# Graduate Programs

Financial aid is available for students pursuing a doctorate. Fellowships, scholarships, teaching assistantships, and graduate research assistantships are awarded on a competitive basis. The Aptitude Test of the Graduate Record Examination is required of candidates for admission to the department and for financial aid; the Advanced Tests are recommended.

M.S. students must complete the professional development and leadership course, ENGI E4000, as a graduation requirement. Ph.D. candidates are strongly encouraged to complete ENGI E6001–6004 and should consult their program for PDL requirements. Ethics training is required of all students pursuing a doctorate.

### M.S. Program in Applied Physics

The program of study leading to the degree of Master of Science, while emphasizing continued work in basic physics, permits many options in several applied physics specialties. The program may be considered simply as additional education in areas beyond the bachelor’s level, or as preparatory to doctoral studies in the applied physics fields of plasma physics, laser physics, or solid-state physics. Specific course requirements for the master’s degree are determined in consultation with the program adviser, but must include four of the six core courses listed below.

The core courses provide a student with a solid foundation in the fundamentals of applied physics, but with the approval of the faculty adviser, other graduate-level courses with APPH designators not listed below may also count as core courses.

**APPH E4100:** Quantum physics of matter

**APPH E4110:** Modern optics

**APPH E4112:** Laser physics

**APPH E4200:** Physics of fluids

**APPH E4300:** Applied electrodynamics

**APPH E4301:** Introduction to plasma physics

### M.S. Program in Applied Mathematics

This 30-point program leads to a Master of Science degree. Students must complete five core courses and five electives. All degree requirements must be completed within five years. A candidate is required to maintain at least a 2.5 grade point average. If a student admitted to the Applied Mathematics M.S. only program is interested in the Ph.D. program, the student must reapply for admission. The core courses provide a student with a foundation in the fundamentals of applied mathematics and contribute 15 points of graduate credit toward the degree. Students must complete five of the following nine courses:

**APMA E4001: **Principles of applied mathematics**
APMA E4101: **Intro to dynamical systems

**Applied functional analysis**

APMA E4150:

APMA E4150:

**Partial differential equations**

APMA E4200:

APMA E4200:

**Functions of a complex variable**

APMA E4204:

APMA E4204:

**Intro to numerical methods**

APMA E4300:

APMA E4300:

**Numerical methods for partial differential equations**

APMA E4301:

APMA E4301:

**Analytic methods for partial differential equations**

APMA E6301:

APMA E6301:

**Numerical analysis for partial differential equations**

APMA E6302:

APMA E6302:

Students must also take a required Research Seminar course, APMA E6100 x or y.

A student must select five elective courses from those listed below (or any of those not used to satisfy the core requirements from the list above) for a total of 15 points of graduate credit. Additional courses not listed below can be applied toward the elective requirements, subject to the approval of the faculty adviser.

*Courses from the Department of Economics, School of Business, School of International and Public Affairs, or quantitative courses offered by the School of Professional Studies may not be counted as electives toward the degree.

Computer science elective courses include:

**CSOR W4231: **Analysis of algorithms, I**
COMS W4236:** Intro to computational complexity

**Numerical algorithms and complexity**

COMS W4241:

COMS W4241:

**Computational learning theory**

COMS W4252:

COMS W4252:

Industrial engineering/operations research elective courses include**:

**IEOR E4003: **Industrial economics**
IEOR E4004: **Intro to operations research: deterministic models

**Optimization: models and methods**

IEOR E4007:

IEOR E4007:

**Stochastic models**

IEOR E4106:

IEOR E4106:

**Intro to probability and statistics**

SIEO W4150:

SIEO W4150:

**Advanced engineering and corporate economics**

IEOR E4403:

IEOR E4403:

**Game theoretic models of operations**

IEOR E4407:

IEOR E4407:

**Elementary stochastic processes**

STAT W4606:

STAT W4606:

**Intro to financial engineering**

IEOR E4700:

IEOR E4700:

**Please check IEOR website for registration procedures required of non-IEOR students.

Other elective courses may be chosen from other departments in SEAS and Arts and Sciences, e.g., the Departments of Mechanical Engineering, Mathematics, and Statistics.

### M.S. Program in Materials Science and Engineering

For more information click **here**.

### M.S. Program in Medical Physics

This CAMPEP-approved 36-point program in medical physics leads to the M.S. degree. It is administered by faculty from the School of Engineering and Applied Science in collaboration with faculty from the College of Physicians and Surgeons and the Mailman School of Public Health. It provides preparation toward certification by the American Board of Radiology. The program consists of a core curriculum of medical and nuclear physics courses, anatomy, lab, seminar, a tutorial, one elective, and two practicums. Specific course requirements are *APPH E4010*, *E4330*, *E4710*, *E4500*, *E4550*, *E4600*, *E6319*, *E6330*, *E6335*, and *APBM E4650*. Approved electives include *APPH E4711*, *APPH E6336*, *APAM E6650*, and a third practicum. Up to 6 points of this 36-point program may be waived based on prior equivalent academic work. A student who enters the 36-point M.S. Program in Medical Physics having satisfactorily completed, prior to beginning the Program, a course determined by the faculty to be equivalent in content to a required course within the Program may be

considered to have satisfied that content requirement, may be allowed to have that requirement waived, and may be permitted to graduate from the M.S. Program in Medical Physics with fewer than 36 points, but not fewer than the 30-point minimum required by the School of Engineering and Applied Science. Evaluation of prior coursework may include review of syllabi, comparison of textbooks, consultation with instructors, and/or written or oral examination administered by Program faculty. A passing grade on a comprehensive examination is required for graduation. This examination, on subjects covered in the curriculum, is taken after two terms of study.

### Certification of Professional Achievement in Medical Physics

This graduate program of instruction leads to the Certification of Professional Achievement and requires satisfactory completion of six of the following courses:

**APPH E4330: **Radiobiology**
APPH E4500: **Health physics

**Dosimetry**

APPH E4600:

APPH E4600:

**Anatomy for physicists and engineers**

APBM E4650:

APBM E4650:

**Clinical nuclear medicine physics**

APPH E6319:

APPH E6319:

**or APPH E6330:**Diagnostic radiology physics

**Radiation therapy physics**

APPH E6335:

APPH E6335:

**or APPH E6336:**Advanced topics in radiation therapy

This is a part-time nondegree program. Students are admitted to the department as certificate-track students.

### PH.D. AND ENG.SC.D. PROGRAMS

After completing the M.S. program, doctoral students specialize in one area of their programs. Some specializations have specific course requirements for the doctorate; elective courses are determined in consultation with the program adviser. Successful completion of an approved 30-point program of study is required in addition to successful completion of a written qualifying examination taken after two semesters of graduate study. An oral examination, taken within one year after the written qualifying examination, and a thesis proposal examination, taken within two years after the written qualifying examination, are required of all doctoral candidates, as is training in research and professional ethics in the first two years of the doctoral program.

### Applied Mathematics

Applied mathematics deals with mathematical concepts and techniques used in various fields of science and engineering. Historically, mathematics was first applied with great success in astronomy and mechanics. Then it developed into a main tool for physics, other physical science, and engineering. It is now becoming important in the biological, geological, economics, business, etc. With the coming of age of the computer, applied mathematics transcended its traditional style and now assumes an even greater importance and a new vitality.

Compared with the pure mathematician, the applied mathematician is more interested in problems coming from other fields. Compared with the engineer and the physical scientist, the applied mathematician is more concerned with the formulation of problems and the nature of solutions. Compared with the computer scientist, the applied mathematician is more concerned with the accuracy of approximations and the interpretation of results. Needless to say, even in this age of specialization, the work of mathematicians, scientists, and engineers frequently overlaps. Applied mathematics, by its very nature, has occupied a central position in this interplay and has remained a field of fascination and excitement for active minds.

The Applied Analysis specialty includes research on analytical and numerical partial differential equations; mathematical foundations of data analytics, uncertainty quantification, stochastic analysis; large-scale scientific computation; fluid dynamics and continuum mechanics; dynamical systems and chaos; and applications to various fields of data and physical and life sciences.

The Atmospheric, Oceanic, and Earth Physics specialty includes research on dynamics of the atmosphere and the ocean; climate modeling; cloud physics; radiation transfer; remote sensing; geophysical/geological fluid dynamics; and geochemistry.

Courses:

**APMA E4001:** Principles of applied mathematics

**APMA E4101:** Intro to dynamical systems

**APMA E4150:** Applied functional analysis

**APMA E4200:** Partial differential equations

**APMA E4204:** Functions of a complex variable

**APMA E4300:** Intro to numerical methods

**APMA E4301:** Numerical methods for partial differential equations

**APMA E4302:** Intro to computational science

**APMA E4400:** Intro to biophysical modeling

**APMA E6301:** Analytic methods for partial differential equations

**APMA E6302:** Numerical analysis for partial differential equations

Related courses of specialization:

**APMA E6901:** Special topics in applied mathematics

**APMA E8308:** Asymptotic methods in applied mathematics

**APPH E4210:** Geophysical fluid dynamics

**APMA E9815:** Geophysical fluid dynamics seminar

**COMS E4205:** Combinatorial theory

**COMS E4241:** Numerical algorithms and complexity

**EESC GU4008:** Introduction to atmospheric science

**PHYS GU4019:** Mathematical methods in physics

**MATH GU4032:** Fourier analysis

### Materials Science and Engineering Program

For more information click **here**.

### Applied Physics

This doctoral program has three specialties. APPH E4018 Applied physics laboratory is required for each of the three specialties in the first year of the doctoral program, in addition to any specific courses required of each specialty.

#### Plasma Physics

This graduate specialty is designed to emphasize preparation for professional careers in plasma research, controlled fusion, and space research. This includes basic training in relevant areas of applied physics, with emphasis on plasma physics and related areas leading to extensive experimental and theoretical research in the Columbia University Plasma Physics Laboratory. Specific course requirements for the plasma physics doctoral program are *APPH E4018, E4200, E4300, E6101, E6102,* and *E9142 *or *E9143*, or equivalents taken at another university.

#### Optical and Laser Physics

This graduate specialty involves a basic training in relevant areas of applied physics with emphasis in quantum mechanics, quantum electronics, and related areas of specialization. Some active areas of research in which the student may concentrate are laser modification of surfaces, optical diagnostics of film processing, inelastic light scattering in nanomaterials, nonlinear optics, ultrafast optoelectronics photonic switching, optical physics of surfaces, and photon integrated circuits. Specific course requirements for the optical and laser physics doctoral specialization are set with the academic adviser.

#### Solid-State Physics

This graduate specialty encompasses the study of the electrical, optical, magnetic, thermal, high-pressure, and ultrafast dynamical properties of solids, with an aim to understanding them in terms of the atomic and electronic structure. The field emphasizes the formation, processing, and properties of thin films, low-dimensional structures—such as one- and two-dimensional electron gases, nanocrystals, surfaces of electronic and optoelectronic interest, and molecules. Facilities include a microelectronics laboratory, high-pressure diamond anvil cells, a molecular beam epitaxy machine, ultrahigh vacuum systems, lasers, equipment for the study of optical properties and transport on the nanoscale, and the instruments in the shared facilities overseen by the Columbia Nano Initiative (CNI). There are also significant resources for electrical and optical experimentation at low temperatures and high magnetic fields. Specific course requirements for the solid-state physics doctoral specialization are set with the academic adviser, in consultation with the Committee on Materials Science and Engineering/Solid-State Science and Engineering.