## Department of Mathematics

Last updated: October 19, 2023 at 9:49 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 2019-2020, 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 Majors 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, Master of Science, and Doctor of Philosophy degrees. The Master of Arts and Master of Science programs give 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 of Arts, Master of Science, 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 and applied 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 them 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

Postbaccalaureate Program in Mathematics

The postbac program in Mathematics is a non-degree program aimed at students wishing to complete and expand their knowledge of mathematics at the undergraduate level and get prepared for more advanced knowledge.

**Knowledge**

- Postbac students are required to demonstrate knowledge in linear algebra and multivariate calculus.
- Postbac students need to complete six Mathematics classes above the calculus level (the linear algebra and multivariate calculus can count toward this requirement, or can be replaced by more advanced classes).
- A wide array of rigorous classes are available ranging from the proof-based (introduction to proof, abstract algebra, analysis and topology) to the applied (mathematical modeling, optimization, simulation and big data, probability and stochastic analysis).
- For the most advanced students, graduate level classes in mathematics are also available.

**Core Skills**

Students completing the Postbac Program in Mathematics acquire a solid foundation in mathematics well beyond the calculus level. They acquire a working knowledge of standard techniques and results in mathematics which are key in a wide range of applications.

**Outcome**

Postbac students exit the Program ideally prepared to apply their mathematical skills in the workplace, or apply for more advanced degrees in mathematics such as a Master or a PhD.

Graduate Program in Mathematics

Master of Arts and Master of Science in Mathematics

**Knowledge**

- MA and MS students are required to demonstrate a broad and deep knowledge of algebra, topology, geometry, and analysis by passing their required core courses.
- A wide array of more advanced or more specialized elective courses is also offered, as well as reading courses.
- Seminars, colloquia, 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. Students are required to take at least one seminar course.
- Students of the Master of Science degree engage with mathematical topics at the frontier of Mathematical knowledge, and are prepared for mathematical research through both classes and seminars.

**Core Skills**

Students graduating with a Master's in Mathematics at Brandeis possess a broad and rigorous foundation in modern mathematics. Students in the Master of Science degree go beyond foundational courses through topics classes and research seminars. Students graduating with a Master of Science who receive approval for a thesis are able to craft an original thesis paper and present it.

**Outcome**

Students graduating with a Master's in Mathematics are ideally prepared to apply for a PhD 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**

- PhD students are required, before they begin to work on their dissertation, to demonstrate a broad and deep knowledge in algebra, topology, geometry, and analysis by passing their required courses.
- 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 PhD 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 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 mathematics successfully 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 PhD dissertation.

**Outcome**

Students graduating with a PhD 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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Faculty

**Olivier Bernardi, Chair **Combinatorics.

**Carolyn Abbott**Geometric group theory. Topology.

**Assaf Bar-Natan**

Teichmüller theory. Low-dimensional topology. Geometric group theory. Differential geometry.

**Theo Douvropoulos**

Combinatorics. Reflection groups.

**Thomas Fai, Director of Graduate Study**

Scientific computing. Fluid dynamics. Mathematical biology.

**Promit Ghosal**

Probability theory. Mathematical physics and statistics.

**Steven Gindi**

Differential geometry. Geometric analysis.

**An Huang**

Algebraic geometry. Graph theory.

**Kiyoshi Igusa**

Differential topology. Representations of quivers.

**Dmitry Kleinbock, Undergraduate Advising Head ** Dynamical systems. Ergodic theory. Number theory.

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

**Tyler Maunu**

Algorithms. Optimization. Statistics and data science.

**Keith Merrill**

Ergodic theory. Dynamical systems. Number theory.

**Thomas Ng**

Number theory. Representation theory.

**Omer Offen ** Number theory. Representation theory.

**Tariq Osman**

Homogeneous dynamics. Number theory.

**Lam Pham**

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

**Daniel Ruberman ** Geometric topology. Gauge theory.

**Yun Shi**

Algebraic geometry. Donaldson-Thomas theory. Stability conditions.

**Rebecca Torrey **

Number theory.

**Jonathan Touboul**

Probability. Stochastic processes. Bifurcation theory. Mathematical modeling in biology.

**Yangyang Wang**

Mathematical biology. Differential equations. Dynamical systems. Multiple timescale dynamics.

**Hongsheng Wu**

Biostatistics. Clinical trials. Statistics education. Health economics. Business analytics.

**Arthur Zhang**

Machine learning. Natural language processing.

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Requirements for the Minor in Mathematics

- MATH 22a or 15a; MATH 22b or 20a.
- Three additional semester courses, either MATH 9b or MATH courses numbered 23 or higher or cross-listed courses in Mathematics. Only cross-listed courses in Mathematics and not in Applied Mathematics may be used. Note that most Math courses numbered 27 or higher require Math 23b as a prerequisite, but the Math courses 35a, 36a, 36b, 37a, 39a, 40a, 121a, 122a, 123a, 124a, and 125a 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, MATH 47a, or any course approved for the major with the Writing Intensive designation. See the full list of WI-designated courses under Courses of Instruction.
- Fulfill the oral communication requirement by successfully completing: MATH 16b, MATH 40a, or any course approved for the major with the Oral Communication designation. See the full list of OC-designated courses under Courses of Instruction.
- Fulfill the digital literacy requirement by successfully completing: MATH 16b, MATH 40a, MATH 124a, COSI 10a, COSI 12b, COSI 21a, or any course approved for the major with the Digital Literacy designation. See the full list of DL-designated courses under Courses of Instruction.

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.

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'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.

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 (excluding Math 92a) 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, 123a, and 125a. (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 201a 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. A written thesis proposal must be prepared at the beginning of the first semester, and be approved by the committee and the Undergraduate Advising Head, prior to registration for the course.

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, MATH 125a, 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 132a, 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 201a 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. A written thesis proposal must be prepared at the beginning of the first semester, and be approved by the committee and the Undergraduate Advising Head, prior to registration for the course.

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Combined BA/MA Program

Undergraduate students are eligible for the BA/MA program in mathematics if they have completed MATH 201a and b; MATH 211a and b; and MATH 221a and b; 225a; plus one other MATH course (or readings course) numbered 201a 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 their program. This must be done no later than May 1 preceding their 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, 16b, 20a, and 23b.
- Offered once each year are MATH 3a, 22a and b, 31a, 35a, 36a and b, 37a, 40a, 100a, 110a, 115a, 121a, and 122a.
- In addition, the following semester courses are usually offered every second year: MATH 28a, 28b, 39a, 47a, 100b, 102a, 104a, 108b, 110b, 123a, 124a, 125a, and 126a.

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Requirements for the Postbaccalaureate Program in Mathematics

- Two core courses: MATH 15a and MATH 20a.
- A grade below a B- 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

- The requirements for the Master of Arts degree include four core classes:
- MATH 201a
- MATH 211a
- MATH 211b
- MATH 221a

- Additionally, students will take two courses selected from MATH 201b, 221b, 225a, 231a, 232a, 234a, or 235a. With the permission of the Director of Graduate Study, 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. A student may request up to two of the listed classes to be substituted by math classes in the 100-199 range upon getting the permission of the DGS.
- Students will also enroll in either two seminar courses (MATH 298 or MATH 299) OR one higher level math elective course (numbered above 200) or a reading course (or another course approved by the DGS). This requirement is optional for PhD students receiving a master’s in passing. The MS/MA requirement of completing the seminar classes MATH 298a or MATH 299a is waived for PhD students who have taken MATH 240a (Second-Year Seminar).

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 core classes MATH 201a, 211a, 211b, and 221a are all required.

(b) At least three of the following courses: MATH 201b, 225a, 221b, 231a, 232a, 234a, or 235a.

(c) At least two additional higher level graduate mathematics courses or reading courses (or another elective approved by the Director of Graduate Study).

With the permission of the Director of Graduate Study, a student with superior preparation may omit one or more of the required 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. A student may request up to 2 of the listed classes to be substituted by Math classes in the 100-199 range upon getting the permission of the DGS.

(d) Two Seminar Courses (either MATH 298 or MATH 299). This requirement is not required for PhD students receiving a master’s in passing.

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 Director of Graduate Study by December 1st of their second year. Students who have received approval of a thesis advisor and the Director of Graduate Study 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 Director of Graduate Study 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 higher level graduate mathematics courses or reading courses (or another elective approved by the Director of Graduate Study). This brings the total number of required classes to 11 for students who are not completing a thesis.

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 two years to complete. The minimum residence requirement is three semesters.

During the first three semesters, 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 201a, 211a and b, and 221a
- In addition, students are required to take at least three of the following courses: MATH 201b, 221b, 225a, 231a, 232a, 234a, or 235a, 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 240a) as well as preparing for the qualifying examinations.
- During this year, students also begin taking reading courses (MATH 290a), which are arranged with a professor to allow students to broaden the scope of their studies, explore possible thesis areas and use them 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 serve as a Teaching Fellow and 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. For the duration of years 1-5 in the program, students who are not serving as a Teaching Fellow are required to become a “.75 TA” for one undergraduate course per semester. Please see the GSAS section on Teaching Requirements and the program handbook for more details.

Residence Requirement

The minimum in-person residence requirement is three years.

Language Requirement

Proficiency in reading one of French, German, or Russian. This requirement must be completed before the end of the degree, typically in the 2nd, 3rd or 4th year of the PhD.

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.

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Annual Academic Performance Review and Progress to the Graduate Degree

Every student, whether or not currently in residence, must register at the beginning of each term. All graduate students will be evaluated by the program each spring. At this evaluation the records of all graduate students will be carefully reviewed with reference to the timely completion of coursework and non-course degree requirements, the quality of the work and research in progress and the student’s overall academic performance in the program.

### Courses of Instruction

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(1-99) Primarily for Undergraduate Students

**
MATH
3a
Explorations in Math: A Course for Educators
**

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.

**
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.

**
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.

**
MATH
9a
Science and Science Fiction
**

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Have you ever read a story and wondered if it could be real life? Or read a story and imagined yourself as a character in its fictional world? Remember when you were a child experiencing many firsts, and asking why, or how? Works of fiction have a magic of revitalizing the reader’s curiosity and imagination. Science fiction, particularly, gets the readers excited about understanding how the world works and imagining their place in it. In this course, we use science fiction stories to build a fresh relationship with math and science and feel empowered to stay curious and ask questions. We attempt to answer some of the scientific questions by a powerful tool: mathematical modeling. As we embark on this journey, we learn how to become better friends with our computers by learning a little about their language. By the end of the course, MATLAB might become your favorite smart pocket calculator. The word ‘science fiction’ might give you mixed feelings: if you have ever tried to imagine yourself in a science fiction world but it felt like an exclusive club, prejudiced based on gender identity, sexual orientation, race, disability, and more, this course gives you the chance to reimagine a more inclusive science fiction world. Every two weeks, we read a science fiction story by a diverse group of writers, including Octavia Butler, Nnedi Okorafor, Ken Liu, and Rebecca Roanhorse, who strive to make the science fiction community more inclusive. We also hear from a diverse group of speakers who discuss topics in race, gender, culture, and equity in scientific or science fiction communities. If the pandemic era has sparked your curiosity about the spread of diseases and you have found yourself watching documentaries on infectious diseases back-to-back, rejoice. In the first half of the course, we focus on topics in epidemiology and population growth, and ask questions like how long would it take for a vicious fungus to take over the world? Or if an alien parasite infected our planet, how could we model its spread as it takes over more human hosts? In the second half, we focus on bio-engineering models that inevitably lead us to questions surrounding climate change. Can we engineer our planet and control climate? As climate change has affected the frequency of natural disasters, can we predict earthquakes, wildfires, and hurricanes accurately? If all fails, is space going to be a second home for us? By the end of the course, we reevaluate our definition of science. What are our responsibilities towards, the consequences of, and limitations of using science to progress humanity? We will feel empowered and inspired to shape science into a more inclusive home for all. Special one-time offering, spring 2022.

**
MATH
9b
Math Puzzles and Games
**

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Prerequisite: MATH 5a.

Introduces mathematical reasoning and proofs. The focus is on the art of problem solving using mathematical abstraction. Mathematical concepts and techniques are introduced through the lens of puzzles and games. Emphasis on teamwork, problem-solving, and analytical reasoning skills. Topics include: logic operations, pathfinding, graph coloring, matching, and other topics related to graph theory. Applications include: board games, scheduling, and public transit. Usually offered every year.

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

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Prerequisite: 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.

**
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.

**
MATH
15a
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.

**
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 semester.

**
MATH
20a
Multi-variable Calculus
**

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Prerequisites: MATH 10b 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.

**
MATH
22a
Honors Linear Algebra
**

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Prerequisite: MATH 22a placement exam and permission of the instructor. Students may take MATH 15a or MATH 22a for credit, but not both.

MATH 22a covers linear algebra. The material is similar to MATH 15a but with some additional content, a more theoretical emphasis, and more attention to proofs. Usually offered every fall.

**
MATH
22b
Honors Multi-variable Calculus
**

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Prerequisite: MATH 15a and permission of the instructor, or MATH 22a. Students may take MATH 20a or 22b for credit, but not both.

Covers the calculus in several variables. The material is similar to MATH 20a but with some additional content, a more theoretical emphasis, and more attention to proofs. Usually offered every spring.

**
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.

**
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.

**
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.

**
MATH
31a
Advanced Linear Algebra
**

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Prerequisite: MATH 23b or equivalent.

Fields. Vector spaces. Linear maps between vector spaces. Linear forms and duality. Multi-linear forms and determinants. Endomorphisms of a vector space (eigenvalues, eigenvectors, minimal and characteristic polynomials, classification). Bilinear forms. Hilbert spaces theory. Tensor products and tensors. Emphasis on concepts and proofs. Spectral theory of self-adjoint operators. Usually offered every year.

**
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.

**
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.

**
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.

**
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 spring.

**
MATH
39a
Introduction to Combinatorics
**

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Prerequisite: COSI 29a or 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.

**
MATH
40a
Introduction to Applied Mathematics
**

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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.

**
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 second year.

**
MATH
48a
Quiver Representations
**

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Prerequisites: MATH 15a and Math 23b or equivalent, or instructor’s permission.

An introduction to representations of quivers and category theory. Topics to be covered are representations of quivers, projective and injective representations, categories, functors, projective resolutions, and examples of Auslander-Reiten quivers. Special one-time offering, spring 2023.

**
MATH
91g
Introduction to Research Practice
**

Prerequisite: Student must complete training relevant to the research group. Yields quarter-course credit. Offered exclusively on a credit/no-credit basis. May be repeated for credit. Does not meet the requirements for the major or minor in Math or Applied Math.

Students engage in mathematics research by working with a faculty member for a minimum of 3 hours per week for one semester. Students who have declared a mathematics major must receive permission from the mathematics Undergraduate Advising Head as well as the faculty sponsor to enroll in MATH 91g. Students who have not yet declared a major must receive permission from their academic advisor as well as the faculty sponsor. Usually offered every year.

**
MATH
98a
Independent Research
**

Usually offered every year.

**
MATH
98b
Independent Research
**

Usually offered every year.

**
MATH
99a
Senior Research
**

Usually offered every year.

**
MATH
99b
Senior Research
**

Usually offered every year.

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(100-199) For Both Undergraduate and Graduate Students

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

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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 year.

**
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.

**
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.

**
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.

**
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.

**
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 year.

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

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Prerequisite: MATH 110a or permission of the instructor.

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

**
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.

**
MATH
121a
Mathematics for Natural Sciences
**

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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 year.

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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 year.

**
MATH
123a
Principles of Mathematical Modeling
**

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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 modeling 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.

**
MATH
124a
Optimization
**

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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 second year.

**
MATH
125a
Mathematics for Machine Learning
**

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*Prerequisites: MATH 10b and either MATH 15a or MATH 22a.*

Serves as a first course in machine learning and general data science, with a focus on the mathematics underlying the various modern machine learning algorithms. The course covers the fundamental concepts of statistical distribution, information theory, statistical learning, optimization and matrix factorizations, as well as classic algorithms such as tree methods, kernel methods and various neural network models. A few important real world examples of current interest will be considered such as computer vision, natural language processing, search engine, recommendation systems, finance, and biology. Usually offered every second year.

**
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.

**
MATH/MUS
121b
Math and Music
**

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Does not satisfy the SN requirement. MATH/MUS121b can count toward the minor in mathematics (replacing one of the 3 electives in the MATH 27+ range). MATH/MUS 121b can count as a cross-listed course for the applied math major. It cannot count toward the pure math major.

Mathematical patterns, symmetries, sequences, modular relationships, and order are ubiquitous in music. In fact, mathematics and music have inspired each other for centuries, with music providing inspiration for some mathematical discoveries and mathematical concepts providing a conceptual framework for thinking about musical expression, tuning, composition, and musical analysis. With the advent of computers and mathematical methods in recent years, new concepts have been implemented into algorithmic music composition. The purpose of this class is to provide students with an introduction to the deep relationship between mathematics and music, to present in depth a collection of selected topics that highlight the influence of symmetries, patterns, stochastic structures and geometrical analysis, and to encourage the students to explore those links in a creative final project. Special one-time offering, spring 2023.

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(200 and above) Primarily for Graduate Students

**
MATH
201a
Algebra I
**

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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.

**
MATH
201b
Algebra II
**

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Prerequisite: MATH 201a or permission of the instructor.

Continuation of MATH 201a. Usually offered every spring.

Olivier Bernardi

**
MATH
211a
Real Analysis
**

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.

**
MATH
211b
Complex Analysis
**

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Prerequisite: MATH 211a 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.

**
MATH
221a
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.

**
MATH
221b
Topology II
**

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Prerequisite: MATH 221a or permission of the instructor.

Continuation of MATH 221a. Manifolds and orientation, cup and cap products, Poincaré duality. Other topics as time permits. Usually offered every spring.

**
MATH
225a
Geometry of Manifolds
**

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.

**
MATH
225b
Differential Geometry
**

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Prerequisite: MATH 225a 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.

**
MATH
231a
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.

**
MATH
232a
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.

**
MATH
234a
Partial Differential Equations
**

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 the equations and variational methods to study them. Usually offered every second year.

**
MATH
235a
Probability Theory
**

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Prerequisite: MATH 211a. 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.

**
MATH
238a
Topics in Applied Mathematics
**

Prerequisite: Some background in Analysis and permission of the instructor.

Techniques from applied mathematics are used increasingly in many branches of research, and are vitally important for studying a wide range of physical and social phenomena. MATH 238a is designed to cover a variety of advanced topics and active research areas in applied mathematics. Through this course students will get the opportunity to explore ideas introduced in other graduate-level courses at a deeper and more rigorous level. The course will help students obtain the necessary background knowledge to build a solid foundation for their research. Usually offered every year.

**
MATH
239a
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.

**
MATH
239b
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.

**
MATH
240a
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.

**
MATH
244a
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.

**
MATH
251a
Topics in Algebra
**

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

**
MATH
252a
Algebraic Geometry I
**

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

**
MATH
252b
Algebraic Geometry II
**

Continuation of MATH 252a. Usually offered every second year.

**
MATH
253a
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.

**
MATH
253b
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.

**
MATH
255a
Topics in Mathematical Physics
**

Prerequiste: First year graduate level math courses, or first year graduate level physics courses, or permission of the instructor.

An introduction to a selection of research topics in mathematical physics. Usually offered every second year.

**
MATH
261a
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.

**
MATH
262b
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.

**
MATH
271a
Topology III: Symplectic Topology and Morse Theory
**

Symplectic topology and Morse theory. Basic symplectic and contact topology with emphasis on the generating function approach. Darboux’s theorem, Gromov’s nonsqueeze theorem, Arnold’s conjecture. Usually offered every fall.

**
MATH
271b
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.

**
MATH
273a
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.

**
MATH
280a
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.

**
MATH
290a
Readings in Mathematics
**

**
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.

**
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.

**
MATH
300a
Master's Thesis
**

Students who have selected and received approval from the faculty member supervising the thesis and the Director of Graduate Study, 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.

**
MATH
393g
Math Internship
**

Permission of the Director of Graduate Study 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.

**
MATH
399a
Advanced Readings in Mathematics
**

**
MATH
401d
Research
**

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

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MATH Digital Literacy

**
COSI
10a
Introduction to Problem Solving in Python
**

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Open only to students with no previous programming background. Students may not take COSI 10a if they have received a satisfactory grade in COSI 12b or COSI 21a. May not be taken for credit by students who took COSI 11a in prior years. Does not meet the requirements for the major or minor in Computer Science.

Introduces computer programming and related computer science principles. Through programming, students will develop fundamental skills such as abstract reasoning and problem solving. Students will master programming techniques using the Python programming language and will develop good program design methodology resulting in correct, robust, and maintainable programs. Usually offered every semester.

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COSI
12b
Advanced Programming Techniques in Java
**

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Prerequisite: COSI 10a or successful completion of the COSI online placement exam.

Studies advanced programming concepts and techniques utilizing the Java programming language. The course covers software engineering concepts, object-oriented design, design patterns and professional best practices. This is a required foundation course that will prepare you for more advanced courses, new programming languages, and frameworks. Usually offered every year.

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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.

**
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 semester.

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MATH
40a
Introduction to Applied Mathematics
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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.

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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 year.

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MATH
124a
Optimization
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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 second year.

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MATH Oral Communication

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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 semester.

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MATH
40a
Introduction to Applied Mathematics
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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.

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MATH Writing Intensive

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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.

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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 second year.

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MATH Cross-Listed

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BIOL
251a
Project Laboratory in Protein Biochemistry
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Features experiments in protein biochemistry that are fundamental to the field of biotechnology. These include protein purification, characterization and quality assessment. Focus is placed on designing purification protocols for both tagged and untagged proteins using biochemical knowledge. The designed protocols are tested by purifying known proteins. As part of the course, students will contribute to research projects of unknown outcome by purifying and assaying novel proteins. Usually offered every year.

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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.

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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.

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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, ECON 213a, 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.

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NPHY
115a
Dynamical Systems
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Prerequisites: MATH 10a, b or equivalent; MATH 15a and/or some coding experience would be helpful.

An introduction to the theory of nonlinear dynamical systems, including bifurcations, limit cycles, chaos, and coupled oscillators. Covers analytical, computational, and graphical methods of solving sets of nonlinear ordinary differential equations, as well as mathematical modeling of natural phenomena. Examples will be drawn from physics, chemistry, population biology, and neuroscience. Usually offered every third year.

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PHIL
106b
Mathematical Logic
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We continue our rigorous investigation of logic that we began in Phil6A by studying the metatheory of formal systems. We begin with an introduction to sets, relations, and functions, after which we prove the Soundness, Completeness, and Löwenheim-Skolem Theorems for First-Order Logic. We end by examining Turing machines in order to introduce students to the notions of computability and undecidability, and to prepare them for the more advanced study of Gödel's Incompleteness Theorems. Usually offered every second year.

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PHYS
100a
Classical Mechanics
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Prerequisites: PHYS 20a or permission of the instructor.

The goal of this course is to engage students in an exploration of classical mechanics from a modern perspective. Students are expected to have familiarity with Newtonian Mechanics, and have taken a calculus-based mechanics course. Usually offered every second year.

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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.

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MATH Cross-Listed in Applied Mathematics

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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.

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BCHM
104a
Physical Chemistry of Macromolecules I
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Prerequisites: MATH 10a,b or equivalent, PHYS 11 or 15.

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

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CHEM
141a
Chemical Thermodynamics
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Prerequisites: Satisfactory grade in CHEM 11a, 15a and CHEM 11b, 15b or the equivalent; MATH 10b 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.

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CHEM
142a
Quantum Mechanics and Spectroscopy
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Prerequisites: A satisfactory grade in CHEM 11b or 15b or the equivalent; MATH 10b or the equivalent; PHYS 10b, 11b, or 15b 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.

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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.

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

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COSI
123a
Statistical Machine Learning
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Prerequisite: COSI 21a, MATH 8a, and either MATH 22a or the combination of MATH 15a and MATH 20a.

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.

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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.

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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.

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COSI
180a
Algorithms
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Prerequisites for undergraduates and combined BA/MA students: COSI 21a and COSI 29a. Graduate students with the appropriate background may request an override for some or all of the prerequisites.

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

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ECON
80a
Microeconomic Theory
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Prerequisite: ECON 2a or 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.

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ECON
132a
Normative Economics
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Prerequisites: ECON 80a and ECON 82b.

The overarching question of normative economics is, what is the objective that policymakers should pursue? We will begin by exploring the role that individual preferences play in shaping the desirable social objective. We will then discuss approaches for balancing competing interests of different individuals. Finally, we will focus on the extent to which policymakers’ decisions should account for the interests of future generations. Usually offered every year.

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ECON
136b
Economics of Digitization
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Prerequisites: ECON 80a and ECON 184b.

Studies how technological advances fundamentally change how markets function, leading to novel firm strategies and consumer harms. Topics include: pricing digital goods, review/ratings platforms, advertising, search platforms, resale of digital goods, etc. Usually offered every year.

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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.

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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.

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

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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, ECON 213a, 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.

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MATH/MUS
121b
Math and Music
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Does not satisfy the SN requirement. MATH/MUS121b can count toward the minor in mathematics (replacing one of the 3 electives in the MATH 27+ range). MATH/MUS 121b can count as a cross-listed course for the applied math major. It cannot count toward the pure math major.

Mathematical patterns, symmetries, sequences, modular relationships, and order are ubiquitous in music. In fact, mathematics and music have inspired each other for centuries, with music providing inspiration for some mathematical discoveries and mathematical concepts providing a conceptual framework for thinking about musical expression, tuning, composition, and musical analysis. With the advent of computers and mathematical methods in recent years, new concepts have been implemented into algorithmic music composition. The purpose of this class is to provide students with an introduction to the deep relationship between mathematics and music, to present in depth a collection of selected topics that highlight the influence of symmetries, patterns, stochastic structures and geometrical analysis, and to encourage the students to explore those links in a creative final project. Special one-time offering, spring 2023.

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NBIO
136b
Computational Neuroscience
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Prerequisites: MATH 10a or higher and one of the following: NBIO 140b/240b, PHYS 10b/11b/15b, BIOL 107a, or any COSI course.

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 limited prior coding experience, with reading assignments and preparation as homework. Usually offered every second year.

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NPHY
115a
Dynamical Systems
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Prerequisites: MATH 10a, b or equivalent; MATH 15a and/or some coding experience would be helpful.

An introduction to the theory of nonlinear dynamical systems, including bifurcations, limit cycles, chaos, and coupled oscillators. Covers analytical, computational, and graphical methods of solving sets of nonlinear ordinary differential equations, as well as mathematical modeling of natural phenomena. Examples will be drawn from physics, chemistry, population biology, and neuroscience. Usually offered every third year.

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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.

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MATH Courses of Related Interest

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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.

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BIOL
107a
Data Analysis and Statistics Workshop
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Prerequisites: BIOL51a, high school statistics, or similar course.

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. Usually offered every year.

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PHIL
138b
Philosophy of Mathematics
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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.