# Ph.D. Model Program: Engineering Sciences

The programs below form a starting point for a discussion with the faculty about areas of interest. Students should work in close consultation with their advisors to develop an appropriate program plan. Courses provide the background knowledge that is often needed to successfully complete research and allow students to learn more broadly about a field or related fields in a structured fashion.

### Bioengineering

This is a diverse and growing field for the Harvard John A. Paulson School of Engineering and Applied Sciences. It encompasses many topics including biomaterials, biomechanics including robotics, biophysics and neuromotor control. *Students should consult their field advisor and Professors Howe or Mooney for guidance in constructing a program in a track other than *Cells, Tissues and Biomaterials.

#### Cells, Tissues, and Biomaterials track

Rationale: Students must achieve graduate-level competence in the following topics:

- Cell biology
- Organ-level physiology
- Chemistry, at least through one year of organic chemistry
- Transport mechanics

**Core Courses**

- ES 222 Advanced Cellular Engineering
- ES 230 Advanced Tissue Engineering
- ES 221 Drug Delivery
- ES 228 Biologically-Inspired Materials

**Depth**

Two courses from the following; at least one must be an Applied Math course.

- AM 201 Physical Mathematics I
- AM 205 Advanced Scientific Computing: Numerical Methods
- AM 121 Introduction to Optimization: Models and Methods
- AP 225 Introduction to Soft Matter
- CS 289 Biologically-inspired Multi-agent Systems
*(formerly CS 266)* - ES 220 Fluid Dynamics
- MCB 212 Topics in Biophysics

**Electives**

Five courses selected for the student’s research and career plans.

- Additional courses from the “Depth” list above; other 200-level technical courses from SEAS, Physics, Applied Physics, Chemistry, Biology; or MIT graduate-level technical courses (subject to CHD approval).
- Up to one “Innovation” style course that broadens a student's perspective (e.g. ES 239), or relevant courses at a suitable level in non-science departments (e.g. technology transfer).
- Up to two may be ES 299r independent study courses, if undertaken with different faculty.
- At least eight of the ten courses must be graduate-level technical courses (200-level or equivalent). At most two courses in the entire plan of study may be undergraduate (100-level).

NOTES:

*1) Required Background*

- Students without prior background in organ-level physiology are required to take at least one appropriate course from an FAS biology department, Harvard Medical School, Harvard T. H. Chan School of Public Health, or the Harvard-MIT Division of Health Sciences and Technology.
- Similarly, students without preparation in probability and biostatistics are required to take at least one course in this subject.
- These courses should be selected in consultation with the advisor to match the student's background and research area.
- Students whose preparation does not include prerequisites for graduate-level study in this area will take more than ten courses.

**2) Biophysics**

Students with a biophysics focus may substitute suitable biophysics courses for the core courses above, e.g. AP 225.

**Typical Plan of Study**

##### Year 1: Fall

- ES 222 Advanced Cellular Engineering
- AM 201 Physical Mathematics I
- HST 030/031 Human Pathology
- ES 299r Special Topics in Engineering Sciences (Independent Study)

##### Year 1: Spring

- ES 230 Advanced Tissue Engineering
- ES 228 Biologically-Inspired Materials
- ES 221 Drug Delivery

##### Year 2: Fall

- ES 220 Fluid Dynamics
- AM 205 Practical Scientific Computing

##### Year 2: Spring

- ES 240 Solid Mechanics
- ES 249 Advanced Neural Control of Movement

### Electrical Engineering

We have 5 distinct subfields of electrical engineering at SEAS. Each subfield has its own recommendation for a model program. For all programs, generally approved programs will contain two or fewer 100 level courses. In addition to AP 298r and ES 299r, programs will often contain courses from SEAS, mathematics, physics, and chemistry.

#### Computer Engineering track

**Core** **courses**

- CS 248 Advanced Design of VLSI Circuits and Systems
- CS 246 Advanced Computer architecture

**Breadth**

Students are asked to take at least one course in each of circuits and devices, EE systems and statistics, and CS software systems. The goal of the CS and EE systems courses is to provide technical breadth across the discipline and to provide exposure to emerging application areas that are of interest to designers of next-generation computing systems (interfaces with communication/networking, signal/image processing, etc).

The breadth requirement is met by taking one course from each of the following three topics:

**Circuits and Devices (One of the following) **

**Circuits and Devices (One of the following)**

- ES 271r Topics in Mixed-Signal Integrated Circuits
- ES 272 RF and High-Speed Integrated Circuits
- Physics 223 Electronics for Scientists

**EE Systems and Statistics (One of the following) **

**EE Systems and Statistics (One of the following)**

- ES 202 Estimation and Control of Dynamic Systems
- ES 250 Information Theory
- ES 255 Statistical Inference with Engineering Applications
- CS 283 Computer Vision

**CS Software Systems (One of the following) **

**CS Software Systems (One of the following)**

- CS 244 Networks Design Projects
- CS 261 Research Topics in Operating Systems
- CS 262 Introduction to Distributed Computing
- CS 265 Big Data Systems
- CS 252r Advanced Topics in Programming Languages

The remaining 5 courses should be selected in consultation with the field adviser. Appropriate courses come from those listed above and courses in Architecture and VLSI-CAD.

#### Control and Robotics track

**Core Courses **

- ES 202 Estimation and Control of Dynamic Systems
- ES 203 Stochastic Control
- ES 209 Nonlinear Control Systems
- ES 259 Advanced Introduction to Robotics
- CS 283 Computer Vision

At least one course in decision theory or estimation

- ES 201 Decision Theory
- ES 255 Statistical Inference with Engineering Applications

At least one course in Information and Communications

- ES 250 Information Theory

The remaining three courses may be 100- or 200-level courses from other EE tracks in Solid State Physics / Electromagnetism / Circuit Design or other 200-level courses in SEAS, Physics, Math, Chemistry, Biology, or graduate-level courses from MIT (with CHD approval) as suits the student's individual preparation, taste and research program.

#### Electronics track

The goal of the program is to provide graduate students with deep and broad understanding of modern solid-state electronics and closely related topics (e.g., lasers). The core courses in the program come from circuits and devices.

**Core Courses**

At least 2 Circuits courses from:

- ES 271r Topics in Mixed-Signal Integrated Circuits
- ES 272 RF and High-Speed Integrated Circuits
- CS 248 Advanced Design of VLSI Circuits and Systems

*Physics 223 Electronics for Scientists is also an extremely useful course in understanding electronics, but given that it is not focused on integrated circuits, students are advised to still take 2 courses from the above list).*

At least one course in Devices from:

- AP 195 Introduction to Solid State Physics
- AP 218 Electrical, Optical, and Magnetic Properties of Materials
- AP 295a Introduction to Quantum Theory of Solids
- AP 295b Quantum Theory of Solids

**Breadth **

General & Essential EE knowledge outside electronics

At least 1 course from

- ES 250 Information Theory
- MIT 6.341 Discrete-Time Signal Processing
- ES 202 Estimation and Control of Dynamic Systems
- CS 246 Computer Architecture

At least 1 course in Electromagnetism, Photonics & Microwaves

- ES 273 Optics and Photonics
- ES 274 Quantum Devices
- AP 216 Electromagnetic Interactions with Matter
- AP 217 Applications of Modern Optics
- Physics 285a Modern Atomic and Optical Physics I

Students can choose the remaining 5 courses from the 200-level technical courses SEAS, Physics, Applied Physics, Chemistry, Biology programs or from MIT. These could include up to two AP 299r or ES 299r courses.

#### Photonics track

Photonics is a broad and diverse field which straddles Electrical Engineering and Applied Physics. The central focus of the proposed Ph.D. model program in Photonics is on *core*** **courses**, **which capture the main thrusts of this field and provide some flexibility in choosing either a more EE oriented or a more Applied Physics oriented program. These core courses require knowledge of fundamentals such as *Electromagnetism*, *Quantum Mechanics *and *Solid-state Physics***. **In the *depth *category advanced (200-level) or less advanced (100-level) course options are given, to account for the different students’ backgrounds. A list of electives to provide additional *breadth*** **complements the program.

**Core Courses **

Three of five:

- ES 273 Optics and Photonics
- ES 274 Quantum Devices
- ES 275 Nanophotonics
- AP 216 Electromagnetic Interactions with Matter
- AP 217 Applications of Modern Optics

**Depth **

One of:

- AP195 Introduction to Solid State Physics
- AP 295a Introduction to Quantum Theory of Solids

One of:

- Physics 143b Quantum Mechanics II
- Chemistry 243 Applied Quantum Mechanics
- Physics 251a Advanced Quantum Mechanics I

One of:

- Physics 232 Advanced Classical Electromagnetism
- ES 151 Applied Electromagnetism

**Breadth courses (Electives) **

Four courses required.* A minimum of two from this suggested list: *

- ES 250 Information Theory
- ES 272 RF and High Speed Integrated Circuits
- ES 173 Introduction to Electronic and Photonic Devices
- ES 277 Microfabrication Laboratory
- AP 218 Electrical, Optical and Magnetic Properties of Materials
- AP 225 Introduction to Soft Matter
- AP 284 Statistical Thermodynamics
- AP 291 Electron Microscopy Lab
- Physics 223 Electronics for Scientists
- Physics 262 Statistical Physics

This list of electives is indicative and alternative course choices, as long as they broaden the student’s knowledge are acceptable, in consultation with the advisors and with the CHD committee approval. For further breadth up to two of the minimum number of required courses (10) can come from “298r” or “299r” courses.** **A maximum of three 100-level courses is allowed.

#### Signal Processing and Communications track

**Core Courses **

- ES 201 Decision Theory
- ES 202 Estimation and Control of Dynamic Systems
- ES 250 Information Theory
- ES 255 Statistical Inference with Engineering Applications

Students will take two courses in an application subject:

*Digital Communications*.

- CS 283 Computer Vision
- MIT 6.450 Principals of Digital Communications

*Applied Probability and Applied Statistics*. Two of the following:

- Statistics 210 Probability I
- Statistics 211 Statistical Inference I
- CS 223 Probabilistic Analysis and Algorithms
- Statistics 220 Bayesian Data Analysis
- MIT 6.262 Discrete Stochastic Processes

*Math and Applied Mathematics*. Two of the following:

- Math 112 Introductory Real Analysis
- Math 136 Differential Geometry
- AM 201 Physical Mathematics I
- ES 220 Fluid Dynamics
- MIT 6.255J Optimization Methods

Note: Many students will replace one or two of these courses with ES 299r.

### Environmental Science and Engineering

*Note*: Many courses in ESE are taught by SEAS faculty jointly with the Earth and Planetary Sciences (EPS) department, and are listed as EPS courses.

**Faculty members and instructors**

The following individuals offer courses in this area:

- James G. Anderson
- Brian F. Farrell
- Peter Huybers
- Daniel J. Jacob
- David Keith
- Zhiming Kuang
- Scot T. Martin
- Michael B. McElroy
- James Rice
- Daniel Schrag
- Elsie Sunderland
- Eli Tziperman
- Patrick Ulrich
- Steven C. Wofsy
- Robin Wordsworth

For more information about this area of study and the affiliated faculty, visit the ESE Teaching Area pages.

**Intellectual Scope**

The Harvard/SEAS ESE is broadly conceived, covering science and related engineering applications for the atmosphere, oceans, land, cryosphere, hydrosphere, and biosphere. Areas of study include dynamics (weather, ocean circulation, physical climate, atmosphere-ocean interactions, glaciology, geomorphologic and hydrogeologic processes), chemistry (atmospheric composition, global and local pollution, aerosols, marine biogeochemistry), biology (atmosphere-biosphere interactions, microbial transformations in the environment), and engineering (water technology, geologic hazards).

**Graduate Programs in ESE**

Graduate programs in ESE are diverse and flexible, as needed for the wide range of subject matter, but nonetheless there are common elements shared by many students. Programs are built around introductory courses that apply principles of physics and chemistry to the atmosphere, oceans, surface and near-surface earth processes. Students build on a strong foundation in mathematics, physics, chemistry, and computational science, and study both the fundamentals and research frontiers of atmospheric, oceanic, and land surface processes and dynamics.

Climate phenomena and global changes in atmospheric and ocean composition are a strong part of the curriculum, including advanced applications of statistics and large-scale data analysis, and including transient/accelerated glacial processes, air and water pollution, the science of geologic hazards (earthquakes, volcanic eruptions, and landslides), energy-related geoscience, and engineering applications are a growing component.

**Requirements **

- Students take 10 total courses
- Students take at least 8 disciplinary courses which do not include a 298r and with at most one 299r, at least 6 of which are at the 200 level (not including any 299r)
- at least one course is chosen from EPS 200, EPS 202, and EPS 208.

Most students take two or more of EPS 200, 202 and 208, and 9 to 10 disciplinary courses. Seminar, research, reading courses (e.g., 299r), and courses in other Harvard schools (e.g. Kennedy School of Government, School of Public Health, Business School, School of Design) fill out the balance of the program requirements (10 total courses)

Students normally take two introductory graduate courses and two more advanced courses in their areas, two applied mathematics courses, and two graduate level courses in engineering sciences, earth science, biology, physics, or other areas where technical breadth of preparation is needed. Many students participate in the Harvard Consortium on Energy and Environment to provide complementary interdisciplinary training and experience (http://energy.harvard.edu/graduate-consortium/energy_and_environment_about_program).appl

**Core Courses**

**Mathematics**

Preparation in mathematics and statistics is required for all ESE students. Since students have diverse backgrounds and a broad range of educational goals, they undertake mathematics at different levels. Generally students will take 2 courses in the mathematical sciences which include math, applied math, and statistics. Note that ordinarily a maximum of 2 100-level courses can be applied to a Ph.D. program of study.

*Minimum levels*

- AM 101 Statistical Inference for Scientists and Engineers
- AM 105 Ordinary and Partial Differential Equations
- AM 115 Mathematical Modeling

*Typical programs include at least one of the following courses*

- AM 201 Physical Mathematics I
- AM 202 Physical Mathematics II (partial differential equations)
- AM 203 Introduction to Disordered Systems and Stochastic Processes
- AM 205 Advanced Scientific Computing: Numerical Methods
- AM 221 Advanced Optimization, AM 222 Stochastic Modeling

**Physics, Chemistry, Biology**

Generally students must take at least one of the following courses. In most cases students take two of the following.

- EPS 200 Atmospheric Chemistry and Physics (includes computer laboratory)
- EPS 202 Mechanics in Earth and Environmental Science
- EPS 208 Physics of Climate

Sub-areas of ESE each have additional educational needs for foundation courses covering various aspects of physics, chemistry, engineering sciences or biology, such as fluid mechanics, spectroscopy, laser physics, ecosystem dynamics, etc. In consultation with his/her advisor, each student will develop a program that includes the relevant graduate courses of this type.

**Physics/Dynamics-oriented courses**

Students divide roughly by physics and chemistry foci, but many students do some of each.

- EPS 231 Climate Dynamics
- EPS 232 Dynamic Meteorology

**Chemistry-oriented courses**

- EPS 236 Environmental Modeling (includes computer laboratory)
- ES 268 Chemical Kinetics
- ES 267 Aerosol Science and Technology

**Technical Breadth Courses **

** **Most programs will have courses outside of the students direct research area. These courses ensure that there is technical breadth in the student’s education. A typical program will contain two courses which provide technical breadth. The following is a list of courses which will often satisfy the technical breadth requirement:

- ES 123 Introduction to Fluid Mechanics and Transport Processes, or ES 220 Fluid Dynamics
- ES 231 Energy Technology
- ES 237 Planetary Radiation and Climate
- ES 240 Solid Mechanics
- ES 265 Advanced Water Treatment
- ES 267 Aerosol Science and Technology
- ES 269 Environmental Nanotechnology
- OEB 157 Global Change Biology
- Physics 223 Electronics for Scientists

This is list is by no means comprehensive. Courses at the graduate level in Physics and Chemistry (e.g. quantum mechanics, molecular structure, electronics) are commonly included.

Please note that many students studying Geophysical Fluid Dynamics take 1 or 2 MIT courses.

All plans of study must also comply with the SEAS overall Ph.D. course requirements documented in the “Policies of the Committee on Higher Degrees” document.

### Materials Science and Mechanical Engineering

The goal of the program is for our students to develop strong foundations in the mathematical and physical principles, and applications, underlying mechanics as a branch of continuum mechanics and applied physics. Students can choose to undertake additional course work in related areas of Engineering Sciences and Applied Physics/Material Sciences/Geophysics. The scope of the program is quite wide and students enter with varying backgrounds. The following is a model. It is hoped that this will serve as a starting point for a discussion with your faculty adviser.

Graduate students specializing in Materials **can ****earn a Ph.D. in Applied Physics (Materials Sciences Track) or Engineering Sciences (Applied Mechanics Track)**. From their course work they will acquire a broad, rigorous background in the foundational elements of structure and defects; chemistry; thermodynamics and kinetics; mechanical, electrical, magnetic, and optical properties of materials.

#### Core Courses

- ES 220 Fluid Dynamics
- ES 240 Solid Mechanics
- AM 201 Physical Mathematics I

It may be necessary for students to take prerequisite courses prior to the core courses listed above. In order to successfully complete ES 220, ES 240, and AM 201 students should have completed ES 123, ES 120 and AM 105. Many students will have completed equivalent courses prior to matriculation. For those who have not, generally one of these courses may be counted towards the 10.

#### Electives

Most students will take 7 electives. Some will only take 6 because they are using ES 123, ES 120 or AM 105 as part of their programs. The choice of electives should be discussed with the advisor and generally students will concentrate in an field of application but they will take courses in different applications in order to provide breadth in the program.

Soft matter and fluids options include:

- AP 225 Introduction to Soft Matter

Solid mechanics options include:

- ES 241 Advanced Elasticity
- ES 242r Solid Mechanics: Advanced Seminar
- ES 246 Plasticity
- ES 247 Fracture Mechanics
- AP 282 Solids: Structure and Defects
- AP 293 Dislocations and Deformation Behavior of Materials
- ES 252r Advanced Topics in Robotics Research
- ES 259 Advanced Introduction to Robotics

Earth science options include:

- EPS 202 Mechanics in Earth and Environmental Science
- EPS 232 Dynamic Meteorology
- 200-level course in climate dynamics or air pollution
- other 200-level EPS courses

Applied Mathematics options include:

- AM 205 Advanced Scientific Computing: Numerical Methods
- other 200-level Applied Math courses
- AM 299r or ES 299r (one or two; if twice then with different instructors)
- one 100-level course of advanced content will be considered on an ad hoc basis.

#### MODEL PLAN OF STUDY I

For students who enter with graduate-level preparation in solid mechanics

##### YEAR 1 Fall:

- AM 201 Physical Mathematics I
- AM 205 Advanced Scientific Computing: Numerical Methods
- ES 220 Fluid Dynamics
- AP 225 Introduction to Soft Matter

##### YEAR 1 Spring:

- AP 298r Interdisciplinary Chemistry, Engineering and Physics: Seminar
- ES 299r Special Topics in Engineering Sciences
- Elective
- Elective

##### YEAR 2 Fall:

- Electives

#### MODEL PLAN OF STUDY II

##### YEAR 1 Fall:

- AM 201 Physical Mathematics I
- AM 205 Advanced Scientific Computing: Numerical Methods
- ES 240 Solid Mechanics
- AP 282 Solids: Structure and Defects

##### YEAR 1 Spring:

- AP 298r Interdisciplinary Chemistry, Engineering and Physics: Seminar
- ES 299r Special Topics in Engineering Sciences
- ES 241 Advanced Elasticity
- Elective

##### YEAR 2 Fall:

- ES 220 Fluid Dynamics
- Elective

*Note that, for Program Plans in Engineering Sciences, Physics 223*Electronics for Scientists

*is considered to be a 200-level SEAS-equivalent technical course.*