This course introduces the basic concepts, ideas and tools involved in doing mathematics. As such, its main focus is on presenting informal logic, and the methods of mathematical proof. These subjects are closely related to the application of mathematics in many areas, particularly computer science. Topics discussed include a basic introduction to elementary number theory, induction, the algebra of sets, relations, equivalence relations, congruences, partitions, and functions, including injections, surjections, and bijections. A basic introduction to the real numbers, rational and irrational numbers. Supremum and infimum of a set. 3 hrs. lec., 2 hrs. rec.
21-128 Mathematical Concepts and Proofs
Intermittent: 12 units
This course is intended for MCS first-semester students who are interested in pursuing a major in mathematical sciences. The course introduces the basic concepts, ideas and tools involved in doing mathematics. As such, its main focus in on presenting informal logic, and the methods of mathematical proof. These subjects are closely related to the application of mathematics in many areas, particularly computer science. Topics discussed include a basic introduction to elementary number theory, induction, the algebra of sets, relations, equivalence relations, congruences, partitions, and functions, including injections, surjections, and bijections. A basic introduction to the real numbers, rational and irrational numbers. Supremum and infimum of a set. This course is a superset of 21-127, with additional out of class time devoted to proofs and additional topics in math. 3 hrs. lec., 2 hrs. rec.
21-228 Discrete Mathematics
Spring: 9 units
The techniques of discrete mathematics arise in every application of mathematics, which is not purely continuous, for example in computer science, economics, and general problems of optimization. This course introduces two of the fundamental areas of discrete mathematics: enumeration and graph theory. The introduction to enumeration includes permutations, combinations, and topics such as discrete probability, combinatorial distributions, recurrence relations, generating functions, Ramsey's Theorem, and the principle of inclusion and exclusion. The introduction to graph theory includes topics such as paths, walks, connectivity, Eulerian and Hamilton cycles, planar graphs, Euler's Theorem, graph coloring, matchings, networks, and trees. 3 hrs. lec, 1 hr. rec.
Prerequisites: 21-128 or 21-127
21-235 Mathematical Studies Analysis I
Fall: 12 units
An honors version of 21-355 for students of greater aptitude and motivation. Topics to be covered include: The Real Number System: sups and infs, completeness, integers and rational numbers. Metric spaces, normed spaces, inner product spaces and their specialization to the Euclidean space. Topological properties of metric spaces (open sets, closed sets, density, compactness, Heine-Borel Theorem). Sequences and convergence; completeness. Baire Category Theorem. Real sequences: limsup and liminf, subsequences, monotonic sequences, Bolzano-Weierstrass Theorem. Real series (criteria for convergence). Continuity, limits of functions, attainment of extrema, Intermediate Value Theorem, uniform continuity. Differentiation of functions of one variable: Chain Rule, local extrema, Mean-Value Theorems, L'Ho'pital's Rule, Taylor's Theorem. Riemann Integration: Partitions, upper and lower integrals, sufficient conditions for integrability, Fundamental Theorem of Calculus. 3 hrs. lec.
Prerequisites: (21-127 or 21-128) and 21-269
21-236 Mathematical Studies Analysis II
Spring: 12 units
An honors version of 21-356 for students of greater aptitude and motivation. Topics to be covered include: Vector differential calculus: differentiability, partial derivatives, directional derivatives, gradients, Jacobians, the chain rule, implicit function theorem, inverse function theorem. Local extrema, constrained problems (Lagrange multipliers). Integration of differential forms: Manifolds, Differential forms (properties, differentiation, change of variables), partition of unity, integration, volume form, area form, Stokes' theorem. Sequences of Functions: Pointwise convergence, uniform convergence, Arzela-Ascoli, Weierstrass approximation theorem. Series of functions: Power series, Fourier series, orthonormal bases. 3 hrs. lec.
Prerequisites: 21-242 and 21-235
21-237 Mathematical Studies Algebra I
Fall: 12 units
An honors version of 21-373 Algebraic structures for students of greater aptitude and motivation. Abstract algebra is the study of algebraic systems by the axiomatic method, and it is one of the core areas of modern mathematics. This course is a rigorous and fast-paced introduction to the basic objects in abstract algebra. Topics to be covered include: Homomorphisms. Subgroups, cosets, Lagrange's theorem. Conjugation. Normal subgroups, quotient groups, first isomorphism theorem. Automorphisms, the automorphism group, characteristic subgroups. Group actions, Cauchy's Theorem, Sylow's theorem. Normalisers and centralisers, class equation, finite p-groups. Dihedral and alternating groups. The second and third isomorphism theorems. Simple groups, statement of Jordan-Holder theorem, semidirect product of groups. Subrings, ideals, quotient rings, first isomorphism theorem. Polynomial rings. Zorn's Lemma. Prime and maximal ideals, prime and irreducible elements. PIDs and UFDs. Noetherian domains. Hilbert Basis Theorem. Gauss' lemma. Eisenstein criterion. Field of fractions of an integral domain. k a field implies k[x] a PID, R a UFD implies R[x] a UFD. Finite fields and applications. 3 hrs. lec.
Prerequisites: (21-127 or 21-128) and 21-269
21-238 Mathematical Studies Algebra II
Spring: 12 units
An honors version of 21-341 Linear Algebra for students of greater aptitude and motivation. Linear algebra is a crucial tool in pure and applied mathematics. This course aims to introduce the main ideas at a high level of rigour and generality. The course starts with the study of (potentially) infinite-dimensional vector spaces over an arbitrary field, continues with the theory of modules (where the role of the field is now played by an arbitrary ring), and concludes with the development of real and complex inner product spaces. Topics to be covered include: Review of fields. Review of Zorn's Lemma. Vector spaces (possibly in finite dimensional) over an arbitrary field. Independent sets, bases, existence of a basis, exchange lemma, dimension. Linear transformations, dual space. Multilinear maps, tensor product, exterior power, determinant of a transformation. Eigenvalues, eigenvectors, characteristic and minimal polynomial of a transformation, Cayley-Hamilton theorem. Review of commutative rings. R-modules. Sums and quotients of modules. Free modules. Structure theorem for fg modules over a PID and applications (Jordan and rational canonical form, structure theory of fg abelian groups). Review of real and complex numbers. Real and complex inner product spaces. Orthonormal sets, orthonormal bases, Gram-Schmidt. Examples: F^n and l^2(F) for F = R; C. Operators: Symmetric/Hermitian and Orthogonal/Unitary operators. Spectral theorem. Quadratic forms. Singular value decomposition. Possible additional topics (time permitting): applications to combinatorics, category theory, representations of finite groups, normed spaces. 3 hrs. lec.
Prerequisites: 21-242 and 21-237
21-240 Matrix Algebra with Applications
Fall and Spring: 10 units
Vectors and matrices, the solution of linear systems of equations, vector spaces and subspaces, orthogonality, determinants, real and complex eigenvalues and eigenvectors, linear transformations. The course is intended for students in Economics, Statistics, Information Systems, and it will focus on topics relevant to these fields. 3 hrs. lec., 1 hr. rec.
21-241 Matrices and Linear Transformations
Fall and Spring: 10 units
A first course in linear algebra intended for scientists, engineers, mathematicians and computer scientists. Students will be required to write some straightforward proofs. Topics to be covered: complex numbers, real and complex vectors and matrices, rowspace and columnspace of a matrix, rank and nullity, solving linear systems by row reduction of a matrix, inverse matrices and determinants, change of basis, linear transformations, inner product of vectors, orthonormal bases and the Gram-Schmidt process, eigenvectors and eigenvalues, diagonalization of a matrix, symmetric and orthogonal matrices. 21-127 is strongly recommended. 3 hrs. lec., 1 hr. rec
21-242 Matrix Theory
Fall and Spring: 10 units
An honors version of 21-241 (Matrix Algebra and Linear Transformations) for students of greater aptitude and motivation. More emphasis will be placed on writing proofs. Topics to be covered: complex numbers, real and complex vectors and matrices, rowspace and columnspace of a matrix, rank and nullity, solving linear systems by row reduction of a matrix, inverse matrices and determinants, change of basis, linear transformations, inner product of vectors, orthonormal bases and the Gram-Schmidt process, eigenvectors and eigenvalues, diagonalization of a matrix, symmetric and orthogonal matrices, hermitian and unitary matrices, quadratic forms. 3 hrs. lec., 1 hr. rec.
21-256 Multivariate Analysis
Fall and Spring: 9 units
This course is designed for students in Economics or Business Administration. Matrix algebra: vectors, matrices, systems of equations, dot product, cross product, lines and planes. Optimization: partial derivatives, the chain rule, gradient, unconstrained optimization, constrained optimization (Lagrange multipliers and the Kuhn-Tucker Theorem). Improper integrals. Multiple integration: iterated integrals, probability applications, triple integrals, change of variables. 3 hrs lec., 1 hr rec.
Prerequisites: 21-112 or 21-120
21-257 Models and Methods for Optimization
Fall and Spring: 9 units
Introduces basic methods of operations research and is intended primarily for Business Administration and Economics majors. Review of linear systems; linear programming, including the simplex algorithm, duality, and sensitivity analysis; the transportation problem; the critical path method; the knapsack problem, traveling salesman problem, and an introduction to set covering models. 3 hrs. lec., 1 hr. rec.
Prerequisites: 21-242 or 21-256 or 21-241 or 21-240 or 18-202 or 06-262
21-259 Calculus in Three Dimensions
Fall and Spring: 9 units
Vectors, lines, planes, quadratic surfaces, polar, cylindrical and spherical coordinates, partial derivatives, directional derivatives, gradient, divergence, curl, chain rule, maximum-minimum problems, multiple integrals, parametric surfaces and curves, line integrals, surface integrals, Green-Gauss theorems. 3 hrs. lec., 1 hr. rec.
Prerequisite: 21-122
21-260 Differential Equations
Fall and Spring: 9 units
Ordinary differential equations: first and second order equations, applications, Laplace transforms; partial differential equations: partial derivatives, separation of variables, Fourier series; systems of ordinary differential equations; applications. 21-259 or 21-268 or 21-269 are recommended. 3 hrs. lec., 1 hr. rec.
Prerequisite: 21-122
21-261 Introduction to Ordinary Differential Equations
Spring: 10 units
A first course in ordinary differential equations intended primarily for math majors and for those students interested in a more conceptual treatment of the subject. One of the goals of this course is to prepare students for upper level courses on differential equations, mathematical analysis and applied mathematics. Students will be required to write rigorous arguments. Topics to be covered: Ordinary differential equations: first and second order equations, applications, Laplace transform, systems of linear ordinary differential equations; systems of nonlinear ordinary differential equations, equilibria and stability, applications. Note: courses 21-259, or 21-268, or 21-269 are recommended. 21-128 can replace 21-127 as a corequisite. 3 hrs. lec., 1 hr. rec.
Prerequisite: 21-122
21-268 Multidimensional Calculus
Fall and Spring: 10 units
A serious introduction to multidimensional calculus that makes use of matrices and linear transformation. Results will be stated carefully and rigorously. Students will be expected to write some proofs; however, some of the deeper results will be presented without proofs. Topics to be covered include: functions of several variables, regions and domains, limits and continuity, partial derivatives, linearization and Jacobian matrices, chain rules, inverse and implicit functions, geometric applications, higher derivatives, Taylor's theorem, optimization, vector fields, multiple integrals and change of variables, Leibnitz's rule, line integrals, Green's theorem, path independence and connectedness, conservative vector fields, surfaces and orientability, surface integrals, divergence theorem and Stokes's theorem. 3 hrs. lec.
Prerequisites: 21-122 and (21-241 or 21-242)
21-269 Vector Analysis
Spring: 10 units
An honors version of 21-268 for students of greater aptitude and motivation. More emphasis will be placed on writing proofs. Topics to be covered include: basic geometry and topology of Euclidean space, curves in space, arclength, curvature and torsion, functions on Euclidean spaces, limits and continuity, partial derivatives, gradients and linearization, chain rules, inverse and implicit function theorems, geometric applications, higher derivatives, Taylor's theorem, optimization, vector fields, multiple integrals and change of variables, Leibnitz's rule, conservative and solenoidal vector fields, divergence and curl, surfaces and orientability, surface integrals, Gauss-Green theorems and Stokes's theorem. A grade of B or better in 21-242 is required. 3 hrs. lec.
Prerequisites: 21-122 and 21-242 Min. grade B
21-270 Introduction to Mathematical Finance
Spring: 9 units
This is a first course for those considering majoring or minoring in Computational Finance. The theme of this course is pricing derivative securities by replication. The simplest case of this idea, static hedging, is used to discuss net present value of a non-random cash flow, internal rate of return, and put-call option parity. Pricing by replication is then considered in a one-period random model. Risk-neutral probability measures, the Fundamental Theorems of Asset Pricing, and an introduction to expected utility maximization and mean-variance analysis are presented in this model. Finally, replication is studied in a multi-period binomial model. Within this model, the replicating strategies for European and American options are determined. 3 hrs. lec.
Prerequisites: 21-120 or 21-112
21-272 Introduction to Partial Differential Equations
Fall: 9 units
A Partial Differential Equation (PDE for short) is a differential equation involving derivatives with respect to more than one variable. These arise in numerous applications from various disciplines. A prototypical example is the heat equation, governing the evolution of temperature in a conductor. This course will serve as a first introduction to PDE's, and will focus on the most important model equations. It will cover both analytical methods (e.g. separation of variables, Green's functions), numerical methods (e.g. finite elements) and the use of a computer to approximate and visualize solutions. Time permitting, it will touch upon the mathematical ideas behind phenomena observed in nature (e.g. speed of wave propagation, and/or shocks in traffic flow).
Prerequisites: (21-269 or 21-268 or 21-259) and (21-260 or 21-261 or 33-231)
21-292 Operations Research I
Spring: 9 units
Operations research offers a scientific approach to decision making, most commonly involving the allocation of scarce resources. This course develops some of the fundamental methods used. Linear programming: the simplex method and its linear algebra foundations, duality, post-optimality and sensitivity analysis; the transportation problem; the critical path method; non-linear programming methods. 3 hrs. lec., 1 hr. rec.
Prerequisites: 21-122 and (21-241 or 21-242)
21-295 Putnam Seminar
Fall: 3 units
A problem solving seminar designed to prepare students to participate in the annual William Lowell Putnam Mathematical Competition. Students solve and present their solutions to problems posed.
21-296 Millennium Problems Seminar
Intermittent: 3 units
This seminar course will discuss some of the most important unsolved problems of mathematics (as deemed in 2000 by an international committee of mathematicians): The Riemann Hypothesis; Yang-Mills Theory and the Mass Gap Hypothesis; the P. vs. NP Problem; smoothness of solutions of the Navier-Stokes Equations; the Hodge Conjecture; the Birch and Swinnerton-Dyer Conjecture. If the time allows, the Poincare conjecture will also be discussed. 1 hr. lec.
21-300 Basic Logic
Fall: 9 units
Propositional and predicate logic: Syntax, proof theory and semantics up to completeness theorem, Lowenheim Skolem theorems, and applications of the compactness theorem. 3 hrs. lec.
Prerequisites: 21-373 or 15-251 or 21-228
21-301 Combinatorics
Fall and Spring: 9 units
A major part of the course concentrates on algebraic methods, which are relevant in the study of error correcting codes, and other areas. Topics covered in depth include permutations and combinations, generating functions, recurrence relations, the principle of inclusion and exclusion, and the Fibonacci sequence and the harmonic series. Additional topics may include existence proofs, partitions, finite calculus, generating combinatorial objects, Polya theory, codes, probabilistic methods. 3 hrs. lec
Prerequisites: 21-122 and (15-251 or 21-228)
21-302 Lambda Calculus
Spring: 9 units
An introductory course in classical lambda calculus, with an emphasis on syntax. The course will describe many research problems which are suitable topics for senior theses or master's theses. Topics will include: Basic properties of reduction and conversion; Reduction and conversion strategies; Calculability and representation of data types; Elementary theory of Ershov numberings; Bohm's theorem, easy terms, and other exotic combinations; Solvability of functional equations (unification); Combinatorics and bases; Simple and algebraic types; Labelled reduction and intersection types; Extensionality and the omega rule.
Prerequisites: (15-150 or 80-310 or 21-300) and 21-301
21-320 Symbolic Programming Methods
Spring: 9 units
The objective of this course is to learn to program in Maple, a powerful symbolic mathematics package available on many platforms at Carnegie Mellon. After learning what Maple can do with the commands provided with the package, students will learn to develop their own Maple functions to accomplish extended mathematical computations. Grades in the course will be based mostly on project work. Projects may come from any relevant field and may be graphical, numerical, or symbolic or all three. The course will involve online demonstrations in most classes. 3 hrs. lec.
Prerequisites: (21-127 or 21-128) and 21-122
21-325 Probability
Fall: 9 units
This course focuses on the understanding of basic concepts in probability theory and illustrates how these concepts can be applied to develop and analyze a variety of models arising in computational biology, finance, engineering and computer science. The firm grounding in the fundamentals is aimed at providing students the flexibility to build and analyze models from diverse applications as well as preparing the interested student for advanced work in these areas. The course will cover core concepts such as probability spaces, random variables, random vectors, multivariate densities, distributions, expectations, sampling and simulation; independence, conditioning, conditional distributions and expectations; limit theorems such as the strong law of large numbers and the central limit theorem; as well as additional topics such as large deviations, random walks and Markov chains, as time permits. 3 hrs. lec.
Prerequisites: 21-259 or 21-268 or 21-269
21-329 Set Theory
Spring: 9 units
Set theory was invented about 110 years ago by George Cantor as an instrument to understand infinite objects and to compare different sizes of infinite sets. Since then set theory has come to play an important role in several branches of modern mathematics, and serves as a foundation of mathematics. Contents: Basic properties of natural numbers, countable and uncountable sets, construction of the real numbers, some basic facts about the topology of the real line, cardinal numbers and cardinal arithmetic, the continuum hypothesis, well ordered sets, ordinal numbers and transfinite induction, the axiom of choice, Zorn's lemma. Optional topics if time permits: Infinitary combinatorics, filters and large cardinals, Borel and analytic sets of reals. 3 hrs. lec.
Prerequisites: 21-128 or 21-127
21-341 Linear Algebra
Fall and Spring: 9 units
21-341 Linear Algebra. A mathematically rigorous treatment of Linear Algebra over an arbitrary field. Topics studied will include abstract vector spaces, linear transformations, determinants, eigenvalues, eigenvectors, inner products, invariant subspaces, canonical forms, the spectral theorem and the singular value decomposition. 21-373 recommended. 3 hrs. lec.
Prerequisites: (21-241 and 21-373) or 21-242
21-350 History of Mathematics
Intermittent: 9 units
Mathematics has a long and interesting history, and there is much insight into both mathematics and history to be gained from its study. The emphasis here will be on learning the mathematics with the added value of appreciating it in historical context. Selected topics may range from early number systems, the development of geometry, the emergence of the ideas of analysis, through to the origins of modern set theory. 3 hrs. lec.
21-355 Principles of Real Analysis I
Fall and Spring: 9 units
This course provides a rigorous and proof-based treatment of functions of one real variable. The Real Number System: Field and order axioms, sups and infs, completeness, integers and rational numbers. Real Sequences: Limits, cluster points, limsup and liminf, subsequences, monotonic sequences, Cauchy's criterion, Bolzano-Weierstrass Theorem. Topology of the Real Line: Open sets, closed sets, density, compactness, Heine-Borel Theorem. Continuity: attainment of extrema, Intermediate Value Theorem, uniform continuity. Differentiation: Chain Rule, local extrema, Mean-Value Theorems, L'Hospital's Rule, Taylor's Theorem. Riemann Integration: Partitions, upper and lower integrals, sufficient conditions for integrability, Fundamental Theorem of Calculus. Sequences of Functions: Pointwise convergence, uniform convergence, interchanging the order of limits. The course presumes some mathematical sophistication including the ability to recognize, read, and write proofs. 3 hrs lec.
Prerequisites: (21-128 or 21-127) and 21-122
21-356 Principles of Real Analysis II
Spring: 9 units
This course provides a rigorous and proof-based treatment of functions of several real variables. Topology in metric spaces, specialization to finite dimensional normed linear spaces. Vector differential calculus: continuity and the total derivative, partial derivatives, directional derivatives, gradients, Jacobians, the chain rule, implicit function theorem. Vector integral calculus: double and triple integrals, arclength and surface area, line integrals, Green's Theorem, surface integrals, Divergence and Stokes Theorems. If time permits: trigonometric series, Fourier series for orthonormal bases, minimization of square error. The course presumes some mathematical sophistication including the ability to recognize, read, and write proofs. 21-268 or 21-269 are strongly recommended rather than 21-259. 3 hrs lec.
Prerequisites: (21-259 or 21-269 or 21-268) and 21-241 and 21-355
21-360 Differential Geometry of Curves and Surfaces
Intermittent: 9 units
The course is a rigorous introduction to the differential and integral calculus of curves and surfaces. Topics to be covered include: Parameterized and regular curves Frenet equations canonical coordinate system, local canonical forms, global properties of plane curves Regular surfaces, differential functions on surfaces, the tangent plane and differential of a map, orientation of surfaces, characterization of compact orientable surfaces, classification of compact surfaces The geometry of the Gauss map, isometries and conformal maps, parallel transport, geodesics, the Gauss-Bonnet theorem and applications. More topics may be covered, as time allows. Students should be prepared to write proofs and perform computations. 21-356 or 21-236 are recommended. 3 hrs. lec.
Prerequisites: 21-269 or 21-268
21-365 Projects in Applied Mathematics
Intermittent: 9 units
This course provides students with an opportunity to solve problems posed by area companies. It is also designed to provide experience working as part of a team to solve problems for a client. The background needed might include linear programming, simulation, data analysis, scheduling, numerical techniques, etc.
21-366 Topics in Applied Mathematics
Intermittent: 9 units
Typical of courses that might be offered from time to time are game theory, non-linear optimization, and dynamic programming. Prerequisites will depend on the content of the course. 3 hrs. lec.
Prerequisites: 21-128 or 21-127
21-369 Numerical Methods
Fall and Spring: 9 units
This course provides an introduction to the use of computers to solve scientific problems. Methods for the computational solution of linear algebra systems, nonlinear equations, the interpolation and approximation of functions, differentiation and integration, and ordinary differential equations. Analysis of roundoff and discretization errors and programming techniques. 21-268 or 21-269 are recommended rather than 21-259. 3 hrs. lec.
Prerequisites: (15-110 or 15-112) and (21-259 or 21-269 or 21-268) and (21-240 or 21-241 or 21-242) and (21-260 or 33-231 or 21-261)
21-370 Discrete Time Finance
Fall: 9 units
This course introduces the Black-Scholes option pricing formula, shows how the binomial model provides a discretization of this formula, and uses this connection to fit the binomial model to data. It then sets the stage for Continuous-Time Finance by discussing in the binomial model the mathematical technology of filtrations, martingales, Markov processes and risk-neutral measures. Additional topics are American options, expected utility maximization, the Fundamental Theorems of Asset Pricing in a multi-period setting, and term structure modeling, including the Heath-Jarrow-Morton model. Students in 21-370 are expected to read and write proofs. 3 hrs lec.
Prerequisites: (21-270 or 70-492) and (21-259 or 21-268 or 21-269 or 21-256)
21-371 Functions of a Complex Variable
Intermittent: 9 units
This course provides an introduction to one of the basic topics of both pure and applied mathematics and is suitable for those with both practical and theoretical interests. Algebra and geometry of complex numbers; complex differentiation and integration. Cauchy's theorem and applications; conformal mapping; applications. 21-268 or 21-269 are recommended rather than 21-259. 3 hrs. lec.
Prerequisites: 21-355 or 21-235
21-372 Partial Differential Equations and Fourier Analysis
Intermittent: 9 units
This course provides an introduction to partial differential equations and is recommended for majors in mathematics, physical science, or engineering. Boundary value problems on an interval, Fourier series, uniform convergence, the heat, wave, and potential equations on bounded domains, general theory of eigenfunction expansion, the Fourier integral applied to problems on unbounded domains, introduction to numerical methods. 21-268 and 21-269 are recommended rather than 21-259; and 21-261 is recommended rather than 21-260. 3 hrs. lec.
Prerequisites: (21-268 or 21-269 or 21-259) and (21-260 or 21-261)
21-373 Algebraic Structures
Fall and Spring: 9 units
Groups: Homomorphisms. Subgroups, cosets, Lagrange's theorem. Conjugation. Normal subgroups, quotient groups, first isomorphism theorem. Group actions, Cauchy's Theorem. Dihedral and alternating groups. The second and third isomorphism theorems. Rings: Subrings, ideals, quotient rings, first isomorphism theorem. Polynomial rings. Prime and maximal ideals, prime and irreducible elements. PIDs and UFDs. Noetherian domains. Gauss' lemma. Eisenstein criterion. Fields: Field of fractions of an integral domain. Finite fields. Applications to coding theory, cryptography, number theory. 3 hrs lec.
Prerequisites: (21-128 or 21-127) and (21-241 or 21-242)
21-374 Field Theory
Spring: 9 units
The purpose of this course is to provide a successor to Algebraic Structures, with an emphasis on applications of groups and rings within algebra to some major classical problems. These include constructions with a ruler and compass, and the solvability or unsolvability of equations by radicals. It also offers an opportunity to see group theory and basic ring theory "in action", and introduces several powerful number theoretic techniques. The basic ideas and methods required to study finite fields will also be introduced. These ideas have recently been applied in a number of areas of theoretical computer science including primality testing and cryptography. 3 hrs. lec.
Prerequisite: 21-373
21-377 Monte Carlo Simulation for Finance
Intermittent: 9 units
first course in Monte Carlo simulation, with applications to Mathematical Finance. Students will put into practice many of the theoretical ideas introduced in Continuous Time Finance. Topics to be covered: random variable/stochastic process generation; options pricing; variance reduction; Markov chain Monte Carlo Methods.
Prerequisites: 21-325 Min. grade B or 21-420
21-378 Mathematics of Fixed Income Markets
Fall: 9 units
A first course in fixed income. Students will be introduced to the most common securities traded in fixed income markets and the valuation methods used to price them. Topics covered include discount factors; interest rates basics; pricing of coupon bonds; identifying the yield to maturity, as well as bond sensitivities to interest rates; term structure modeling; forward and swap rates; fixed income derivatives (including mortgage backed securities) and their valuation through backwards induction; fixed income indexes and return attribution. For a co-requisite, 36-225 can be accepted as an alternative for 21-325.
Prerequisite: 21-270 Min. grade B
21-380 Introduction to Mathematical Modeling
Intermittent: 9 units
This course shall examine mathematical models, which may be used to describe natural phenomena. Examples, which have been studied include: continuum description of highway traffic, discrete velocity models of a monotonic gas, chemotactic behavior in biological systems, European options pricing, and cellular-automata. Systems such as the first four are described by partial differential equations; the last involves discrete-time and discrete-phase dynamical systems, which have been used to successfully represent both physical and biological systems. The course will develop these models and then examine the behavior of the underlying systems, both analytically and numerically. The mathematical tools required will be developed in the course.
Prerequisites: (21-241 or 21-242) and (21-261 or 21-260)
21-393 Operations Research II
Fall: 9 units
Building on an understanding of Linear Programming developed in 21-292 Operations Research I, this course introduces more advanced topics. Integer programming, including cutting planes and branch and bound. Dynamic programming. An introduction to Combinatorial Optimization including optimal spanning trees, shortest paths, the assignment problem and max-flow/min-cut. The traveling salesman problem and NP-completeness. An important goal of this course is for the student to gain experience with the process of working in a group to apply operations research methods to solve a problem. A portion of the course is devoted to a group project based upon case studies and the methods presented. 36-410 recommended. 3 hrs. lec.
Prerequisites: (15-251 or 21-228) and 21-292
21-400 Intermediate Logic
Spring: 9 units
The course builds on the proof theory and model theory of first-order logic covered in 21-300. These are applied in 21-400 to Peano Arithmetic and its standard model, the natural numbers. The main results are the incompleteness, undefinability and undecidability theorems of Godel, Tarski, Church and others. Leading up to these, it is explained how logic is formalized within arithmetic, how this leads to the phenomenon of self-reference, and what it means for the axioms of a theory to be computably enumerable. Related aspects of computability theory are included to the extent that time permits.
Prerequisite: 21-300
21-420 Continuous-Time Finance
Spring: 9 units
This course begins with Brownian motion, stochastic integration,and Ito's formula from stochastic calculus. This theory is used to develop the Black-Scholes option pricing formula and the Black-Scholes partial differential equation. Additional topics may include models of credit risk, simulation, and expected utility maximization. 3 hrs lec.
Prerequisites: (21-260 or 18-202) and 21-370 and (21-325 or 36-225 or 36-217)
21-435 Applied Harmonic Analysis
Spring: 9 units
This course serves as a broad introduction to harmonic analysis and its applications, particularly in 1-dimensional signal processing and in image processing, for undergraduate students in mathematics, engineering, and the applied sciences. Topics include: Discrete Fourier transform and fast Fourier transform; Fourier series and the Fourier transform; Hilbert spaces and applications; Shannon sampling theorem, bandlimited functions, uncertainty principle; Wavelets and multi-resolution analysis; Applications in image processing.
Prerequisites: (21-355 or 21-235) and (21-241 or 21-242)
21-441 Number Theory
Fall: 9 units
Number theory deals with the integers, the most basic structures of mathematics. It is one of the most ancient, beautiful, and well-studied branches of mathematics, and has recently found surprising new applications in communications and cryptography. Course contents: Structure of the integers, greatest common divisiors, prime factorization. Modular arithmetic, Fermat's Theorem, Chinese Remainder Theorem. Number theoretic functions, e.g. Euler's function, Mobius functions, and identities. Diophantine equations, Pell's Equation, continued fractions. Modular polynomial equations, quadratic reciprocity. 3 hrs. lec.
Prerequisites: (21-242 or 21-241) and 21-373
21-465 Topology
Fall: 9 units
Metric spaces. Topological spaces. Separation axioms. Open, closed and compact sets. Continuous functions. Product spaces, subspaces, quotient spaces. Connectedness and path-connectedness. Homotopy. Fundamental group of a pointed space. Simply connected spaces. Winding number, the fundamental group of the circle. Functorial property of the fundamental group. Brouwer fixed point theorem. Covering spaces. van Kampen's theorem. 2-manifolds. Triangulations. Euler characteristic. Surgery, classification of compact 2-manifolds. 3 hrs lec.
Prerequisites: 21-355 and 21-373
21-467 Differential Geometry
9 units
This course will provide a thorough and rigorous introduction to differential geometry on manifolds. Contents: Differentiable manifolds; tangent spaces; vector fields and n-forms; integral curves; cotangent vectors; tensors; Riemannian metrics; connection; parallel transport; geodesics and convex neighborhoods; sectional, Ricci, scalar curvatures; tensors on Riemannian manifolds; Lie groups; transformation groups.
Prerequisites: 21-373 and 21-356
21-469 Numerical Methods II: Scientific Computing
Spring: 9 units
This course is the continuation for 21-369 and is centered on the mathematics of scientific computing and advanced numerical methods. The focus of this course is on numerical methods for partial differential equations, with an emphasis on computing and applications. This course is intended for undergraduate students in mathematics, engineering, and the applied sciences. Topics will include: Numerical methods for dynamical systems; numerical methods for linear partial differential equations; computational linear algebra; data fitting and approximation; computational methods for nonlinear problems.
Prerequisites: (21-235 or 21-355) and (21-261 or 21-260) and (21-242 or 21-241) and 21-272 and 21-369
21-470 Selected Topics in Analysis
Intermittent: 9 units
Typical of courses, which are offered from time to time are finite difference equations, calculus of variations, and applied control theory. The prerequisites will depend on the content of the course. 3 hrs. lec.
Prerequisites: 21-241 and 21-259 and 21-260
21-476 Introduction to Dynamical Systems
Intermittent: 9 units
This course is an introduction to differentiable dynamical systems. The material includes basic properties of dynamical systems, including the existence and uniqueness theory, continuation, singular points, orbits, and their classification. The Poincare'-Bendixson theorem and typical applications, like Lienard equations and Lotka-Volterra are also covered. An introduction to chaos as time permits. 3 hrs. lec.
Prerequisites: (21-241 or 21-242) and 21-261
21-484 Graph Theory
Spring: 9 units
Graph theory uses basic concepts to approach a diversity of problems and nontrivial applications in operations research, computer science and other disciplines. It is one of the very few mathematical areas where one is always close to interesting unsolved problems. Topics include graphs and subgraphs, trees, connectivity, Euler tours and Hamilton cycles, matchings, graph colorings, planar graphs and Euler's Formula, directed graphs, network flows, counting arguments, and graph algorithms. 3 hrs. lec.
Prerequisites: (15-251 or 21-228) and (21-242 or 21-241)
21-499 Undergraduate Research Topic
Fall: 9 units
This course affords undergraduates to pursue elementary research topics in the area of expertise of the instructor. The prerequisites will depend on the content of the course.
21-590 Practicum
All Semesters
Students in this course gain experience with the application of mathematical models to business and/or industrial problems during an internship. The internship is set up by the student in consultation with a faculty member. The students must also have a mentor at the firm providing the internship, who together with the faculty member develops a description of the goals of the internship. The internship must include the opportunity to learn about problems which have mathematical content.
21-599 Undergraduate Reading and Research
Fall and Spring
Individual reading courses or projects in mathematics and its applications. Prerequisites and units to be negotiated with individual instructors.
21-600 Mathematical Logic I
Fall: 12 units
The study of formal logical systems, which model the reasoning of mathematics, scientific disciplines, and everyday discourse. Propositional Calculus and First-order Logic. Syntax, axiomatic treatment, derived rules of inference, proof techniques, computer-assisted formal proofs, normal forms, consistency, independence, semantics, soundness, completeness, Lowenheim-Skolem Theorem, compactness, equality. 3 hrs. lec.
Prerequisites: 21-373 Min. grade B or 21-484 Min. grade B or 21-228 Min. grade B
21-602 Introduction to Set Theory I
Fall: 12 units
First order definability and the Zermelo-Fraenkel axioms; cardinal arithmetic, ordered sets, well-ordered sets (axiom of choice), transfinite induction, the filter of closed unbounded sets (Fodor, Ulm and Solovay's theorems), Delta systems, basic results in partition calculus (e.g., Ramsey's Theorem and the Erdos-Rado Theorem); small to medium large cardinals; applications to general topology (e.g., Alexandroff's conjecture), and the basic ideas of descriptive set theory. The independence of Suslin conjecture from the usual axioms. Godel's axiom of constructibility. Time permitting, the Galvin-Hajnal-Shelah inequality will be proved. 3 hrs. lec.
21-603 Model Theory I
Intermittent: 12 units
Similarity types, structures; downward Lowenheim Skolem theorem; construction of models from constants, Henkin's omitting types theory, prime models; elementary chains of models, basic two cardinal theorems, saturated models, basic results on countable models including Ryll-Nardzewski's theorem; indiscernible sequences, Ehrenfeucht-Mostowski models; introduction to stability, rank functions, primary models, and a proof of Morley's catagoricity theorem; basic facts about infinitary languages, computation of Hanf-Morley numbers.
21-604 Introduction to Recursion Theory
Intermittent: 12 units
Models of computation, computable functions, solvable and unsolvable problems, reducibilities among problems, recursive and recursively enumerable sets, the recursion theorem, Post's problem and the Friedberg-Muchnik theorem, general degrees and r.e. degrees, the arithmetical hierarchy, the hyper-arithmetical hierarchy, the analytical hierarchy, higher type recursion. 3 hrs. lec.
21-610 Algebra I
Spring: 12 units
The structure of finitely generated abelian groups, the Sylow theorems, nilpotent and solvable groups, simplicity of alternating and projective special linear groups, free groups, the Neilsen-Schreier theorem. Vector spaces over division rings, field extensions, the fundamental Galois correspondence, algebraic closure. The Jacobson radical and the structure of semisimple rings. Time permitting, one of the following topics will be included: Wedderburn's theorem on finite division rings, Frobenius' Theorem. Prerequisite: Familiarity with the content of an undergraduate course on groups and rings. 3 hrs. lec.
21-620 Real Analysis
Fall: 6 units
A review of one-dimensional, undergraduate analysis, including a rigorous treatment of the following topics in the context of real numbers: sequences, compactness, continuity, differentiation, Riemann integration. (Mini-course. Normally combined with 21-621.) 3 hrs. lec.
21-621 Introduction to Lebesgue Integration
Fall: 6 units
Construction of Lebesgue measure and the Lebesgue integral on the real line. Fatou's Lemma, the monotone convergence theorem, the dominated convergence theorem. (Mini-course. Normally combined with 21-620.) 3 hrs. lec.
21-630 Ordinary Differential Equations
All Semesters: 12 units
Basic concepts covered are existence and uniqueness of solutions, continuation of solutions, continuous dependence, and stability. For autonomous systems, topics included are: orbits, limit sets, Liapunov's direct method, and Poincar-Bendixson theory. For linear systems, topics included are: fundamental solutions, variation of constants, stability, matrix exponential solutions, and saddle points. Time permitting, one or more of the following topics will be covered: differential inequalities, boundary-value problems and Sturm-Liouville theory, Floquet theory.
21-632 Introduction to Differential Equations
Fall: 12 units
This course serves as a broad introduction to Ordinary and Partial Differential Equations for beginning graduate students and advanced undergraduate students in mathematics, engineering, and the applied sciences. Mathematical sophistication in real analysis at the level of 21-355/356 is assumed. Topics include: essentials of Ordinary Differential Equations, origins of Partial Differential Equations, the study of model problems including the Poisson and Laplace equations, the heat equation, the transport equation, and the wave equation. 3 hrs. lec.
21-640 Introduction to Functional Analysis
Spring: 12 units
Linear spaces: Hilbert spaces, Banach spaces, topological vector spaces. Hilbert spaces: geometry, projections, Riesz Representation Theorem, bilinear and quadratic forms, orthonormal sets and Fourier series. Banach spaces: continuity of linear mappings, Hahn-Banach Theorem, uniform boundedness, open-mapping theorem. Closed operators, closed graph theorem. Dual spaces: weak and weak-star topologies (Banach-Alaoglu Theorem), reflexivity. Space of bounded continuous functions and its dual. Linear operators and adjoints: basic properties, null spaces and ranges. Compact operators. Sequences of bounded linear operators: weak, strong and uniform convergence. Introduction to spectral theory: Notions of spectrum and resolvent set of bounded operators, spectral theory of compact operators. Time permitting: Fredholm Alternative. Time permitting: Stone-Weierstrass Theorem.
Prerequisites: 21-651 and (21-720 or 21-621)
21-651 General Topology
Fall: 12 units
Metric spaces: continuity, compactness, Arzela-Ascoli Theorem, completeness and completion, Baire Category Theorem. General topological spaces: bases and subbases, products, quotients, subspaces, continuity, topologies generated by sets of functions, homeomorphisms. Convergence: nets, filters, and the inadequacy of sequences. Separation: Hausdorff spaces, regular spaces, completely regular spaces, normal spaces, Urysohn's Lemma, Tietze's Extension Theorem. Connectedness. Countability conditions: first and second countability, separability, Lindelof property. Compactness: Tychonoff's Theorem, local compactness, one-point compactification. 3 hrs. lec.
21-660 Introduction to Numerical Analysis I
Spring: 12 units
Finite precision arithmetic, interpolation, spline approximation, numerical integration, numerical solution of linear and nonlinear systems of equations, optimization in finite dimensional spaces. 3 hrs. lec.
21-690 Methods of Optimization
Fall: 12 units
An introduction to the theory and algorithms of linear and nonlinear programming with an emphasis on modern computational considerations. The simplex method and its variants, duality theory and sensitivity analysis. Large-scale linear programming. Optimality conditions for unconstrained nonlinear optimization. Newton's method, line searches, trust regions and convergence rates. Constrained problems, feasible-point methods, penalty and barrier methods, interior-point methods.
21-700 Mathematical Logic II
Spring: 12 units
Higher-order logic (type theory). Syntax, Lambda-notation, Axioms of Description and Choice, computer-assisted formal proofs, semantics, soundness, standard and non-standard models, completeness, compactness, formalization of mathematics, definability of natural numbers, representability of recursive functions, Church's Thesis. Godel's Incompleteness Theorems, undecidability, undefinability.
Prerequisites: 21-600 or 21-300
Course Website: http://gtps.math.cmu.edu/description-700.txt
21-701 Discrete Mathematics
Intermittent: 12 units
Combinatorial analysis, graph theory with applications to problems in computational complexity, networks, and other areas.
21-720 Measure and Integration
Spring: 12 units
The Lebesgue integral, absolute continuity, signed measures and the Radon-Nikodym Theorem, Lp spaces and the Riesz Representation Theorem, product measures and Fubini's Theorem.
21-721 Probability
All Semesters: 12 units
Probability spaces, random variables, expectation, independence, Borel-Cantelli lemmas. Kernels and product spaces, existence of probability measures on infinite product spaces, Kolmogorov's zero-one law. Weak and strong laws of large numbers, ergodic theorems, stationary sequences. Conditional expectation: characterization, construction and properties. Relation to kernels, conditional distribution, density. Filtration, adapted and predictable processes, martingales, stopping times, upcrossing inequality and martingale convergence theorems, backward martingales, optional stopping, maximal inequalities. Various applications of martingales: branching processes, Polya's urn, generalized Borel-Cantelli, Levy's 0-1 law, martingale method, strong law of large numbers, etc. Weak convergence of probability measures, characteristic functions of random variables, weak convergence in terms of characteristic functions. Central limit theorem, Poisson convergence, Poisson process. Large deviations, rate functions, Cramer's Theorem.
Prerequisite: 21-720
21-723 Advanced Real Analysis
Intermittent: 12 units
This course is a sequel to 21-720 (Measure and Integration). It is meant to introduce students to a number of important advanced topics in analysis.Topics include: distributions, Fourier series and transform, Sobolev spaces, Bochner integration, basics of interpolation theory, integral transforms. 3 hrs. lec.Prerequisites: 21-720 Corequisites: 21-640
Prerequisite: 21-720
21-724 Sobolev Spaces
All Semesters: 12 units
Weak derivatives, Sobolev spaces of integer order, embedding theorems, interpolation inequalities, traces.
21-732 Partial Differential Equations I
All Semesters: 12 units
An introduction to the modern theory of partial differential equations. Including functional analytic techniques. Topics vary slightly from year to year, but generally include existence, uniqueness and regularity for linear elliptic boundary value problems and an introduction to the theory of evolution equations.
21-737 Probabilistic Combinatorics
Intermittent: 12 units
This course covers the probabilistic method for combinatorics in detail and introduces randomized algorithms and the theory of random graphs. Methods covered include the second moment method, the Rödl nibble, the Lovász local lemma, correlation inequalities, martingale's and tight concentration, Janson's inequality, branching processes, coupling and the differential equations method for discrete random processes. Objects studied include the configuration model for random regular graphs, Markov chains, the phase transition in the Erdős-Rényi random graph, and the Barabási-Albert preferential attachment model.
21-738 Extremal Combinatorics
All Semesters: 12 units
Classical problems and results in extremal combinatorics including the Turán and Zarankiewicz problems, the Erdős-Stone theorem and the Erdos-Simonovits stability theorem. Extremal set theory including the Erdős-Rado sunflower lemma and variations, VC-dimension, and Kneser's conjecture. The Szemeredi regularity lemma. Algebraic methods including finite field constructions and eigenvalues and expansion properties of graphs. Shannon capacity of graphs. Chromatic number of Rn and Borsuk's conjecture. Graph decomposition including Graham-Pollack and Baranyai's theorem.
21-832 Partial Differential Equations II
Spring: 12 units
Elliptic boundary value problems, Green's theorem calculations, integral equation methods, variational formulations and Galerkin's method, regularity theory, parabolic problems and semigroups.
21-901 Masters Degree Research
All Semesters
Missing Course Description - please contact the teaching department.

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, Associate Professor – Ph.D., Princeton University; Carnegie Mellon, 2012–.

CHARLES V. COFFMAN, Emeritus – Ph.D., Johns Hopkins University; Carnegie Mellon, 1962–.

CLINTON CONLEY, Assistant Professor – Ph.D., University of California Los Angeles; Carnegie Mellon, 2009–.

GERARD CORNUEJOLS, University Professor – Ph.D., Cornell University; Carnegie Mellon, 1978–.

JAMES CUMMINGS, Professor – Ph.D., Cambridge University; Carnegie Mellon, 1996–.

HASAN DEMIRKOPARAN, Associate Teaching Professor – Ph.D., Michigan State University; Carnegie Mellon, 2005–.

IRENE M. FONSECA, University Professor – Ph.D., University of Minnesota; Carnegie Mellon, 1987–.

TIMOTHY FLAHERTY, Associate Teaching Professor – Ph.D., University of Pittsburgh,; Carnegie Mellon, 1999–.

ALAN M. FRIEZE, University Professor – Ph.D., University of London; Carnegie Mellon, 1987–.

IRINA GHEORGHICIUC, Associate 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–.

YU GU, Assistant Professor – Ph.D., Columbia University; Carnegie Mellon, 2017–.

MORTON E. GURTIN, Emeritus – Ph.D., Brown University; Carnegie Mellon, 1966–.

DAVID HANDRON, Associate Teaching Professor – Ph.D., Rice University; Carnegie Mellon, 1999–.

JASON HOWELL, Associate Teaching Professor – Ph.D., Clemson University; Carnegie Mellon, 2017–.

WILLIAM J. HRUSA, Professor – Ph.D., Brown University; Carnegie Mellon, 1982–.

GAUTAM IYER, Associate Professor – Ph.D., University of Chicago; Carnegie Mellon, 2009–.

GREGORY JOHNSON, Assistant Teaching Professor – Ph.D., University of Maryland; Carnegie Mellon, 2009–.

NIRAJ KHARE, Visiting Assistant Professor – Ph.D., Ohio State University; Carnegie Mellon, 2014–.

DAVID KINDERLEHRER, Professor – Ph.D., University of California at Berkeley; Carnegie Mellon, 1990–.

DMITRY KRAMKOV, Professor – Ph.D., Steklov Mathematical Institute; Carnegie Mellon, 2000–.

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, Associate Professor – Ph.D., Princeton University; Carnegie Mellon, 2009–.

JOHN MACKEY, Teaching Professor – Ph.D., University of Hawaii; Carnegie Mellon, 2003–.

DANIELA MIHAI, Associate Teaching Professor – Ph.D., University of Pittsburgh; Carnegie Mellon, 2007–.

RICHARD A. MOORE, Emeritus – Ph.D., Washington University; Carnegie Mellon, 1956–.

JOHANNES MUHLE-KARBE, Associate Professor – PhD, Technical University of Munich; Carnegie Mellon, 2017–.

ROY A. NICOLAIDES, Professor – Ph.D., University of London; Carnegie Mellon, 1984–.

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, Associate Professor – Ph.D., Rutgers University; Carnegie Mellon, 2013–.

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–.

MARY RADCLIFFE, Shelly Postdoctoral Teaching Fellow – Ph.D., University of California at San Diego; Carnegie Mellon, 2015–.

HAYDEN SCHAEFFER, Assistant Professor – Ph.D., University of California at Los Angeles; Carnegie Mellon, 2015–.

JOHN W. SCHAEFFER, Professor – Ph.D., Indiana University; Carnegie Mellon, 1983–.

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, University Professor – Ph.D., University of Illinois; Carnegie Mellon, 1980–.

DEJAN SLEPCEV, 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 of Mathematics Emeritus – Ph.D., University of Paris; Carnegie Mellon, 1987–.

IAN TICE, Assistant Professor – Ph.D., New York University; Carnegie Mellon, 2012–.

TOMASZ TKOCZ, Assistant Professor – Ph.D., University of Warwick; Carnegie Mellon, 2017–.

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–.

ZELEALEM YILMA, Assistant Teaching Professor – Ph.D., Carnegie Mellon University; Carnegie Mellon, 2015–.