## Department of Mathematics

Last updated: February 3, 2021 at 10:17 AM

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Programs of Study

- Minor
- Major (BA)
- Majors (BS)
- Postbaccalaureate Program
- Combined BA/MA
- Master of Arts
- Master of Science
- Doctor of Philosophy

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Objectives

As our society becomes more technological, it is increasingly affected by mathematics. Quite sophisticated mathematics is now central to the natural sciences, to ecological issues, to economics, and to our commercial and technical life. A student who takes such general-level courses as MATH 5a, 8a, 10a, 10b, 15a, or 20a will better be prepared to engage with the modern world. To major in Mathematics or Applied Mathematics, one needs to take more advanced courses. Starting from the academic year 2020-2021, the Department of Mathematics offers three degrees: Bachelor of Arts in Mathematics, Bachelor of Science in Mathematics, and Bachelor of Science in Applied Mathematics. This is a testament to the fact that mathematics is, at the same time, both a subject of the greatest inherent depth and beauty with a history extending from antiquity, and also a powerful tool for understanding our world.

Undergraduate Major in Mathematics

The undergraduate major introduces students to some fundamental fields of mathematics—algebra, real and complex analysis, geometry, and topology—and to the habit of mathematical thought. Mathematics majors may go on to graduate school, scientific research, finance, actuarial science, or mathematics teaching, but many choose the major for its inherent interest.

Undergraduate Major in Applied Mathematics

Applications of mathematics to physics, biology, chemistry, economics and social sciences have proved particularly fruitful, and have led to the development of new mathematical tools and methods. The Applied Mathematics major will introduce students to the essential tools used in such applications. It will prepare students for professional careers in public institutions, research centers or private companies using quantitative methods (such as modeling, data analysis or optimization) to understand and solve complex real-world problems.

Postbaccalaureate Program in Mathematics

The mathematics department offers a postbaccalaureate program for students with a bachelor’s degree in a different field who wish to prepare for graduate school or a career requiring enhanced mathematical skills.

Graduate Program in Mathematics

The graduate program in mathematics offers the Master of Arts and Doctor of Philosophy degrees. The Master's program gives students a rigorous foundation in graduate-level mathematics. The doctoral program, in addition to coursework, includes seminar participation, teaching and research experience, and is designed to lead to a broad understanding of the subject.

Entering students may be admitted to either the master's or the doctoral program. The courses offered by the department, participation in seminars, and exposure to a cutting-edge research environment provide the students with a broad foundation for work in modern pure mathematics and prepare them for careers as mathematicians in academia, industry, or government.

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Learning Goals

Students may study mathematics for several reasons: for its own intrinsic interest, for its applications to other fields such as economics, computer science, and physical and life sciences, and for the analytical skills that it provides for such fields of study as law, medicine, and business. The Mathematics Department at Brandeis serves a diverse audience, consisting of students with all of these reasons.

Learning Goals for Non-majors

Non-majors who take mathematics courses include pre-medical students, education minors, many science and economics majors, and mathematics minors. Although their mathematical goals may vary depending on their interests, the following are among the most important:

- Improved analytical reasoning skills
- Enhanced basic computational skills
- Familiarity with basic mathematical terms and their physical meanings
- The ability to model real-world problems mathematically
- An appreciation for the power of mathematical thinking

Note that the Mathematics Department at Brandeis offers a Minor in Mathematics, but not in Applied Mathematics.

Undergraduate Major

Knowledge

Students completing the major in mathematics will:

- Understand the fundamental concepts of mathematical proof, logic, abstraction and generalization.
- Achieve a basic knowledge of the following areas of mathematics:

- Matrices, linear algebra, and multivariable calculus.
- Analysis in one and several variables, including properties of the real numbers and of limits.
- Axiomatically defined algebraic structures, such as groups, rings, fields, and vector spaces.

Mathematics majors will know the basic ideas of some, but not necessarily all, of the following areas: differential equations, probability and statistics, number theory, combinatorics, real and complex analysis, topology, and differential geometry.

Students completing the major in applied mathematics will:

- Gain knowledge on the fundamental objects, frameworks and theorems in applied mathematics, including the fields of probability, mathematical modeling, numerical analysis and differential equations.
- Understand the main connections between the mathematical sciences and other scientific or humanistic disciplines.
- Acquire the principles specific to applications of mathematics and use then in developing models and analyzing them rigorously. Be able to formalize and abstract a concrete problem into mathematical models, and apply mathematical concepts and reasoning to solve problems arising in other sciences or in industry.

Core Skills

Mathematics majors will be able to read and write mathematical proofs, abstract general principles from examples, and distinguish correct from fallacious arguments. Majors will learn to apply general principles to specific cases, solve non-routine mathematical problems, and to apply mathematics to the real world.

Applied Mathematics majors will develop their ability to:

- Understand, modify or construct mathematical models of systems arising in natural or social sciences.
- Assess their relevance, accuracy and usefulness.
- Analyze formally these models and provide relevant information on the application domain.
- Clearly communicate the results of mathematical analysis to various audiences.

Upon Graduation

Mathematics majors with appropriate backgrounds and preparation may:

- Pursue graduate study and a scholarly career in mathematics
- Work as actuaries
- Teach mathematics at the K-12 level
- Work in fields such as computer science, operations research, economics, finance, biology, physics, or other sciences
- Attend medical, law, or business school

Graduate Program in Mathematics

Master of Arts in Mathematics

**Knowledge**

- M.A. students are required to demonstrate a broad and deep knowledge of algebra, topology, geometry, and analysis. This is done by passing with a high mark the exams of the seven fundamental courses offered every year in those fields (Algebra I and II, Topology I and II, Real Analysis, Complex Analysis, and Geometric Analysis).
- A wide array of more advanced or more specialized elective courses is also offered, as well as reading courses, and M.A. students are required to take at least two of them.
- Seminars, colloquium and special lectures are also regularly given by scholars from all over the world, and allow the students to be exposed to current-research mathematics.

**Core Skills**

Students graduating with a Master's in Mathematics at Brandeis possess a rigorous foundation in modern mathematics.

**Outcome**

Students graduating with a Master's in Mathematics are ideally prepared to apply for a Ph.D. program in pure or applied mathematics, physics, and other sciences. They also have competencies in mathematics that are in high demand in many industries, or for certain jobs in the government.

**Doctor of Philosophy in Mathematics**

**Knowledge**

- Ph.D. students are required, before they begin to work on their dissertation, to demonstrate a broad and deep knowledge in algebra, topology, geometry, and analysis. This is done by passing with a high mark the exams of the seven courses offered every year in those fields (Algebra I and II, Topology I and II, Real Analysis, Complex Analysis, and Geometric Analysis).
- A wide array of more advanced or more specialized elective courses is also offered, and students are required to take a certain number of them, according to their taste and to the needs of their progress towards their dissertation.
- Many reading courses, where one or a small group of students read a research paper or a mathematical book under the guidance of a professor, are offered, often on demand. They allow the students to acquire progressively the knowledge necessary to enter current research.
- Seminars, colloquium, and special lectures are also regularly given by scholars from all over the world, and allow the students to learn more current-research mathematics.

**Core Skills**

Students graduating with a Ph.D. in Mathematics at Brandeis:

- Have learned to read and understand research papers, both in English and another language of their choice;
- Have learned how to present mathematical materials, in particular their own results, in seminars and other expositions destined to fellow graduate students and researchers;
- Have participated, as a teaching fellow, in a structured program of undergraduate teaching, giving them the skills and experience necessary to teach successfully mathematics at various undergraduate levels;
- Have attained research expertise and completed a significant body of original research that advances a specific field of study in mathematics;
- Have written and defended a Ph.D. dissertation.

**Outcome**

Students graduating with a Ph.D. have been trained to be effective teachers and cutting-edge researchers. They may work in academia, either in a research-oriented institution or in a teaching-oriented one, in many industries, or in the government.

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How to Become a Major

Students who enjoy mathematics are urged to consider majoring in either Mathematics or Applied Mathematics. Note that a student can declare a Major in Mathematics or a Major in Applied Mathematics but not both. Brandeis offers a wide variety of mathematics courses, and majors will have the benefits of small classes and individual faculty attention. For either of the majors a student should have completed either MATH 15a and 20a, or MATH 22a and b by the end of the sophomore year—these courses are prerequisites to the higher-level offerings. Therefore, it is important for students to start calculus and linear algebra (MATH 10a, 10b, 15a, 20a, or 22a and 22b) in the first year.

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How to Be Admitted to the Postbaccalaureate and Graduate Programs

The general requirements for admission are the same as those for the Graduate School as a whole. The department has a variety of fellowships and scholarships available for well-qualified PhD students. The application deadline for PhD students is January 15. Admission of postbaccalaureate and M.A. students is rolling through May 1, beginning January 15. PhD applications must contain three letters of recommendation; M.A. applications must contain two letters, and certificate applications require one letter of recommendation. Graduate Record Exam (GRE) general and subject tests are recommended, but not required.

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Faculty

**Joël Bellaïche, Chair ** Number theory.

**Mark Adler ** Analysis. Differential equations. Completely integrable systems.

**Olivier Bernardi, Graduate Advisor**

Combinatorics.

**Ruth Charney**

Geometric group theory. Topology.

**Charlie Cosnier-Horeau**

Vision. Field theory. Automorphic forms in string theory.

**Thomas Fai**

Scientific computing. Fluid dynamics. Mathematical biology.

**An Huang, Undergraduate Advising Head**

Algebraic Geometry and Graph Theory.

**Kiyoshi Igusa**

Differential topology. Representations of quivers.

**Yoosik Kim**

Symplectic geometry. Floer theory. Mirror symmetry.

**Dmitry Kleinbock ** Dynamical systems. Ergodic theory. Number theory.

**Rahul Krishna**

Number theory. Representation theory.

**Bong Lian ** Representation theory. Calabi-Yau geometry. String theory.

**Alan Mayer ** Classical algebraic geometry and related topics in mathematical physics.

**Keith Merrill**

Ergodic theory. Dynamical systems. Number theory.

**Gleb Nenashev (on leave spring 2021)**

Combinatorics. Commutative algebra.

**Omer Offen, Study Abroad Liaison ** Number theory. Representation theory.

**Youngmin Park**

Dynamical Systems. Mathematical neuroscience.

**Denis Patterson**

Integral equations. Dynamical systems. Mathematical modeling.

**Gail Peretti**

Statistics.

**Lam Pham (on leave academic year 2020-2021)**

Discrete subgroups of lie groups. Arithmetic groups. Dynamical systems.

**Daniel Ruberman ** Geometric topology and gauge theory.

**Rebecca Torrey **

Number theory.

**Jonathan Touboul**

Mathematical Neuroscience.

**John Wilmes**

Algorithms and Combinatorics.

**Ying Zhang**

Scientific computing. Fluid dynamics. Mathematical biology.

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Requirements for the Minor

- MATH 22a or 15a; MATH 22b or 20a.
- Three additional semester courses, either MATH courses numbered 27 or higher or cross-listed courses in Mathematics. Only cross-listed courses in Mathematics and not in Applied Mathematics may be used. Most Math courses numbered 27 or higher require Math 23b as a prerequisite, but Math 35a, 36a, 36b, 37a, and 39a do not.
- No grade below a C- will be given credit toward the minor.
- No course taken pass/fail may count towards the minor requirements.
- No more than one cross-listed course may be used to satisfy the requirements for the minor. Only cross-listed courses in Mathematics and not in Applied Mathematics may be used.

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Requirements for the Major

Required of All Majors

**Foundational Literacies:** As part of completing the major, students must:

- Fulfill the writing intensive requirement by successfully completing one of the following: MATH 23b or MATH 47a.
- Fulfill the oral communication requirement by successfully completing: MATH 16b (in conjunction with either MATH 15a or MATH 22a) or MATH 40a.
- Fulfill the digital literacy requirement by successfully completing: MATH 16b in conjunction with MATH 15a or MATH 22a, or COSI 10a, COSI 12b, or COSI 21a.

No grade below a C- will be given credit toward the majors, honors, or the teacher preparation track.

No course taken pass/fail may count towards the majors, honors, or the teacher preparation track requirements.

Bachelor's Degrees in Mathematics

All Bachelor's degrees in Mathematics require the following core classes:

- MATH 15a or 22a; MATH 20a or 22b.
- MATH 23b or exemption. See item E in Special Notes Relating to Undergraduates.
- MATH 35a, 110a, or 115a.
- MATH 28a, 28b, or 100a.
- No grade below a C- will be given credit toward the major, honors, or the teacher preparation track.
- No course taken pass/fail may count towards the major, honors, or the teacher preparation track requirements.

Cross-Listed Courses: All degrees in Mathematics have the following restrictions on cross-listed courses:

- No more than two cross-listed courses may be used to satisfy major requirements
- Only cross-listed courses in Mathematics (and not in Applied Mathematics) may be used

Bachelor of Arts in Mathematics

In addition to the requirements for all degrees, a degree of Bachelor of Arts in Mathematics requires four additional semester courses, either MATH courses numbered 27 or higher or cross-listed courses in Mathematics.

Bachelor of Science in Mathematics

In addition to the requirements for all degrees, a degree of Bachelor of Science in Mathematics requires seven additional semester courses, either MATH courses numbered 27 or higher or cross-listed courses in Mathematics.

Honors Standards

Honors in Mathematics: Along with the additional courses required, all candidates for a

degree with honors must satisfy the following:

- All courses used to satisfy major requirements must be passed with a grade of B or higher.
- At least four of the courses used to satisfy the major requirements must be MATH courses numbered 100 or higher, excluding MATH 121a, 122a, and 123a. (Cross-listed courses do not count toward this requirement.)

Bachelor of Arts in Mathematics with Honors

In addition to the requirements for all degrees, a degree of Bachelor of Arts in Mathematics with Honors with requires six additional semester courses, either MATH courses numbered 27 or higher or cross-listed courses in Mathematics, and courses must meet the additional honors standards.

Bachelor of Science in Mathematics with Honors

In addition to the requirements for all degrees, a degree of Bachelor of Science in Mathematics with honors requires seven additional semester courses, either MATH courses numbered 27 or higher or cross-listed courses in Mathematics, which meets the honors standards. In addition to the seven courses, one of the following must be completed:

- Two MATH courses numbered 130 or higher. (Cross-listed courses do not count toward this requirement). These two courses count toward the four courses required to satisfy the Honors Standards.
- Or, completion and defense of a senior honors thesis. Students considering this option should enroll in MATH 99a and MATH 99b.

Teacher Preparation Track

Students who complete the Brandeis program for Massachusetts High School Teacher Licensure (see the Education Program section in this *Bulletin*) may earn a bachelor's degree in mathematics by satisfying major requirements A, B, C, and D above and the following:

F. Two additional courses, either MATH courses numbered 27 or higher or cross-listed courses in Mathematics. Only cross-listed courses in Mathematics and not in Applied Mathematics may be used.

G. A computer science course numbered 10 or higher.

H. Completion of the High School Teacher Licensure Program.

Bachelor of Science in Applied Mathematics

At least twelve semester courses are required, including the following:

- Three foundational courses: MATH 15a or MATH 22a, MATH 20a or MATH 22b, and MATH 36a.
- MATH 23b or an exemption.
- MATH 36b or MATH 40a.
- Two of the following analysis courses: MATH 35a, MATH 37a, MATH 110a or MATH 115a.
- Two of the following: MATH 121a, MATH 122a, MATH 123a, MATH 124a or MATH 126a.
- One additional mathematics course numbered 27 or higher or a course cross-listed in Applied Mathematics.
- Two courses must be taken from another department from the following list: BCHM 102a, BCHM 104a, BCHM 145a, CHEM 141a, CHEM 142a, CHEM 146b, COSI 21a, COSI 112a, COSI 123a, COSI 130a, COSI 177a, COSI 180a, ECON 80a, ECON 161a, ECON 181b, ECON 182a, ECON 184b, NBIO 136b, NPHY 115a, any PHYS course numbered 20 or higher, and QBIO 110a.
- No grade below a C- will be given credit toward the Bachelor of Science degree.
- No course taken pass/fail may count towards the Bachelor of Science degree.

Bachelor of Science in Applied Mathematics with Honors

A degree in Applied Mathematics with honors requires satisfactory completion of all of the above requirements, as well as one of the following:

- Two MATH courses numbered 130 or higher. (Cross-listed courses do not count toward this requirement)
- Or, completion and defense of a senior honors thesis. Students considering this option should enroll in MATH 99a and MATH 99b.

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Combined BA/MA Program

Undergraduate students are eligible for the BA/MA program in mathematics if they have completed MATH 131a and b; 140a; 141a and b; and 151a and b; plus one other MATH course (or readings course) numbered 130 or higher, with a grade of B- or better. In addition, students must fulfill a minimum of three years' residence on campus. A student must make an appointment with the Undergraduate Advising Head in the Department of Mathematics in order to add the BA/MA to his/her program. This must be done no later than May 1 preceding his/her final year of study on campus.

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Special Notes Relating to Undergraduates

- With permission of the Undergraduate Advising Head, courses taken in other Brandeis departments or taken at other universities may be substituted for required mathematics courses.
- Students planning to take MATH 10a or 10b or to place into MATH 15a or 20a should take the Calculus and Linear Algebra Placement Exam. This online exam can be found, along with instructions for scoring and interpreting the results, on Placement Testing. Students planning to take MATH 22a must take the MATH 22a Placement Exam, which can be found at the same place.

Students with AP Mathematics credit should consult the chart in this Bulletin to see which Brandeis mathematics courses are equivalent to their AP credit. Note: Students who want to use their AP score to place into an upper level course must still take the Calculus Placement Exam or the MATH 22a Placement Exam to make sure that their preparation is sufficient. Questions about placement should be directed to the elementary mathematics coordinator or the Undergraduate Advising Head. - The usual calculus sequence is MATH 10a, 10b, 15a, and 20a. Students may precede this sequence with MATH 5a. Starting fall 2019 students must take Math 15a or Linear Algebra Placement Exam in order to enroll in Math 20a. Students with a strong interest in mathematics and science are encouraged to take MATH 22a,b in place of MATH 15a and 20a.
- A student may not receive credit for more than one of MATH 15a and 22a; or MATH 20a and 22b; or ECON 184b and 185a. Similarly, a student may not receive credit for more than one of MATH 28a and 100a or MATH 28b or 100b.
- Students should normally take MATH 23b before taking upper-level pure mathematics courses (i.e., those which require 23b as a prerequisite). For many students this means taking MATH 23b concurrently with MATH 15a or MATH 20a or MATH 22a or b. Students may also take MATH 23b concurrently with MATH 35a and MATH 36a as these do not have MATH 23b as a prerequisite. A student may be exempted from the requirement of taking MATH 23b by satisfactory performance on an exemption exam. The exemption exam will be given at the beginning of the fall semester.
- Students interested in graduate school or a more intensive study of mathematics are urged to include all of the following courses in their program:
- MATH 22a and b.
- MATH 100a and b.
- MATH 35a or 110a and b.
- MATH 115a.
- Other courses numbered 100 or higher.

- The following schedule determines course offerings in mathematics:
- Offered every semester are MATH 5a, 8a, 10a and b, 15a, 20a, and 23b.
- Offered once each year are MATH 3a, 22a and b, 35a, 36a and b, 37a, 40a, 47a, 100a, 110a, and 115a.
- In addition, the following semester courses are usually offered every second year according to the following schedule:
- even-odd years (e.g., 2020-2021): MATH 28b, 102a, 108b, 121a, 124a, and 126a.
- odd-even years (e.g., 2021-2022): MATH 28a, 39a, 104a, 100b, 104a, 122a, and 123a.

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Requirements for the Postbaccalaureate Program in Mathematics

- Two core courses: MATH 15a and MATH 20a.
- A grade below a C will not count towards the post-baccalaureate program.
- Elective courses: At least four additional MATH courses. Students who have taken linear algebra and/or multivariable calculus prior to entering the program may substitute additional electives for these two courses. At most one cross-listed course may be used to fulfill the elective requirement.

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Requirements for the Degree of Master of Arts

Course Requirement

Beginning in fall 2021, the requirements for the Master of Arts degree include four core classes: MATH 131a, 141a and b, and 151a. Additionally, students will take two courses selected from MATH 131b, 140a, 151b, 161a, 162a, 164a or 165a. With the permission of the Graduate Advising Head, a student with superior preparation may omit one or more of these courses and elect higher-level courses instead. In this case, the student must take an examination in the equivalent material during the first two weeks of the course.

Students will also enroll in at least one MATH 298a Specialized Mathematics Seminar Class or MATH 299a Mathematics Seminar Class.

To complete the degree, students will either take one more graduate math elective (or other approved elective) or two additional MATH 298a or MATH 299a seminars.

The typical time to degree is one year (2 semesters). During each semester, full-time students should be enrolled in at least 12 credits approved by the department.

Residence Requirement

The minimum residence requirement is one year and students typically take one year to complete the program. Students still completing requirements after this may complete the program as Extended Master's students upon approval by the Department.

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Requirements for the Degree of Master of Science

Course Requirement

Students must complete nine courses as follows:

(a) The classes MATH 131a, 141a, 141b, and 151a are all required.

(b) At least three of the following courses: MATH 131b, 140a, 151b, 161a, 162a, 164a or 165a.

(c) At least two additional graduate mathematics courses or reading courses (or another elective approved by the Graduate Advising Head).

With the permission of the Graduate Advising Head, a student with superior preparation may omit one or more of the require courses and elect higher-level courses instead. In this case, the student must take an examination in the equivalent material during the first two weeks of the course.

Option 1: Students who are interested in writing a thesis are encouraged to start thinking of a potential thesis advisor early on (the advisor will be mathematics faculty or faculty in another Brandeis department upon approval). Before students can select this option, they will need to find a thesis advisor and receive approval from the thesis advisor and Graduate Advising Head by December 1st of their second year. Students who have received approval of a thesis advisor and the Graduate Advising Head should enroll in a Master's Thesis course under the supervision of their advisor.

The MS thesis consists of reading advanced mathematics material in books and research articles, writing a thesis on a topic, and presenting the results of your reading and research during an oral presentation at the end of the semester. The discovery of a new mathematical result is encouraged, but is not a necessary condition to pass the class. The oral presentation should be given in front of the Graduate Advising Head and the professor supervising the MS Thesis, and is open to the other members of the department.

Option 2: A student may complete the Master's by taking two additional graduate mathematics courses or reading courses (or another elective approved by the Graduate Advising Head). This brings the total number of required classes to 11 for students who are not completing a thesis.

In addition to the above requirements, each Master's student must enroll in MATH 298a Specialized Mathematics Seminar Class or MATH 299a Mathematics Seminar Class during the fall or spring semester of their first year.

The typical time to degree is 2 years (4 semesters). During each semester, full-time students should be enrolled in at least 12 credits approved by the department.

Residence Requirement

The Master of Science is designed to take 2 years to complete. The minimum residence requirement is 3 semesters.

During the first 3 semester, the student is registered as a full time student. The fourth semester is completed as an Extended Master's student.

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Requirements for the Degree of Doctor of Philosophy

Program of Study

During all fall and spring semesters, students should be enrolled in at least 12 credits approved by the department. The normal first year of study includes the core classes MATH 131a, 141a and b, and 151a. In addition, students are required to take at least three of the following courses: MATH 131b, 140a, 151b, 161a, 162a, 164a or 165a one or two of which are typically taken during the first year. With the permission of the graduate advisor, a student with superior preparation may omit one or more of these courses and elect higher-level courses instead. In this case, the student must take an examination in the equivalent material during the first two weeks of the course. The second year's work will normally consist of the remaining required courses, higher-level courses and the second-year seminar (MATH 200a) as well as preparing for the qualifying examinations. During this year, students also begin taking reading courses (Math 299a), which are arranged with a professor to allow students to broaden the scope of their studies, explore possible thesis areas and use as a vehicle for their major and minor exams. By the end of their second year, students should select a dissertation advisor. Students who are ready to commence their dissertation, typically in their third year, start registering for 12 credits of 401d Dissertation Research every semester. By the end of their third year, students should complete their major exam. Students are encouraged to complete their minor exam in their second or third year but must pass it no later than their fourth year. During their third year and beyond, students also continue to take advanced courses and seminars. In addition, all PhD students are required to take the Division of Science Responsible Conduct of Research (RCR) workshop, offered in the spring. Students in their first year of study may wait until their second year to fulfill this requirement.

Teaching Requirements

An important part of the doctoral program is participation, as a teaching fellow, in a structured program of undergraduate teaching. During the spring semester of the first year, every student takes part in our teaching apprenticeship program to learn basic classroom teaching skills. All graduate students are then expected to teach a section of calculus or pre-calculus for at least four semesters, usually beginning in the second year of study. Teaching fellows must also enroll every fall semester in the Teaching Practicum, in which their teaching is evaluated and discussed.

Residence Requirement

The minimum academic residence requirement is three years.

Language Requirement

Proficiency in reading one of French, German, or Russian.

Qualifying Examination

The qualifying examination consists of two parts: a major examination and a minor examination. For the major examination, the student will choose a limited area of mathematics (e.g., differential topology, several complex variables, or ring theory) and will work with his/her advisor to form a faculty committee of three that includes the advisor. Together they will plan a program of study and a subsequent examination in that material. The aim of this study is to prepare the student for research toward the PhD. The minor examination will be more limited in scope and less advanced in content. Its subject matter should be significantly different from that of the major examination. Usually preparation for the exam takes the form of a reading course, in which the student will present a talk on the topic of the course and the examiner will administer an oral exam at the end of the semester.

Dissertation and Defense

The doctoral degree will be awarded only after the submission and acceptance of an approved dissertation and the successful defense of that dissertation.

### Courses of Instruction

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(1-99) Primarily for Undergraduate Students

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MATH
3a
Explorations in Math: A Course for Educators
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An in-depth exploration of the fundamental ideas underlying the mathematics taught in elementary and middle school. Emphasis is on problem solving, experimenting with mathematical ideas, and articulating mathematical reasoning. Usually offered every spring.

Staff

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MATH
5a
Precalculus Mathematics
**

*Does not satisfy the School of Science requirement. Students may not take MATH 5a if they have received a satisfactory grade in any math class numbered 10 or higher.*

Brief review of algebra followed by the study of functions. Emphasis on exponential, logarithmic, and trigonometric functions. The course's goal is to prepare students for MATH 10a. The decision to take this course should be guided by the results of the mathematics placement exam. Usually offered every semester.

Rebecca Torrey

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MATH
8a
Introduction to Probability and Statistics
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Discrete probability spaces, random variables, expectation, variance, approximation by the normal curve, sample mean and variance, and confidence intervals. Does not require calculus; only high school algebra and graphing of functions. Usually offered every semester.

Staff

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MATH
10a
Techniques of Calculus (a)
**

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*Prerequisite: A satisfactory grade of C- or higher in MATH 5a or placement by examination. Students may not take MATH 10a if they have received a satisfactory grade in MATH 10b or MATH 20a.*

Introduction to differential (and some integral) calculus of one variable, with emphasis on techniques and applications. Usually offered every semester in multiple sections.

Staff

**
MATH
10b
Techniques of Calculus (b)
**

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*Prerequisite: A satisfactory grade of C- or higher in MATH 10a or placement by examination. Continuation of 10a. Students may not take MATH 10a and MATH 10b simultaneously. Students may not take MATH 10b if they have received a satisfactory grade in MATH 20a.*

Introduction to integral calculus of one variable with emphasis on techniques and applications. Usually offered every semester in multiple sections.

Staff

**
MATH
15a
Applied Linear Algebra
**

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*Prerequisites: MATH 5a and permission of the instructor, placement by examination, or any mathematics course numbered 10 or above. Students may take MATH 15a or 22a for credit, but not both.*

Matrices, determinants, linear equations, vector spaces, eigenvalues, quadratic forms, linear programming. Emphasis on techniques and applications. Usually offered every semester.

Tarakaram Gollamudi and John Wilmes (fall), Staff (spring)

**
MATH
16b
Applied Linear Algebra Practicum
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*Prerequisite: MATH 15a or MATH 22a. Yields half-course credit.*

Introduces fundamental skills for both computing and oral communication in the context of applied linear algebra problems. Includes basics of Python, numpy, and matplotlib. Usually offered every year.

John Wilmes

**
MATH
20a
Multi-variable Calculus
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*Prerequisites: MATH 10a and b and MATH 15a, or placement by examination. Students may take Math 20a or 22b for credit, but not both. Students may not take MATH 10a or 10b or 15a concurrently with MATH 20a.*

Among the topics treated are functions of several variables, vector-valued functions, partial derivatives and multiple integrals, extremum problems, line and surface integrals, Green's and Stokes's theorems. Emphasis on techniques and applications. Usually offered every semester.

Keith Merrill (fall), Rahul Krishna (spring)

**
MATH
22a
Honors Linear Algebra and Multi-variable Calculus, Part I
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*Prerequisite: MATH 22 placement exam and permission of the instructor. Students may take MATH 15a or 22a for credit, but not both.*

MATH 22a and b cover linear algebra and calculus of several variables. The material is similar to that of MATH 15a and MATH 20b, but with a more theoretical emphasis and with more attention to proofs. Usually offered every fall.

Bong Lian

**
MATH
22b
Honors Linear Algebra and Multi-variable Calculus, Part II
**

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sn
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*Prerequisite: MATH 22a or permission of the instructor. Students may take MATH 20a or 22b for credit, but not both.*

See MATH 22a for course description. Usually offered every spring.

Daniel Ruberman

**
MATH
23b
Introduction to Proofs
**

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*Prerequisites: MATH 15a, 20a, or 22a, or permission of the instructor.*

Emphasizes the analysis and writing of proofs. Various techniques of proof are introduced and illustrated with topics chosen from set theory, calculus, algebra, and geometry. Usually offered every semester.

Rahul Krishna and Gleb Nenashev (fall), An Huang (spring)

**
MATH
28a
Introduction to Groups
**

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*Prerequisites: MATH 23b and either MATH 15a or 22a, or permission of the instructor. Students may take MATH 28a or 100a for credit, but not both.*

Groups. Lagrange's theorem. Modulo n addition and multiplication. Matrix groups and permutation groups. Homomorphisms, normal subgroups, cosets, and factor groups. Usually offered every second year.

Staff

**
MATH
28b
Introduction to Rings and Fields
**

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*Prerequisites: MATH 23b and either MATH 15a, 22a, or permission of the instructor. Students may take MATH 28b or 100b for credit, but not both.*

Fields. Z/p and other finite fields. Commutative rings. Polynomial rings and subrings of C. Euclidean rings. The quotient ring A/(f). Polynomials over Z. Usually offered every second year.

Staff

**
MATH
35a
Advanced Calculus and Fourier Analysis
**

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*Prerequisites: MATH 15a or 22a and MATH 20a or 22b. *

Complex numbers. Fourier series and Fourier integrals. Introduction to ODE and PDE. Classical PDE from physics: wave and string equations. Application of Fourier decomposition to the solution of linear PDEs. Generalization of the method with other orthogonal sets of functions time permitting: Introduction to Bessel and Legendre functions, and the Sturm-Liouville theory. Usually offered every fall.

Ying Zhang

**
MATH
36a
Probability
**

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*Prerequisite: MATH 20a or 22b.*

Sample spaces and probability measures, elementary combinatorial examples. Conditional probability. Random variables, expectations, variance, distribution and density functions. Independence and correlation. Chebychev's inequality and the weak law of large numbers. Central limit theorem. Markov and Poisson processes. Usually offered every fall.

Jonathan Touboul

**
MATH
36b
Mathematical Statistics
**

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*Prerequisite: MATH 36a or permission of the instructor.*

Probability distributions, estimators, hypothesis testing, data analysis. Theorems will be proved and applied to real data. Topics include maximum likelihood estimators, the information inequality, chi-square test, and analysis of variance. Usually offered every spring.

Keith Merrill

**
MATH
37a
Differential Equations
**

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*Prerequisites: MATH 15a or 22a and MATH 20a or 22b.*

A first course in ordinary differential equations. Study of general techniques, with a view to solving specific problems such as the brachistochrone problem, the hanging chain problem, the motion of the planets, the vibrating string, Gauss's hypergeometric equation, the Volterra predator-prey model, isoperimetric problems, and the Abel mechanical problem. Usually offered every fall.

Thomas Fai

**
MATH
39a
Introduction to Combinatorics
**

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*Prerequisite: COSI 29a, MATH 23b, or permission of the instructor.*

Topics include graph theory (trees, planarity, coloring, Eulerian and Hamiltonian cycles), combinatorial optimization (network flows, matching theory), enumeration (permutations and combinations, generating functions, inclusion-exclusion), and extremal combinatorics (pigeonhole principle, Ramsey's theorem). Usually offered every second year.

Staff

**
MATH
40a
Introduction to Applied Mathematics
**

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oc
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*Prerequisites: MATH 15a or MATH 22a and MATH 20a or MATH 22b.*

Introduces the problems and issues of applied mathematics, with emphasis on how mathematical ideas can have a major impact on diverse fields of human inquiry. Usually offered every fall.

Staff

**
MATH
47a
Introduction to Mathematical Research
**

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*Prerequisite: MATH 23b or permission of the instructor.*

Students work on research projects that involve generating data, making conjectures, and proving theorems, and present their results orally and in writing. Introduces applications of computers in mathematical research: symbolic computation, typesetting, and literature search. Usually offered every fall.

Kiyoshi Igusa

**
MATH
98a
Independent Research
**

Usually offered every year.

Staff

**
MATH
98b
Independent Research
**

Usually offered every year.

Staff

**
MATH
99a
Senior Research
**

Usually offered every year.

Staff

**
MATH
99b
Senior Research
**

Usually offered every year.

Staff

###
(100-199) For Both Undergraduate and Graduate Students

Courses numbered 131 and above are ordinarily taken by graduate students; interested undergraduates should consult with the instructor regarding the required background for each course.

**
MATH
100a
Introduction to Algebra, Part I
**

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sn
]

*Prerequisite: MATH 23b and MATH 15a or 22a, or permission of the instructor. Students may take MATH 28a or 100a for credit, but not both. *

An introduction to the basic notions of modern algebra—rings, fields, and linear algebra. Usually offered every spring.

Staff

**
MATH
100b
Introduction to Algebra, Part II
**

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*Prerequisite: MATH 100a or permission of the instructor. Students may take MATH 28b or 100b for credit, but not both.*

A continuation of MATH 100a, culminating in Galois theory. Usually offered every second year.

An Huang

**
MATH
102a
Introduction to Differential Geometry
**

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*Prerequisites: MATH 23b and either MATH 20a or 22b or permission of the instructor.*

Introduces the classical geometry of curves and surfaces. Topics include the Frenet equations and global properties of curves, local surface theory, including the fundamental forms and the Gauss map, intrinsic geometry of surfaces, Gauss's fundamental theorem and the Gauss-Bonnet Theorem. Usually offered every second year.

Mark Adler

**
MATH
104a
Introduction to Topology
**

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*Prerequisites: MATH 15a or 22a and MATH 20a or 22b, and MATH 23b, or permission of the instructor. *

An introduction to point set topology, covering spaces, and the fundamental group. Usually offered every second year.

Staff

**
MATH
108b
Introduction to Number Theory
**

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*Prerequisites: MATH 23b and MATH 15a or 22a, or permission of the instructor. *

Congruences, finite fields, the Gaussian integers, and other rings of numbers. Quadratic reciprocity. Such topics as quadratic forms or elliptic curves will be covered as time permits. Usually offered every second year.

Omer Offen

**
MATH
110a
Introduction to Real Analysis, Part I
**

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*Prerequisites: MATH 15a or 22a and MATH 20a or 22b, and MATH 23b, or permission of the instructor. *

MATH 110a and b give a rigorous introduction to metric space topology, continuity, derivatives, and Riemann and Lebesgue integrals. Usually offered every fall.

Staff

**
MATH
110b
Introduction to Real Analysis, Part II
**

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*Prerequisite: MATH 110a or permission of the instructor. May not be taken for credit by students who took MATH 40b in prior years.*

See MATH 110a for course description. Usually offered every second year.

Staff

**
MATH
115a
Introduction to Complex Analysis
**

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*Prerequisites: MATH 15a or 22a and MATH 20a or 22b, and MATH 23b or permission of the instructor. *

An introduction to functions of a complex variable. Topics include analytic functions, line integrals, power series, residues, conformal mappings. Usually offered every spring.

Alan Mayer

**
MATH
121a
Mathematics for Natural Sciences
**

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sn
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*Prerequisites: MATH 15a or MATH 22a and MATH 20a or MATH 22b.*

Introduces a set of mathematical tools of great applicability to the natural sciences. It will prepare students to use these tools in concrete applications. Topics include complex numbers, power series, calculus of variations, and Laplace transform. Usually offered every second year.

Jonathan Touboul

**
MATH
122a
Numerical Methods and Big Data
**

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*Prerequisites: MATH 15a or MATH 22a and MATH 20a or MATH 22b, and basic proficiency with a programming language such as Python or Matlab.*

Introduces fundamental techniques of numerical linear algebra widely used for data science and scientific computing. The purpose of this course is to introduce methods that are useful in applications and research. Usually offered every second year.

Staff

**
MATH
123a
Principles of Mathematical Modeling and Applications to Biology
**

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*Prerequisites: MATH 15a or MATH 22a, MATH 20a or MATH 22b, and MATH 37a.*

Provides the basic concepts and approaches for modelling in physics and biology. The course will be developed around examples of central research interest in biology and related fields. Usually offered every second year.

Jonathan Touboul

**
MATH
124a
Optimization
**

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sn
]

*Prerequisites: MATH 15a or MATH 22a, MATH 20a or MATH 22b, MATH 23b, and basic proficiency with a programming language such as Python or Matlab, or permission of the instructor. *

Explores the theory of mathematical optimization and its fundamental algorithms, emphasizing problems arising in machine learning, economics, and operations research. Topics include linear and integer programming, convex analysis, and duality. Usually offered every spring.

John Wilmes

**
MATH
126a
Introduction to Stochastic Processes and Models
**

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]

*Prerequisites: MATH 15a, 20a, and 36a. *

Basic definitions and properties of finite and infinite Markov chains in discrete and continuous time, recurrent and transient states, convergence to equilibrium, Martingales, Wiener processes and stochastic integrals with applications to biology, economics, and physics. Usually offered every second year.

Jonathan Touboul

**
MATH
131a
Algebra I
**

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sn
]

*Prerequisites: MATH 100a and 100b or permission of the instructor. *

Groups, rings, modules, Galois theory, affine rings, and rings of algebraic numbers. Multilinear algebra. The Wedderburn theorems. Other topics as time permits. Usually offered every fall.

Olivier Bernardi

**
MATH
131b
Algebra II
**

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sn
]

*Prerequisite: MATH 131a or permission of the instructor. *

Continuation of MATH 131a. Usually offered every spring.

Kiyoshi Igusa

**
MATH
140a
Geometry of Manifolds
**

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sn
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*Prerequisites: MATH 110a and 110b or permission of the instructor. *

Manifolds, tensor bundles, vector fields, and differential forms. Frobenius theorem. Integration, Stokes's theorem, and de Rham's theorem. Usually offered every fall.

Daniel Ruberman

**
MATH
140b
Differential Geometry
**

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]

*Prerequisite: MATH 140a or permission of the instructor. *

Riemannian metrics, parallel transport, geodesics, curvature. Introduction to Lie groups and Lie algebras, vector bundles and principal bundles. Usually offered every second year.

Daniel Ruberman

**
MATH
141a
Real Analysis
**

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*Prerequisites: MATH 110a and 110b or permission of the instructor. *

Measure and integration. Lp spaces, Banach spaces, Hilbert spaces. Radon-Nikodym, Riesz representation, and Fubini theorems. Fourier transforms. Usually offered every fall.

Rahul Krishna

**
MATH
141b
Complex Analysis
**

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sn
]

*Prerequisite: MATH 141a or permission of the instructor. *

The Cauchy integral theorem, calculus of residues, and maximum modulus principle. Harmonic functions. The Riemann mapping theorem and conformal mappings. Other topics as time permits. Usually offered every spring.

Mark Adler

**
MATH
151a
Topology I
**

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*Prerequisite: MATH 104a or permission of the instructor. *

Fundamental group, covering spaces. Cell complexes, homology and cohomology theory, with applications. Usually offered every fall.

Ruth Charney

**
MATH
151b
Topology II
**

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]

*Prerequisite: MATH 151a or permission of the instructor. *

Continuation of MATH 151a. Manifolds and orientation, cup and cap products, Poincaré duality. Other topics as time permits. Usually offered every spring.

Staff

**
MATH
161a
Advanced Bifurcation Analysis in Dynamical Systems
**

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*Prerequisites: MATH 20a, MATH 23b, and MATH 37a.*

Exposes tools from modern theory of dynamical systems and bifurcations for general nonlinear differential equations (including infinite dimensional delayed or integral equations). Such systems are increasingly used in research or advanced models of natural and social phenomena. Usually offered every second year.

Staff

**
MATH
162a
Numerical Methods for Scientific Computing
**

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*Prerequisites: MATH 37a and MATH 122a, or permission of the instructor. A basic proficiency with a programming language such as Python or Matlab is required.*

Studies numerical methods for linear algebra, ordinary and partial differential equations, and optimization. Equal emphasis will be placed on theory (stability, accuracy, and convergence) and practical problem-solving using a programming language such as Python. Usually offered every second year.

Staff

**
MATH
164a
Partial Differential Equations
**

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*Prerequisites: MATH 35a and MATH 37a. *

This course will introduce students to mathematical aspects of partial differential equations (PDE's). It will aim to strike a balance between theory, such as conditions granting the existence and uniqueness of solutions, methods to solve these equations in practice, such as Green's functions, and physical intuition e.g. conservation laws that give rise to th_e equations and variational methods to study them. Usually offered every second year.

Mark Adler

**
MATH
165a
Probability Theory
**

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*Prerequisite: MATH 141a. MATH 20a, MATH 23b, MATH 36a, and MATH 110a may be accepted for the prerequisite.*

Exposes tools from modern theory of probability and stochastic processes, as well as modern applications. Such systems are increasingly used in mathematical research and become essential in advanced studies of natural and social phenomena. Usually offered every second year.

Staff

**
MATH
180a
Combinatorics
**

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Emphasis on enumerative combinatorics. Generating functions and their applications to counting graphs, paths, permutations, and partitions. Bijective counting, combinatorial identities, Lagrange inversion and Möbius inversion. Usually offered every second year.

Staff

**
MATH
180b
Topics in Combinatorics
**

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Possible topics include symmetric functions, graph theory, extremal combinatorics, combinatorial optimization, coding theory. Usually offered every second year.

Olivier Bernardi

###
(200 and above) Primarily for Graduate Students

All graduate-level courses will have organizational meetings the first week of classes.

**
MATH
200a
Second-Year Seminar
**

A course for second-year students in the PhD program designed to provide exposure to current research and practice in giving seminar talks. Students read recent journal articles and preprints and present the material. Usually offered every spring.

Staff

**
MATH
201a
Topics in Algebra
**

Introduction to a field of algebra. Possible topics include representation theory, vertex algebras, algebraic groups. Usually offered every year.

Staff

**
MATH
202a
Algebraic Geometry I
**

Varieties and schemes. Cohomology theory. Curves and surfaces. Usually offered every second year.

Alan Mayer

**
MATH
202b
Algebraic Geometry II
**

Continuation of MATH 202a. Usually offered every second year.

Alan Mayer

**
MATH
203a
Introduction to Algebraic Number Theory
**

Basic algebraic number theory (number fields, ramification theory, class groups, Dirichlet unit theorem), zeta and L-functions (Riemann zeta function, Dirichlet L-functions, primes in arithmetic progressions, prime number theorem). Usually offered every second year.

Omer Offen

**
MATH
203b
Topics in Number Theory
**

Possible topics include class field theory, cyclotomic fields, modular forms, analytic number theory, ergodic number theory. Usually offered every second year.

Joël Bellaïche

**
MATH
204a
T.A. Practicum
**

Teaching elementary mathematics courses is a subtle and difficult art involving many skills besides those that make mathematicians good at proving theorems. This course focuses on the development and support of teaching skills. The main feature is individual observation of the graduate student by the practicum teacher, who provides written criticism of and consultation on classroom teaching practices. May not be counted toward one of the lecture courses that is required in the second and third years. Usually offered every fall.

Rebecca Torrey

**
MATH
211a
Topics in Differential Geometry and Analysis I
**

Possible topics include complex manifolds, elliptic operators, index theory, random matrix theory, integrable systems, dynamical systems, ergodic theory. Usually offered every spring.

Dmitry Kleinbock

**
MATH
212b
Functional Analysis
**

Banach and Hilbert spaces, linear operators, operator topologies, Banach algebras. Convexity and fixed point theorems, integration on locally compact groups. Spectral theory. Other topics as time permits. Usually offered every second year.

Dmitry Kleinbock

**
MATH
221a
Topology III: Vector Bundles and Applications
**

Vector bundles and characteristic classes. Elementary homotopy theory and obstruction theory. Cobordism and transversality; other topics as time permits. Usually offered every fall.

Kiyoshi Igusa

**
MATH
221b
Topics in Topology
**

Topics in topology and geometry. In recent years, topics have included knot theory, symplectic and contact topology, gauge theory, and three-dimensional topology. Usually offered every year.

Staff

**
MATH
223a
Lie Algebras: Representation Theory
**

Theorems of Engel and Lie. Semisimple Lie algebras, Cartan's criterion. Universal enveloping algebras, PBW theorem, Serre's construction. Representation theory. Other topics as time permits. Usually offered every second year.

An Huang

**
MATH
224b
Lie Groups
**

Basic theory of Lie groups and Lie algebras. Homogeneous spaces. Haar measure. Compact Lie groups, representation theory, Peter-Weyl theorem, differential slice theorem. Complex reductive groups. Other topics as time permits. Usually offered every second year.

Staff

**
MATH
250a
Complex Algebraic Geometry I
**

Riemann surfaces, Riemann-Roch theorems, Jacobians. Complex manifolds, Hodge decomposition theorem, cohomology of sheaves, Serre duality. Vector bundles and Chern classes. Other topics as time permits. Usually offered every second year.

Staff

**
MATH
250b
Complex Algebraic Geometry II
**

Continuation of MATH 250a. Usually offered every second year.

Staff

**
MATH
290a
Readings in Mathematics
**

Staff

**
MATH
298a
Specialized Mathematics Seminar Class
**

Gives graduate students a repeated exposure to current research in a specific mathematical field. This will train students to absorb ideas and concepts pitched at a higher level of abstraction than in a typical graduate class. Attendance at one of the specialized seminars in the mathematics department, such as the Topology Seminar, Combinatorics Seminar, New England Dynamics and Number Theory Seminar, or the Mathematical Biology Seminar is required. Usually offered every year.

Olivier Bernardi

**
MATH
299a
Mathematics Seminar Class
**

*Yields half-course credit.*

Exposes students to mathematical research over a diverse set of mathematical topics. This course also trains students to absorb ideas and concepts pitched at a higher level of abstraction than in typical graduate classes. The requirement for the class is a regular attendance at seminars in the mathematics department, such as the Everytopic seminar, Topology Seminar, Combinatorics seminar, etc. Students are encouraged to ask questions and engage with the speakers. In addition, students will be required to write a one-page reflection on a one-hour seminar of their choosing. Usually offered every year.

Olivier Bernardi

**
MATH
300a
Master's Thesis
**

Students who have selected and received approval from the faculty member supervising the thesis and the Graduate Advising Head, may enroll in this thesis course with their faculty supervisor. The thesis consists of reading some advanced mathematics material in the form of topics books or a series of research articles, writing a thesis on a topic, and presenting the results of your reading and research during an oral presentation at the end of the semester. Instructor's Signature Required.

Staff

**
MATH
393g
Math Internship
**

*Permission of the Director of Graduate Studies required. Yields quarter-course credit. May be repeated for credit. For Ph.D. students only.*

A real-world workplace experience that is approved and monitored by a faculty member. Students have the opportunity to complete a paid or unpaid internship in an area such as education, data science or software engineering. The internship is an opportunity to develop professional skills, explore career paths, and make connections with employers. Usually offered every year.

Staff

**
MATH
399a
Advanced Readings in Mathematics
**

Staff

**
MATH
401d
Research
**

Independent research for the PhD degree. Specific sections for individual faculty members as requested.

Staff

###
Cross-Listed in Mathematics

**
COSI
130a
Introduction to the Theory of Computation
**

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*Prerequisite: COSI 29a or the equivalent. *

Introduces topics in the theory of computation, including: finite automata and regular languages, pushdown automata and context-free languages, context-sensitive languages and Type 0 languages, Turing machines and Church's thesis, the halting problem and undecidability, and introduction to NP and PSPACE complete problems. Usually offered every year.

Antonella Di Lillo or James Storer

**
COSI
190a
Introduction to Programming Language Theory
**

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*Prerequisite: COSI 121b or familiarity with a functional programming language, set theory and logic.*

An introduction to the mathematical semantics of functional programming languages. Principles of denotational semantics; lambda calculus and its programming idiom; Church-Rosser theorem and Böhm's theorem; simply typed lambda calculus and its model theory: completeness for the full type frame, Statman's 1-section theorem and completeness of beta-eta reasoning; PCF and full abstraction with parallel operations; linear logic, proofnets, context semantics and geometry of interaction, game semantics, and full abstraction. Usually offered every second year.

Staff

**
ECON
184b
Econometrics
**

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*Prerequisites: ECON 83a. Corequisite: ECON 80a or permission of the instructor. Students must earn a C- or higher in MATH 10a, or otherwise satisfy the calculus requirement, to enroll in this course. This course may not be taken for credit by students who have previously taken or are currently enrolled in ECON 185a or ECON 311a.*

An introduction to the theory of econometric regression and forecasting models, with applications to the analysis of business and economic data. Usually offered every year.

Mahsa Akbari, Elizabeth Brainerd and Yinchu Zhu

**
NPHY
115a
Dynamical Systems
**

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sn
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*Prerequisites: MATH 10b and MATH 15a or PHYS 20a or equivalent. *

Covers analytic, computational and graphical methods for solving systems of coupled nonlinear ordinary differential equations. We study bifurcations, limit cycles, coupled oscillators and noise, with examples from physics, chemistry, population biology and many models of neurons. Usually offered every third year.

Irving Epstein

**
PHIL
106b
Mathematical Logic
**

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Covers in detail several of the following proofs: the Gödel Incompleteness Results, Tarski's Undefinability of Truth Theorem, Church's Theorem on the Undecidability of Predicate Logic, and Elementary Recursive Function Theory. Usually offered every year.

Staff

**
PHYS
100a
Classical Mechanics
**

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sn
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*Prerequisites: PHYS 20a or permission of the instructor.*

Lagrangian dynamics, Hamiltonian mechanics, planetary motion, general theory of small vibrations. Introduction to continuum mechanics. Usually offered every second year.

Richard Fell

**
PHYS
110a
Mathematical Methods in Continuum Mechanics
**

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*Prerequisite: PHYS 30a, PHYS 31a, or permission of the instructor.*

Studies mathematical techniques that arise in the context of continuum mechanical (fluids and elastic media). Subjects include vector and tensor calculus, differential geometry, differential equations, and dimensional analysis. Usually offered every other year.

Bulbul Chakraborty

**
QBIO
110a
Numerical Modeling of Biological Systems
**

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*Prerequisite: MATH 10a and b or equivalent.*

Modern scientific computation applied to problems in molecular and cell biology. Covers techniques such as numerical integration of differential equations, molecular dynamics and Monte Carlo simulations. Applications range from enzymes and molecular motors to cells. Usually offered every second year.

Michael Hagen

###
Cross-Listed in Applied Mathematics

All undergraduate PHYS courses numbered 20a or higher are cross-listed in Applied Mathematics.

**
BCHM
102a
Quantitative Approaches to Biochemical Systems
**

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*Prerequisite: BCHM 100a or equivalent and Math 10a and b or equivalent.*

Introduces quantitative approaches to analyzing macromolecular structure and function. Emphasizes the use of basic thermodynamics and single-molecule and ensemble kinetics to elucidate biochemical reaction mechanisms. Also discusses the physical bases of spectroscopic and diffraction methods commonly used in the study of proteins and nucleic acids. Usually offered every year.

Maria-Eirini Pandelia

**
BCHM
104a
Classical and Statistical Thermodynamics
**

[
sn
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*Prerequisites: MATH 10a,b or equivalent, PHYS 11 or 15.*

Covers basics of physical chemistry underpinning applications in BCHM 104b. Focus is placed on quantitative treatments of the probabilistic nature of molecular reality: molecular kinetic theory, basic statistical mechanics, and chemical thermodynamics in aqueous solution. Usually offered every second year.

Daniel Oprian

**
BCHM
145a
How to Decide: Bayesian Inference and Computational Statistics
**

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*Prerequisites: Math 10a and b.*

A calculus-based courses that teaches the theory and practice of modern statistical methods used by experimental scientists. Topics include Bayesian inference, maximum likelihood estimation, and computational resampling methods. The course consists of a mixture of small lectures and in-class computational exercises. Usually offered ever third year.

Jeff Gelles and Douglas Theobald

**
CHEM
141a
Classical and Statistical Thermodynamics
**

[
sn
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*Prerequisites: Satisfactory grade in CHEM 11a, 15a and CHEM 11b, 15b or the equivalent; MATH 10a,b or the equivalent; PHYS 10a,b, 11a,b or 15a,b or the equivalent. Organic chemistry is also recommended.*

Thermodynamic principles, tools, and applications in chemistry and biology. Usually offered every year.

Klaus Schmidt-Rohr

**
CHEM
142a
Quantum Mechanics and Spectroscopy
**

[
sn
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*Prerequisites: A satisfactory grade in CHEM 11a,b or 15a,b or the equivalent; MATH 10a,b or the equivalent; PHYS 10a,b, 11a,b, or 15a,b or the equivalent. Organic chemistry is also recommended.*

Solutions to the Schrodinger equation of relevance to molecular structure, reactivity and spectroscopy; introduction to quantum mechanical calculations and computational methods. Usually offered every year.

Rebecca Gieseking

**
CHEM
146b
Advanced NMR Spectroscopy
**

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sn
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*Prerequisites: A satisfactory grade in PHYS 10a,b, 11a,b, or 15a,b or the equivalent; MATH 10a,10b. *

A detailed discussion of modern NMR methods will be presented. The course is designed so as to be accessible to nonspecialists, but still provide a strong background in the theory and practice of modern NMR techniques. Topics include the theory of pulse and multidimensional NMR experiments, chemical shift, scalar and dipolar coupling, NOE, spin-operator formalism, heteronuclear and inverse-detection methods, Hartmann-Hahn and spin-locking experiments. Experimental considerations such as pulse sequence design, phase cycling, and gradient methods will be discussed. Guest lecturers will provide insight into particular topics such as solid-state NMR and NMR instrumental design. Usually offered every second year.

Thomas Pochapsky

**
COSI
21a
Data Structures and the Fundamentals of Computing
**

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*Prerequisite: COSI 12b. Graduate students may take this course concurrently with COSI 12b with permission of the Director of Graduate Studies.*

Focuses on the design and analysis of algorithms and the use of data structures. Through the introduction of the most widely used data structures employed in solving commonly encountered problems. Students will learn different ways to organize data for easy access and efficient manipulation. The course also covers algorithms to solve classic problems, as well as algorithm design strategies; and computational complexity theory for studying the efficiency of the algorithms. Usually offered every year.

Antonella Di Lillo

**
COSI
112a
Modal, Temporal, and Spatial Logic for Language
**

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sn
]

*Prerequisites: COSI 29a, COSI 121b, LING 130a, or PHIL 6a.. *

Examines the formal and computational properties of logical systems that are used in AI and linguistics. This includes (briefly) propositional logic and first order logic, and then an in-depth study of modal logic, temporal logic, spatial logic, and dynamic logic. Throughout the analyses of these systems, focuses on how they are used in the modeling of linguistic data. Usually offered every second year.

James Pustejovsky

**
COSI
123a
Statistical Machine Learning
**

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*Prerequisite: COSI 29a, MATH 10a, and MATH 15a.*

Focuses on learning from data using statistical analysis tools and deals with the issues of designing algorithms and systems that automatically improve with experience. This course is designed to give students a thorough grounding in the methodologies, technologies, mathematics, and algorithms currently needed by research in learning with data. Usually offered every year.

Pengyu Hong

**
COSI
130a
Introduction to the Theory of Computation
**

[
sn
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*Prerequisite: COSI 29a or the equivalent. *

Introduces topics in the theory of computation, including: finite automata and regular languages, pushdown automata and context-free languages, context-sensitive languages and Type 0 languages, Turing machines and Church's thesis, the halting problem and undecidability, and introduction to NP and PSPACE complete problems. Usually offered every year.

Antonella Di Lillo or James Storer

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COSI
177a
Scientific Data Processing in Matlab
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*Prerequisite: COSI 12b. MATH 15a is recommended.*

Introduces scientific computing using Matlab. Programming concepts such as data types, vectors, conditional execution, loops, procedural abstraction, modules, APIs are presented. The course will present scientific techniques relevant to computational science, with an emphasis on image processing. Usually offered every second year.

Antonella DiLillo

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COSI
180a
Algorithms
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*Prerequisites for undergraduates and combined BA/MA students: COSI 21a, COSI 29a, and COSI 131a. Graduate students with the appropriate background do not need to satisfy the course prerequisites.*

Basic concepts in the design and analysis of algorithms. Usually offered every second year.

James Storer

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ECON
80a
Microeconomic Theory
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*Prerequisite: ECON 10a. Students must earn C- or higher in MATH 10a, or otherwise satisfy the calculus requirement, to enroll in this course.*

Analysis of the behavior of economic units within a market economy. Emphasis upon individuals' decisions as demanders of goods and suppliers of resources, and firms' decisions as suppliers of goods and demanders of resources under various market structures. Usually offered every semester.

Geoff Clarke, Kathryn Graddy, Nader Habibi, Benjamin Shiller, and Nelson Sa

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ECON
161a
International Macroeconomics and Finance
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*Prerequisites: ECON 82b. Corequisite: ECON 184b or permission of the instructor.*

Applications of international economic theory – regarding trade, the balance of payments, investments, and exchange rates – to the management of import/export firms and multinational corporations. Usually offered every year.

George Hall

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ECON
181b
Game Theory and Economic Applications
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*Prerequisites: ECON 80a, ECON 83a, and MATH 10a or equivalent.*

Analysis of decision making in multiperson settings. Studies models of equilibrium and various kinds of games under perfect and imperfect information. The applications include business strategy and competition, auctions, and risk sharing. Usually offered every year.

Nelson Sa

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ECON
182a
Topics in Advanced Macroeconomics
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*Prerequisite: ECON 80a, 82b, and 83a.*

Contemporary theories of economic growth, business cycles, monetary economics, and financial crises and their implications for monetary and fiscal policy. Emphasis on empirical work and computer modeling. Usually offered every year.

George Hall

**
ECON
184b
Econometrics
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*Prerequisites: ECON 83a. Corequisite: ECON 80a or permission of the instructor. Students must earn a C- or higher in MATH 10a, or otherwise satisfy the calculus requirement, to enroll in this course. This course may not be taken for credit by students who have previously taken or are currently enrolled in ECON 185a or ECON 311a.*

An introduction to the theory of econometric regression and forecasting models, with applications to the analysis of business and economic data. Usually offered every year.

Mahsa Akbari, Elizabeth Brainerd and Yinchu Zhu

**
NBIO
136b
Computational Neuroscience
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*Prerequisite: MATH 10a and either NBIO 140b or PHYS 10a or approved equivalents.*

An introduction to concepts and methods in computer modeling and analysis of neural systems. Topics include single and multicompartmental models of neurons, information representation and processing by populations of neurons, synaptic plasticity and models of learning, working memory, decision making and neural oscillations. The course will be based on in-class computer tutorials, assuming no prior coding experience, with reading assignments and preparation as homework. Usually offered every second year.

Paul Miller

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NPHY
115a
Dynamical Systems
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*Prerequisites: MATH 10b and MATH 15a or PHYS 20a or equivalent. *

Covers analytic, computational and graphical methods for solving systems of coupled nonlinear ordinary differential equations. We study bifurcations, limit cycles, coupled oscillators and noise, with examples from physics, chemistry, population biology and many models of neurons. Usually offered every third year.

Irving Epstein

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QBIO
110a
Numerical Modeling of Biological Systems
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*Prerequisite: MATH 10a and b or equivalent.*

Modern scientific computation applied to problems in molecular and cell biology. Covers techniques such as numerical integration of differential equations, molecular dynamics and Monte Carlo simulations. Applications range from enzymes and molecular motors to cells. Usually offered every second year.

Michael Hagen

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Courses of Related Interest

Note: the following courses do not count as credit toward the major or the minor in Mathematics and the major in Applied Mathematics.

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BIOL
51a
Biostatistics
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*Prerequisites: BIOL 14a and BIOL 15b.*

An introductory level biostatistics class providing an overview to statistical methods used in biological and medical research. Topics include descriptive statistics, elementary probability theory, commonly observed distributions, basic concepts of statistical inference, hypothesis testing, regression, as well as analysis of variance. Basic statistical analysis using the R software package will be introduced. Usually offered every semester.

Kene Piasta

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BIOL
107a
Data Analysis and Statistics Workshop
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The interpretation of data is key to making new discoveries, making optimal decisions, and designing experiments. Students will learn skills of data analysis and computer coding through hands-on, computer-based tutorials and exercises that include experimental data from the biological sciences. Knowledge of very basic statistics (mean, median) will be assumed. Usually offered every year.

Stephen Van Hooser

**
PHIL
138b
Philosophy of Mathematics
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*Prerequisite: A course in logic or permission of the instructor.*

Basic issues in the foundations of mathematics will be explored through close study of selections from Frege, Russell, Carnap, and others, as well as from contemporary philosophers. Questions addressed include: What are the natural numbers? Do they exist in the same sense as tables and chairs? How can "finite beings" grasp infinity? What is the relationship between arithmetic and geometry? The classic foundational "programs," logicism, formalism, and intuitionism, are explored. Usually offered every second year.

Palle Yourgrau