Applied and Computational Mathematics Jump To: Jump To: Overview There has never been a better time to be a mathematician. The combination of mathematics and computer modeling has transformed science and engineering, and is changing the nature of research in the biological sciences, data science and many other areas.The minor in the Program in Applied and Computational Mathematics (PACM) is designed for students from engineering, from the physical, biological and social sciences, and from the humanities who wish to broaden their mathematical and computational skills. It is also an opportunity for mathematics majors whose primary focus is in pure mathematics to discover the challenges presented by applications from the sciences and engineering. The interaction between mathematical foundations and applications lies at the core of the program.Princeton does not offer a separate major in Applied and Computational Mathematics. Students seeking to pursue an academic program whose primary focus lies in applied mathematics may either major in mathematics with a course of study geared toward applications, or major in another discipline and combine their major with the PACM minor. Program Offerings Minor Offering type Minor There has never been a better time to be a mathematician. The combination of mathematics and computer modeling has transformed science and engineering, and is changing the nature of research in the biological sciences, data science and many other areas.The minor in the Program in Applied and Computational Mathematics (PACM) is designed for students from engineering, from the physical, biological and social sciences, and from the humanities who wish to broaden their mathematical and computational skills. It is also an opportunity for mathematics majors whose primary focus is in pure mathematics to discover the challenges presented by applications from the sciences and engineering. The interaction between mathematical foundations and applications lies at the core of the program.Princeton does not offer a separate major in Applied and Computational Mathematics. Students seeking to pursue an academic program whose primary focus lies in applied mathematics may either major in mathematics with a course of study geared toward applications, or major in another discipline and combine their major with the PACM minor. Goals for Student Learning Mathematics is the language of science, and almost every discipline of science and engineering is ultimately founded on mathematics. At the same time, state-of-the-art computational methods are enabling the study of increasingly complex systems. The aim of the Program in Applied and Computational Mathematics minor is to enable students from a broad range of disciplines to develop a stronger mathematical and computational foundation, and to promote dialogue between mathematics and its applications.At the same time, the minor aims to encourage students whose primary concentration is in pure mathematics to explore the impact of mathematics on science, engineering and technology.A core tenet of the PACM minor is the bilateral interaction between mathematical foundations and applications. A strong background in mathematics and computational methods provides students with powerful tools to address problems that arise in other disciplines, and may lead them to view these from a new perspective. Conversely, challenging mathematical and computational questions that arise in applications strongly motivate the study and development of mathematics. The PACM minor promotes such interactions:By requiring all students to take both mathematical foundations and applications courses;By requiring all students to perform an independent research project that connects mathematics with its applications;By requiring all students to attend and present their independent work in the minor seminar, where they are exposed to a broad range of applications of mathematics to different disciplines and have the opportunity to interact with students with applied mathematical interests from many different majors across campus.At the same time, the flexible program is designed to allow each student to tailor their courses and independent work to their own interests, and to make mathematics as accessible as possible for students with diverse interests and backgrounds. Prerequisites The PACM minor has no formal prerequisites. However, the PACM course requirements (as detailed below) can be fulfilled only with 300+ level courses. While the PACM minor does not have any formal course requirements at the 200 level, students should note that most 300+ level mathematical foundations courses are likely to require either MAT 201 or MAT 202 as prerequisites. Admission to the Program Students interested in the PACM minor must contact the program's undergraduate representative on or before February 1 of their junior year to discuss their interests, and to lay out a plan for their course selection and research component. Program of Study The requirements for the PACM minor consist of:A total of five courses at 300 level or higher (requires letter grade; pass/D/fail not accepted), at least three of which are not included in the requirements for the candidate’s major.An independent research project consisting of a paper in one of the following formats: (a) a project that you are working on with a professor; or (b) a summer research project. The research project may not be used to satisfy any requirements of your major or of any other minor or certificate. Most (but not all) majors require a senior thesis and/or junior independent work, which therefore cannot be used as such for PACM; however, a significant extension of such independent work could be used for PACM subject to approval of the PACM undergraduate representative.Students are required to participate during the spring semester of their junior and senior years in a not-for-credit colloquium offered by PACM. This will provide a forum for presentation and discussion of research projects among all students in the minor and will introduce them to a broad range of areas within applied mathematics.The PACM course requirement may be satisfied by a broad range of courses that place a particular emphasis on applied mathematics, which are offered by the mathematics department as well as the science, engineering and economics departments. The five required courses must be distributed between the following two areas, with at least two from each area:Mathematical foundations and techniques, including differential equations, real and complex analysis, discrete mathematics, probability, numerical methods, etc.Mathematical applications in diverse areas offered by the applied and computational mathematics program and by science, engineering and economics departments.An extensive list of courses that meet the minor requirements can be found on the program website. Courses that do not appear on this list may be approved by the PACM undergraduate representative. Specific programs should be tailored in consultation with the PACM undergraduate representative to meet the individual needs and interests of each student.The PACM research project is typically done under the supervision of a PACM core or associated faculty member, but external advisers are regularly accommodated. In the latter case, a second reader from PACM is asked to verify that the paper contains enough applied mathematics to satisfy the minor requirements. In any case, plans for the research project must be approved by the undergraduate representative. Additional Information Normally, students may not pursue both the PACM minor and the minor offered by the mathematics department. However, students who believe they have a compelling curricular reason to pursue both minors may apply to the directors of both programs for permission to do so. Students who are majoring in mathematics are welcome to combine their studies with the PACM minor. Faculty Director Amit Singer Director of Undergraduate Program Paul Seymour Executive Committee Noga M. Alon, Mathematics René A. Carmona, Oper Res and Financial Eng Emily Ann Carter, Mechanical & Aerospace Eng Maria Chudnovsky, Mathematics Peter Constantin, Mathematics Amit Singer, Mathematics Howard A. Stone, Mechanical & Aerospace Eng Romain Teyssier, Astrophysical Sciences Jeroen Tromp, Geosciences Ramon van Handel, Oper Res and Financial Eng Associated Faculty Ryan P. Adams, Computer Science Amir Ali Ahmadi, Oper Res and Financial Eng Michael Aizenman, Physics Yacine Aït-Sahalia, Economics William Bialek, Physics Mark Braverman, Computer Science Carlos D. Brody, Princeton Neuroscience Inst Adam S. Burrows, Astrophysical Sciences Roberto Car, Chemistry Bernard Chazelle, Computer Science Jianqing Fan, Oper Res and Financial Eng Jason W. Fleischer, Electrical & Comp Engineering Mikko P. Haataja, Mechanical & Aerospace Eng Gregory W. Hammett, PPPL Theory Isaac M. Held, Atmospheric & Oceanic Sciences Sergiu Klainerman, Mathematics Naomi E. Leonard, Mechanical & Aerospace Eng Simon A. Levin, Ecology & Evolutionary Biology Luigi Martinelli, Mechanical & Aerospace Eng William A. Massey, Oper Res and Financial Eng Assaf Naor, Mathematics Jonathan W. Pillow, Princeton Neuroscience Inst H. Vincent Poor, Electrical & Comp Engineering Frans Pretorius, Physics Herschel A. Rabitz, Chemistry Peter J. Ramadge, Electrical & Comp Engineering Jennifer Rexford, Computer Science Clarence W. Rowley, Mechanical & Aerospace Eng Szymon M. Rusinkiewicz, Computer Science Frederik J. Simons, Geosciences Jaswinder P. Singh, Computer Science Ronnie Sircar, Oper Res and Financial Eng Mete Soner, Oper Res and Financial Eng John D. Storey, Integrative Genomics Sankaran Sundaresan, Chemical and Biological Eng Ludovic Tangpi, Oper Res and Financial Eng Robert E. Tarjan, Computer Science Corina E. Tarnita, Ecology & Evolutionary Biology Salvatore Torquato, Chemistry Olga G. Troyanskaya, Computer Science Matt Weinberg, Computer Science Professor Noga M. Alon Maria Chudnovsky Peter Constantin Amit Singer Romain Teyssier Jeroen Tromp Associate Professor Ramon van Handel Lecturer Henry F. Schreiner Visiting Professor Ehud Yariv For a full list of faculty members and fellows please visit the department or program website. Courses APC 199 - Math Alive (also MAT 199) Spring QCR An exploration of some of the mathematical ideas behind important modern applications, from banking and computing to listening to music. Intended for students who have not had college-level mathematics and are not planning to major in a mathematically based field. The course is organized in independent two-week modules focusing on particular applications, such as bar codes, CD-players, population models, and space flight. The emphasis is on ideas and mathematical reasoning, not on sophisticated mathematical techniques. Two 90-minute classes, one computer laboratory. Staff APC 350 - Introduction to Differential Equations (also MAT 322) Spring QCR This course will introduce the basic theory, models and techniques for ordinary and partial differential equations. Emphasis will be placed on the connection with other disciplines of science and engineering. We will try to strike a balance between the theoretical (e.g. existence and uniqueness issues, qualitative properties) and the more practical issues such as analytical and numerical approximations. Prerequisites: Multivariable calculus and linear algebra. M. Gilles ECE 486 - Transmission and Compression of Information (also APC 486) Not offered this year An introduction to lossless data compression algorithms, modulation/demodulation of digital data, error correcting codes, channel capacity, lossy compression of analog and digital sources. Three hours of lectures. Prerequisites: 301, ORF 309. Staff EGR 192 - An Integrated Introduction to Engineering, Mathematics, Physics (also APC 192/MAT 192/PHY 192) Not offered this year QCR Taken concurrently with EGR/MAT/PHY 191. An integrated course that covers the material of PHY 103 and MAT 201 with the emphasis on applications to engineering. Math topics include: vector calculus; partial derivatives and matrices; line integrals; simple differential equations; surface and volume integrals; and Green's, Stokes's, and divergence theorems. One lecture, two preceptorials. C. Kelleher GEO 441 - Computational Geophysics (also APC 441) Spring An introduction to weak numerical methods used in computational geophysics. Finite- and spectral-elements, representation of fields, quadrature, assembly, local versus global meshes, domain decomposition, time marching and stability, parallel implementation and message-passing, and load-balancing. Parameter estimation and "imaging" using data assimilation techniques and related "adjoint" methods. Labs provide experience in meshing complicated surfaces and volumes as well as solving partial differential equations relevant to geophysics. Prerequisites: MAT 201; partial differential equations and basic programming skills. Two 90-minute lectures. J. Tromp MAT 323 - Topics in Mathematical Modeling (also APC 323) Not offered this year QCR Draws problems from the sciences and engineering for which mathematical models have been developed and analyzed to describe, understand and predict natural and man-made phenomena. Emphasizes model building strategies, analytical and computational methods, and how scientific problems motivate new mathematics. This interdisciplinary course in collaboration with Molecular Biology, Psychology and the Program in Neuroscience is directed toward upper class undergraduate students and first-year graduate students with knowledge of linear algebra and differential equations. Staff MAT 377 - Combinatorial Mathematics (also APC 377) Fall QCR Combinatorics is the study of enumeration and structure of discrete objects. These structures are widespread throughout mathematics, including geometry, topology and algebra, as well as computer science, physics and optimization. This course will give an introduction to modern techniques in the field, and how they relate to objects such as polytopes, permutations and hyperplane arrangements. N. Alon
