Areas of Application

Combining applied mathematics with astronomy or another similar physical science allows delving deeper into mathematical foundations, while maintaining a strong overview of the major concepts and methods.

Note: Only one of Physics 15a, 15b, 15c, PS 12a, 12b can be used within this application area.

Example Plan of Study

The biological sciences application area combines the foundation of a life sciences concentration with a focus on biological data analysis and/or modeling using computation, statistics, and mathematics.

At least one course should be a broad ("gateway", "portal", or "foundation") intermediate-level course in a life sciences concentration: 

• MCB 60, 63, 64, 65, 68 for molecular and cellular biology or for chemical and physical biology; 
• OEB 50-59 for integrative biology; 
• MCB 80, 81,105,115,125 for neuroscience; 
• SCRB 10 for human developmental & regenerative biology; 
• HEB 30, 39, 45, 50, 55 for human evolutionary biology;
• ES 53 for bioengineering.

At least three of the five application area courses should be "quantitative", meaning that they have substantial mathematical or computational content. The quantitative application area courses can be drawn from any area relevant to quantitative biology, not just life sciences courses but also including computer science, statistics, and applied math courses, provided they form a coherent plan. Some examples of life sciences courses that currently count as quantitative include: 

LS50      Integrated science 
MCB111    Mathematics in biology
MCB112    Biological data analysis 
MCB131    Computational neuroscience
MCB195    Foundations of systems biology
MCB198    Advanced mathematical techniques for modern biology
MCB199    Statistical thermodynamics and quantitative biology
Neuro120   Introductory computational neuroscience
Neuro140  Biological and artificial intelligence
Neuro141  Physics of sensory systems in biology
Neuro1401 Computational cognitive neuroscience
OEB242    Population genetics
OEB252    Coalescent theory

Some examples of other courses suitable as quantitative AM/bio
application area courses include:

CS1810     Machine learning
CS1820    Artificial intelligence
Stat 111    Introduction to statistical inference
Stat115   Computational biology and bioinformatics
Stat117   Data analysis in modern biostatistics
Stat 139   Linear models
AM 120   Applied linear algebra and big data
AM 121   Introduction to optimization: models and methods
AM 226   Neural Computation
ES 157   Biological signal processing
ES 201   Decision theory

Notes: 

Introductory-level life sciences courses including LS1a/b and OEB10 do not count toward AM/bio concentration requirements, although they may be prerequisites for other courses.

The special full-year freshman course LS50 Integrated Science can be counted as two courses: once in the application area (where it counts both as an intermediate-level foundational course and as quantitative), and also toward the modeling category in the breadth requirement.

Theoretical chemistry provides an opportunity for a potential area of application focusing in physical chemistry.  Three quantitative chemistry courses (three of PS 10, Chem 60, Chem 160, Chem 161, Chem 163) and two additional related courses.

Example Plan of Study

Applied Mathematics concentrators specializing in computer science will build a broader base of applicable mathematics and focus on those aspects of the subject which depend most directly on such mathematics.

For an application in computer science, students will take CS 1200, 1210 or CS 1240, as well as at least two more core courses drawn from the 1200s, 1300s, 1500s, 1610, 1750, or the 1800s.

The other two courses in the application area should be chosen to be coherent with the rest of the plan.  These two courses can be CS courses (CS 20, CS 51, CS 61, etc), but are not required to be.  As examples, students with interests in computational linguistics have incorporated linguistics coursework into their plan, along with core courses in natural language processing.  Students with interests in computer graphics have incorporated drawing or animation coursework into their plan, along with core courses in computer graphics.

Note: Ordinarily, AM enforces recommended preparation for courses; check the recommended preparation for the courses you are considering via my.harvard.  Include those courses in your plan of study if you have not yet completed them (courses can be removed from the plan once they are complete).

Applied Mathematics concentrators specializing in data science will satisfy a quantitative methods requirement by taking AM 120, Stat 111, and either CS 1810 or CS 1820.  The application area also involves two electives.  Courses that are recommended preparation for the quantitative methods courses do also need to be completed (and can be counted either in the breadth or as an elective in the application area).

Recommended electives include CS 1710 (data visualization), CS 1200 (background for CS 1820), CS 51 (background for CS 1810), CS 20 (background for CS 1200), MCB 112, econometrics, and courses treating data and data science from other perspectives

Note: Ordinarily, AM enforces recommended preparation for courses; check the recommended preparation for the courses you are considering via my.harvard.  Include those courses in your plan of study if you have not yet completed them (courses can be removed from the plan once they are complete).

The Decision and Control area is concerned with topics that are sometimes called operations research and/or systems engineering. The common theme is optimization, in various forms and contexts, both to understand natural systems and to design man made systems.

Example Plans of Study

Digital Humanities comprises a rapidly expanding area of research on texts, images, sounds, performances, and more, using techniques such as network analysis, GIS, and topic modeling. Any program in Arts & Humanities can serve as the inspiration for a student's inquiries.

Applied Mathematics concentrators with an application area in digital humanities should typically take a foundation course in the humanities (a lecture) and an advanced course (a seminar or seminar-equivalent) focused on interpretation and scholarship.  For quantitative methods courses, students should take one digital humanities methods course, and two mathematical courses drawn from the quantitative methods requirements for the data science application area.

Options for the digital humanities methods courses:

Hist1993: Introduction to Digital History
Anthro2020: GIS & Spatial Analysis in Archaeology
CHNSHIS 202: Digital Methods for Chinese History
Complit249: Cartography and Early Modern Literature
EASTD 110: East Asian Digital Humanities
Engl298dh: Methods in Digital Humanities
Hist 92r: History Lab
Hist1952: Mapping History
Music 167: Introduction to Electroacoustic Music

AM students planning to pursue an application to the earth sciences should take the Physical Sciences 12 or Physics 15 sequence as important background knowledge. Courses in EPS that have both physics and 21a/21b prerequisites are appropriate for the application area.

Note: Only one of Physics 15a, 15b, 15c, PS 12a, 12b can be counted within this application area.

Example Plan of Study

Mathematical modeling is used extensively in economics, and it is generally agreed that the foundation of economic theory is formed on a mathematical basis. The requirements for applied mathematics and economics are made and continuously updated in cooperation with the Economics Department.

See full list of courses and details

The birth of internet technology has strengthened the argument for combining computer science and economics into a single track. The core part of such a program should include Ec 1011a; CS 1360; and one of CS 1810, CS 1820, CS 2810.  Ec 1011a is forming the core of the economics and CS 1810/1820 is forming the core of the computer science in the plan.  As for every application area, the overall program of five courses should form a coherent set of five courses.

Notes: Ordinarily, AM enforces recommended preparation for courses; check the recommended preparation for the courses you are considering via my.harvard.  Include those courses in your plan of study if you have not yet completed them (courses can be removed from the plan once they are complete).

In years when CS 1360 is not offered, Econ 1071 can be accepted in its place.

Example Plans of Study

Some other courses of interest may include CS 2370/Econ 2070, CS 2380, Econ 2355

Three important general paths of study involve circuit design, signal processing/communications, and the mathematics of intelligent machines.

Note: Only one of Physics 15a, 15b, 15c, PS 12a, 12b can be used within this application area.

Applied Math concentrators on a Government Track are required to take at least 2 substantive (i.e., primarily about politics) Government classes and 3 Government courses that are more mathematically oriented.

Substantive courses (two Gov courses): Gov foundational courses (10, 20, 30, 40), Gov 97 Sophomore Tutorial (recommended), and other courses with Gov course numbers that focus on politics.  

Note: Gov 50, 51, 52, 61, 62 do not count for the application area.

Core statistics course (one course): Gov 2001 Quantitative social science methods. Note: students may petition to substitute Stat 111 or Stat 139 for this core statistics requirement.  The Gov 2000-level courses are open to AM/Gov students

Core formal theory course (one course): Gov 2005 Formal Theory 1

One additional mathematical course:

Statistics: Gov 2002 Quantitative social science methods II, Gov 2003 Causal Inference, Gov 2017 Advanced Topics in Political Methodology, Gov 2018 Applied machine learning for the social sciences, Gov 2019 Advanced topics in political methodology

Formal Theory: Gov 2006 Formal Theory 2

Example Plan of Study

There are two main options in this area: macroscopic (or classical) physics and microscopic (or quantum) physics.  Students with an application area in physics take at least one of Physics 143a (quantum mechanics) or Physics 181 (Statistical mechanics).

Note: Only one of Physics 15a, 15b, PS 12a, 12b can be counted within this application area.  Students ordinarily count Physics 15b and 15c towards the application area.

Example Plan of Study

Applied Mathematics concentrators specializing in psychology will build a psychology foundation while developing an understanding of complementary mathematics.

For an application in psychology, students will take one foundational psychology course, one advanced psychology course from within the Psychology Department, and three quantitative courses.  One of the three quantitative courses must be a statistical inference course (Stat 111, Stat 139, Psych 1950, Econ 1126, Gov 2001).  The other two can be drawn from Statistics, Computer Science, Economics, Engineering Sciences, or Mathematical Biology.  Psych 1401 also counts as a quantitative course for the psychology application area, as does Psych 1952.  As with all applications, the five courses should form a coherent plan.

Note: Undergraduates do not have priority for enrollment in Psych 1950.  Most AM/Psych students have at least two courses from outside of Psych in their five-course plan.

Example Plan of Study

This area is concerned with the design, implementation and study of algorithms for the approximate solution of continuous mathematical problems on digital computers: problems posed in the language of calculus and linear algebra, including differential and integral equations, root finding, and optimization.

Applied Math concentrators on a Sociology Track are expected to take at least 2 substantive classes in Sociology and 3 courses that are methodologically oriented.  Two of the three methodological courses should be in sociology

Substantive courses: two courses drawn from foundational sociology courses (Soc 1000-1089) or from the tutorial in social theory (Soc 97) 

Core statistics course (one course): Soc 2202 Intermediate quantitative research methods, Soc 2203 Advanced quantitative research methods, Stat 111 Introduction to statistical inference, or Stat 139 Linear models.

Two advanced methodological courses: Soc 90 Quantitative research labs, Soc 2203 Advanced quantitive research methods, Soc 2211 Analysis of longitudinal data, Soc 2272 Computational sociology, other Soc 22xx quantitative methods courses, Stat 140 Design of experiments, Stat 160 Design and analysis of sample surveys, Gov 2003 Causal inference, Gov 2017 Advanced topics in political methodology

Example Plan of Study

Applied Mathematics concentrators specializing in the Study of the Human Past will learn how to apply quantitative methods to historical datasets drawn from archaeological, genetic, historical, linguistic/philological, or textual sources. Students should expect to take 2 foundational courses and at least 1 quantitive course concerned with the human past in one or more of the participating departments (including Anthropology, East Asian Languages and Civilizations, Government, History, English) along with additional quantitatively oriented courses. For more information, contact am-advising@seas.harvard.edu