The goal of these lectures is to provide an introduction to Fourier analysis. The first topic is Fourier series, in particular, the Gibbs phenomenon and the dependence of their convergence properties on the suitability method used. The second topic is the Fourier transform, first the L¹ theory and then, using Hermite functions, the L² theory. The third topic is L. Schwartz’s theory of tempered distributions. The fourth topic is the theory of weak convergence of probability measures and application to a derivation of the Lévy-Khinchine formula for the Fourier transform of infinitely divisible probability measures. The final topic is the elementary theory of singular integral operators, starting with the Hilbert transform and then using the method of rations to prove the Lᵖ boundedness of even Calderòn-Zygmund kernels.

This lesson introduces students to the “Tragedy of the Commons,” an extended metaphor for problems of shared environmental or man-made resources that are overused and eventually depleted. In this metaphor, shared resources are compared to a common grazing pasture, or “commons,” on which any dairy farmer can graze as many cows as he/she wishes. If too many cows are added to the commons, they will overeat the grass in the pasture and the shared resource will become depleted – a disadvantage to everyone. In this lesson, students will be inspired to think about possible solutions to this problem. To get there, they will use basic math to frame the problem and will discover how useful this can be in considering consequences of various actions. Most importantly, they will become comfortable with the concept of problems of shared resources – and will learn to recognize, and seek out, examples all around them. An exposure to algebra 1 and basic functions is the only math prerequisite necessary. The lesson will take around 50 minutes to complete and the required materials for this lesson are paper and pens or pencils, as well as some sort of prize to provide the winning team with in the final activity. For all five activities, students are asked to work in groups of 4, but groups of 3 or 5 would also be okay. Students will work with their groups to discuss the logic behind the tragedy of the commons, to consider some options for preventing this tragedy and to examine examples of problems of shared resources that are relevant to them. They will also come up with functions that fit behavior described in the video, and be asked to think about the behavior of functions provided in the video and accompanying materials.

This class serves as an introduction to mass transport in environmental flows, with emphasis given to river and lake systems. The class will cover the derivation and solutions to the differential form of mass conservation equations. Class topics to be covered will include: molecular and turbulent diffusion, boundary layers, dissolution, bed-water exchange, air-water exchange and particle transport.

This Guide for Educators was developed by the MIT Environmental Solutions Initiative as an extension of our TILclimate (Today I Learned: Climate) podcast, to make it easier for you to teach climate change, earth science, and energy topics in the classroom. It is an extension of the TILclimate episode "TIL about everyday travel."

15.975 U-Lab: Leading Profound Innovation for a More Sustainable World is an interactive and experiential class about leading profound innovation for pioneering a more sustainable economy and society. The class is organized around personal reflection practices, relational practices, and societal practices. It focuses on the intertwined relationship between the evolution of capitalism, multi-stakeholder innovation, and presencing.

This course provides a comprehensive introduction to the making of US foreign and national security policy. It examines the laws that guide policy-making, studies the actors and organizations involved in the inter-agency process, and explores how interaction between the executive, legislative, and judicial branches shapes policy development and implementation. Students acquire practical experience through policy writing and a crisis simulation. This course is designed for students interested in international relations, security, and public policy.

“Designing a dream city is easy. Rebuilding a living one takes imagination.”    -Jane Jacobs
This course examines the challenges that cities will face and strategies they can use to prepare for the impacts of climate change. Particular attention will be paid to the presence of global disparities, the needs of vulnerable populations and resource constrained locales, and the ways in which local government and community-based activities can achieve equitable levels of climate-readiness.

This class is about figuring out together what cities and users can do to reduce their energy use and carbon emissions. Many other classes at MIT focus on policies, technologies, and systems, often at the national or international level, but this course focuses on the scale of cities and users. It is designed for any students interested in learning how to intervene in the energy use of cities using policy, technology, economics, and urban planning.

This course examines the policy, politics, planning, and engineering of transportation systems in urban areas, with a special focus on the Boston area. It covers the role of the federal, state, and local government and the MPO, public transit in the era of the automobile, analysis of current trends and pattern breaks; analytical tools for transportation planning, traffic engineering, and policy analysis; the contribution of transportation to air pollution, social costs, and climate change; land use and transportation interactions, and more. Transportation sustainability is a central theme throughout the course, as well as consideration of if and how it is possible to resolve the tension between the three E’s (environment, economy, and equity). The goal of this course is to elicit discussion, stimulate independent thinking, and encourage students to understand and challenge the “conventional wisdom” of transportation planning.

it would be ideal if students already have learned that DNA is the genetic material, and that DNA is made up of As, Ts, Gs, and Cs. It also would help if students already know that each human has two versions of every piece of DNA in their genome, one from mom and one from dad. The lesson will take about one class period, with roughly 30 minutes of footage and 30 minutes of activities.

This video is meant to be a fun, hands-on session that gets students to think hard about how machines work. It teaches them the connection between the geometry that they study and the kinematics that engineers use -- explaining that kinematics is simply geometry in motion. In this lesson, geometry will be used in a way that students are not used to. Materials necessary for the hands-on activities include two options: pegboard, nails/screws and a small saw; or colored construction paper, thumbtacks and scissors. Some in-class activities for the breaks between the video segments include: exploring the role of geometry in a slider-crank mechanism; determining at which point to locate a joint or bearing in a mechanism; recognizing useful mechanisms in the students' communities that employ the same guided motion they have been studying.

Visualizing Cultures was launched at MIT in 2002 to explore the potential of the Web for developing innovative image-driven scholarship and learning. The VC mission is to use new technology and hitherto inaccessible visual materials to reconstruct the past as people of the time visualized the world (or imagined it to be).

Topical units to date focus on Japan in the modern world and early-modern China. The thrust of these explorations extends beyond Asia per se, however, to address "culture" in much broader ways—cultures of modernization, war and peace, consumerism, images of "Self" and "Others," and so on.

This course, which examines ways of resolving conflicts over the allocation of water resources, is designed to raise student awareness of the state of freshwater resources globally and the need for more effective water governance. It builds on several case studies of transboundary water conflicts in different parts of the world while also helping students develop the negotiation and mediation skills they will need to resolve water disputes.

This subject is concerned with quantitative methods for analyzing large-scale water resource problems. Topics covered include the design and management of facilities for river basin development, flood control, water supply, groundwater remediation, and other activities related to water resources. Simulation models and optimization methods are often used to support analyses of water resource problems. In this subject we will be constructing simulation models with the MATLAB® programming language and solving numerical optimization problems with the GAMS optimization package.

This course deals with the principles of infrastructure planning in developing countries, with a focus on appropriate and sustainable technologies for water and sanitation. It also incorporates technical, socio-cultural, public health, and economic factors into the planning and design of water and sanitation systems. Upon completion, students will be able to plan simple, yet reliable, water supply and sanitation systems for developing countries that are compatible with local customs and available human and material resources. Graduate and upper division students from any department who are interested in international development at the grassroots level are encouraged to participate in this interdisciplinary subject.
Acknowledgment
This course was jointly developed by Earthea Nance and Susan Murcott in Spring 2006.

This course is an overview of engineering approaches to protecting water quality with an emphasis on fundamental principals. Theory and conceptual design of systems for treating municipal wastewater and drinking water are discussed, as well as reactor theory, process kinetics, and models. Physical, chemical, and biological processes are presented, including sedimentation, filtration, biological treatment, disinfection, and sludge processing. Finally, there is discussion of engineered and natural processes for wastewater treatment.

The major purpose of this lesson is to promote the learning of eye function by associating eye problems and diseases to parts of the eye that are affected. Included in this module are discussions and activities that teach about eye components and their functions. The main activity is dissecting a cow eye, which in many high schools is part of the anatomy curriculum. This lesson extends the curriculum by discussing eye diseases that students might be familiar with. An added fun part of the lesson is discussion of what various animals see.

This video lesson will introduce students to algorithmic thinking through the use of a popular field in graph theory—social networking. Specifically, by acting as nodes in a graph (i.e. people in a social network), the students will experientially gain an understanding of graph theory terminology and distance in a graph (i.e. number of introductions required to meet a target person). Once the idea of distance in a graph has been built, the students will discover Dijkstra's Algorithm. The lesson should take approximately 90 minutes and can be comfortably partitioned across two class sessions if necessary (see the note in the accompanying Teacher Guide). There are no special supplies needed for this class and all necessary hand-outs can be downloaded from this website.

Beavers are generally known as the engineers of the animal world. In fact the beaver is MIT's mascot! But honeybees might be better engineers than beavers! And in this lesson involving geometry in interesting ways, you'll see why! Honeybees, over time, have optimized the design of their beehives. Mathematicians can do no better. In this lesson, students will learn how to find the areas of shapes (triangles, squares, hexagons) in terms of the radius of a circle drawn inside of these shapes. They will also learn to compare those shapes to see which one is the most efficient for beehives. This lesson also discusses the three-dimensional shape of the honeycomb and shows how bees have optimized that in multiple dimensions. During classroom breaks, students will do active learning around the mathematics involved in this engineering expertise of honeybees. Students should be conversant in geometry, and a little calculus and differential equations would help, but not mandatory.

[The author] asked immigrants from 28 countries why they're serving in the U.S. military — and it's not primarily to gain citizenship.