Mathematics
Program Offerings
Most firstyear students and sophomores interested in science, engineering or finance take courses from the standard calculus and linear algebra sequence 103104201202, which emphasizes concrete computations over more theoretical considerations. Note that 201 and 202 can be taken in either order.
Students who are not prepared to begin with 103 may take 100, an introduction to calculus with precalculus review.
Prospective economics majors can minimally fulfill their mathematics prerequisites with (100)103175. Note that 175 covers selected topics from 201, with biology and economics applications in mind. It is recommended that prospective mathtrack economics/finance majors take the standard sequence 103104201202 instead of 175.
More mathematically inclined students, especially prospective physics majors, may opt to replace 201202 with 203204, for greater emphasis on theory and more challenging computational problems.
Prospective mathematics majors must take at least one course introducing formal mathematical argument and rigorous proofs. The recommended firstyear sequence for prospective majors is 215217300. Prospective majors who already have substantial experience with universitylevel proofbased analysis courses may consider the accelerated sequence 216218 instead. Other possible sequences for prospective majors include 210215217300 or 214204203 and 203204215, although the latter two are relatively rare. Note that 203 and 204 can be taken in either order. Some of our successful majors became interested in mathematics as a major after taking courses at the 100 level. As an alternative to 104, and as a leadin to MAT215, students may opt to take 210 in the fall, followed by 215 in the spring. Such students should consult with the junior adviser or the associate director of undergraduate studies as soon as possible to plan a course of study.
Placement Guidelines
Students who need placement advice about 214, 215 or 216 should consult the Department of Mathematics homepage (the "Undergraduate" tab has a section on placement) or contact Professor McConnell, the junior adviser.
Students with little or no background in calculus, but with strong precalculus skills, should take 103. Alternatively, 100 offers intensive precalculus review, along with an introduction to the main ideas of calculus, as preparation for 103. To succeed in 104 or 175, a background in differential calculus at the level of a 5 on the BC Advanced Placement Examination is advised. Students with a stronger calculus preparation in differential and integral calculus, as well as infinite series, may opt to start in 201. Students who possess, in addition, a particularly strong interest in mathematics as well as an SAT mathematics score of at least 750 may opt for 203 or 210 or 214 or 215 or 216 instead.
The placement workshops for incoming students at orientation are designed to help students consider all these issues in order to make a good initial selfplacement in the mathematics curriculum, which can then be adjusted during drop/add if necessary.
Goals for Student Learning
Mathematics is a discipline inseparable from scientific and philosophical inquiry. The rigorous and logical thinking that characterizes mathematics is an essential tool for theorybuilding of any kind because its clarity and precision expose hidden assumptions, inner inconsistencies and deep structural similarities in problems that seem unrelated on the surface. Our courses cover a wide variety of wellestablished mathematical knowledge that is actively under development by today’s mathematicians and that offers fundamental tools for scientists and engineers of all kinds.
Students begin their work in the department with a thorough training in rigorous logical reasoning and mathematical proofs in the context of analysis and linear algebra. Next, they complete a survey of the main areas of modern mathematics by completing core courses in real and complex analysis, in algebra and in geometry/topology or discrete mathematics. Then students are free to take courses exploring a wide variety of topics in both pure and applied mathematics to acquire a good general knowledge of the main areas of current mathematical work.
In the independent work, students learn how to move beyond the classical knowledge found in textbooks to explore contemporary research literature through collaboration with their peers and with active researchers in mathematics or applied fields. Through this collaboration, students:
 Learn how to join a scholarly discussion in progress to orient themselves in a rapidly developing area of research.
 Build on their broad general knowledge of mathematics and logical reasoning skills in order to develop a working knowledge of a significant area of contemporary mathematics via the research literature.
 Learn to identify interesting problems they want to investigate and develop their own ideas about how to carry out those investigations.
 Learn to come up with a complete argument of their own, adapting and expanding ideas and techniques from various sources, as needed.
 Develop mastery of rigorous logical thinking by constructing complete and correct arguments.
 Develop mastery of clear mathematical exposition in a manner that allows and invites dialog with other scholars in the intended audience, where important definitions and theorems are clearly explained and contributions of other scholars are properly acknowledged.
The program produces critical and creative thinkers with a broad general knowledge of contemporary mathematics and with the analytic and expository skills needed for collaborative problemsolving in any quantitative setting.
Advanced Placement
Incoming firstyear students who report a score of 5 on the BC Advanced Placement Examination (or a 7 on the IC [higher level] math examination, or an A on the British Alevel math exam) will receive one unit of advanced placement credit for MAT103. For more information, please consult the website of the Office of the Dean of the College.
Prerequisites
To major in mathematics, students need a strong foundation in linear algebra and analysis, in both one and several variables, as well as experience in understanding and writing rigorous mathematical proofs. Generally, prospective majors with a very strong background in calculus are strongly recommended to start in 215, followed by 217 and 300. Prospective majors with extensive prior experience with calculus and rigorous proofs can start in 216, followed by 218. Students with an interest in rigorous proofs may start in 210 or 214 and then continue to the 215217300 sequence.
Students who are undecided, or students who are initially planning to major in a different quantitative discipline, may opt instead to begin their calculus and linear algebra work in the 203204 or the 103104201202 sequences, where proofs are not an emphasis. From these sequences, prospective mathematics majors should consult with the junior adviser or the associate director of undergraduate studies early on, in order to plan a program that includes as much of the 215217300 sequence as possible.
Most majors begin taking courses at the 300 level by the second semester of sophomore year, in preparation for their junior independent work. The junior adviser meets with students individually to help them plan a course of study. A student’s path through the upperdivision courses will naturally depend on their longterm goals and prior experience.
Further information for prospective majors is available on the department home page.
Program of Study
Students must complete four core requirements:
 one course in real analysis (e.g., 300 or 320 or 325 or 425 or 385)
 one course in complex analysis (e.g., 330 or 335)
 one course in algebra (e.g., 340 or 345)
 one course in geometry or topology (e.g., 355 or 365 or 560)
It is recommended that students complete some of these core requirements by the end of their sophomore year. Completing these core courses early allows for more options for junior and senior independent work.
Note: One course in discrete mathematics (e.g., 375, 377 or 378) can replace the geometry/topology core requirement, if desired.
In addition to the four core requirements, students must complete an additional four courses at the 300 level or higher, up to three of which may be cognate courses outside the mathematics department, with permission from the junior or senior advisers or the director of undergraduate studies. No more than two of the eight courses may be reading courses.
The departmental grade (the average grade of the eight departmental courses) together with grades and reports on independent work form the basis on which honors and prizes are awarded upon graduation.
Students should refer to Course Offerings to determine which courses are offered in a given term. Programs of study in various fields of pure mathematics and applied mathematics are available. Appropriate plans of study may be arranged for students interested in specialized related fields such as physics, the biological sciences, finance and computer science, for example. For students interested in these areas, a coherent program containing up to three courses in a cognate field may be approved.
Independent Work
All departmental students engage in independent work, supervised by a member of the department chosen in consultation with a departmental adviser. The junior year independent work generally consists of participating actively in a junior seminar in both the fall and spring semesters. Alternatively, a student may opt to replace one junior seminar with supervised reading in a special subject and then writing a paper based on that reading. The independent work in the senior year centers on writing a senior thesis. A substantial percentage of our majors work with faculty in other departments on their senior project.
Senior Departmental Examination
Each senior takes an oral examination based on the senior thesis and the broader subfield to which it contributes. A departmental committee conducts the examination in May.
The mathematics minor provides a structure that encourages students to explore mathematical ideas for their own sake in the openminded, intellectually curious way that is the main goal of a liberal arts education. The program allows both for a minor in mathematics that supplements students’ work in their major field of concentration, and also for a minor in mathematics that allows students to explore new directions. A student who decides to minor in mathematics might be someone who wants to learn more math because they have always been curious and enthusiastic math learners, and they want to build a deeper understanding of the mathematical ideas that appear throughout their work in their chosen disciplines. We also welcome students from the humanities or social sciences who have enjoyed learning math and would like to continue to explore more advanced ideas in mathematics, whether or not they are directly relevant to their major work.
Goals for Student Learning
The goal of the minor in mathematics is to allow students to explore contemporary mathematics, either as a topic for its own sake or to gain a deeper knowledge of a specific area most relevant for the students’ work in their major field of concentration. In consultation with the minor program director of the mathematics department, the student should develop a plan that complements their work in their chosen field of concentration. The plan can be to gain a broad knowledge of mathematics, in which case we recommend courses in algebra, complex analysis and real analysis. Alternatively, students with a strong interest in a more specialized area of mathematics can choose a sequence of courses in that field.
Prerequisites
The recommended preparation for the minor is the same as for the major in math: knowledge of calculus (both single and several variable), linear algebra and familiarity with proofs. These criteria are satisfied by taking one each of the following:
• MAT201/203/218 (several variable calculus);
• MAT202/204/217 (linear algebra); and
• MAT214/215/216/217/218 (familiarity with proofs).
(Note that 217 and 218 each cover two of the recommended preparations.) With the prior approval of the minor program director, students may count one course taken in another department toward the minor prerequisites. These courses should be completed by the end of sophomore year.
Admission to the Program
The student should contact the minor program director of the mathematics department during the spring of sophomore year to have a course plan approved. Later admission to the Program is allowed during the junior year, though this offers less flexibility in the choice of the required courses.
Program of Study
In addition to completing the required prerequisite courses, the minor requires four mathematics department courses at the 300 level or higher. By permission, one of these can be a cognate if it fits into the student’s plan and is approved in advance. These courses cannot be counted for the student’s major or for other minors or certificates.
In addition to the four mathematics department courses, one Junior Seminar in the mathematics department is required, which can be taken in the senior year. Exceptionally, if the plan of the student justifies it, the Junior Seminar can be replaced by a Junior Paper, which can be done in the senior year or as a summer project.
Additional Information
Normally, students may not pursue both the minor in PACM and the minor in MAT. However, students who believe they have a compelling curricular reason to pursue both may apply to the directors of both programs for permission to do so.
Faculty

Chair
 Igor Rodnianski

Associate Chair
 János Kollár

Director of Undergraduate Studies
 Jennifer M. Johnson (associate)
 János Kollár

Director of Graduate Studies
 Lue Pan (associate)
 Chenyang Xu

Professor
 Michael Aizenman
 Noga M. Alon
 Manjul Bhargava
 SunYung A. Chang
 Maria Chudnovsky
 Fernando Codá Marques
 Peter Constantin
 Mihalis Dafermos
 Zeev Dvir
 Charles L. Fefferman
 David Gabai
 June E. Huh
 Alexandru D. Ionescu
 Nicholas M. Katz
 Sergiu Klainerman
 János Kollár
 Assaf Naor
 Peter Steven Ozsváth
 Igor Rodnianski
 Peter C. Sarnak
 Will Sawin
 Paul Seymour
 Amit Singer
 Christopher M. Skinner
 Allan M. Sly
 Zoltán Szabó
 Chenyang Xu
 Paul C. Yang
 ShouWu Zhang

Assistant Professor
 Bjoern Bringmann
 Matija Bucic
 Marc Aurèle Tiberius Gilles
 Jonathan Hanselman
 Susanna Haziot
 Ana Menezes
 Lue Pan
 Ravi Shankar
 Jacob Shapiro
 Jakub Witaszek
 Ruobing Zhang

Associated Faculty
 John P. Burgess, Philosophy
 René A. Carmona, Oper Res and Financial Eng
 Bernard Chazelle, Computer Science
 Hans P. Halvorson, Philosophy
 William A. Massey, Oper Res and Financial Eng
 Frans Pretorius, Physics
 Robert E. Tarjan, Computer Science
 Ramon van Handel, Oper Res and Financial Eng

Instructor
 Louis Esser
 Tangli Ge
 Sepehr Hajebi
 Lili He
 Tongmu He
 Kimoi Kemboi
 Dmitry Krachun
 Hongyi Liu
 Anubhav Mukherjee
 Sung Gi Park
 Semon Rezchikov
 Joshua X. Wang
 Mingjia Zhang

University Lecturer
 Jennifer M. Johnson

Senior Lecturer
 Jonathan M. Fickenscher
 Mark W. McConnell

Lecturer
 Fraser M. Binns
 Tatyana Chmutova
 Tatiana K. Howard
 Justin Lacini
 Tristan J. Leger
 Jennifer Li
 Andrew O'Desky
 Stan Palasek
 John T. Sheridan
 Sahana Vasudevan
 David Villalobos
 Liyang Yang
 Bogdan Zavyalov

Visiting Professor
 Bhargav B. Bhatt
 Alex Kontorovich

Visiting Lecturer with Rank of Professor
 Camillo De Lellis
 Helmut H. Hofer
 Aaron Naber
 Akshay Venkatesh
For a full list of faculty members and fellows please visit the department or program website.