Undergraduate Technical Electives

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.”

Applications of Physics

Courses that will give a student a broad background in applications of physics are:

ELEN E3000x: Circuits, systems, and electronics (J)
MSAE E3103x: Elements of materials science (J)
APPH E4010x:
Introduction to nuclear science
PHYS G4018y:
Solid-state physics
APMA E4101y: Introduction to dynamical systems
APPH E4110x: Modern optics
APPH E4112y:
Laser physics
APPH E4200x:
Physics of fluids
APPH E4301y:
Introduction to plasma physics

back to top

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.

Atmosphere, Oceans and Climate

APPH E4008y: Introduction to atmospheric science
APPH E4200x: Physics of fluids
APPH E4210y:
Geophysical fluid dynamics
EESC W4925x: Principles of physical oceanography
EESC W4930y
: Earth's oceans and atmosphere

Solid Earth Geophysics

EESC W4001x: Advanced general geology
EESC W4113x:
Introduction to mineralogy
APPH E4200x: Physics of fluids
EESC W4701y:
Introduction to igneous petrology
EESC W4941y:
Principles of geophysics
EESC W4950x:
Mathematical methods in the Earth sciences
(See also courses listed under Scientific Computation and Computer Science)

back to top

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 (J)
ASTR G4001y:
Astrophysics, I
APMA E4101x:
Introduction to dynamical systems

back to top

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.


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


MATH W4071x: Mathematics of finance
IEOR E4106y:
Introduction to operations research: stochastic models (J)
SIEO W4150x,y:
Probability and statistics (J)
ECON W4280:
Corporate finance
IEOR E4700:
Introduction to financial engineering

back to top

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.

MATH V3386x:
Differential geometry
APMA E4001y:
Principles of applied mathematics
APMA E4101x:
Introduction to dynamical systems
APMA E4301x:
Numerical methods for partial differential equations
APMA E4302x:
Parallel scientific computing
PHYS G4019y:
Mathematical methods of physics

back to top

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.

MATH V3386x:
Differential geometry
APMA E4101x:
Introduction to dynamical systems
APMA E4150x:
Applied functional analysis
MATH W4032x:
Fourier analysis
MATH W4062y:
Modern analysis, II
SIEO W4150x,y:
Introduction to probability and statistics (J)
PHYS W4386x- W4387y:
Geometrical concepts in physics

back to top

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

BIOL C2005x-C2006y:
Introduction to biology, I & II
APMA E4400y:
Introduction to biophysical modeling

Other Technical Electives (a course in a least two areas recommended):

Biological Materials

BIOL W4070x:
The biology and physics of single molecules
CHEN E4650x:


BMEN E3320y:
Fluid biomechanics (J)
BMEN E4300y:
Solid biomechanics (J)

Genomics and Bioinformatics

BIOL W3037y: Whole genome bioinformatics (J)
ECBM E3060x:
Introduction to genomic information science and technology (J)
CBMF W4761y:
Computational genomics


BIOL W3004x: Neurobiology, I (J)
BIOL W3005y:
Neurobiology, II (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.

back to top

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:

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
APMA E4300y:
Introduction to numerical methods
APMA E4301:
Numerical methods for partial differential equations
AMCS E4302:
Parallel scientific computing
COMS W4701x,y:
Artificial intelligence
COMS W4771y:
Machine learning

back to top

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:

PHYS W3083y: Electronics laboratory (J)
MSAE E3103x:
Elements of material science (J)
ELEN E3106x:
Solid-state devices and materials (J)
MSAE E4100x:
PHYS G4018y:
Solid-state physics
MSAE E4206x:
Electronic and magnetic properties of solids
MSAE E4207y:
Lattice vibrations and crystal defects

back to top

500 W. 120th St., Mudd 200, MC 4701 New York, NY 10027 / Phone: 212-854-4457 / Fax: 212-854-8257 / Email: seasinfo.apam@columbia.edu

©2012 Columbia University