• A Message From The Deans
• Mission of the School
• About the School and University
• Undergraduate Studies
• Graduate Studies
• Faculty Administration
• Departments and Academic Programs
• Undergraduate Minors
• Interdisiplinary Courses and Courses in Other Divsions of the University
  • Interdisciplinary Engineering Courses
  • Courses In Other Divisions of the University
• Campus and Student Life
• Scholarships, Fellowships, Awards, and Prizes
• University and School Policies Procedures, and Regulations
• Directory of University Resources

Courses In Other Divisions of the University

This listing of courses has been selected with specific engineering program requirements in mind. For information on these courses and additional courses offered by these departments, please consult the bulletins of Columbia College, the School of Continuing Education, the School of General Studies, and the Graduate School of Arts and Sciences.

Biological Sciences

BIOL C2005x Introductory biology, I: biochemistry, genetics, and molecular biology
Lect: 3. Recit: 2. 4 pts. L. Chasin and D. Mowshowitz.
Prerequisite: One year of college chemistry, or a strong high school chemistry background. Recommended introductory biology course for biology and related majors and for premedical students. Fundamental principles of biochemistry, molecular biology, and genetics.

BIOL C2006y Introductory biology, II: cell biology, development, and physiology
Lect: 3. Recit: 2. 4 pts. D. Mowshowitz.
Prerequisite: ENVB W2001 or BIOL C2005 or the instructor’s permission. The recommended second term of biology for biology and related majors and for premedical students. Cellular biology and development; physiology of cells and organisms.

BIOL W2501x or y Contemporary biology
laboratory
Lab: 4. 3 pts. C. Hazen.
Each section limited to twenty-four students. Early registration is advised. Students must come to the first day of class to secure their place. Strongly recommended prerequisite or required corequisite: BIOL C2005 or F2401. Laboratory fee: $150. Emphasis on experimental techniques and data analysis in a variety of biological disciplines.

BIOC C3501 Biochemistry: structure and metabolism
Lect: 3. Recit: 1. 4 pts. B. Stockwell and L. Tong.
Prerequisites: One year of college-level biology and one year of organic chemistry, or the instructor’s permission. Biochemistry, organic chemistry, and structural biology. Structure and function of both proteins and small molecules in biological systems. The first half of the course covers protein structure and enzyme kinetics. The second half of the course will focus on the organic chemistry involved in metabolic pathways.

Business

BUSI W3021x and y Introduction to marketing management
3 pts. Instructor to be announced.
No previous background in marketing is required for the course. Introduction to the basic concepts of marketing. Students develop an understanding of, and the decision-making capabilities for, formulating marketing strategies for the complex situations that characterize real-life marketing problems.   

Chemistry

Courses of Instruction

Pre-engineering students should refer to the First Year–Sophomore Program to determine the chemistry requirements for admission to particular Junior-Senior Programs. Special attention should be given to the requirements for admission to chemical engineering, biomedical engineering, materials science and metallurgical engineering, and other related fields.

Laboratory Fee

The laboratory fee covers the cost of nonreturnable items, chemicals, and reasonable breakage. In addition, students may be charged for lab handouts and excessive breakage, for cleaning of equipment returned dirty, and for checking out late.

CHEM C1403x-C1404y General chemistry
Lect: 3.5 pts. L. Brus, R. Friesner, R. Gonzalez, G. Parkin, and J. Valentini.
Only students with scheduling conflicts need report to the Chemistry Department (340 Havemeyer) during registration. Preparation equivalent to one year of high school chemistry is assumed and concurrent registration in MATH V1101. Students lacking such preparation should plan independent study of chemistry over the summer or take CHEM F0001 before taking C1403. Topics include stoichiometry, states of matter, chemical equilibria, acids and bases, chemical thermodynamics, nuclear properties, electronic structures of atoms, periodic properties, chemical bonding, molecular geometry, introduction to organic and biological chemistry, solid-state and materials science, polymer science and macromolecular structures, chemical kinetics, coordination chemistry, and electrochemistry. Although C1403 and C1404 are separate courses, students are expected to take currently in CHEM C2507, the intensive general chemistry laboratory course. Recitation section required.

CHEM C1500x or y General chemistry laboratory
Lab: 3 pts. S. Hansen and J. Ulichny.
Prerequisite or corequisite: CHEM C1403. Fee $140. An introduction to basic techniques of modern experimental chemistry, including quantitative procedures and chemical analysis.

CHEM C1604x Second semester general chemistry (intensive)
Lect: 3.5 pts. G. Flynn
Prerequisite: A grade of B or better in CHEM C1403 or F1403 or acceptable performance on the department placement exam. Corequisite: MATH V1102. Topics include gases (kinetic theory of gases); binary collision model for chemical reactions; chemical kinetics; acid-base equilibria; thermochemistry (thermodynamics I); spontaneous processes (thermodynamics II); chemical bonding in polyatomic molecules. Recitation section required.

CHEM C2507y Intensive general chemistry laboratory
Lab: 3 pts. L. Avila.
Prerequisite: CHEM C1604 or C3045 and the instructor’s permission. Fee: $140. An introduction to basic techniques and practices of modern experimental chemistry, including qualitative procedures and chemical analysis. This course differs from CHEM C1500 in its emphasis on instrumentation and methods.

HEM C3045x-C3046y Intensive organic chemistry for first-year students (lecture)
Lect: 3.5 pts. R. Breslow and J. Leighton.
Prerequisite: A grade of 5 on the Chemistry Advanced Placement Examination and an acceptable grade on the department placement exam.  Not open to students who have taken other courses in college-level chemistry. Premedical students may take CHEM C3045, C3046, and C3545 to meet the minimum requirements for admission to medical school. This course covers the same material as CHEM C3443-C3444 but is intended for students who have learned the principles of general chemistry in high school. The level of instruction will be appropriate for those who have not had a college course in general chemistry. Students enrolled in CHEM C3045-C3046 are expected to enroll concurrently in CHEM C2507, the intensive general chemistry laboratory course. Recitation section required

CHEM C3071y Introduction to inorganic chemistry
Lect: 3 pts. Not given in 2009–2010.
Prerequisite: CHEM C3444 (or F3444) or C3046. Principles governing the structure and reactivity of inorganic compounds surveyed from experimental and theoretical viewpoints. Topics include inorganic solids, aqueous and nonaqueous solutions, the chemistry of selected main group elements, transition metal chemistry, metal clusters, metal carbonyls, and organometallic chemistry.

CHEM C3079x-C3080y Physical chemistry, I and II
Lect: 4 pts. A. Cacciuto and D. Reichman.
Prerequisites: CHEM C1403-C1404 or C3045-C3046; PHYS C1401-C1402, or the equivalent; MATH V1101-V1102 or V1207-V1208. Recommended parallel: CHEM C3085-C3086. Elementary but comprehensive treatment of the fundamental laws governing the behavior of individual atoms and molecules and collections of them. C3079: The thermodynamics of chemical systems at equilibrium and the chemical kinetics of nonequilibrium systems. C3080: The quantum mechanics of atoms and molecules, the quantum statistical mechanics of chemical systems and the connection of statistical mechanics to thermodynamics. Recitation section required.

CHEM C3085x-C3086y Physical and analytical chemistry laboratory
Lab: 4 pts. L. Avila.
Prerequisite or corequisite: CHEM C3079-C3080. C3085 is prerequisite to C3086. Fee: $125 per term. Techniques of experimental physical chemistry and instrumental analysis, including infrared and ultraviolet spectrophotometry, magnetic resonance, electroanalytical methods, calorimetry, reaction kinetics, hydrodynamic methods, and applications of digital computers to the analysis of experimental data.

CHEM C3098x and y Supervised independent research
Lab: 4 pts. J. Valentini.
Prerequisite: Tthe permission of the professor in charge for entrance, and the permission of the departmental representative for aggregate points in excess of 12 or less than 4. Laboratory fee: $105 per term. This course may be repeated for credit (see major and concentration requirements). Individual research under the supervision of a member of the staff. Research areas include organic, physical, inorganic, analytical, and biological chemistry.

CHEM C3443x-C3444y Organic chemistry (lecture)
Lect: 3.5 pts. D. Sames and S. Snyder.
Prerequisites: CHEM C1404 (or C1604) and C1500 or their equivalents. The principles of organic chemistry. The structure and reactivity of organic molecules are examined from the standpoint of modern theories of chemistry. Topics include stereochemistry, reactions of organic molecules, mechanisms of organic reactions, syntheses and degradations of organic molecules, and spectroscopic techniques of structure determination. Recitation section required.

CHEM C3543x and y Organic chemistry (laboratory)
Lab: 3 pts. A. Ghurbanyan.
Prerequisite: CHEM C1500. Corequisite: CHEM C3443. Students planning to take a full year of laboratory should enroll in CHEM C3543 and C3546. Laboratory fee: $125. Techniques of experimental organic chemistry, with emphasis on understanding fundamental principles underlying the experiments and methodology of solving laboratory problems involving organic molecules.

CHEM C3545x Organic chemistry (laboratory)
Lab: 3 pts. A. Ghurbanyan.
Prerequisites: CHEM C3045, C3046, and C2407. Laboratory fee: $125. The course covers the same material as CHEM C3543, but is intended for those students who have taken CHEM C3045-C3046.

CHEM C3546y Advanced organic chemistry (laboratory)
Lab: 3 pts. A. Ghurbanyan.
Prerequisite: CHEM C3543 or C3545. Corequisite: C3444. Laboratory fee: $125.
A project laboratory with emphasis on complex synthesis and advanced techniques, including qualitative organic analysis and instrumentation.

Earth and Environmental Sciences

Undergraduates in the four-year course of study in the School of Engineering and Applied Science may take courses numbered up to 4999 but may enter courses of higher numbers only if

1. the course is expressly included in the prescribed curriculum or
2. special permission is obtained from the Department of Earth and Environmental Sciences.

EESC V1011x Earth, origin, evolution, processes, future
Lect: 3. Lab: 3.4 pts. Professors Mutter and Nettles.
Students who wish to take only the lectures should register for EESC V1411. What is the nature of our planet and how did it form? From geochemical and geophysical perspectives we explore Earth’s internal structure, its dynamical character expressed in plate tectonics, and ask if its future behavior can be known.

EESC V1030x Oceanography
3 pts. Professor Hoenisch.
Explore the geology of the sea floor, understand what drives ocean currents and how ocean ecosystems operate. Case studies and discussions centered on ocean-related issues facing society.

EESC V1201y Environmental risks and disasters
3 pts. Professor Ekstrom.
Prerequisites: High-school science and mathematics. An introduction to risks and hazards in the environment. Different types of hazards are analyzed and compared: natural disasters, such as tornados, earthquakes, and meteorite impacts; acute and chronic health effects caused by exposure to radiation and toxic substances such as radon, asbestos, and arsenic; long-term societal effects due to environmental change, such as sea level rise and global warming. Emphasizes the basic physical principles controlling the hazardous phenomena and develops simple quantitative methods for making scientifically reasoned assessments of the threats (to health and wealth) posed by various events, processes, and exposures. Discusses methods of risk mitigation and sociological, psychological, and economic aspects of risk control and management.

EESC W3018y Weapons of mass destruction
3 pts. Professor Richards.
Prerequisite: one semester of a lab science or permission of the instructor. A review of the history and environmental consequences of nuclear, chemical, and biological weapons of mass destruction (WMD); of how these weapons work, what they cost, how they have spread, how they might be used, how they are currently controlled by international treaties and domestic legislation, and what issues of policy and technology arise in current debates on WMD. What aspects of the manufacture of WMD are easily addressed, and what aspects are technically challenging? It may be expected that current events/headlines will be discussed in class.

EESC W4001x Advanced general geology
3 pts. Lab: 3. 4 pts. Professors Scholz and Anders.
Prerequisite: One semester of college-level calculus, physics, and chemistry. Fee: $35. A concentrated introduction to the solid Earth, its interior, and near-surface geology. Intended for students with good backgrounds in the physical sciences but none in geology. Laboratory and field trips.

EESC W4008x Introduction to atmospheric science
Lect: 3. 3 pts. Professor Del Genio.
Prerequisite: Advanced calculus and general physics, or the instructor’s permission. Basic physical processes controlling atmospheric
structure: thermodynamics; radiation physics
and radiative transfer; principles of atmospheric dynamics; cloud processes; applications to Earth’s atmospheric general circulation, climatic variations, and the atmospheres of the other planets.

EESC W4009x. Chemical geology
4 pts. Offered in alternate years. Professor Walker.
Prerequisite: Physical chemistry or the instructor’s permission. Thermodynamics as applied to earth systems.

EESC W4050x Global assessment and monitoring using remote sensing
3 pts. Offered in alternate years. Professor Small.
Prerequisite: permission of the instructors. Recommended preparation: some college-level physics or math. Enrollment limited to 24 students. General introduction to fundamentals of remote sensing and image processing. Example applications in the Earth and environmental
sciences are explored through the analysis of remote sensing imagery in a state-of-the-art
visualization laboratory. Lab required.

EESC W4076y Geologic mapping
3 pts. Professors Walker and Anders.
Field work on weekends in April and for two weeks in mid-May, immediately following the end of examinations. Estimated expenses: $250. The principles and practice of deciphering geologic history by observing rocks in the field, making geological maps, constructing geological cross-sections, and writing short reports.

EESC W4085x Geodynamics
3 pts. Offered in alternate years. Professor Buck.
Prerequisites: Calculus, differential equations, introductory physics. Physical processes that control plate tectonics and the evolution of planetary interiors and surfaces; analytical descriptions of these processes; weekly physical model demonstrations.

EESC W4113x Introduction to mineralogy
3. Lab: 3. 4 pts. Offered in alternate years. Professor Walker.
Prerequisites: Introductory geology or the equivalent and elementary college physics and chemistry, or the instructor’s permission. Elementary crystallography and crystal structures, optical properties of minerals, mineral associations and phase equilibria, economic minerals. Laboratory: identification of minerals in hand specimens, chemical and physical tests, and use of the petrographic microscope.

EESC W4230y Crustal deformation
3 pts. Professors Anders and Scholz.
Prerequisites: Introductory geology and one year of calculus. Recommended preparation: higher levels of mathematics. Introduction to the deformation processes in the Earth’s crust. Fundamental theories of stress and strain; rock behavior in both brittle and ductile fields; earthquake processes; ductile deformation; large-scale crustal contractional and extensional events.

EESC W4300x The Earth’s deep interior
3 pts. Professor Ekstrom.
Prerequisites: Calculus, differential equations, one year of college physics, and EESC W4950 or its equivalent. An introduction to properties of the Earth’s mantle, fluid outer core, and solid inner core. Current knowledge of these features is explored, using observations of seismology, heat flow, gravity, and geomagnetism, plus information on the Earth’s bulk composition.

EESC W4701x or y. Introduction to igneous petrology
4 pts. Offered in alternate years. Professor Kelemen.
Prerequisites: EESC V1011-V1012 or the equivalent. Recommended preparation: EESC W4113 and knowledge of chemistry. Fee: $15. Students not enrolled in terrestrial geology may elect to write a substantial term paper in lieu of the laboratory course. Compositional characteristics of igneous and metamorphic rocks and how they can be used as tools to investigate earth processes. Development of igneous and metamorphic rocks in a plate-tectonic framework.

EESC W4885y The chemistry of continental waters
3 pts. Offered in alternate years. Instructors to be announced.
Recommended preparation: A solid background in basic chemistry. Introduction to geochemical cycles involving the atmosphere, land, and bio-sphere; chemistry of precipitation, weathering reactions, rivers, lakes, estuaries, and ground-waters; stable isotopes and radioactive tracers of transport processes in continental waters.

EESC W4924y Introduction to atmospheric chemistry
3 pts. Offered in alternate years. Professor Shindell.
A survey of trace gas photochemistry important
in the Earth’s atmosphere. Major topics are composition, including biogenic and anthropogenic inputs, and chemical processes, including reaction kinetics and photochemistry. Specific applications to tropospheric air quality, including smog, acid rain, and stratospheric ozone, including the Antarctic ozone hole, are covered, with an emphasis on the response to anthropogenic pollutants and climate change.

EESC W4925x Principles of physical oceanography
3 pts. Professor Gordon.
Recommended preparation: A solid background in mathematics, physics, and chemistry. Physical properties of seawater, water masses and their distribution, sea-air interaction influence on the ocean structure, basic ocean circulation pattern, relation of diffusion and advection with respect to distribution of ocean properties, and introduction to ocean dynamics.

EESC W4926y. Principles of chemical oceanography
3 pts. Offered in alternate years.
Professors Anderson and Hoenisch.
Recommended preparation: A solid background in mathematics, physics, and chemistry. Given in alternate years. Factors controlling the concentration and distribution of dissolved chemical species within the sea. Application of tracer and natural radioisotope methods to large-scale mixing of the ocean, the geological record preserved in marine sediments, the role of ocean processes in the global carbon cycle, and biogeochemical processes influencing the distribution and fate of elements in the ocean.

EESC W4930y Earth’s oceans and atmosphere
3 pts. Professor Gordon.
Recommended preparation: A good background in the physical sciences. Physical properties of water and air. Overview of the stratification and circulation of Earth’s oceans and atmosphere and their governing processes; ocean-atmosphere interaction; resultant climate system; natural and anthropogenic forced climate change.

EESC W4941x or y Principles of geophysics
3 pts. Offered in alternate years. Instructor to be announced.
Prerequisite: Calculus through MATH V1202 and physics through PHYS C1007. The structure and properties of the Earth as inferred from geophysical investigations: gravity, isostasy, earthquakes, seismic exploration, geomagnetism, marine geophysics, satellite observations, tides. Recommended for nongeophysics majors or those with little previous geophysics background.

EESC W4947y Plate tectonics
3 pts. Professor Abers.
Prerequisites: Physical geology. Prepares students for research and oral exams. Evolution of the interiors and surfaces of Earth, Venus, Mars and the moons of Jupiter. Planetary accretion, tidal heating, convection, magma oceans, formation of continents, mantle plumes, sea-floor spreading, kinematics of triple junctions, surface repaving, subduction, sedimentation, catastrophic impacts and floods, and the building of mountain chains.

EESC W4949x Introduction to seismology
3 pts. Offered in alternate years. Professors Ekstrom and Menke.
Prerequisites: Elementary college physics and mathematics (including calculus). Basic methods of seismogram analysis. Classification of seismic waves and elementary theory of body waves and normal modes. Elementary aspects of seismic prospecting, earthquake source theory, instrumentation, discrimination between explosions and earthquakes, inversion of seismic data to infer Earth structure, earthquake engineering, earthquake insurance and hazards mitigation, estimation of seismic risk, and earthquake prediction.  

Humanities and Social Sciences

For listings of additional courses of interest to engineering students, consult
the bulletins of Columbia College; the School of General Studies; the Graduate School of Architecture, Planning, and Preservation; the Graduate School of Business; and the Graduate School of Arts and Sciences.

COCI C1101-C1102 Introduction to contemporary civilization in the West
4 pts.
Taught by members of the Departments of Anthropology, Architecture, Classics, English and Comparative Literature, French, German, History, Italian, Journalism, Middle East and Asian Languages and Cultures, Philosophy, Political Science, Religion, Slavic Languages, Sociology, and Spanish; members of the Society of Fellows in the Humanities; and Senior Scholars. Major works by over twenty authors, ranging from Plato to modern writers. Students are expected to write at least three papers, to complete two examination
each semester, and to participate actively in class discussions.

ECON W1105x or y Principles of economics
4 pts. Recitation section required (W1155).
How a market economy determines the relative prices of goods, factors of production, and the allocation of resources, and the circumstances under which it does so efficiently. Why such an economy has fluctuations and how they may be controlled.

ENGL C1010x or y University writing
3 pts. The staff.
Teaches general techniques and strategies for academic reading and writing. Students read and discuss a range of published essays, complete regular reading and writing exercises, write several longer essays, and undertake a collaborative research and writing project designed by the class. Students placed in C1010 whose names fall in the first part of the alphabet must take the course in the fall. Students whose names fall in the second part of the alphabet take the course in the spring. The alphabet will be split somewhere between K and O. The exact place for the split will be posted before fall registration.

Global Core
The Global Core requirement consists of courses that examine areas not the primary focus of Literature Humanities and Contemporary Civilization and that, like other Core courses, are broadly introductory, interdisciplinary, and temporally or spatially expansive. Courses in the Global Core are organized around a set of primary texts or artifacts, which may range from texts of literate traditions to media (e.g. film), ritual performances or oral sources, produced in the regions of the world in question. Global Core courses fall into two categories: those that focus on a specific culture or civilization, tracing its appearance and/or existence across a significant span of time and sometimes across more than one present-day country or region; and those that address several world settings or cultures comparatively (and may include Europe and the West), in terms of a common theme, a set of analytic questions, or interactions between different world regions. Students must complete two courses from the Global Core List of Approved Courses for a letter grade.

HUMA C1001x-C1002y Masterpieces of Western literature and philosophy
4 pts.
Taught by members of the Departments of Classics, English and Comparative Literature, French, German, Italian, Middle East and Asian Languages and Cultures, Philosophy, Religion, Slavic Languages, and Spanish; and members of the Society of Fellows in the Humanities. Major works by over twenty authors, ranging in time, theme, and genre from Homer to Virginia Woolf. Students are expected to write at least two papers, to complete two examinations each semester, and to participate actively in class discussions.

HUMA W1121x or y Masterpieces of Western art
3 pts.
Popularly known as “Art Hum,” this course teaches students how to look at, think about, and engage in critical discussion of the visual arts. Not a historical survey, but an analytical study of a limited number of monuments and artists ranging from early Athens to the present, the course focuses on the formal structure of works of architecture, sculpture, painting, and other media, as well as the historical contexts in which these works were made and understood.

HUMA W1123x or y Masterpieces of Western music
3 pts.
Popularly known as “Music Hum,” this course aims to instill in students a basic comprehension of the many forms of the Western musical imagination. The course involves students actively in the process of critical listening, both in the classroom and in concerts. Although not a history of Western music, the course is taught in chronological format and includes masterpieces by Josquin des Prez, Monteverdi, Bach, Handel, Mozart, Haydn, Beethoven, Verdi, Wagner, Schoenberg, Stravinsky, Louis Armstrong, and Duke Ellington, among others.  

Mathematics

Courses for First-Year Students

Depending on the program, completion of Calculus III or IV satisfies the basic mathematics requirement. Normally students who have taken an AP Calculus course begin with either Calculus II or Calculus III. Refer to the AP guidelines on page 14 for placement information. The sequence ends with MATH E1210: Ordinary differential equations.

Students who wish to transfer from one calculus course to another are allowed to do so beyond the date specified on the Academic Calendar. They are considered to be adjusting their level, not changing their program. They must, however, obtain the approval of the new instructor and the Center for Student Advising before reporting to the Registrar.  

MATH V1101 Calculus I
Lect: 3 pts.
Functions, limits, derivatives, introduction to
integrals.

MATH V1102 Calculus II
Lect. 3 pts.
Prerequisite: Calculus I or the equivalent. Methods of integration, applications of integrals, series, including Taylor’s series.

MATH V1201 Calculus III
Lect. 3 pts.
Prerequisite: Calculus II or the equivalent. Vector algebra, complex numbers and exponential,
vector differential calculus.

MATH V1202 Calculus IV
Lect: 3 pts.
Prerequisite: Calculus II and III. Multiple integrals, line and surface integrals, calculus of vector fields, Fourier series.

MATH V1207x-V1208y Honors math A-B
Lect. and recit. 4 pts. M. Thaddeus.
Prerequisite: Score of 5 on the Advanced Placement BC calculus exam. The second term of this course may not be taken without the first. Multivariable calculus and linear algebra from a rigorous point of view.

MATH E1210x or y Ordinary differential equations
Lect: 3 pts. T. Perutz.
Prerequisite: MATH V1201 or the equivalent. Special differential equations of order one.
Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications. 

MATH V2010 x and y Linear algebra
Lect: 3 pts. Prerequisite: MATH VI201 or the equivalent. Vector spaces, linear transformations, matrices, quadratic and hermitian forms, reduction to canonical forms. 

MATH V2500y Analysis and optimization
Lect: 3 pts. H. Pinkham.
Prerequisites: MATH V1102 and V1201 or the equivalent, and MATH V2010. Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control.

MATH V3007y Complex variables
Lect: 3 pts. Chiu-chu Liu.
Prerequisite: MATH V1202. An elementary course in functions of a complex variable. Fundamental properties of the complex numbers, differentiability, Cauchy-Riemann equations, Cauchy integral theorem, Taylor and Laurent series, poles, and essential singularities. Residue theorem and conformal mapping.

MATH V3027x Ordinary differential equations
Lect: 3 pts. P. Daskalopoulos.
Prerequisite: MATH V1201 or the equivalent. Equations of order one, linear equations, series solutions at regular and singular points, boundary value problems. Selected applications.

MATH V3028y Partial differential equations
Lect: 3 pts. P. Daskalopoulos.
Prerequisite: MATH V3027 or the equivalent. Introduction to partial differential equations.
First-order equations. Linear second-order equations, separation of variables, solution by series expansions. Boundary value problems.

MATH W4032x Fourier analysis
Lect: 3 pts. M. Lipyanskiy.
Prerequisite: MATH V1201 and linear algebra, or MATH V1202. Fourier series and integrals, discrete analogues, inversion and Poisson summation, formulae, convolution, Heisenberg uncertainty principle. Emphasis on the application of Fourier analysis to a wide range of disciplines.

MATH W4041x-W4642y Introduction to modern algebra
Lect: 3 pts. P. Gallagher.
The second term of this course may not be taken without the first. Prerequisite: MATH V1202 and V2010 or the equivalent. Groups, homomorphisms, rings, ideals, fields, polynominals, and field extensions. Galois theory.

MATH W4061x-W4062y Introduction to modern analysis
Lect: 3 pts. D. De Silva.
The second term of this course may not be taken without the first. Prerequisite: MATH V1202 or the equivalent. Real numbers, metric spaces, elements of general topology. Continuous and differentiable functions. Implicit functions. Integration, change of variables. Function spaces. Further topics chosen by the instructor.

MATH W4065x Honors complex variables
Lect: 3 pts. K. Tignor.
Prerequisite: MATH V1207, V1208, or W4061.
A theoretical introduction to analytic functions. Holomorphic functions, harmonic functions, power series, Cauchy-Riemann equations, Cauchy’s integral formula, poles, Laurent series, residue theorem. Other topics as time permits: elliptic functions, the gamma and zeta functions, the Riemann mapping theorem, Riemann surfaces, Nevanlinna theory.

Physics

The general four-term pre-engineering physics sequence consists of PHYS C1401, C1402, C1403, and C1494 (laboratory); or PHYS C1601, C1602, C2601, and C2699 (laboratory).

PHYS C1401x Introduction to mechanics and thermodynamics
Lect: 3 pts. Professors Dodd and Hughes.
Corequisite: MATH V1101 or the equivalent. Fundamental laws of mechanics, kinematics and dynamics, work and energy, rotational dynamics, oscillations, gravitation, fluids, temperature and heat, gas laws, the first and second laws of thermodynamics.

PHYS C1402y Introduction to electricity, magnetism, and optics
Lect: 3 pts. Professors Dodd and Hughes.
Prerequisite: PHYS C1401. Corequisite: MATH V1102 or the equivalent. Electric fields, direct currents, magnetic fields, alternating currents, electromagnetic waves, polarization, geometrical optics, interference and diffraction. 

PHYS C1403x Introduction to classical and quantum waves
Lect: 3 pts. Professor Brooijmans.
Prerequisite: PHYS C1402. Corequisite: MATH V1201 or the equivalent. Classical waves and the wave equation, Fourier series and integrals, normal modes, wave-particle duality, the uncertainty principle, basic principles of quantum mechanics, energy levels, reflection and transmission coefficients, applications to atomic physics.

PHYS C1493x Introduction to experimental physics
Lab and lecture: 3 pts. Lect: Tues., 3:10–4:00 p.m. Lab: 3 hours weekly to be arranged. Instructor to be announced.
Prerequisites: PHYS C1401 and C1402. Laboratory work associated with the two prerequisite lecture courses. Experiments in mechanics, thermodynamics, electricity, magnetism, optics, and wave motion. (Students cannot receive credit for both PHYS C1493 and C1494.)

PHYS C1494y Introduction to experimental physics
Lab and lecture: 3 pts. Lect: Tues., 3:10–4:00 p.m.  Lab: 3 hours weekly to be arranged. Instructor to be announced.
Prerequisites: PHYS C1401, C1402, and C1403. Laboratory work associated with the three prerequisite lecture courses. Experiments in mechanics, thermodynamics, electricity, magnetism, optics, wave motion, atomic and nuclear physics. (Students cannot receive credit for both PHYS C1493 and C1494.)

PHYS C1601x Physics, I: mechanics and relativity
Lect: 3.5 pts. Rec: 1 hour weekly to be arranged. Professor Christ.
Corequisite: MATH V1102 or the equivalent. Fundamental laws of mechanics, kinematics and dynamics, work and energy, rotational dynamics, oscillations, gravitation, fluids, introduction to special relativity and relativistic kinematics. The course is preparatory for advanced work in physics and related fields.

PHYS C1602y Physics, II: thermodynamics, electricity, and magnetism
Lect: 3.5 pts. Professor Weinberg.
Prerequisite: PHYS C1601. Corequisite: MATH V1201 or the equivalent. Temperature and heat, gas laws, the first and second laws of thermodynamics, kinetic theory of gases, electric fields, direct currents, magnetic fields, alternating currents, electromagnetic waves. The course Is preparatory for advanced work in physics and related fields.

PHYS C2601x Physics, III: classical and quantum waves
Lect: 3.5 pts. Rec: 1 hour weekly to be arranged. Professor Kim.
Prerequisite: PHYS C1602 or C1402. Corequisite: MATH V1202 or the equivalent. Classical waves and the wave equation, geometrical optics, interference and diffraction, Fourier series and integrals, normal modes, wave-particle duality, the uncertainty principle, basic principles of quantum mechanics, energy levels, reflection and transmission coefficients, the harmonic oscillator. The course is preparatory for advanced work in physics and related fields.

PHYS C2699y Experiments in classical and modern physics
Lab and lecture: 3 pts. Lect: Tues., 3:10–4:00 p.m.  Lab: 3 hours weekly to be arranged. Instructor to be announced.
Prerequisites: PHYS C1601 (or C1401), C1602 (or C1402), and C2601. Laboratory work associated with the three prerequisite lecture courses. Experiments in mechanics, thermodynamics, electricity, magnetism, optics, wave motion, atomic and nuclear physics.

PHYS C2801x-C2802y Accelerated physics, I and II
Lect: 4.5 pts. Rec: 1 hour weekly to be arranged. Professor Cole.
Prerequisite: Advanced placement in physics and mathematics, or the equivalent, and the instructor’s permission. (A special placement meeting is held during Orientation.) This accelerated two-semester sequence covers the subject matter of PHYS C1601, C1602, and C2601 and is intended for students who have an exceptionally strong background in both physics and mathematics. The course is preparatory for advanced work in physics and related fields. There is no accompanying laboratory; however, students are encouraged to take the intermediate laboratory, PHYS W3081, in the following year.

PHYS W3002y From quarks to the cosmos: applications of modern physics
Lect: 3.5 pts. Professor Marka.
Prerequisites: PHYS C2601 or C2802. This course reinforces basic ideas of modern physics through applications to nuclear physics, high-energy physics, astrophysics, and cosmology. The ongoing Columbia research programs in these fields are used as practical examples. The course is preparatory for advanced work in physics and related fields.

PHYS W3003x Mechanics
Lect: 3 pts. Professor Blaer.
Prerequisite: General physics; differential and integral calculus. Newtonian mechanics, oscillations and resonance, conservative forces and potential energy, central forces, noninertial frames of reference, rigid body motion, an introduction to Lagrange’s formulation of mechanics, coupled oscillators, and normal modes.

PHYS W3007y Electricity and magnetism
Lect: 3 pts. Professor Nicolis.
Prerequisite: General physics; differential and integral calculus. Electrostatics and magnetostatics, Laplace’s equation and boundary-value problems, multipole expansions, dielectric and magnetic materials, Faraday’s law, AC circuits, Maxwell’s equations, Lorentz covariance, and special relativity.

PHYS W3008x Electromagnetic waves and optics
Lect: 3 pts. Professor Marka.
Prerequisite: PHYS W3007. Maxwell’s equations and electromagnetic potentials, the wave equation, propagation of plane waves, reflection and refraction, geometrical optics, transmission lines, wave guides, resonant cavities, radiation, interference of waves, and diffraction.

PHYS W3081x or y Intermediate laboratory work
Lab: 2 pts. Professors May and Aprile.
Primarily for junior and senior physics majors. Other majors require the instructor’s permission. May be repeated for credit by performing different experiments. The laboratory has 13 individual experiments available, of which two are required per 2 points. Each experiment is chosen by the student in consultation with the instructor. Each section meets one afternoon per week, with registration in each section limited by the laboratory capacity. Experiments (classical and modern) cover topics in electricity, magnetism, optics, atomic physics, and nuclear physics.

PHYS W3083y Electronics laboratory
Lab: 3 pts. Professor Parsons.
Registration is limited to the capacity of the laboratory. Corequisite or prerequisite: PHYS W3003 or W3007. A sequence of experiments in solid-state electronics, with introductory lectures.

PHYS G4003y Advanced mechanics
Lect: 3 pts. Professor Pontón.
Prerequisite: Differential and integral calculus,
differential equations, and PHYS W3003 or the equivalent. Lagrange’s formulation of mechanics, calculus of variations and the Action Principle, Hamilton’s formulation of mechanics, rigid body motion, Euler angles, continuum mechanics, Introduction to chaotic dynamics.

PHYS G4018y Solid-state physics
Lect: 3 pts. Professor Uemura.
Prerequisites: PHYS G4021 and G4023, or the equivalent. Introduction to solid-state physics: crystal structures, properties of periodic lattices, electrons in metals, band structure, transport properties, semiconductors, magnetism, and superconductivity.

PHYS G4019x Mathematical methods of physics
Lect: 3 pts. Professor Halpin-Healy.
Prerequisite: Differential and integral calculus. Highlights of complex analysis, differential equations, integral equations, Green’s functions,
special functions, Fourier and other transforms, approximation methods, group theory and representations, differential geometry and manifolds. Emphasis is placed on applications to physical problems.

PHYS G4021x-G4022y Quantum mechanics, I and II
Lect: 3 pts. Professor Mueller.
Prerequisite: PHYS C2601 or C2802, or the equivalent. The formulation of quantum mechanics in terms of state vectors and linear operators, three-dimensional spherically symmetric potentials, the theory of angular momentum and spin, time-independent and time-dependent perturbation theory, scattering theory, identical particles. Selected phenomena from atomic physics, nuclear physics, and elementary particle physics are described and then interpreted using quantum mechanical models.

PHYS G4023x Thermal and statistical physics
Lect: 3 pts. Professor Heinz.
Prerequisite: PHYS G4021 or the equivalent. Thermodynamics, kinetic theory, and methods
of statistical mechanics; energy and entropy; Boltzmann, Fermi, and Bose distributions; ideal and real gases; blackbody radiation; chemical equilibrium; phase transitions; ferromagnetism.

PHYS G4040x General relativity
Lect: 3 pts. Not given in 2009–2010.
Prerequisites: PHYS G4003 and W3007. Tensor algebra, tensor analysis, introduction to Riemann geometry. Motion of particles, fluid, and fields in curved spacetime.  Einstein equation. Schwarzschild solution; test-particle orbits and light bending.  Introduction to black holes, gravational waves, and cosmological models.  

Statistics

Engineering students interested in a survey of the mathematical theory of probability and statistics should consider the pair STAT W3105: Probability theory and W3107: Statistical inference. Students seeking a quicker overview that focuses more on probability theory should consider SIEO W4150. STAT W4105 and W4107 are the equivalent of W3105 and W3107, respectively; but graduate students may not register for W3105 and W3107. STAT W4109 (6 pts) covers the same material as W3105 and W3107 in a single semester. STAT W4315: Linear regression models takes W3105 and SEAS 2009–2010 W3107 as prerequisites; like other advanced offerings in statistics, it covers both theory and practical aspects of modeling and data analysis. Advanced offerings in probability theory, stochastic processes, and mathematical finance generally take STAT W3105 as a prerequisite; advanced offerings in statistical theory and methods generally take STAT W4107 and, in several cases, W4315 as prerequisites; an exception is STAT W4220: Data mining, which has a course in computer programming as prerequisite and STAT W3107 as corequisite. STAT 4201 is a high-level survey of applied statistical methods.

Please note that STAT W3000 has been renumbered as W3105 and STAT W3659 has been renumbered as W3107. For a description of the following course offered jointly by the Departments of Statistics and Industrial Engineering and Operations Research, see ‘‘Industrial Engineering and Operations Research."

Prerequisites: MATH V1101 and V1102 or the equivalent. A quick calculus-based tour of the fundamentals of probability theory and statistical inference. Probabilistic models, random variables, useful distributions, expectations, laws of large numbers, central limit theorem. Statistical inference: point and confidence interval estimation, hypothesis tests, linear regression. Students seeking a more thorough introduction to probability and statistics should consider STAT W3105 and W3107.

Prerequisites: MATH V1101 and V1102 or the equivalent. A calculus-based introduction to probability theory.Topics covered include random variables, conditional probability, expectation, independence, Bayes’ rule, important distributions, joint distributions, moment-generating functions, central limit theorem, laws of large numbers, and Markov’s inequality.

3 pts. Instructor to be announced.
Prerequisite: STAT W3105 or W4105, or the equivalent. Calculus-based introduction to the theory of statistics. Useful distributions, law of large numbers and central limit theorem, point estimation, hypothesis testing, confidence intervals maximum likelihood, likelihood ratio tests, nonparametric procedures, theory of least squares, and analysis of variance.

Prerequisite: A one-term introductory statistics course. This is a course on getting the most out of data. The emphasis will be on hands-on experience, involving case studies with real data and using common statistical packages. The course covers, at a very high level, exploratory data analysis, model formulation, goodness-of-fit testing, and other standard and nonstandard statistical procedures, including linear regression, analysis of variance, nonlinear regression, generalized linear models, survival analysis, time series analysis, and modern regression methods. Students will be expected to propose a data set of their choice for use as case study material.

Prerequisite: COMS W1003, W1004, W1005, W1007, or the equivalent. Corequisite: STAT W3107. Data mining is a dynamic and fast-growing field at the interface of statistics and computer science. The emergence of massive datasets containing millions or even billions of observations provides the primary impetus for the field. Such datasets arise, for instance, in large-scale retailing, telecommunications, astronomy, computational and statistical challenges.This course will provide an overview of current research in data mining and will be suitable for graduate students from many disciplines. Specific topics covered include databases and data warehousing, exploratory data analysis and visualization, descriptive modeling, predictive modeling, pattern and rule discovery, text mining, Bayesian data mining, and causal inference.

Prerequisites: STAT W4105 and W4107. This is a master-level course introducing statistical methodologies in quantitative finance. Financial applications and statistical methodologies are intertwined in all lectures, with several research topics being introduced through problems in a term project. Lecture notes by the instructor will be distributed. The course will cover linear regression with applications to single and multifactor pricing models, multivariate analysis and their applications in Markowitz’s portfolio management, estimation and modeling of volatilities, calculation of value-at-risk, nonparametric methods with applications to option pricing and interest rate markets.

Prerequisites: STAT W3107 or the equivalent, MATH V2110 or the equivalent. Corequisites: MATH V1101, V1102, and V2110. Simple and multiple regression, including testing, estimation and confidence procedures, modeling, regression diagnostics and plots, polynomial regression, fixed effects ANOVA and ANCOVA models, nonlinear regression, multiple comparisons, co-linearity and confounding, model selection. Emphasis on geometric approach to the theory and the use of a statistical package to analyze data.

Statistical methods for rates and proportions, ordered and nominal categorical responses, contingency tables, odds-ratios, exact inference, logistic regression, Poisson regression, generalized linear models.

Introductory course on the design and analysis of sample surveys. How sample surveys are conducted, why the designs are used, how to analyze survey results, and how to derive from first principles the standard results and their generalizations. Discussions include detail surveys from areas including public health, social work, opinion polling, and other topics of interest.

Prerequisite: STAT W4107. Statistical inference without parametric model assumption. Hypothesis testing using ranks, permutations, and order statistics. Semi-parametric analysis of censored survival data. Analysis of discrete outcomes. Contingency tables. Applications.

Prerequisite: STAT W4315 or the equivalent. Leastsquares smoothing and prediction, linear systems, Fourier analysis and spectral estimation. Impulse response and transfer function. Fourier series, the fast Fourier transform algorithm, autocorrelation function, and spectral density. Univariate Box- Jenkins modeling and forecasting. Emphasis on practical applications in examples from the physical sciences, social sciences, and business. Computing is an integral part of the course.

Prerequisite: STAT W4315 or the equivalent. Survival distributions, types of censored data, estimation for various survival models, nonparametric estimation of survival distributions, the proportional hazard and accelerated lifetime models for covariate data, regression analysis with lifetime data. Extensive use of the computer to analyze data. Applications in clinical trials and acturial science.

Prerequisite: STAT W4105 or the equivalent. Review of elements of probability theory. Poisson processes. Exponential distribution. Renewal theory. Wald’s equation. Introduction to discrete time Markov chains and applications to queueing theory, inventory models, branching processes.

Prerequisites: STAT W3105 or the equivalent. This course covers theory of stochastic processes applied to finance. It covers concepts of martingales, Markov chain models, Brownian motion. Stochastic integration, Ito’s formula as a theoretical foundation of processes used in financial modeling. It also introduces basic discrete and continuous time models of asset price evolutions in the context of the following problems in finance: portfolio optimization, option pricing, spot rate interest modeling.

No prerequisite. Introduction to the mathematical theory of interest as well as the elements of economic and financial theory of interest. Topics include rates of interest and discount; simple, compound, real, nominal, effective, dollar (time)- weighted; present, current, future value; discount function; annuities; stocks and other financial instruments; definitions of key terms of modern financial analysis; yield curves; spot (forward) rates; duration; immunization; and short sales. The course will cover determining equivalent measures of interest, discounting, accumulating, determining yield rates, and amortization.