The applied mathematics program is flexible and intensive. A student must take the required courses listed below, or prove equivalent standing, and then may elect the other courses from mathematics, computer science, physics, Earth and environmental sciences, biophysics, economics, business and finance, or other application fields. Each student tailors his or her own program in close collaboration with an adviser. He or she must also register for the applied mathematics seminar during both the junior and senior years. During the junior year, the student attends the seminar lectures for 1 point; during the senior year, he or she attends the seminar lectures as well as tutorial problem sessions for 4 points.
While it is common for students in the program to go on to graduate school, many graduating seniors will find employment directly in industry, government, education, or other fields.
Of the 33 points of elective content in the third and fourth years, at least 21 points of technical courses approved by the adviser must be taken. The remaining points of electives are intended primarily as an opportunity to complete the four-year, 27-point nontechnical requirement, but any type of course work can satisfy them.
Undergraduate Double Major in Applied Physics and Applied Mathematics
Students satisfy all requirements for both majors, except for the seminar requirements. They are required to take both senior seminars, APMA E493 and APPH E4903 (taking one in the junior year and one in the senior year, due to timing conflicts), but not the junior seminars, APMA E4901 and APPH E4901. A single course may be used to fulfill a requirement in both majors. Students must maintain a GPA at or above 3.75, and must graduate with at least 143 points, 15 above the regular 128-point requirement. These extra 15 points should be technical electives appropriate for one or both majors.
To apply, a student first obtains the approval of both the general undergraduate AP advisor and the general undergraduate AM advisor, and then the approval of the Dean.
Specialty Areas in Applied Physics and Applied Mathematics
Both applied physics and applied mathematics students can focus their
technical electives and develop a strong base of knowledge in a
specialty area. There is no requirement to focus electives, so students
may take as many or as few of the recommended courses in a specialty
area as is appropriate to their schedules and interests. Some
specialties are given below, but this is not an exclusive list and
others can be worked out in coordination with the student's adviser.
The courses that are often taken, or in some cases need to be taken, in
the junior year are denoted with a "J." Applied Mathematics Undergraduate Curriculum
Technical Electives
Applications of Physics
Courses that will give a student a broad background in applications of physics are:
MSAE E3103x: Elements of materials science (J)
ELEN E3000x:
Introduction to circuits, systems, and electronics (J)
APPH E4010x:
Introduction to nuclear science
APPH E4110x:
Modern optics
APPH E4112y:
Laser physics
APPH E4200x:
Physics of fluids
APPH E4301y:
Introduction to plasma physics
PHYS G4018y:
Solid-state physics
APMA E4101y:
Introduction to dynamical systems
The Earth sciences provide a wide range of problems of interest to physicists and mathematicians ranging from the dynamics of the Earth's climate to earthquake physics to dynamics of Earth's deep interior. The Lamont-Doherty Earth Observatory, which is part of Columbia University, provides enormous resources for students interested in this area.
Atmosphere, Oceans and Climate
APPH E4200x: Physics of fluids
APPH E4210y: Geophysical fluid dynamics
EESC W4008y: Introduction to atmospheric science
ESC W4925: Introduction to physical oceanography
EES W4930y: Earth's oceans and atmosphere
Solid Earth Geophysics
APPH E4200x: Physics of fluids
EESC W4941y: Principles of geophysics
ESC W4001x: Advanced general geology
EESC W4113x: Introduction to mineralogy
EESC W4701y: Introduction to igneous petrology
EESC W4950x: Mathematical methods in the Earth sciences
(See also courses listed under scientific computation and computer science.)
Basic Physics and Astrophysics
Fundamental physics and astrophysics can be emphasized. Not only is astrophysics providing a deeper understanding of the universe, but it is also testing the fundamental principles of physics.
PHYS W3002y: From quarks to the cosmos: applications of modern physics
ASTR C3601x: General relativity, black holes, and cosmology (J)
ASTR C3602y: Physical cosmology and extragalactic astronomy (J)
APMA E4101x: Introduction to dynamical systems
ASTR G4001y: Astrophysics, I
Business and Finance
The knowledge of physics and mathematics that is gained in the applied physics and applied mathematics programs is a strong base for a career in business or finance.Economics
ECON W3211x,y: Intermediate microeconomics (J)
ECON W3213x,y: Intermediate macroeconomics (J)
Industrial Engineering and Operations Research
IEOR E4003x: Industrial economics
IEOR E4201x: The engineering of management, I
IEOR E4202y: The engineering of management, II
Finance
SIEO W4150x,y: Probability and statistics (J)
IEOR E4106y: Introduction to operations research: stochastic models (J)
IEOR E4700: Introduction to financial engineering
MATH W4071x: Mathematics of finance
ECIE W4280: Corporate finance
Mathematics Applicable to Physics
Applied physics students can specialize in the mathematics that is
applicable to physics. This specialization is particularly useful for
students interested in theoretical physics.
APMA E4101x:
Introduction to dynamical systems
APMA E4001y:
Principles of applied mathematics
APMA E4301x: Numerical methods for partial differential equations
APMA E4302x: Parallel scientific computing
MATH V3386x:
Differential geometry
MATH W4386x-W4387y:
Geometrical concepts in physics
Fundamental Mathematics in Applied Mathematics
This specialization is intended for students who desire a more solid
foundation in the mathematical methods and underlying theory. For
example, this specialization could be followed by students with an
interest in graduate work in applied mathematics.
APMA E4101x:
Introduction to dynamical systems
APMA E4150x: Applied functional analysis
SIEO W4150x,y:
Introduction to probability and statistics (J)
MATH V3386x:
Differential geometry
MATH W4386x-W4387y:
Geometrical concepts in physics
MATH W4032x:
Fourier analysis
MATH W4062y:
Mathematical analysis, II
Quantitative Biology
Traditionally biology was considered a descriptive science in contrast
to the quantitative sciences that are based on mathematics, such as
physics. This view no longer coincides with reality. Researchers from
biology as well as from the physical sciences, applied mathematics, and
computer science are rapidly building a quantitative base of biological
knowledge. Students can acquire a strong base of knowledge in
quantitative biology, both biophysics and computational biology, while
completing the applied physics or applied mathematics programs.
Professional-level Course:
APPH E1300y:
Physics of the human body
Recommended:
BIOL C2005x-C2006y:
Introduction to molecular and cellular biology, I & II
APMA E4400y:
Introduction to biophysical modeling
Other Technical Electives (a course in a least two areas recommended):
Biological Materials
CHEN E4650x: Biopolymers
BIOL W4070x: The biology and physics of single molecules
Biomechanics
BMEN E3320y: Fluid biomechanics (J)
BMEN E4300y: Solid biomechanics (J)
Genomics and Bioinformatics
ECBM E3060x: Introduction to genomic information science and technology (J)
BIOL W3037y: Whole genome bioinformatics (J)
CBMF W4761y: Computational genomics
Neurobiology
BIOL W3004x: Cellular and molecular neurobiology (J)
BIOL W3005y: Systems neurobiology (J)
ELEN G4011x: Computational neuroscience
The second term of biology will be considered a technical elective if a
student has credits from at least two other of the recommended courses
in quantitative biology at the 3000 level or above.
Scientific Computation and Computer Science
Advanced computation has become a core tool in science, engineering,
and mathematics and provides challenges for both physicists and
mathematicians. Courses that build on both practical and theoretical
aspects of computing and computation include:
APMA E4300y:
Introduction to numerical methods
APMA E4301:
Numerical methods for partial differential equations
AMCS E4302:
Parallel scientific computing
MATH V3020x:
Number theory and cryptography (J)
COMS W3137x,y:
Data structures and algorithms (or
COMS W3139y:
Honors data structures and algorithms) (J)
COMS W3157x,y:
Advanced programming (J)
COMS W3203x,y:
Discrete mathematics: introduction to combinatorics and graph theory (J)
COMS W4203y:
Graph theory
COMS W4701x,y:
Artificial intelligence
COMS W4771y:
Machine learning
Solid-State Physics
Much of modern technology is based on solid-state physics, the study of
solids and liquids. Courses that will build a strong base for a career
in this area are:
MSAE E3103x:
Elements of material science (J)
ELEN E3106x:
Solid-state devices and materials (J)
MSAE E4206x:
Electronic and magnetic properties of solids
PHYS G4018y:
Solid-state physics
MSAE E4207y:
Lattice vibrations and crystal defects
PHYS W3083y:
Electronics laboratory (J)
For more information, please see: Applied Mathematics Advising FAQ page
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