Undergraduate Programs
The Department of Applied Physics and Applied Mathematics offers three undergraduate programs: applied physics, applied mathematics, and materials science and engineering. The materials science and engineering program is described here.
The applied physics and applied mathematics programs provide an excellent preparation for graduate study or for careers in which mathematical and technical sophistication are important. Using the large number of electives in these programs, students can tailor their programs to fit their personal and career interests. By focusing their technical electives, students can obtain a strong base of knowledge in a specialized area. In addition to formal minors, some areas of specialization that are available are described here. All technical electives are normally at the 3000 level or above.
Undergraduate Program in Applied Physics
The applied physics program stresses the basic physics that underlies most developments in engineering and the mathematical tools that are important to both physicists and engineers. Since the advances in most branches of technology lead to rapid changes in state-of-the-art techniques, the applied physics program provides the student with a broad base of fundamental science and mathematics while retaining the opportunity for specialization through technical electives.
The applied physics curriculum offers students the skills, experience, and preparation necessary for several career options, including opportunities to minor in economics and to take business-related courses. In recent years, applied physics graduates have entered graduate programs in many areas of applied physics or physics, enrolled in medical school, or been employed in various technical or financial areas immediately after receiving the B.S. degree.
Several areas of applied physics are represented by active research programs in the department for graduate instruction. These include fusion and space plasma physics, optical and laser physics, and condensed matter physics. Undergrad-uate students can receive course credit for research or an independent project with a faculty member. Opportunities also exist for undergraduate students in the applied physics program to participate in this research through part-time employment during the academic year and full-time employment during the summer, either at Columbia or as part of the NSF REU program nationwide. Practical research experience is a valuable supplement to the formal course of instruction. Applied physics students participate in an informal undergraduate seminar to study current and practical problems in applied physics, and obtain hands-on experience in at least two advanced laboratory courses.
Majors are introduced to two areas of application of applied physics (AP) by a course in each of two areas. Approved areas and courses are:
DYNAMICAL SYSTEMS
APMA E4101 or PHYS G4003
OPTICAL OR LASER PHYSICS
APPH E4110 or E4112
NUCLEAR SCIENCE
APPH E4010
PLASMA PHYSICS
APPH E4301
PHYSICS OF FLUIDS
APPH E4200
CONDENSED MATTER PHYSICS
PHYS G4018
BIOPHYSICAL MODELING
APMA E4400
In addition to these courses, courses listed in the Specialty Areas in Applied Physics can be used to satisfy this requirement with preapproval of the applied physics adviser.
All students must take 30 points of electives in the third and fourth years, of which 17 points must be technical courses approved by the adviser. The 17 points include 2 points of an advanced laboratory in addition to APPH E4018. Technical electives must be at the 3000 level or above unless prior approval is obtained from the department. A number of approved technical electives are listed in the section on specialty areas following. 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. .
back to topUndergraduate Program in Applied Mathematics
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.
back to topUndergraduate 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 E4903 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 adviser and the general undergraduate AM adviser, and then the approval of the Dean.
back to topSpecialty 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.”
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
• Earth and Atmospheric Sciences
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.
A. Atmosphere, Oceans and Climate
APPH E4200x:
Physics of fluids
APPH E4210y:
Geophysical fluid dynamics
EESC W4008y: Introduction to atmospheric science
ESC W4925x: Introduction to
physical oceanography
EES W4930y: Earth's oceans and
atmosphere
B. Solid Earth Geophysics
APPH E4200x: Physics of fluids
EESC W4941y: Principles of geophysics
EESC W4001x: Advanced general geology
EESC W4113x: Introduction to mineralogy
EESC W4701y: Introduction to igneous petrology
EESC W4950x: Mathematical
methods in the Earth sciences
• 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.
A. Economics
ECON W3211x,y:
Intermediate
microeconomics (J)
ECON W3213x,y: Intermediate
macroeconomics (J)
B. Industrial Engineering and Operations Research
IEOR E4003x: Industrial economics
IEOR E4201x: The engineering of
management, I
IEOR E4202y: The engineering of
management, II
C. Finance
SIEO W4150x,y: Probability and
statistics (J)
IEOR E4106y: Introduction to
operations research: stochastic models (J)
IEOR E4700x: 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 W4150x: 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 at least two areas recommended):
A. Biological Materials
CHEN E4650x:
Biopolymers
BIOL W4070x:
The biology and physics of single molecules
B. Biomechanics
BMEN E3320y:
Fluid biomechanics (J)
BMEN E4300y:
Solid biomechanics (J)
C. Genomics and Bioinformatics
ECBM E3060x:
Introduction to genomic information science and technology (J)
BIOL W3037y:
Whole genome bioinformatics (J)
CBMF W4761y:
Computational genomics
D. 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)