Princeton University Mathematics Program: A Legacy of Excellence and Innovation
Princeton University's Department of Mathematics, founded in 1760, stands as one of the world's preeminent centers for mathematical education and research. Its illustrious history, distinguished faculty, and diverse student body contribute to the vibrant academic and intellectual atmosphere within the department. The curriculum is designed to provide students with a strong foundation in mathematical reasoning, while also encouraging independent research and collaboration.
Historical Overview
The first mathematics courses at Princeton, including algebra, trigonometry, geometry, and conic sections, were offered in 1760. Walter Minto became one of the earliest teachers of mathematics beginning in 1787. By the dawn of the twentieth century, the department had risen to become "one of the world's great centers of mathematical teaching and research."
In 1903, President Woodrow Wilson appointed Henry Burchard Fine as dean of the faculty, and in 1905, as the first chairman of the Department of Mathematics. The university extended invitations to numerous leading mathematicians, including Luther P. Eisenhart, Solomon Lefschetz, James W. Alexander II, James Jeans, J.H.M. Wedderburn, George David Birkhoff, and Oswald Veblen, to conduct research at Princeton. In 1928, Princeton established the first research professorship in mathematics in the United States.
The establishment of the Institute for Advanced Study (IAS) in Princeton, New Jersey, in 1930 further propelled mathematical research in the area. Despite remaining separate entities, the IAS and Princeton have maintained close relationships and collaborative projects due to their proximity. The political climate in Europe during the 1930s led to an influx of scholars into the United States, including Emil Artin, Valentine Bargmann, and William Feller, further enriching the department.
In 1968, the department relocated to a new Fine Hall (the former building was renamed Jones Hall). The new building comprises three stories of classrooms and graduate student offices, with an additional ten floors housing faculty offices, a seminar room, and a professors' lounge. Fine Hall, often described as the "anchor of mathematics," served as the home of the School of Mathematics. This move strengthened the collaboration between the Institute for Advanced Study and Princeton University. Fine Hall is connected to Jadwin Hall, which houses additional classroom and academic facilities.
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The Albert Einstein Memorial Lecture, a free public event, is held annually at Princeton around Einstein's birthday on March 14.
Curriculum and Program Structure
Undergraduate Program
At the undergraduate level, approximately 70-75 students concentrate in mathematics. The curriculum emphasizes a strong foundation in linear algebra and analysis, as well as experience in understanding and writing rigorous mathematical proofs.
Coursework: Students complete required courses in real analysis, complex analysis, algebra, geometry, and topology. Prospective majors with a strong background in calculus are recommended to start in MAT215 (Single Variable Analysis with an Introduction to Proofs), followed by MAT217 and MAT300. Students with extensive prior experience with calculus and rigorous proofs can start in MAT216 (Multivariable Analysis and Linear Algebra I), followed by MAT218. The sequences MAT215-217-300 or MAT216-218 are strongly recommended for admission to the department.
Independent Work: All departmental students engage in independent work, supervised by a faculty member. In the junior year, this typically involves participating actively in a junior seminar in both the fall and spring semesters. Alternatively, a student may replace one junior seminar with supervised reading in a special subject and then writing a paper based on that reading. The senior year focuses on writing a senior thesis.
Degree Requirements: Degree requirements include completion of eight upper-division courses: four core courses (one from each of real analysis, complex analysis, algebra, and geometry/topology/discrete math) and four additional courses chosen to create a program of study for particular fields of pure and applied mathematics. Up to three of the eight departmental courses may be cognate courses outside the Mathematics Department, with permission.
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Bridge Courses: The Mathematics Department offers some introductory graduate courses, called "bridge" courses, that are accessible to undergraduates with sufficient background.
The department encourages those interested in pursuing careers to participate in the Principia: The Princeton Undergraduate Mathematics Journal.
Graduate Program
The graduate program emphasizes independent research from the start. It is designed to equip students with the skills and knowledge necessary to become leading researchers in their fields.
Core Training: Students begin 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.
Independent Research: 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. First-year students are set on the fast track of research by choosing two advanced topics of research, beyond having a strong knowledge of three more general subjects: algebra, and real and complex analysis, as part of the required General Examination. It is the hope that one, or both, of the advanced topics will lead to the further discovery of a thesis problem.
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Supportive Environment: While independent work and research are emphasized, there exists a real sense of camaraderie among the graduate students. The atmosphere created is one of excitement and stimulation as well as of mentoring and support. The program has minimal requirements and maximal research and educational opportunities.
Advanced Courses and Seminars: The department offers a broad variety of advanced research topics courses as well as more introductory level courses in algebra, analysis, and geometry, which help first-year students strengthen their mathematical background and get involved with faculty through basic course work. In addition to the courses, there are several informal seminars specifically geared toward graduate students:
- Colloquium Lunch Talk: Experts who have been invited to present at the Department Colloquium give introductory talks, which allows graduate students to understand the afternoon colloquium more easily.
- Graduate Student Seminar (GSS): Organized and presented by graduate students for graduate students, creating a vibrant mathematical interaction among them.
- What’s Happening in Fine Hall (WHIFH) seminar: Faculty give talks in their own research areas specifically geared towards graduate students.
Foreign Language Requirement: Students must demonstrate proficiency in one foreign language (French, German, or Russian), proving their ability to work with mathematical texts from scholars around the world.
Financial Support: Graduate students are offered full tuition and student health plan coverage for all five years of the program.
Rankings and Recognition
Princeton's Mathematics Department consistently ranks among the top programs in the world. U.S. News & World Report ranks it as the No. 1 program nationwide, tied with the Massachusetts Institute of Technology. Its particular strengths are Algebra, Number Theory, Algebraic Geometry, Geometry, and Topology, as those are the specialties ranked as No. 1. Times Higher Education also places the department as No. 1.
The department is home to a number of scholars, researchers, and professors who have continued to be recognized by the international community. The university has been affiliated with 15 Fields Medalists, behind Harvard University (#1 with 18) and the University of Paris (#2 with 16).
Notable Individuals
The Department of Mathematics has been associated with numerous influential figures in the field, including:
- John Nash: Former faculty member and winner of the 1994 Nobel Memorial Prize in Economic Sciences.
- Alan Turing: Received his doctorate from the department.
- Albert Einstein: Frequently gave lectures at Princeton and had an office in the building.
- Fields Medalists: Manjul Bhargava, Charles Fefferman, Gerd Faltings, Michael Freedman, Elon Lindenstrauss, Andrei Okounkov, Terence Tao, William Thurston, Akshay Venkatesh, and Edward Witten (who began graduate study in the mathematics department before transferring to the physics department).
Other noteworthy Princeton mathematicians include Ralph Fox, Donald C. Spencer, and John R. Stallings Jr.
Research and Special Programs
The department runs the Program in Applied and Computational Mathematics (PACM), an interdisciplinary and interdepartmental program for scholars interested in the application of mathematics to other fields. The Women and Mathematics program is co-directed by the department and the IAS.
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