Department of Mathematical Sciences

Prasad Tetali, Department Head

Jason Howell, Associate Head

David Offner, Director of Undergraduate Studies

Ian Tice, Director of the Honors Degree Program

Location: Wean Hall 6113

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 optimization. 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

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, the department offers 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.

21-242 Matrix Theory is an honors version of 21-241 Matrices and Linear Transformations .
21-269 Vector Analysis is an honors version of 21-268 Multidimensional Calculus .

Admission to 21-242 Matrix Theory is based on an assessment exam taken at the start of the freshman year. Admission to 21-269 Vector Analysis is based on a student's performance in 21-242 Matrix Theory, and on other courses taken in the fall semester.

Mathematical Studies

The sequence of undergraduate honors courses continues with the Mathematical Studies courses, aimed primarily at sophomores. These highly demanding courses provide excellent preparation for graduate study, with many of the participants taking graduate courses as early as their junior year. Students will be expected to master material at a high level of abstraction, and to work on very challenging problems. The typical enrollment of about 15 students allows for close contact with the instructors.

21-235 Mathematical Studies Analysis I is an honors version of 21-355 Principles of Real Analysis I.
21-237 Mathematical Studies Algebra I is an honors version of 21-373 Algebraic Structures.
21-236 Mathematical Studies Analysis II is an honors version of 21-356 Principles of Real Analysis II.
21-238 Mathematical Studies Algebra II) is an honors version of 21-341 Linear Algebra.

Admission to Mathematical Studies is by invitation. Interested students should apply during the spring of their freshman year. Applicants are not absolutely required to have taken 21-242 Matrix Theory or 21-269 Vector Analysis, and may be admitted on the basis of exceptionally strong performance in non-honors mathematics courses.

It is possible to take only the algebra courses or only the analysis courses. Admission to 21-236 Mathematical Studies Analysis II requires a grade of B or better in 21-235 Mathematical Studies Analysis I, and similarly, admission to 21-238 Mathematical Studies Algebra II requires a grade of B or better in 21-237 Mathematical Studies Algebra I.

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 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.

Curriculum

We provide a list of the requirements for each concentration in the B.S. degree in Mathematical Sciences. Any exceptions to the elective requirements require prior approval from the student's academic advisor.

A student preparing for graduate study should also consider undertaking independent work. The Department offers 21-410 Research Topics in Mathematical Sciences and 21-599 Undergraduate Reading and Research for this purpose. At most, nine units of 21-410 or 21-599 credit can be applied toward depth elective requirements, and to do so requires prior approval from the student's academic advisor. Courses numbered 21-600 and above carry graduate credit, with courses at the 600-level designed as transitional courses to graduate study.

For each concentration, a suggested schedule that includes general education requirements can be found here. For a list of courses required for all Mellon College of Science students, see the MCS General Education Requirements.

By default, students must fulfill all of the requirements of the catalog of the year they entered CMU. Students who wish to be considered for a subsequent catalog may submit a request to their academic advisor.

B.S. in Mathematical Sciences

This program is the most flexible available to our majors, where students choose eight electives within the major and at least seven free electives, giving them the opportunity to design a program to suit their individual interests and goals.

The requirements for the B.S. in Mathematical Sciences are:

Mathematical Sciences Courses (required)

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

Courses		Units
21-120	Differential and Integral Calculus	10
21-122	Integration and Approximation	10
21-127	Concepts of Mathematics	12
or 21-128	Mathematical Concepts and Proofs
21-201	Undergraduate Colloquium	1
21-228	Discrete Mathematics	9-12
or 15-251	Great Ideas in Theoretical Computer Science
21-241	Matrices and Linear Transformations	11
or 21-242	Matrix Theory
21-259	Calculus in Three Dimensions	10-12
or 21-266	Vector Calculus for Computer Scientists
or 21-268	Multidimensional Calculus
or 21-269	Vector Analysis
21-260	Differential Equations	9-10
or 21-261	Introduction to Ordinary Differential Equations
or 33-231	Physical Analysis
21-325	Probability	9-12
or 15-259	Probability and Computing
or 36-218	Probability Theory for Computer Scientists
21-341	Linear Algebra	9
21-355	Principles of Real Analysis I	9-12
or 21-455	Intermediate Real Analysis I
21-356	Principles of Real Analysis II	9-10
or 21-456	Intermediate Real Analysis II
21-373	Algebraic Structures	9
		117-130

Computer Science Courses (required)

Courses		Units
15-110	Principles of Computing	10-12
or 15-112	Fundamentals of Programming and Computer Science
or 02-120	Undergraduate Programming for Scientists
		10-12

DEPTH ELECTIVES (REQUIRED)

Seventy-two total units

Forty-five units of Mathematical Sciences Electives (at the 21-300 level or above or 21-270 or 21-292).
Twenty-seven units of 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 be at the 36-300 level or above and have at least 36-225 as a prerequisite) electives.

MCS General Education (required)

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

Minimum number of units required for degree:360

Mathematical Sciences Electives for Students Intending Graduate Study

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
21-301	Combinatorics	9
21-326	Markov Chains: Theory, Simulation and Applications	9
21-360	Differential Geometry of Curves and Surfaces	9
21-371	Functions of a Complex Variable	9
21-374	Field Theory	9
21-441	Number Theory	9
21-484	Graph Theory	9
21-602	Introduction to Set Theory I	12
21-603	Model Theory I	12
21-610	Algebra I	12
21-623	Complex Analysis	12
21-624	Descriptive Set Theory	12
21-630	Ordinary Differential Equations	12
21-632	Introduction to Differential Equations	12
21-640	Introduction to Functional Analysis	12
21-651	General Topology	12
21-660	Introduction to Numerical Analysis I	12
21-701	Discrete Mathematics	12
21-720	Measure and Integration	12
21-721	Probability	12
21-723	Advanced Real Analysis	12
21-737	Probabilistic Combinatorics	12
21-738	Extremal Combinatorics	12

B.S. in Mathematical Sciences (Operations Research and Statistics)

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 curriculum typically includes group projects. Students choosing this concentration may not pursue an additional minor in Statistics in the Dietrich College of Humanities and Social Sciences.

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

Mathematical Sciences Courses (required)

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

Courses		Units
21-120	Differential and Integral Calculus	10
21-122	Integration and Approximation	10
21-127	Concepts of Mathematics	12
or 21-128	Mathematical Concepts and Proofs
21-201	Undergraduate Colloquium	1
21-228	Discrete Mathematics	9-12
or 15-251	Great Ideas in Theoretical Computer Science
21-241	Matrices and Linear Transformations	11
or 21-242	Matrix Theory
21-259	Calculus in Three Dimensions	10-12
or 21-266	Vector Calculus for Computer Scientists
or 21-268	Multidimensional Calculus
or 21-269	Vector Analysis
21-260	Differential Equations	9-10
or 21-261	Introduction to Ordinary Differential Equations
or 33-231	Physical Analysis
21-292	Operations Research I	9
21-369	Numerical Methods	12
21-393	Operations Research II	9
		102-108

Statistics Courses (required)

Courses		Units
21-325	Probability	9-12
or 15-259	Probability and Computing
or 36-218	Probability Theory for Computer Scientists
36-226	Introduction to Statistical Inference	9
36-401	Modern Regression	9
36-402	Advanced Methods for Data Analysis	9
36-410	Introduction to Probability Modeling	9
		45-48

Economics, Business, and Computer Science Courses (required)

Courses		Units
15-110	Principles of Computing	10
70-122	Introduction to Accounting	9
73-102	Principles of Microeconomics	9
73-103	Principles of Macroeconomics	9
73-230	Intermediate Microeconomics	9
or 73-240	Intermediate Macroeconomics
		46

Depth Electives (required)

Forty-five units of depth electives, to be chosen from the list below, or any other Mathematical Sciences Elective (at the 21-300 level or above or 21-270 or 21-292). At least nine of these units must be Mathematical Sciences Electives (21-XXX)

The courses 21-355 and 21-455 are particularly recommended for a student planning to pursue graduate work.

Courses		Units
10-301	Introduction to Machine Learning	12
or 10-315	Introduction to Machine Learning (SCS Majors)
15-122	Principles of Imperative Computation	12
15-150	Principles of Functional Programming	12
15-210	Parallel and Sequential Data Structures and Algorithms	12
21-236	Mathematical Studies Analysis II	12
21-270	Introduction to Mathematical Finance	9
21-301	Combinatorics	9
21-321	Interactive Theorem Proving	9
21-326	Markov Chains: Theory, Simulation and Applications	9
21-341	Linear Algebra	9
21-355	Principles of Real Analysis I	9-12
or 21-455	Intermediate Real Analysis I
21-356	Principles of Real Analysis II	9-10
or 21-456	Intermediate Real Analysis II
21-370	Discrete Time Finance	9
21-373	Algebraic Structures	9
21-378	Mathematics of Fixed Income Markets	9
21-420	Continuous-Time Finance	9
21-484	Graph Theory	9
36-46X	Special Topics (Statistics)	9-12
36-47X	Special Topics (Statistics)	9-12
70-371	Operations Management	9
70-460	Mathematical Models for Consulting	9
70-467	Machine Learning for Business Analytics	9
70-469	End to End Business Analytics	9
70-471	Supply Chain Management	9

MCS General Education (required)

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

Note that 73-102, 73-103, 73-230, and 73-240 satisfy Nontechnical Elective requirements from the MCS general education core.

Minimum number of units required for degree:360

B.S. in Mathematical Sciences (Statistics)

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 Statistics concentration is jointly administered by the Department of Mathematical Sciences and the Department of Statistics and Data Science. Students choosing this concentration may not pursue an additional minor in Statistics in the Dietrich College of Humanities and Social Sciences.

The requirements for the Statistics concentration are:

Mathematical Sciences Courses (required)

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

Courses		Units
21-120	Differential and Integral Calculus	10
21-122	Integration and Approximation	10
21-127	Concepts of Mathematics	12
or 21-128	Mathematical Concepts and Proofs
21-201	Undergraduate Colloquium	1
21-228	Discrete Mathematics	9-12
or 15-251	Great Ideas in Theoretical Computer Science
21-241	Matrices and Linear Transformations	11
or 21-242	Matrix Theory
21-259	Calculus in Three Dimensions	10-12
or 21-266	Vector Calculus for Computer Scientists
or 21-268	Multidimensional Calculus
or 21-269	Vector Analysis
21-260	Differential Equations	9-10
or 21-261	Introduction to Ordinary Differential Equations
or 33-231	Physical Analysis
21-292	Operations Research I	9
21-369	Numerical Methods	12
21-393	Operations Research II	9
		102-108

Statistics Courses (required)

Courses		Units
21-325	Probability	9-12
or 15-259	Probability and Computing
or 36-218	Probability Theory for Computer Scientists
36-226	Introduction to Statistical Inference	9
36-401	Modern Regression	9
36-402	Advanced Methods for Data Analysis	9
36-410	Introduction to Probability Modeling	9
		45-48

Economics and Computer Science Courses (required)

Courses		Units
15-112	Fundamentals of Programming and Computer Science	12
or 02-120	Undergraduate Programming for Scientists
15-122	Principles of Imperative Computation	12
73-102	Principles of Microeconomics	9
		33

Depth Electives (required)

The courses 21-355 and 21-455 are particularly recommended for a student planning to pursue graduate work.

Courses		Units
10-301	Introduction to Machine Learning	12
or 10-315	Introduction to Machine Learning (SCS Majors)
15-150	Principles of Functional Programming	12
15-210	Parallel and Sequential Data Structures and Algorithms	12
21-270	Introduction to Mathematical Finance	9
21-321	Interactive Theorem Proving	9
21-326	Markov Chains: Theory, Simulation and Applications	9
21-341	Linear Algebra	9
21-355	Principles of Real Analysis I	9-12
or 21-455	Intermediate Real Analysis I
21-356	Principles of Real Analysis II	9-10
or 21-456	Intermediate Real Analysis II
21-370	Discrete Time Finance	9
21-373	Algebraic Structures	9
21-378	Mathematics of Fixed Income Markets	9
21-420	Continuous-Time Finance	9
21-484	Graph Theory	9
36-46X	Special Topics (Statistics)	9-12
36-47X	Special Topics (Statistics)	9-12

MCS General Education (required)

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

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

Minimum number of units required for degree:360

B.S. in Mathematical Sciences (Discrete Mathematics and Logic)

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 rigorous 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.

Courses		Units
15-122	Principles of Imperative Computation	12
15-150	Principles of Functional Programming	12
15-210	Parallel and Sequential Data Structures and Algorithms	12
21-120	Differential and Integral Calculus	10
21-122	Integration and Approximation	10
21-127	Concepts of Mathematics	12
or 21-128	Mathematical Concepts and Proofs
21-201	Undergraduate Colloquium	1
21-228	Discrete Mathematics	9
or 21-301	Combinatorics
21-241	Matrices and Linear Transformations	11
or 21-242	Matrix Theory
21-300	Basic Logic	9
21-341	Linear Algebra	9
21-355	Principles of Real Analysis I	9-12
or 21-455	Intermediate Real Analysis I
21-373	Algebraic Structures	9
		125-128

Computer Science Electives (required)

Any two courses at the 300 level or above. The following are specifically suggested:
15-312	Foundations of Programming Languages	12
15-317	Constructive Logic	9
15-451	Algorithm Design and Analysis	12

Students pursuing this concentration who minor in Computer Science must take at least 18 units of 15-300 level (or above) courses to avoid excessive double-counting.

Depth Electives (required)

Seventy-two units of depth electives, to be chosen from the two lists below, or any other Mathematical Sciences Elective (at the 21-300 level or above or 21-270 or 21-292). At least thirty-six of these units must be Discrete Mathematics and Logic Electives (List 1).

List 1 (Discrete Mathematics and Logic Electives)

Courses		Units
15-251	Great Ideas in Theoretical Computer Science	12
or 21-301	Combinatorics
21-321	Interactive Theorem Proving	9
21-322	Topics in Formal Mathematics	9
21-325	Probability	9
or 15-259	Probability and Computing
21-329	Set Theory	9
21-374	Field Theory	9
21-400	Intermediate Logic	9
21-441	Number Theory	9
21-484	Graph Theory	9
21-602	Introduction to Set Theory I	12
21-603	Model Theory I	12
21-610	Algebra I	12
21-701	Discrete Mathematics	12
80-305	Game Theory	9
80-311	Undecidability and Incompleteness	9
80-411	Proof Theory	9
80-413	Category Theory	9

List 2 (Mathematics Electives)

Courses		Units
21-259	Calculus in Three Dimensions	10-12
or 21-266	Vector Calculus for Computer Scientists
or 21-268	Multidimensional Calculus
or 21-269	Vector Analysis
21-260	Differential Equations	9-10
or 21-261	Introduction to Ordinary Differential Equations
or 33-231	Physical Analysis
21-270	Introduction to Mathematical Finance	9
21-292	Operations Research I	9
21-326	Markov Chains: Theory, Simulation and Applications	9
21-356	Principles of Real Analysis II	9-10
or 21-456	Intermediate Real Analysis II
21-366	Topics in Applied Mathematics	9
21-369	Numerical Methods	12
21-370	Discrete Time Finance	9
21-371	Functions of a Complex Variable	9
21-393	Operations Research II	9
21-420	Continuous-Time Finance	9
21-410	Research Topics in Mathematical Sciences	9

MCS General Education (required)

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

Minimum number of units required for degree:360

B.S. in Mathematical Sciences (Computational and Applied Mathematics)

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
21-120	Differential and Integral Calculus	10
21-122	Integration and Approximation	10
21-127	Concepts of Mathematics	12
or 21-128	Mathematical Concepts and Proofs
21-201	Undergraduate Colloquium	1
21-228	Discrete Mathematics	9-12
or 15-251	Great Ideas in Theoretical Computer Science
21-241	Matrices and Linear Transformations	11
or 21-242	Matrix Theory
21-259	Calculus in Three Dimensions	10-12
or 21-266	Vector Calculus for Computer Scientists
or 21-268	Multidimensional Calculus
or 21-269	Vector Analysis
21-260	Differential Equations	9-10
or 21-261	Introduction to Ordinary Differential Equations
or 33-231	Physical Analysis
21-325	Probability	9-12
or 15-259	Probability and Computing
or 36-218	Probability Theory for Computer Scientists
21-355	Principles of Real Analysis I	9-12
or 21-455	Intermediate Real Analysis I
21-369	Numerical Methods	12
21-469	Computational Introduction to Partial Differential Equations	12
		114-126

Computer Science Courses (required)

Courses		Units
15-122	Principles of Imperative Computation	12

Depth Electives (required)

Sixty-three total units

Twenty-seven of these units must be Computational and Applied Mathematics (List 1)
Twenty-seven of these units must be Computational and Applied Mathematics Electives (List 1) or Mathematics Electives (List 2) or any other Mathematical Sciences Elective (at the 21-300 level or above or 21-270 or 21-292).
Nine units must be from the two lists below, or 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 be at the 36-300 level or above and have at least 36-225 as a prerequisite).

List 1 (Computational and Applied Mathematics Electives)

Courses		Units
10-301	Introduction to Machine Learning	12
or 10-315	Introduction to Machine Learning (SCS Majors)
21-270	Introduction to Mathematical Finance	9
21-292	Operations Research I	9
21-326	Markov Chains: Theory, Simulation and Applications	9
21-344	Numerical Linear Algebra	9
21-380	Introduction to Mathematical Modeling	9

List 2 (Mathematics Electives)

Courses		Units
21-321	Interactive Theorem Proving	9
21-341	Linear Algebra	9
21-356	Principles of Real Analysis II	9-10
or 21-456	Intermediate Real Analysis II
21-370	Discrete Time Finance	9
21-371	Functions of a Complex Variable	9
21-373	Algebraic Structures	9
21-378	Mathematics of Fixed Income Markets	9
21-393	Operations Research II	9
21-420	Continuous-Time Finance	9
21-484	Graph Theory	9
21-632	Introduction to Differential Equations	12
21-640	Introduction to Functional Analysis	12
21-651	General Topology	12
21-660	Introduction to Numerical Analysis I	12
21-690	Methods of Optimization	12
21-720	Measure and Integration	12
21-721	Probability	12
21-723	Advanced Real Analysis	12
21-732	Partial Differential Equations I	12

MCS General Education (required)

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

Minimum number of units required for degree:360

B.A. in Mathematical Sciences

Mathematical Sciences Courses (required)

21-120	Differential and Integral Calculus	10
21-122	Integration and Approximation	10
21-127	Concepts of Mathematics	12
or 21-128	Mathematical Concepts and Proofs
21-201	Undergraduate Colloquium	1
21-228	Discrete Mathematics	9-12
or 15-251	Great Ideas in Theoretical Computer Science
21-241	Matrices and Linear Transformations	11
or 21-242	Matrix Theory
21-259	Calculus in Three Dimensions	10-12
or 21-266	Vector Calculus for Computer Scientists
or 21-268	Multidimensional Calculus
or 21-269	Vector Analysis
21-260	Differential Equations	9-10
or 21-261	Introduction to Ordinary Differential Equations
or 33-231	Physical Analysis
21-325	Probability	9-12
or 15-259	Probability and Computing
or 36-218	Probability Theory for Computer Scientists

Depth Electives (required)

Seventy-two total units

Forty-five units of Mathematical Sciences Electives (at the 21-300 level or above; or 21-270 or 21-292).
Twenty-seven units of 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 (at the 36-300 level or above; and have at least 36-225 as a prerequisite) electives.

MCS General Education (required)

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

Minimum number of units required for degree:360

Additional Major Requirements

All concentrations within the B.S. in Mathematical Sciences are available as an additional major to students majoring in other departments. The requirements for the additional majors are the same as those for the B.S degrees, except that the MCS General Education requirements are waived, along with the requirement to take 21-201. In order to avoid double-counting issues, students are encouraged to consult with their academic advisor for their primary degree as well as their additional major advisor. Please visit the Department of Mathematical Sciences Undergraduate FAQ website (under "Admissions") for further details

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. Please see below if you are a Computational Finance major.

21-127	Concepts of Mathematics	12
or 21-128	Mathematical Concepts and Proofs
or 15-151	Mathematical Foundations for Computer Science
21-228	Discrete Mathematics	9-12
or 15-251	Great Ideas in Theoretical Computer Science
21-241	Matrices and Linear Transformations	11
or 21-242	Matrix Theory
21-355	Principles of Real Analysis I	9-12
or 21-455	Intermediate Real Analysis I
21-xxx	Mathematical Sciences Elective (300-level or higher)	9-12
21-xxx	Mathematical Sciences Elective (300-level or higher)	9-12

To avoid excessive double-counting, the two mathematical sciences electives may not also count toward any other major or minor requirement.

Computational Finance majors who declare a minor in Mathematical Sciences should take the following six courses:

Required courses are:

21-127	Concepts of Mathematics	12
or 21-128	Mathematical Concepts and Proofs
or 15-151	Mathematical Foundations for Computer Science
21-228	Discrete Mathematics	9-12
or 15-251	Great Ideas in Theoretical Computer Science
21-241	Matrices and Linear Transformations	11
or 21-242	Matrix Theory
21-355	Principles of Real Analysis I	9
21-325	Probability	9-12
or 15-259	Probability and Computing
or 36-218	Probability Theory for Computer Scientists

Nine units of Mathematical Sciences Electives, to be chosen from the following list:

21-300	Basic Logic	9
21-301	Combinatorics	9
21-329	Set Theory	9
21-373	Algebraic Structures	9
21-484	Graph Theory	9

*Students who take 21-325 (or 15-259 or 36-218) to fulfill their BSCF requirements should take an additional 21-3xx elective to avoid excessive double counting.

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:

21-228	Discrete Mathematics ¹	9
or 21-301	Combinatorics
21-300	Basic Logic	9

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

Thirty-six units of Mathematical Sciences Electives, to be chosen from the following two groups (at least nine units from each group).

Logic
21-321	Interactive Theorem Proving	9
21-329	Set Theory	9
21-400	Intermediate Logic	9
21-602	Introduction to Set Theory I	12
21-603	Model Theory I	12
80-305	Game Theory	9
80-311	Undecidability and Incompleteness	9
80-315	Logics for Knowledge and Belief	9
80-411	Proof Theory	9
80-413	Category Theory	9

Algebra and Discrete Mathematics
21-341	Linear Algebra	9
21-373	Algebraic Structures	9
21-374	Field Theory	9
21-441	Number Theory	9
21-484	Graph Theory	9
21-610	Algebra I	12
21-701	Discrete Mathematics	12

To avoid excessive double-counting, at least two mathematical sciences electives may not also count toward any other major or minor requirement.

The Honors Degree Program

This demanding program qualifies the student for an additional degree, the Master of Science in Mathematical Sciences. Admission to the Honors Degree Program is selective and interested students should apply for admission during their junior year. In the application process, the Department will hold to the same high standards which apply to admission to any graduate program. Applicants are not absolutely required to have taken the Mathematical Studies courses and may be admitted on the basis of exceptionally strong performance in non-honors mathematics courses or of accomplishments in research. Applicants are expected to have completed the Mathematical Studies sequences in algebra and analysis or 21-355/21-356 and 21-373/21-341 prior to application. 21-455/21-456 may be taken in place of 21-355/21-356.

In order to complete the Honors Degree Program, students must complete five mathematics graduate courses with grades of B or better and write an honors thesis. At the time of admission, students will declare a timetable on which they plan to take the graduate courses, do the research required for the thesis, and write up their work: this timetable can naturally be adjusted as required. At most, one of these five graduate courses may be applied towards the student's bachelor's degree program.

At least three graduate courses must come from this list of introductory courses:

21-602 Introduction to Set Theory I
21-603 Model Theory I
21-610 Algebra I
21-632 Introduction to Differential Equations
21-640 Introduction to Functional Analysis
21-651 General Topology
21-701 Discrete Mathematics
21-720 Measure and Integration
21-721 Probability
21-737 Probabilistic Combinatorics

By special permission of the department, one graduate course with sufficient mathematical content offered in another department may be counted. The honors thesis may either be research-based or expository: expository theses must be at a high mathematical level, at least that of a second-year graduate course. Students should plan on finding a thesis advisor by the end of their junior year. Students are required to take 21-901 Master's Degree Research during their senior year, subject to the following conditions:

Students must pass a minimum of 15 units of 21-901 to earn the M.S. in Mathematical Sciences.
Students who have not defended their thesis by the Add-Course-Deadline during each of their last two semesters must register for a minimum of three units of 21-901 for that semester.
Students may not overload more than 66 units while taking 21-901.

The Master of Science in Mathematical Sciences may be earned together with a bachelor of science from another department.

Course Descriptions

About Course Numbers:

Each Carnegie Mellon course number begins with a two-digit prefix that designates the department offering the course (i.e., 76-xxx courses are offered by the Department of English). 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. Depending on the department, xx-6xx courses may be either undergraduate senior-level or graduate-level, and xx-7xx courses and higher are graduate-level. Consult the Schedule of Classes each semester for course offerings and for any necessary pre-requisites or co-requisites.

21-090 Precalculus: Fall: 10 units
Extensive treatment of topics chosen to prepare students for the study of calculus. Preliminary topics include algebraic techniques and linear and nonlinear inequalities. Special emphasis is given to polynomial, rational, exponential, logarithmic, trigonometric and inverse trigonometric functions and their graphs, as well as basic and analytic trigonometry. (Three 50 minute lectures, two 50 minute recitations)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-102 Exploring Modern Mathematics: Fall and Spring: 9 units
This course is designed for non-math majors who are interested in learning some contemporary applications of mathematics with minimal prerequisite math knowledge. The course will survey the mathematical concepts centered along various themes, which may include the mathematics of social choice (voting and apportionment systems), topics in management science (optimization and elementary graph theory), modeling growth systems (population and finance), shape and form (symmetry and fractals), basic applications of probability and counting, and basic applications of number theory (cryptography and coding theory). Additional topics may be presented at the discretion of the instructor.

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-108 Introduction to Mathematical Concepts: Fall: 6 units
This course is an introduction to the vocabulary necessary for understanding and proving mathematical statements. The topics in this course include integers, rational numbers, polynomials, divisibility of numbers and polynomials, basic logic, sets, relations, functions, rule of sum, and rule of product. (Three 50 minute lectures, two 50 minute recitations)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-111 Differential Calculus: Fall: 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. (Three 50 minute lectures, two 50 minute recitations)
Prerequisite: 21-090
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures, two 50 minute recitations)
Prerequisite: 21-111
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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 approximations, maximum-minimum problems, inverse functions, definite and indefinite integrals; integration by substitution and by parts. Applications of integration, as time permits. (Three 50 minute lectures, two 50 minute recitations)
Prerequisite: 21-090
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-122 Integration and Approximation: All Semesters: 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. (Three 50 minute lectures, two 50 minute recitations)
Prerequisites: 21-120 or 21-112
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures, two 50 minute recitations)
Prerequisites: 21-120 or 21-112
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-127 Concepts of Mathematics: All Semesters: 12 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. (Three 50 minute lectures, two 50 minute recitations)
Prerequisites: 15-112 Min. grade C or 21-108 Min. grade C or 21-112 Min. grade C or 21-120 Min. grade C or 02-120 Min. grade C

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-128 Mathematical Concepts and Proofs: Fall: 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. (Three 50 minute lectures, two 50 minute recitations)
Prerequisites: 02-120 Min. grade C or 21-120 Min. grade C or 21-108 Min. grade C or 15-112 Min. grade C or 21-112 Min. grade C

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-150 Mathematics and the Arts: Intermittent: 9 units
Mathematics and the creative arts have long, interlinked histories. This course touches upon a broad range of mathematical ideas and the writers, artists, and art movements that were influenced and inspired by them. Topics include the use of geometric patterns in Islamic art, the influence of non-Euclidean geometry on Cubism, the constrained writing experiments of the Oulipo, and literary works exploring the concept of infinity.

21-201 Undergraduate Colloquium: Fall and Spring: 1 unit
The purpose of this course is to introduce math majors to the different degree programs in Mathematical Sciences, and to inform math majors about relevant topics such as advising, math courses, graduate schools, and typical career paths in the mathematical sciences. The Career and Professional Development Center will present modules on professional communication, developing interview and networking skills, and preparing for career fairs. (One 50 minute session)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-228 Discrete Mathematics: Fall and 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. (Three 50 minute lectures, one 50 minute recitation)
Prerequisites: 15-151 or 21-127 or 21-128
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-235 Mathematical Studies Analysis I: Fall: 12 units
A component of the honors program, 21-235 is a more demanding version of 21-355 of greater scope. Topics to be covered typically include: metric spaces, normed spaces, and inner product spaces; further properties of metric spaces such as completions, density, compactness, and connectedness; limits and continuity of maps between metric spaces, homeomorphisms, extension theorems, contraction mappings, extreme and intermediate value theorems; convergence of sequences and series of functions; metric spaces of functions, sequences, and metric subsets; Stone-Weierstrass and Arzela-Ascoli theorems; Baire category and applications; infinite series in normed spaces, convergence tests, and power series; differential calculus of maps between normed spaces, inverse and implicit function theorems in Banach spaces; existence results in ordinary differential equations. The prerequisite sequence 21-128, 21-242, 21-269 is particularly recommended. (Three 50 minute lectures, one 50 minute recitation)
Prerequisites: 21-268 or 21-256 or 21-269 or 21-259
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-236 Mathematical Studies Analysis II: Spring: 12 units
A component of the honors program, 21-236 is a more demanding version of 21-356 of greater scope. Topics to be covered typically include: Lebesgue measure in Euclidean space, measurable functions, the Lebesgue integral, integral limit theorems, Fubini-Tonelli theorem, and change of variables; Lebesgue spaces, completeness, approximation, and embeddings; absolutely continuous functions, functions of bounded variation, and curve lengths; differentiable submanifolds of Euclidean space, tangent spaces, mappings between manifolds, vector and tensor fields, manifolds with boundary and orientations; differential forms, integration of forms, Stokes' theorem; Hausdorff measure, divergence theorem. (Three 50 minute lectures, one 50 minute recitation)
Prerequisite: 21-235 Min. grade B

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-237 Mathematical Studies Algebra I: Fall: 12 units
A component of the honors program, 21-237 is a more demanding version of 21-373 (Algebraic Structures) of greater scope. 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, focusing on groups and rings. Group-theoretic topics to be covered include: homomorphisms, subgroups, cosets, Lagrange's theorem, conjugation, normal subgroups, quotient groups, isomorphism theorems, automorphism groups, characteristic subgroups, group actions, Cauchy's theorem, Sylow's theorem, normalisers, centralisers, class equation, finite p-groups, permutation and alternating groups, direct and semidirect products, simple groups, subnormal series, the Jordan-Holder theorem. Ring-theoretic topics include: subrings, ideals, quotient rings, isomorphism theorems, polynomial rings, Zorn's Lemma, prime and maximal ideals, prime and irreducible elements, factorization, PIDs and UFDs, Noetherian domains, the Hilbert Basis Theorem, Gauss' lemma and the Eisenstein criterion for irreducibility, fields of fractions, properties of polynomial rings over fields and UFDs, finite fields and applications. The prerequisite sequence of 21-128, 21-242, 21-269 is particularly recommended. (Three 50 minute lectures, one 50 minute recitation)
Prerequisites: (21-128 or 21-127 or 15-151) and (21-268 or 21-269) and (21-242 or 21-241)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-238 Mathematical Studies Algebra II: Spring: 12 units
A component of the honors program, 21-238 is a more demanding version of 21-341 (Linear Algebra) of greater scope. 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 covers vector spaces over arbitrary fields and the natural generalization to modules over rings. Vector space topics to be covered include: fields, Zorn's Lemma, vector spaces (possibly infinite dimensional) over an arbitrary field, independent sets, bases, existence of a basis, exchange lemma, dimension. Linear transformations, dual space, multilinear maps, tensor products, exterior powers, the determinant, eigenvalues, eigenvectors, characteristic and minimal polynomial of a transformation, the Cayley-Hamilton theorem. Module-theoretic topics to be covered include: review of (commutative) rings, R-modules, sums and quotients of modules, free modules, the structure theorem for finitely generated modules over a PID, Jordan and rational canonical forms, structure theory of finitely generated abelian groups. Further topics in real and complex inner product spaces include: orthonormal sets, orthonormal bases, the Gram-Schmidt process, symmetric/Hermitian operators, orthogonal/unitary operators, the spectral theorem, quadratic forms, the singular value decomposition. Possible additional topics: applications to combinatorics, category theory, representations of finite groups, unitary representations of infinite groups. (Three 50 minute lectures, one 50 minute recitation)
Prerequisite: 21-237 Min. grade B

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures, one 50 minute recitation)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-241 Matrices and Linear Transformations: All Semesters: 11 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. (Three 50 minute lectures, two 50 minute recitations)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-242 Matrix Theory: Fall: 11 units
A component of the honors program, 21-242 is a more demanding version of 21-241 (Matrix Algebra and Linear Transformations), of greater scope, with increased emphasis placed on rigorous 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. (Three 50 minute lectures, two 50 minute recitations)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-254 Linear Algebra and Vector Calculus for Engineers: All Semesters: 11 units
This course will introduce the fundamentals of vector calculus and linear algebra. The topics include vector and matrix operations, determinants, linear systems, matrix eigenvalue problems, vector differential calculus including gradient, divergence, curl, and vector integral calculus including line, surface, and volume integral theorems. Lecture and assignments will emphasize the applications of these topics to engineering problems. (Three 50 minute lectures, one 50 minute recitation)
Prerequisite: 21-122
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures, one 50 minute recitation)
Prerequisites: 21-120 or 21-112
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-257 Models and Methods for Optimization: Intermittent: 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. (Three 50 minute lectures, one 50 minute recitation)
Prerequisites: 21-256 or 21-242 or 21-241 or 21-240 or 18-202 or 06-262

21-259 Calculus in Three Dimensions: All Semesters: 10 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. (Three 50 minute lectures, two 50 minute recitations)
Prerequisite: 21-122
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-260 Differential Equations: All Semesters: 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. (Three 50 minute lectures, one 50 minute recitation)
Prerequisite: 21-122
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-261 Introduction to Ordinary Differential Equations: Fall: 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. Corequisites: (21-268 or 21-269 or 21-259) and (21-241 or 21-242). (Three 50 minute lectures, one 50 minute recitation)
Prerequisite: 21-122
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-266 Vector Calculus for Computer Scientists: Spring: 10 units
This course is an introduction to vector calculus making use of techniques from linear algebra. Topics covered include scalar-valued and vector-valued functions, conic sections and quadric surfaces, new coordinate systems, partial derivatives, tangent planes, the Jacobian matrix, the chain rule, gradient, divergence, curl, the Hessian matrix, linear and quadratic approximation, local and global extrema, Lagrange multipliers, multiple integration, parametrised curves, line integrals, conservative vector fields, parametrised surfaces, surface integrals, Green's theorem, Stokes's theorem and Gauss's theorem. (Three 50 minute lectures, one 50 minute recitation)
Prerequisites: 21-122 and (21-242 or 21-241 Min. grade C)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-268 Multidimensional Calculus: Spring: 11 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. (Three 50 minute lectures, two 50 minute recitations)
Prerequisites: 21-122 and (21-241 or 21-242) and (15-151 or 21-128 or 21-127)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-269 Vector Analysis: Spring: 12 units
A component of the honors program, 21-269 is a more demanding version of 21-268 of greater scope, with greater emphasis placed on rigorous proofs. Topics to be covered typically include: the real field, sups, infs, and completeness; geometry and topology of metric spaces; limits, continuity, and derivatives of maps between normed spaces; inverse and implicit function theorems, higher derivatives, Taylor's theorem, extremal calculus, and Lagrange multipliers. Integration. Iterated integration and change of variables. (Three 50 minute lectures, two 50 minute recitations)
Prerequisites: 21-122 and (21-127 or 15-151) and (21-241 Min. grade A or 21-242 Min. grade B)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures)
Prerequisites: 21-120 or 21-112
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures, one 50 minute recitation)
Prerequisites: 21-122 and (21-241 or 21-240 or 21-242) and (21-228 or 15-251)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (One 50 minute session)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures)
Prerequisites: 15-251 or 21-228 or 21-373
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures)
Prerequisites: 21-122 and (21-228 or 15-251)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-321 Interactive Theorem Proving: Fall: 9 units
Computational proof assistants now make it possible to work interactively to write mechanically verified definitions, theorems, and proofs. Important theorems have been formalized in this way, and digital libraries are being developed collaboratively by the mathematical community. Formalization of mathematics also serves as a gateway to the use of new technologies for discovery, such as automated reasoning and machine learning. This course will teach you how to formalize mathematics so that you, too, can contribute to the effort. We will explore a logical framework, dependent type theory, which serves as a practical foundation in a number of proof assistants. Finally, as time allows, we will explore ways of automating various aspects of mathematical reasoning. (Three 50 minute lectures)
Prerequisites: 21-127 or 15-151 or 21-128
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-322 Topics in Formal Mathematics: Intermittent: 9 units
Complementary to 21-321 Interactive Theorem Proving, this course is designed to present special topics on the use of formalization and formal methods in mathematics. For example, the course might focus on formalization of a specific branch of mathematics, or on a specific method for automation or computational reasoning.
Prerequisites: (21-128 or 15-151 or 21-127) and (21-259 or 21-269 or 21-268 or 21-266)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-325 Probability: All Semesters: 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. (Three 50 minute lectures)
Prerequisites: (15-150 or 21-128 or 21-127) and (21-269 or 21-268 or 21-256 or 21-259)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-326 Markov Chains: Theory, Simulation and Applications: Intermittent: 9 units
This course is a natural sequel to 21-325 (Probability) covering Markov Chains and several applications. Markov Chains arise in several situations that most of us use in everyday life (e.g. next word prediction on our smartphone keyboards, and ranking results in internet searches). This course covers the basics Markov chains in a mathematically rigorous way, the basics of sampling and MCMC (Markov Chain Monte Carlo), and analyzes several applications that arise in CS (counting matchings, pagerank, graph coloring), statistical physics (Ising model, hard core model) and optimization (traveling salesman). While several applications will be implemented on a computer (using numerical Python), students will be given starter code and only be asked to implement key mathematical steps of the algorithm. (Three 50 minute lectures)
Prerequisites: (15-112 or 02-120) and (15-259 or 21-325) and (21-240 or 21-242 or 21-241)

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. (Three 50 minute lectures)
Prerequisites: 21-128 or 15-151 or 21-127
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-341 Linear Algebra: Fall and Spring: 9 units
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. (Three 50 minute lectures)
Prerequisites: (21-241 or 21-242) and 21-373
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-344 Numerical Linear Algebra: Spring: 9 units
An introduction to algorithms pertaining to matrices and large linear systems of equations. Direct methods for large sparse problems including graph data structures, maximum matchings, row and column orderings, and pivoting strategies. Iterative methods including Conjugate Gradient and GMRES, with a discussion of preconditioning strategies. Additional topics include: computation of eigenvalues and eigenvectors, condition numbers, the QR and singular value decompositions, least-squares systems. (Three 50 minute lectures)
Prerequisites: (15-112 or 02-120) and (21-241 or 21-240 or 21-242) and (21-269 or 21-268 or 21-259)

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 course presumes some mathematical sophistication including the ability to recognize, read, and write proofs. Topics include: The Real Number System: Field and order axioms, sups and infs, completeness. Real Sequences. Bolzano-Weierstrass theorem. Topology of the Real Line: Open sets, closed sets, compactness, Heine-Borel Theorem. Continuity: extreme and intermediate value theorems, uniform continuity. Differentiation: chain rule, local extrema, mean-value theorem, L'Hospital's rule, Taylor's theorem. Riemann integration: sufficient conditions for integrability, fundamental theorems of calculus. (Three 50 minute lectures)
Prerequisites: (21-127 or 21-128 or 15-151) and 21-122
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-356 Principles of Real Analysis II: Fall and Spring: 9 units
This course provides a rigorous and proof-based treatment of functions of several real variables. The course presumes some mathematical sophistication including the ability to recognize, read, and write proofs. Topics include: Metric spaces. Differential calculus in Euclidean spaces: continuity, differentiability, partial derivatives, gradients, differentiation rules, implicit and inverse function theorems. Multiple integrals. Integration on curves and hypersurfaces: arclength, and generalized area. The divergence theorem and the 3D Stokes theorem. Regarding prerequisites, 21-268 or 21-269 are strongly recommended rather than 21-259. (Three 50 minute lectures)
Prerequisites: (21-268 or 21-259 or 21-269) and (21-241 or 21-242) and (21-455 or 21-355)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures)
Prerequisites: 21-269 or 21-268
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-366 Topics in Applied Mathematics: Intermittent: 9 units
This course affords students with the opportunity to study topics which are in the area of expertise of the instructor. This course may taken more than once if content is sufficiently different. Course prerequisites will depend on the content of the course. Please see the course URL for semester-specific topics. (Three 50 minute lectures)
Prerequisites: 21-120 and (21-242 or 21-241)

Course Website: https://www.cmu.edu/math/courses/special-topics.html

21-369 Numerical Methods: Fall and Spring: 12 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 prerequisites, rather than 21-259. (Three 50 minute lectures, one 50 minute recitation)
Prerequisites: (02-120 or 15-110 or 15-112) and (21-259 or 21-269 or 21-268) and (21-242 or 21-241 or 21-240 or 33-232) and (21-630 or 33-231 or 21-260 or 21-261)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. Acceptable corequisites include 21-325 or 36-225 (Three 50 minute lectures)
Prerequisites: (70-492 or 21-270) and (21-268 or 21-269 or 21-259 or 21-256)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-371 Functions of a Complex Variable: Fall: 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 prerequisites, rather than 21-259. (Three 50 minute lectures)
Prerequisites: 21-355 or 21-235 or 21-455
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures)
Prerequisites: (21-127 or 21-128 or 15-151) and (21-241 or 21-242)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures)
Prerequisite: 21-373
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-375 Topics in Algebra: Intermittent: 9 units
Typical of courses that might be offered from time to time are elliptic curves, commutative algebra, and theory of Boolean functions. (Three 50 minute lectures)
Prerequisite: 21-373
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures)
Prerequisite: 21-270 Min. grade B

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-380 Introduction to Mathematical Modeling: Fall: 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. (Three 50 minute lectures)
Prerequisites: (21-241 or 21-242) and (21-261 or 21-260)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures)
Prerequisites: (21-228 or 15-251) and 21-292
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-400 Intermediate Logic: Intermittent: 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. (Three 50 minute lectures)
Prerequisite: 21-300

21-410 Research Topics in Mathematical Sciences: Intermittent: 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. (Three 50 minute lectures)
Prerequisites: (21-242 or 21-240 or 21-241) and (21-261 or 21-260) and (21-259 or 21-268 or 21-269)

Course Website: https://www.cmu.edu/math/courses/special-topics.html

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. (Three 50 minute lectures)
Prerequisites: (21-260 or 18-202) and 21-370 and (36-225 or 36-217 or 21-325 or 15-259 or 36-218)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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 divisors, 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. (Three 50 minute lectures)
Prerequisites: (21-242 or 21-241) and 21-373
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-455 Intermediate Real Analysis I: Intermittent: 12 units
This course provides a rigorous and proof-based treatment of the general theory of functions on metric spaces. The course serves as a more advanced version of 21-355 Principles of Real Analysis and is primarily intended for students who have taken 21-269. Topics include: Metric spaces: Completeness, density, separability, compactness, connectedness. Baire theorem and applications. Contraction maps: fixed points, applications, inverse, and implicit function theorems. Spaces of functions: uniform and pointwise convergence, Stone-Weierstrass theorem, Arzela-Ascoli theorem. (Three 50 minute lectures, one 50 minute recitation)
Prerequisites: 21-269 or 21-268
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-456 Intermediate Real Analysis II: Intermittent: 10 units
This course provides a rigorous and proof-based treatment of the general theory of functions on Euclidean spaces. The course serves as a more advanced version of 21-356 Principles of Real Analysis II and is primarily intended for students who have taken 21-455. Topics include: Lebesgue integration in Euclidean spaces: Lebesgue measure and Lebesgue integration, convergence and integration theorems, Fubini's theorem, change of variables. Curves, arclength, curve integrals. Submanifolds of Euclidean space: applications of the inverse and implicit function theorems, tangent space, normal space, orientation, integration on manifolds. Theorems of vector calculus: divergence theorem, Stokes theorem in 3D. (Three 50 minute lectures, one 50 minute recitations)
Prerequisites: 21-355 or 21-455 or 21-235
Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-469 Computational Introduction to Partial Differential Equations: Intermittent: 12 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. Most PDEs do not have explicit solutions, and hence computational methods are essential for understanding the underlying phenomena. This course will serve as a first introduction to PDEs and their numerical approximation, and will focus on a variety of mathematical models. It will cover both analytical methods, numerical methods (e.g. finite differences) and the use of a computer to approximate and visualize solutions. The mathematical ideas behind phenomena observed in nature will be studied at the theoretical level and in numerical simulations (e.g. speed of wave propagation, and/or shocks in traffic flow). Topics will include: Derivation of PDEs from physical principles, analytical and computational tools for the transport equation and the Poisson equation, Fourier analysis, analytical and numerical techniques for the solution of parabolic equations and if time permits, the wave equation. (Three 50 minute lectures, one 80 minute recitation)
Prerequisites: (21-240 or 21-241 or 21-242) and (21-269 or 21-259 or 21-268) and (21-261 or 21-630 or 21-260 or 33-231) and (02-120 or 15-112 or 15-110)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures)
Prerequisites: (21-228 or 15-251) and (21-241 or 21-242)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-590 Practicum in Mathematical Sciences: All Semesters: 3 units
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. Tuition is charged for this course.

21-599 Undergraduate Reading and Research: Fall and Spring
This course offers individualized reading or project-based learning in mathematics and its applications. Prerequisites, unit requirements, and deliverables will be determined in consultation with the faculty member. To register, please email cgilchri@andrew.cmu.edu, copy the faculty member, and include the number of units that you will be taking.

21-602 Introduction to Set Theory I: Fall: 12 units
The axioms of ZFC, ordinal arithmetic, cardinal arithmetic including Konig's lemma, class length induction and recursion, the rank hierarchy, the Mostowski collapse theorem, the H-hierarchy, the Delta_1 absoluteness theorem, the absoluteness of wellfoundedness, the reflection theorem for hierarchies of sets, ordinal definability, the model HOD, relative consistency, Goedel's theorem that HOD is a model of ZFC, constructibility, Goedel's theorem that L is a model of ZFC + GCH, the Borel and Projective hierarchies and their effective versions, Suslin representations for Sigma^1_1, Pi^1_1 and Sigma^1_2, sets of reals, Shoenfield's absoluteness theorem, the complexity of the set of constructible reals, the combinatorics of club and stationary sets (including the diagonal intersection, the normality of the club filter, Fodor's lemma and its applications), Solovay's splitting theorem, model theoretic techniques commonly applied in set theory (e.g., elementary substructures, chains of models and ultrapowers), club and stationary subsets of [X]^/omega and their combinatorics, Jensen's diamond principles and his proofs that they hold in L, Gregory's theorem and generalizations, constructions of various kinds of uncountable trees (including Aronszajn, special, Suslin, Kurepa), Jensen's square principles and elementary applications, the basic theory of large cardinals (including inaccesssible, Mahlo, weakly compact and measurable cardinals), Scott's theorem that there are no measurable cardinals in L, Kunen's theorem that the only elementary embedding from V to V is the identity. Optional topic: SCH and Silver's theorem. (Three 50 minute lectures)
Prerequisites: (21-235 Min. grade B or 21-355 Min. grade B or 21-455 Min. grade B) and (21-237 Min. grade B or 21-373 Min. grade B) and 21-329 Min. grade B and 21-300 Min. grade B

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-603 Model Theory I: Fall: 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. (Three 50 minute lectures)
Prerequisites: (21-355 Min. grade B or 21-235 Min. grade B or 21-455 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

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. (Three 50 minute lectures)
Prerequisites: (21-235 Min. grade B or 21-355 Min. grade B or 21-455 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

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. (Three 50 minute lectures)
Prerequisites: (21-455 Min. grade B or 21-355 Min. grade B or 21-235 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-623 Complex Analysis: Intermittent: 12 units
The complex plane, holomorphic functions, power series, complex integration, and Cauchy's Theorem. Calculus of residues. Additional topics may include conformal mappings and the application of complex transforms to differential equations. (Three 50 minute lectures)
Prerequisites: (21-456 Min. grade B or 21-236 Min. grade B or 21-356 Min. grade B) and (21-237 Min. grade B or 21-373 Min. grade B)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-624 Descriptive Set Theory: Intermittent: 12 units
The central theme of the course is the study of "definable" subsets of Polish spaces (a class of topological spaces containing many spaces studied across mathematics). In some sense, restricting attention to definable sets makes life nicer; for instance, (properly interpreted) the continuum hypothesis becomes true! And weird Banach-Tarski stuff doesn't happen. However, it comes at a price: equivalence relations can have more classes than elements, and acyclic graphs can become hard to properly color. For the first portion of the course we will work through the basic theory of Borel and analytic subsets of Polish spaces, highlighting their interaction with measure and Baire category. We will then focus on the modern theory of equivalence relations, paying special attention to orbit equivalence relations of group actions and connectedness relations of graphs. Along the way we will establish several classical dichotomy theorems. (Three 50 minute lectures)
Prerequisites: (21-355 Min. grade B or 21-455 Min. grade B or 21-235 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B) and 21-329 Min. grade B

21-630 Ordinary Differential Equations: Intermittent: 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. (Three 50 minute lectures)
Prerequisites: (21-235 Min. grade B or 21-455 Min. grade B or 21-355 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

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. (Three 50 minute lectures)
Prerequisites: (21-356 Min. grade B or 21-236 Min. grade B or 21-456 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

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. (Three 50 minute lectures)
Prerequisites: (21-236 Min. grade B or 21-651 Min. grade B) and (21-720 Min. grade B or 21-236 Min. grade B)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

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. (Three 50 minute lectures)
Prerequisites: (21-455 Min. grade B or 21-355 Min. grade B or 21-235 Min. grade B) and (21-237 Min. grade B or 21-373 Min. grade B)

21-660 Introduction to Numerical Analysis I: Intermittent: 12 units
Finite precision arithmetic, interpolation, spline approximation, numerical integration, numerical solution of linear and nonlinear systems of equations, optimization in finite dimensional spaces. (Three 50 minute lectures)
Prerequisites: (21-356 Min. grade B or 21-236 Min. grade B or 21-456 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

21-670 Linear Algebra for Data Science: Fall: 6 units
This course is designed to present and discuss those aspects of Linear Algebra that are most important in Data Analytics. The emphasis will be on developing intuition and understanding how to use linear algebra, rather than on proofs. (Three 50 minute lectures)
Prerequisites: (21-355 Min. grade B or 21-455 Min. grade B or 21-235 Min. grade B) and 21-373 Min. grade B

21-671 Computational Linear Algebra: Fall: 12 units
This is a survey of methods in computational linear algebra. Topics covered in this course focus around algorithms for solving (dense or large and sparse) linear systems. Regularization and underdetermined systems will be discussed in detail. Rather than assuming prior knowledge in numerical analysis or matrix theory, we will introduce standard methods or results when needed. In this way, much of the material is self-contained. Theoretical and experimental results will be covered accordingly, with an emphasis on cost, stability, and convergence. (Three 50 minute lectures)
Prerequisites: (21-240 or 21-241 or 21-242) and (21-259 or 21-269 or 21-268)

21-681 Stochastic Calculus in Finance: Intermittent: 6 units
This is a graduate-level introduction to continuous-time equilibrium asset pricing models. Using tools from Ito calculus, the first part of the course covers the benchmark case of complete, frictionless markets, for which a fairly general theory and a number of solvable examples have been developed. The second part of the course then provides an overview of cutting-edge research on extensions of the baseline model that account for "flaws and frictions" such as heterogeneous beliefs, trading costs, or asymmetric information. In the third part of the course, students will present a related research paper, chosen together with the instructor in accordance with their background and research interests. (One 80 minute lecture)
Prerequisites: (21-235 Min. grade B or 21-355 Min. grade B or 21-455 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

21-690 Methods of Optimization: Spring: 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. (Three 50 minute lectures)
Prerequisites: (21-456 Min. grade B or 21-236 Min. grade B or 21-356 Min. grade B) and (21-237 Min. grade B or 21-373 Min. grade B)

Course Website: https://www.cmu.edu/math/undergrad/math-course-information.html

21-701 Discrete Mathematics: Fall: 12 units
Combinatorial analysis, graph theory with applications to problems in computational complexity, networks, and other areas. (Three 50 minute lectures)
Prerequisites: (21-355 Min. grade B or 21-235 Min. grade B or 21-455 Min. grade B) and (21-237 Min. grade B or 21-373 Min. grade B)

21-702 Set Theory II: Intermittent: 12 units
This course is a sequel to 21-602 Set Theory. The main goal is to prove Solovay's theorem that Con(ZFC + an inaccessible cardinal) implies Con(ZF + DC + every set of reals is Lebesgue measurable). Topics covered include absoluteness theorems, Borel codes, the Levy collapse, product forcing, relative constructibility, and the basics of iterated forcing up to the consistency of Martin's Axiom. (Three 50 minute lectures)
Prerequisites: (21-355 Min. grade B or 21-455 Min. grade B or 21-235 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B) and 21-602 Min. grade B

21-703 Model Theory II: Intermittent: 12 units
The course concentrates in what is considered "main stream model theory" with is Shelah's classification theory (known also as Stability). Among the topics to be presented are stability, superstability, the theory of various notions of primeness, rank functions, forking calculus, the stability spectrum theorem, finite equivalence relations theorem, stable groups (up to and including the Macintyre-Cherlin-Shelah theorem on super-stable fields), and some elementary geometric model theory. If time permits also: basic facts about infinitary languages, computation of Hanf-Morley numbers; some of the Ax-Kochen-Ershov theory of model theory for fields with valuations (will apply this to solve Artin's conjecture). (Three 50 minute lectures)
Prerequisites: (21-235 Min. grade B or 21-355 Min. grade B or 21-455 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B) and 21-603 Min. grade B

Course Website: http://www.math.cmu.edu/~rami/mt2.11.desc.html

21-720 Measure and Integration: Fall: 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. (Three 50 minute lectures)
Prerequisites: (21-455 Min. grade B or 21-355 Min. grade B or 21-235 Min. grade B) and (21-237 Min. grade B or 21-373 Min. grade B)

21-721 Probability: Spring: 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. (Three 50 minute lectures)
Prerequisites: (21-355 Min. grade B or 21-235 Min. grade B or 21-455 Min. grade B) and (21-237 Min. grade B or 21-373 Min. grade B) and 21-720 Min. grade B

21-723 Advanced Real Analysis: Spring: 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. (Three 50 minute lectures)
Prerequisites: (21-356 Min. grade B or 21-236 Min. grade B or 21-456 Min. grade B) and (21-237 Min. grade B or 21-373 Min. grade B) and 21-720 Min. grade B

21-732 Partial Differential Equations I: Intermittent: 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. (Three 50 minute lectures)
Prerequisites: 21-723 Min. grade B and 21-632 Min. grade B and 21-640 Min. grade B and (21-237 Min. grade B or 21-373 Min. grade B)

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 R and #246;dl nibble, the Lov and #225;sz 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 Erd and #246;s-R and #233;nyi random graph, and the Barab and #225;si-Albert preferential attachment model. (Three 50 minute lectures)
Prerequisites: (21-701 Min. grade B or 21-301) and (21-325 or 36-225 or 36-218 or 15-259)

21-738 Extremal Combinatorics: Intermittent: 12 units
Classical problems and results in extremal combinatorics including the Tur and #225;n and Zarankiewicz problems, the Erdos-Stone theorem and the Erdos-Simonovits stability theorem. Extremal set theory including the Erdos-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. (Three 50 minute lectures)
Prerequisites: (21-455 Min. grade B or 21-235 Min. grade B or 21-355 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

21-742 Calculus Of Variations: Intermittent: 12 units
Classical fixed endpoint examples. Fixed endpoint problems in classes of absolutely continuous functions: existence via lower semicontinuity. Tonelli's existence theorem. Euler-Lagrange and DuBois Reymond equations, transversality conditions, Weierstrass field theory, Hamilton-Jacobi theory. Problems with constraints. (Three 50 minute lectures)
Prerequisites: (21-355 Min. grade B or 21-235 Min. grade B or 21-455 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B) and 21-723 Min. grade B

21-752 Algebraic Topology: Intermittent: 12 units
Topology is a less rigid variant of geometry that studies shapes of spaces. Algebraic topology associates algebraic invariants, such as groups or rings, to such spaces. This is achieved by building a space from simpler ones or by algebraically keeping track of how to map a simple space into a given space. This course will cover the fundamental group and covering spaces, homology theories, and the cohomology ring of a space (time permitting). (Three 50 minute lectures)
Prerequisites: 21-651 Min. grade B and (21-235 Min. grade B or 21-355 Min. grade B or 21-455 Min. grade B) and (21-237 Min. grade B or 21-373 Min. grade B)

21-759 Differential Geometry: Intermittent: 12 units
Manifolds in Euclidean spaces, curves and surfaces, principal curvatures, geodesics. Surfaces with constant mean curvature, minimal surfaces. Abstract differentiable manifolds, tangent spaces, vector bundles, affine connections, parallelisms, covariant gradients, Cartan torsion, Riemann curvature. Riemannian geometry, Lie groups. Familiarity with analysis in finite dimensional spaces will be assumed. (Three 50 minute lectures)
Prerequisites: (21-356 Min. grade B or 21-236 Min. grade B or 21-456 Min. grade B) and (21-237 Min. grade B or 21-373 Min. grade B)

21-762 Finite Element Methods: Intermittent: 12 units
Finite element methods for elliptic boundary value problems. Analysis of errors, approximation by finite element spaces. Efficient implementation of finite element spaces. Efficient implementation of finite element algorithms, finite element methods for parabolic and eigenvalue problems, effects of curved boundaries. Numerical quadrature, non-conforming methods. (Three 50 minute lectures)
Prerequisites: (21-236 Min. grade B or 21-456 Min. grade B or 21-356 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

21-765 Introduction to Parallel Computing and Scientific Computation: Spring: 9 units
Course objectives: to develop structural intuition of how the hardware and the software work, starting from simple systems to complex shared resource architectures; to provide guidelines about how to write and document a software package; to familiarize the audience with the main parallel programming techniques and the common software packages/libraries. (One 110 minute lecture)

Course Website: http://www.math.cmu.edu/~florin/M21-765/index.html

21-770 Introduction to Continuum Mechanics: Intermittent: 12 units
General discussion of the behavior of continuous bodies with an emphasis on those concepts common to the description of all continuous bodies. Specific examples from elasticity and fluid mechanics. Familiarity with analysis in finite dimensional spaces will be assumed. (Three 50 minute lectures)
Prerequisites: (21-356 Min. grade B or 21-236 Min. grade B or 21-456 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

21-800 Advanced Topics in Logic: Intermittent: 12 units
This course affords students with the opportunity to study topics which are in the area of expertise of the instructor. This course may taken more than once if content is sufficiently different. Course prerequisites will depend on the content of the course. Please see the course URL for semester-specific topics. (Three 50 minute lectures)
Prerequisites: (21-455 Min. grade B or 21-355 Min. grade B or 21-235 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

Course Website: https://www.cmu.edu/math/courses/special-topics.html

21-801 Advanced Topics in Discrete Mathematics: Intermittent: 12 units
Course topics will vary depending on the semester and instructor. May be taken more than once if content is sufficiently different. (Three 50 minute lectures)
Prerequisites: (21-355 Min. grade B or 21-235 Min. grade B or 21-455 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B)

21-803 Model Theory III: Intermittent: 12 units
We will concentrate in classification theory for first-order theories. The theory was developed mostly by Saharon Shelah presented in his 1978 (2nd ed 1990) book and in several hundreds of papers. We will present a modern overview of Shelah's theory incorporating few recent innovations and simplifications. The development of the theory was motivated by set-theoretic questions like: "what is the asymptotic behavior of the function I(/aleph_/alpha,T) as a function of /alpha ?" and "what is the first /lambda such that an uncountable first-order stable theory T is stable in /lambda?" Surprisingly the full answer to such combinatorial set-theoretic questions led for a development and discovery of a conceptually rich theory which seems to be related to aspects of commutative algebra and algebraic-geometry. This theory found several applications in the form of solving fundamental problems in classical fields of mathematics among them geometry and number theory. The focus will be on the simplest and most fundamental aspects of the pure theory. Primarily around a notion called forking and various characterizations of classes of theories. (Three 50 minute lectures)
Prerequisites: (21-235 Min. grade B or 21-455 Min. grade B or 21-355 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B) and 21-703 Min. grade B

21-820 Advanced Topics in Analysis: Intermittent: 12 units
Course topics will vary depending on the semester and instructor. May be taken more than once if content is sufficiently different.
Prerequisites: 21-651 Min. grade C and 21-720 Min. grade C and 21-724 and 21-640 Min. grade C

Course Website: https://www.cmu.edu/math/courses/special-topics.html

21-830 Advanced Topics in Partial Differential Equations: Intermittent: 12 units
Course topics will vary depending on the semester and instructor. May be taken more than once if content is sufficiently different.
Prerequisites: (21-235 Min. grade B or 21-455 Min. grade B or 21-355 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B) and 21-632 Min. grade B

21-849 Special Topics: Intermittent: 12 units
This course affords students with the opportunity to study topics which are in the area of expertise of the instructor. This course may taken more than once if content is sufficiently different. Course prerequisites will depend on the content of the course. Please see the course URL for semester-specific topics. (Three 50 minute lectures)
Prerequisites: (21-236 or 21-610) and 21-651
Course Website: https://www.cmu.edu/math/courses/special-topics.html

21-860 Advanced Topics In Numerical Analysis: Intermittent: 12 units
Content varies. May be taken more than once if content is sufficiently different. (Three 50 minute lectures)
Prerequisites: (21-355 Min. grade B or 21-455 Min. grade B or 21-235 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B) and 21-660 Min. grade B

21-880 Stochastic Calculus: Fall: 12 units
This is a first Ph.D.-level course in stochastic calculus for continuous-time processes. It includes martingales and semi-martingales, Brownian motion, the Poisson process, representation of continuous martingales as time-changed Brownian motions, construction of the Ito integral, and Ito's formula. (Two 80 minute lectures)
Prerequisites: (21-355 Min. grade B or 21-235 Min. grade B or 21-455 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B) and 21-721 Min. grade B

21-882 Advanced Topics in Financial Mathematics: Intermittent: 12 units
Content varies. May be taken more than once if content is sufficiently different. (Two 80 minute lectures)
Prerequisites: (21-455 Min. grade B or 21-235 Min. grade B or 21-355 Min. grade B) and (21-373 Min. grade B or 21-237 Min. grade B) and 21-721 Min. grade B

21-901 Master's Degree Research: All Semesters
This course is for students admitted to the Mathematical Sciences Honors Degree Program. It allows for students to engage in research activities related to their Master's thesis, under the supervision of their thesis supervisor. The supervisor should be contacted prior to enrollment in 21-901, as they are required to provide consent for the student's enrollment to the Academic Program Manager, who authorizes registration in this course.

Faculty

NOHA ABDELGHANY, Assistant Teaching Professor – Ph.D., Western Michigan University; Carnegie Mellon, 2022–

THERESA ANDERSON, Associate Professor – Ph.D., Brown University; Carnegie Mellon, 2022–

JEREMY AVIGAD, Professor – Ph.D., University of California, Berkeley; Carnegie Mellon, 1996–

NICHOLAS BOFFI , Assistant Professor – Ph.D., Harvard University ; Carnegie Mellon, 2024–

THOMAS BOHMAN, Professor – Ph.D., Rutgers University; Carnegie Mellon, 1998–

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

CLINTON CONLEY, Associate Professor – Ph.D., University of California Los Angeles; Carnegie Mellon, 2014–

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

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

LAYAN EL HAJJ, Associate Teaching Professor – Ph.D., McGill University; Carnegie Mellon, 2023–

CHRISTOPHER EUR, Assistant Professor – Ph.D., University of California, Berkeley; Carnegie Mellon, 2024–

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

IRENE FONSECA, Kavčić-Moura University Professor of Mathematics – Ph.D., University of Minnesota; Carnegie Mellon, 1987–

FLORIAN FRICK, Associate Professor – Ph.D., Technical University of Berlin; Carnegie Mellon, 2018–

ALAN FRIEZE, Orion Hoch, S 1952, University Professor of Mathematical Sciences – Ph.D., University of London; Carnegie Mellon, 1987–

IRINA GHEORGHICIUC, Teaching Professor – Ph.D., University of Pennsylvania; Carnegie Mellon, 2007–

RAMI GROSSBERG, Professor – Ph.D., Hebrew University of Jerusalem; Carnegie Mellon, 1988–

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

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

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

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

NIRAJ KHARE, Associate Teaching Professor – Ph.D., Ohio State University; Carnegie Mellon, 2014–

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

DMITRY KRAMKOV, Mellon College of Science Professor of Mathematical Finance – Ph.D., Steklov Mathematical Institute; Carnegie Mellon, 2000–

MARTIN LARSSON, Professor – Ph.D., Cornell University; Carnegie Mellon, 2019–

GIOVANNI LEONI, Professor – Ph.D., University of Minnesota; Carnegie Mellon, 2002–

PO-SHEN LOH, Professor – Ph.D., Princeton University; Carnegie Mellon, 2009–

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

SERGEY NADTOCHIY, Professor – Ph.D., Princeton University; Carnegie Mellon, 2025–

ROBIN NEUMAYER, Associate Professor – Ph.D., The University of Texas at Austin; Carnegie Mellon, 2021–

CLIVE NEWSTEAD, Associate Teaching Professor – PhD, Carnegie Mellon University; Carnegie Mellon, 2018–

DAVID OFFNER, Associate Teaching Professor – Ph.D., Carnegie Mellon University; Carnegie Mellon, 2019–

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

AGOSTON PISZTORA, Associate Professor – Ph.D., ETH Zurich; Carnegie Mellon, 1996–

DYLAN QUINTANA, Assistant Teaching Professor – Ph.D., University of Chicago; Carnegie Mellon, 2023–

MATTHEW ROSENZWEIG, Assistant Professor – Ph.D., University of Texas at Austin; Carnegie Mellon, 2023–

ERNEST SCHIMMERLING, Professor – Ph.D., University of California at Los Angeles; Carnegie Mellon, 1998–

MYKHAYLO SHKOLNIKOV, Professor – Ph.D., Stanford University; Carnegie Mellon, 2024–

JONATHAN SIMONE, Assistant Teaching Professor – Ph.D., University of Virginia; Carnegie Mellon, 2024–

DEJAN SLEPČEV, Professor, Mellon College of Science Associate Dean for Faculty and Graduate Affairs – Ph.D., University of Texas at Austin; Carnegie Mellon, 2006–

RICHARD STATMAN, Professor – Ph.D., Stanford University; Carnegie Mellon, 1984–

PRASAD TETALI, Alexander M. Knaster Professor & Department Head – Ph.D., New York University; Carnegie Mellon, 2021–

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

KONSTANTIN TIKHOMIROV, Associate Professor – Ph.D., University of Alberta; Carnegie Mellon, 2022–

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

NOEL WALKINGTON, Professor – Ph.D., University of Texas at Austin; Carnegie Mellon, 1989–

ANTHONY WESTON, Associate Teaching Professor – Ph.D., Kent State University; Carnegie Mellon, 2022–

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

MICHAEL YOUNG, Associate Professor & Mellon College of Science Associate Dean for Community Engagement – Ph.D., Carnegie Mellon University; Carnegie Mellon, 2021–

Emeriti Faculty

MANUEL BLUM, University Professor Emeritus – Ph.D., Massachusetts Institute of Technology; Carnegie Mellon, 1999–

DEBORAH BRANDON, Associate Teaching Professor Emerita – Ph.D., Carnegie Mellon University; Carnegie Mellon, 1991–

GÉRARD CORNUÉJOLS, IBM University Professor of Operations Research Emeritus – Ph.D., Cornell University; Carnegie Mellon, 1978–

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

JOHN LEHOCZKY, Thomas Lord University Professor of Statistics Emeritus – Ph.D., Stanford University; Carnegie Mellon, 1969–

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

MARION OLIVER, Teaching Professor Emeritus – Ph.D., Carnegie Mellon University ; Carnegie Mellon, 2004–

DAVID OWEN, Professor Emeritus – Ph.D., Brown University; Carnegie Mellon, 1967–

ROBERT PEGO, Professor Emeritus – Ph.D., University of California at Berkeley; Carnegie Mellon, 2004–

JACK SCHAEFFER, Professor Emeritus – Ph.D., Indiana University; Carnegie Mellon, 1983–

ROBERT SEKERKA, University Professor Emeritus – Ph.D., Harvard University; Carnegie Mellon, 1969–

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

SHLOMO TA'ASAN, Professor Emeritus – Ph.D., Weizmann Institute; Carnegie Mellon, 1994–

LUC TARTAR, University Professor of Mathematics Emeritus – Ph.D., University of Paris; Carnegie Mellon, 1987–

RUSSELL WALKER, Teaching Professor Emeritus – D.A., Carnegie Mellon University; Carnegie Mellon, 1984–

WILLIAM WILLIAMS, Professor Emeritus – Ph.D., Brown University; Carnegie Mellon, 1966–

Undergraduate Catalog

Department of Mathematical Sciences

Special Options

Matrix Theory and Vector Analysis

Mathematical Studies

Interdisciplinary Programs

Curriculum

B.S. in Mathematical Sciences

Mathematical Sciences Courses (required)

Computer Science Courses (required)

DEPTH ELECTIVES (REQUIRED)

MCS General Education (required)

Minimum number of units required for degree:360

Mathematical Sciences Electives for Students Intending Graduate Study

B.S. in Mathematical Sciences (Operations Research and Statistics)

Mathematical Sciences Courses (required)

Statistics Courses (required)

Economics, Business, and Computer Science Courses (required)

Depth Electives (required)

MCS General Education (required)

Minimum number of units required for degree:360

B.S. in Mathematical Sciences (Statistics)

Mathematical Sciences Courses (required)

Statistics Courses (required)

Economics and Computer Science Courses (required)

Depth Electives (required)

MCS General Education (required)

Minimum number of units required for degree:360

B.S. in Mathematical Sciences (Discrete Mathematics and Logic)

Mathematical Sciences and Computer Science Courses (required)

Computer Science Electives (required)

Depth Electives (required)

List 1 (Discrete Mathematics and Logic Electives)

List 2 (Mathematics Electives)

MCS General Education (required)

Minimum number of units required for degree:360

B.S. in Mathematical Sciences (Computational and Applied Mathematics)

Mathematical Sciences Courses (required)

Computer Science Courses (required)

Depth Electives (required)

List 1 (Computational and Applied Mathematics Electives)

List 2 (Mathematics Electives)

MCS General Education (required)

Minimum number of units required for degree:360

B.A. in Mathematical Sciences

Mathematical Sciences Courses (required)

Depth Electives (required)

MCS General Education (required)

Minimum number of units required for degree:360

Additional Major Requirements

The Minor in Mathematical Sciences

The Minor in Discrete Mathematics and Logic

The Honors Degree Program

Course Descriptions

About Course Numbers:

Faculty

Emeriti Faculty