# Mathematics

## Program Offerings

Most first-year students and sophomores interested in science, engineering or finance take courses from the standard calculus and linear algebra sequence 103-104-201-202, 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 pre-calculus review.

Prospective economics majors can minimally fulfill their mathematics prerequisites with (100)-103-175. Note that 175 covers selected topics from 201, with biology and economics applications in mind. It is recommended that prospective math-track economics/finance majors take the standard sequence 103-104-201-202 instead of 175.

More mathematically inclined students, especially prospective physics majors, may opt to replace 201-202 with 203-204, 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 first-year sequence for prospective majors is 215-217-300. Prospective majors who already have substantial experience with university-level proof-based analysis courses may consider the accelerated sequence 216-218 instead. Other possible sequences for prospective majors include 210-215-217-300 or 214-204-203 and 203-204-215, 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 lead-in 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 home page (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 pre-calculus skills, should take 103. Alternatively, 100 offers intensive pre-calculus 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 a 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 self-placement 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 theory-building 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 well-established 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 problem-solving in any quantitative setting.

### Advanced Placement

Incoming first-year 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 A-level 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.

One unit of advanced placement credit is provisionally granted when a student enrolls in MAT 104 or 175 in the fall term of their first year. Two units of advanced placement credit are provisionally granted when a student enrolls in MAT 201, 203, 215 or 216 in the fall term of their first year. Provisional credit will be converted to advanced placement credit upon successful completion of the relevant course.

### 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 215-217-300 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 203-204 or the 103-104-201-202 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 215-217-300 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 upper-division courses will naturally depend on their long-term 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 is 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.

## 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
- Sun-Yung 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
- Emmy Murphy
- Assaf Naor
- Peter Steven Ozsváth
- John V. Pardon
- Igor Rodnianski
- Peter C. Sarnak
- Paul Seymour
- Amit Singer
- Christopher M. Skinner
- Allan M. Sly
- Zoltán Szabó
- Chenyang Xu
- Paul C. Yang
- Shou-Wu Zhang

### Assistant Professor

- Matija Bucic
- Jonathan Hanselman
- Casey L. Kelleher
- Ana Menezes
- Evita Nestoridi
- Lue Pan
- Jacob Shapiro
- Jakub Witaszek
- Ian Zemke
- 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
- Robert J. Vanderbei, Oper Res and Financial Eng
- Ramon van Handel, Oper Res and Financial Eng

### Instructor

- Louis Esser
- Tangli Ge
- Lili He
- Dmitry Krachun
- Jennifer Li
- Hongyi Liu
- Anubhav Mukherjee
- Jean Pierre Mutanguha
- Semon Rezchikov
- Ravi Shankar
- Artane Siad
- Liyang Yang
- Mingjia Zhang

### University Lecturer

- Jennifer M. Johnson

### Senior Lecturer

- Jonathan M. Fickenscher
- Mark W. McConnell

### Lecturer

- Fraser M. Binns
- David Boozer
- Tatyana Chmutova
- Giorgio Cipolloni
- Federico Glaudo
- Xiaoyu He
- Tatiana K. Howard
- Dominique Kemp
- Justin Lacini
- Samuel Mundy
- Andrew O'Desky
- Samuel Pérez-Ayala
- John T. Sheridan
- David Villalobos
- Ruiyi Yang

### Visiting Professor

- Bhargav B. Bhatt

### Visiting Lecturer with Rank of Professor

- Camillo De Lellis
- Helmut H. Hofer
- Akshay Venkatesh

For a full list of faculty members and fellows please visit the department or program website.

## Courses

### MAT 100 - Calculus Foundations Fall/Spring QCR

### MAT 102 - Survey of Calculus Not offered this year QCR

### MAT 103 - Calculus I Fall/Spring QCR

### MAT 104 - Calculus II Fall/Spring QCR

### MAT 175 - Mathematics for Economics/Life Sciences Fall/Spring QCR

### MAT 191 - An Integrated Introduction to Engineering, Mathematics, Physics (also EGR 191/PHY 191) Not offered this year SEL

### MAT 192 - An Integrated Introduction to Engineering, Mathematics, Physics (also APC 192/EGR 192/PHY 192) Not offered this year QCR

### MAT 199 - Math Alive (also APC 199) Spring QCR

### MAT 201 - Multivariable Calculus Fall/Spring QCR

### MAT 202 - Linear Algebra with Applications Fall/Spring QCR

### MAT 203 - Advanced Vector Calculus Fall QCR

### MAT 204 - Advanced Linear Algebra with Applications Spring QCR

### MAT 214 - Numbers, Equations, and Proofs Fall QCR

### MAT 215 - Single Variable Analysis with an Introduction to Proofs Fall/Spring QCR

### MAT 217 - Honors Linear Algebra Spring QCR

### MAT 218 - Multivariable Analysis and Linear Algebra II Spring QCR

### MAT 300 - Multivariable Analysis I Fall QCR

### MAT 305 - Mathematical Logic Not offered this year QCR

### MAT 306 - Set Theory (also PHI 323) Fall QCR

### MAT 320 - Introduction to Real Analysis Fall QCR

### MAT 323 - Topics in Mathematical Modeling (also APC 323) Not offered this year QCR

### MAT 325 - Analysis I: Fourier Series and Partial Differential Equations Spring QCR

### MAT 330 - Complex Analysis with Applications Spring QCR

### MAT 335 - Analysis II: Complex Analysis Fall QCR

### MAT 345 - Algebra I Fall QCR

### MAT 346 - Algebra II Spring QCR

### MAT 355 - Introduction to Differential Geometry Spring QCR

### MAT 365 - Topology Fall QCR

### MAT 375 - Introduction to Graph Theory (also COS 342) Spring QCR

### MAT 377 - Combinatorial Mathematics (also APC 377) Fall QCR

### MAT 378 - Theory of Games Spring QCR

### MAT 380 - Probability and Stochastic Systems (also EGR 309/ORF 309) Fall/Spring

### MAT 385 - Probability Theory Fall QCR

### MAT 391 - Mathematics in Engineering I (also CBE 305/EGR 305/MAE 305) Fall/Spring QCR

### MAT 392 - Mathematics in Engineering II (also MAE 306) Spring

### MAT 393 - Mathematical Programming Not offered this year QCR

### MAT 407 - Theory of Computation (also COS 487) Fall

### MAT 419 - Topics in Number Theory Fall/Spring QCR

*Course Offerings*listing for topic details. Prerequisites: MAT 215, 345, 346 or equivalent. Staff

### MAT 425 - Analysis III: Integration Theory and Hilbert Spaces Spring QCR

### MAT 427 - Ordinary Differential Equations Not offered this year QCR

### MAT 429 - Topics in Analysis Not offered this year QCR

*Course Offerings*listing for topic details. Staff