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

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. The Department of Mathematics offers three undergraduate degrees: Bachelor of Arts in Mathematics, Bachelor of Science in Mathematics, and Bachelor of Science in Applied Mathematics. Students interested in studying abstract and applied mathematics can, as of 2025-2026, combine these programs (see item A in the Special Notes Relating to Undergraduates).  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.

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:

1. Improved analytical reasoning skills
2. Enhanced basic computational skills
3. Familiarity with basic mathematical terms and their physical meanings
4. The ability to model real-world problems mathematically
5. 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:

1. Understand the fundamental concepts of mathematical proof, logic, abstraction and generalization.
2. 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:

1. Gain knowledge on the fundamental objects, frameworks and theorems in applied mathematics, including the fields of probability, mathematical modeling, numerical analysis and differential equations.
2. Understand the main connections between the mathematical sciences and other scientific or humanistic disciplines.
3. 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:

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

Upon Graduation

Mathematics majors with appropriate backgrounds and preparation may:

1. Pursue graduate study and a scholarly career in mathematics
2. Work as actuaries
3. Teach mathematics at the K-12 level
4. Work in fields such as computer science, operations research, economics, finance, biology, physics, or other sciences
5. 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

1. Postbac students are required to demonstrate knowledge in linear algebra and multivariate calculus. 
2. 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).
3. 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).
4. 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

1. 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.
2. A wide array of more advanced or more specialized elective courses is also offered, as well as reading courses.
3. 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.
4. 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

1. 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.
2. 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.
3. 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.
4. 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:

1. Have learned to read and understand research papers;
2. Have learned how to present mathematical materials, in particular their own results to fellow graduate students and researchers;
3. 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;
4. Have attained research expertise and completed a significant body of original research that advances a specific field of study in mathematics;
5. Have taken advanced courses and seminars.
6. 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.

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.

Carolyn Abbott
Geometric group theory. Topology.

Daniel Álvarez-Gavela
Symplectic topology, h-principles

Olivier Bernardi
Combinatorics. Probability.

Justin Campbell
Algebraic geometry. Langlands Program.

Theo Douvropoulos
Combinatorics. Reflection groups.

Thomas Fai
Scientific computing. Fluid dynamics. Mathematical biology.

An Huang
Algebraic geometry. Graph theory.

Kiyoshi Igusa
Differential topology. Representations of quivers.

Connor Kennedy
Dynamical systems. Machine learning.

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.

 Yun Shi
Algebraic geometry. Donaldson-Thomas theory. Stability conditions.

Rebecca Torrey
Number theory.

Jonathan Touboul, Chair
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.

1. MATH 22a or 15a; MATH 22b or 20a.
2. Three additional semester courses, either MATH 9b or MATH courses numbered 23 or higher (including MATH/MUS 121b) 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, 40a, 121a, 122a, 123a, 124a, 125a, and 126a do not.
3. No grade below a C- will be given credit toward the minor.
4. No course taken pass/fail may count towards the minor requirements.
5. 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.

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 122a, 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.

Bachelor's Degrees in Mathematics

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

1. MATH 15a or 22a; MATH 20a or 22b.
2. MATH 23b or exemption. See item F in Special Notes Relating to Undergraduates.
3. MATH 35a, 110a, or 115a.
4. MATH 31a, 100a, or 108b.
5. No grade below a C- will be given credit toward the major, honors, or the teacher preparation track.
6. 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 MATH courses numbered 27 or higher (excluding Math 92a and MATH/MUS 121b). Up to two of these courses can be replaced with courses cross-listed in Mathematics (not in Applied 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 MATH courses numbered 27 or higher (excluding Math 92a and MATH/MUS 121b). Up to two of these courses can be replaced with courses cross-listed in Mathematics (not in Applied Mathematics).

Honors Standards

Honors in Mathematics

Students interested in pursuing departmental honors should notify the department by completing this form. 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, 124a, 125a, and 126a. (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 requires six additional semester MATH courses numbered 27 or higher (excluding Math 92a and MATH/MUS 121b). Up to two of these courses can be replaced with courses cross-listed in Mathematics (not in Applied Mathematics). 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 MATH courses numbered 27 or higher (excluding Math 92a and MATH/MUS 121b). Up to two of these courses can be replaced with courses cross-listed in Mathematics (not in Applied Mathematics). Courses must meet the additional 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:

E. MATH 8a (Introduction to Probability and Statistics) or 36a (Probability).

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:

1. Three foundational courses: MATH 15a or MATH 22a, MATH 20a or MATH 22b, and MATH 36a.
2. MATH 23b or an exemption.
3. MATH 36b or MATH 40a.
4. Two of the following analysis courses: MATH 35a, MATH 37a, MATH 110a or MATH 115a.
5. Two of the following: MATH 121a, MATH 122a, MATH 123a, MATH 124a, MATH 125a, or MATH 126a.
6. Three courses cross-listed for applied mathematics and offered by another department, or any PHYS courses numbered 20 or higher. A MATH course numbered 27 or higher may be substituted for one of these courses. Note: MATH/MUS 121b can count as a cross-listed course for the applied math major.
7. No grade below a C- will be given credit toward the Bachelor of Science degree.
8. 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. 

Undergraduate students are eligible for the BA/MA program in mathematics if they have completed MATH 201a; MATH 211a and b; MATH 221a; plus two courses selected from MATH 201b, 221b, 225a, 231a, 232a, 234a, or 235a, 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.

1. Students interested in double majoring in Mathematics and Applied Mathematics need to satisfy all requirements for both programs they intend to complete, and may double count a maximum of two classes towards both sets of requirements (in addition to the three foundational classes required by all programs of study: Math15a/Math22a, Math20a/Math22b, and Math23b).
2. 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. 
3. 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.
4. 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.
5. 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.
6. 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.
7. Students interested in graduate school or a more intensive study of mathematics are urged to include all of the following courses in their program:
   1. MATH 22a and b.
   2. MATH 100a and b.
   3. MATH 35a or 110a and b.
   4. MATH 115a.
   5. Other courses numbered 100 or higher.
8. The following schedule determines course offerings in mathematics:
   1. Usually offered every semester are MATH 5a, 8a, 10a and b, 15a, 20a, and 23b.
   2. Usually offered once each year are MATH 9b, 16b, 22a and b, 35a, 36a and b, 37a, 40a, 100a, 110a, 115a, 121a, and 122a.
   3. In addition, the following semester courses are usually offered every second year: MATH 3a, 31a, 39a, 47a, 100b, 102a, 104a, 108b, 110b, 123a, 124a, and 126a.

1. Two core courses: MATH 15a and MATH 20a.
2. A grade below a B- will not count towards the post-baccalaureate program.
3. 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.

Course Requirement

1. The requirements for the Master of Arts degree include four core classes: 
   • MATH 201a 
   • MATH 211a
   • MATH 211b 
   • MATH 221a 
2. 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.
3. Students will also enroll in either three semesters of seminar courses (MATH 298a or MATH 299a) OR one semester of seminar course (MATH 298a or MATH 299a) along with one higher level math elective course (numbered above 200) or a reading course, or another course approved by the Math 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.

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) OR one higher level math elective course (numbered above 200) or a reading course (or another course approved by the DGS). This requirement is not required for PhD students receiving a master’s in passing.

(e) Thesis, or two additional courses.

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.

Program of Study

During all fall and spring semesters, students should be enrolled in at least 12 credits approved by the department. 

1. The normal first year of study includes the core classes MATH 201a, 211a and b, and 221a 
2. 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. 
3. 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. 
4. 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. 
5. 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. 
6. During their third year and beyond, students also continue to take advanced courses and seminars.

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 TA. 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.

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.

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.