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:

     ELEN E3000x: Circuits, systems, and electronics (J)
     MSAE E3103x: Elements of material science (J)
     APPH E4010x: Intro to nuclear science
     PHYS GU4018y: Solid-state physics
     APMA E4101y: Intro to dynamical systems
     APPH E4110x: Modern optics
     APPH E4112y: Laser physics
     APPH E4200x: Physics of fluids
     APPH E4301y: Intro to plasma physics

• 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: Intro to atmospheric science
   APPH E4200x: Physics of fluids
   APPH E4210y: Geophysical fluid dynamics
   EESC GU4925x: Principles of physical oceanography
   EESC GU4930y: Earth’s oceans and atmosphere

   SOLID EARTH GEOPHYSICS
   EESC GU4001x: Advanced general geology
   EESC GU4113x: Intro to mineralogy
   APPH E4200x: Physics of fluids
   EESC GU4701y: Intro to igneous petrology
   EESC GU4941y: Principles of geophysics
   EESC GU4950x: 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 UN3002y:  From quarks to the cosmos: applications of modern physics
   ASTR UN3601x: General relativity, black holes, and cosmology (J)
   ASTR UN3602y: Physical cosmology (J)
   ASTR GU4001y: Astrophysics, I
   APMA E4101x: Intro to dynamical systems

• 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 UN3211x,y: Intermed microeconomics (J)
   ECON UN3213x,y: Intermed macroeconomics (J)

   INDUSTRIAL ENGINEERING AND OPERATIONS RESEARCH
   IEOR E4003x: Industrial economics
   IEOR E4201x: The eng of management, I
   IEOR E4202y: The eng of management, II

   FINANCE
   MATH GU4071x: Mathematics of finance
   IEOR E4106y: Intro to operations research: stochastic models (J)
   STAT GU4001x,y: Probability and statistics (J)
   ECIE GU4280: Corporate finance
   IEOR E4700x: Intro to financial engineering

• 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 UN3386x: Differential geometry
   APMA E4001y: Principles of applied mathematics
   APMA E4101x: Intro to dynamical systems
   APMA E4301x: Numerical methods for partial differential equations
   APMA E4302x: Parallel scientific computing
   PHYS GU4019y: Mathematical methods of 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.

   MATH UN3386x: Differential geometry
   APMA E4101x:
Intro to dynamical systems
   APMA E4150x: Applied functional analysis
   MATH GU4032x: Fourier analysis
   MATH GU4062y: Mathematical analysis, II
   STAT GU4001x,y: Intro to probability and statistics (J)
   PHYS GU4386x-W4387y: Geometrical concepts in physics

• 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 E3400y:
Physics of the human body

   RECOMMENDED:

   BIOL UN2005x-UN2006y:
Intro biology, I and II
   APMA E4400y: Intro to biophysical modeling

   OTHER TECHNICAL ELECTIVES (A COURSE IN AT LEAST TWO AREAS RECOMMENDED):

   BIOLOGICAL MATERIALS
   BIOL GU4070x: The biology and physics of single molecules
   CHEN E4650x: Biopolymers

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

   GENOMICS AND BIOINFORMATICS
   BIOL UN3037y: Whole genome bioinformatics (J)
   ECBM E3060x: Introduction to genomic information science and technology (J)
   CBMF W4761y: Computational genomics

   NEUROBIOLOGY
   BIOL UN3004x: Neurobiology, I (J)
   BIOL UN3005y: Neruobiology, II (J)
   ELEN E4011x: 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:

   MATH UN3020x: 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: intro to combinatorics and graph theory (J)
   COMS W4203y: Graph theory
   APMA E4300y: Intro to numerical methods
   APMA E4301x: Numerical methods for partial differential equations
   AMCS E4302x: Parallel scientific computing
   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

   PHYS UN3083y: Electronics laboratory (J)
   MSAE E3103x: Elements of material science (J)
   ELEN E3106x: Solid-state devices and materials (J)
   MSAE E4100x: Crystallography
   PHYS GU4018y: Solid-state physics
   MSAE E4206x: Electronic and magnetic properties of solids
   MSAE E4207y: Lattice vibrations and crystal defects

Undergraduate Program in Materials Science and Engineering

For more information click here.