" Double affine Hecke algebras (DAHA), also called Cherednik algebras, and their representations appear in many contexts: integrable systems (Calogero-Moser and Ruijsenaars models), algebraic geometry (Hilbert schemes), orthogonal polynomials, Lie theory, quantum groups, etc. In this course we will review the basic theory of DAHA and their representations, emphasizing their connections with other subjects and open problems."

Focuses on the role of the business improvement district (BID) as a popular and contemporary tool for urban revitalization. Explores BID origins, theoretical underpinnings, enabling legislation, and organizational issues. Emphasizes BID service provision including advocacy, marketing, sanitation, streetscape improvement, security and transportation, while examining BID performance using such indicators as crime rates, vacancy rates, and retail sales. Considers BID organizations throughout North America as well as comparable schemes in Australia, Holland, Japan, New Zealand, South Africa, and the United Kingdom. This course focuses on the origins, functions, and implications of downtown management organizations (DMOs), such as business improvement districts, in a variety of national contexts including the United States, Canada, South Africa, and the United Kingdom. It critically examines how a range of urban theories provide a rationale for the establishment and design of DMOs; the evolution and transnational transfer of DMO policy; and the spatial and political externalities associated with the local proliferation of DMOs. Particular emphasis is given to the role of DMOs in securing public space.

Seminar on downtown in US cities from the late nineteenth century to the late twentieth. Emphasis on downtown as an idea, place, and cluster of interests, on the changing character of downtown, and on recent efforts to rebuild it. Subjects considered include subways, skyscrapers, highways, urban renewal, and retail centers. Focus on readings, discussions, and individual research projects. Meets with graduate subject 11.339, but assignments differ.

This is an activity about observing and mapping sunspots by direct solar observation. Learners will use a small telescope, binoculars, or a Sunspotter to create a projected image of the Sun and trace the position of any observed sunspots on a piece of paper. Additionally, learners will mark the direction of the Sun image’s motion. This is Activity 2 of the Space Weather Forecast curriculum.

Seminar on a selected topic from Renaissance architecture. Requires original research and presentation of a report. The aim of this course is to highlight some technical aspects of the classical tradition in architecture that have so far received only sporadic attention. It is well known that quantification has always been an essential component of classical design: proportional systems in particular have been keenly investigated. But the actual technical tools whereby quantitative precision was conceived, represented, transmitted, and implemented in pre-modern architecture remain mostly unexplored. By showing that a dialectical relationship between architectural theory and data-processing technologies was as crucial in the past as it is today, this course hopes to promote a more historically aware understanding of the current computer-induced transformations in architectural design.

The course provides the technological background of treatment processes applied for production of drinking water. Treatment processes are demonstrated with laboratory experiments.

The first two weeks of this course are an overview of performing improvisation with introductory and advanced exercises in the techniques of improvisation. The final four weeks focus on applying these concepts in business situations to practice and mastering these improvisation tools in leadership learning.

"This course focuses on dynamic optimization methods, both in discrete and in continuous time. We approach these problems from a dynamic programming and optimal control perspective. We also study the dynamic systems that come from the solutions to these problems. The course will illustrate how these techniques are useful in various applications, drawing on many economic examples. However, the focus will remain on gaining a general command of the tools so that they can be applied later in other classes."

Deterministic optimization: maximum principle, dynamic programming, calculus of variations, optimal control, dynamic games. Stochastic optimization: stochastic optimal control and dynamic programming, Markov processes, Ito calculus, Markov games. Applications. Dynamical systems: local and global analysis and chaos. The unifying theme of this course is best captured by the title of our main reference book: "Recursive Methods in Economic Dynamics". We start by covering deterministic and stochastic dynamic optimization using dynamic programming analysis. We then study the properties of the resulting dynamic systems. Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. We shall stress applications and examples of all these techniques throughout the course.

The course covers the basic models and solution techniques for problems of sequential decision making under uncertainty (stochastic control). We will consider optimal control of a dynamical system over both a finite and an infinite number of stages. This includes systems with finite or infinite state spaces, as well as perfectly or imperfectly observed systems. We will also discuss approximation methods for problems involving large state spaces. Applications of dynamic programming in a variety of fields will be covered in recitations.

The course addresses dynamic systems, i.e., systems that evolve with time. Typically these systems have inputs and outputs; it is of interest to understand how the input affects the output (or, vice-versa, what inputs should be given to generate a desired output). In particular, we will concentrate on systems that can be modeled by Ordinary Differential Equations (ODEs), and that satisfy certain linearity and time-invariance conditions. We will analyze the response of these systems to inputs and initial conditions. It is of particular interest to analyze systems obtained as interconnections (e.g., feedback) of two or more other systems. We will learn how to design (control) systems that ensure desirable properties (e.g., stability, performance) of the interconnection with a given dynamic system.

Momentum principles and energy principles. Lagrange's equations, Hamilton's principle. Applications to mechanical systems including gyroscopic effects. Study of steady motions and nature of small deviations therefrom. Natural modes and natural frequencies for continuous and lumped parameter systems. Forced vibrations. Dynamic stability theory. Causes of instability. This course reviews momentum and energy principles, and then covers the following topics: Hamilton's principle and Lagrange's equations; three-dimensional kinematics and dynamics of rigid bodies; steady motions and small deviations therefrom, gyroscopic effects, and causes of instability; free and forced vibrations of lumped-parameter and continuous systems; nonlinear oscillations and the phase plane; nonholonomic systems; and an introduction to wave propagation in continuous systems. This course was originally developed by Professor T. Akylas.

This class is an introduction to the dynamics and vibrations of lumped-parameter models of mechanical systems. Topics include kinematics; force-momentum formulation for systems of particles and rigid bodies in planar motion; work-energy concepts; virtual displacements and virtual work; Lagrange's equations for systems of particles and rigid bodies in planar motion; linearization of equations of motion; linear stability analysis of mechanical systems; free and forced vibration of linear multi-degree of freedom models of mechanical systems; and matrix eigenvalue problems. The class includes an introduction to numerical methods and using MATLABĺ¨ to solve dynamics and vibrations problems.

Upon successful completion of this course, students will be able to: * Create lumped parameter models (expressed as ODEs) of simple dynamic systems in the electrical and mechanical energy domains * Make quantitative estimates of model parameters from experimental measurements * Obtain the time-domain response of linear systems to initial conditions and/or common forcing functions (specifically; impulse, step and ramp input) by both analytical and computational methods * Obtain the frequency-domain response of linear systems to sinusoidal inputs * Compensate the transient response of dynamic systems using feedback techniques * Design, implement and test an active control system to achieve a desired performance measureMastery of these topics will be assessed via homework, quizzes/exams, and lab assignments.

Introduction to dynamics and vibration of lumped-parameter models of mechanical systems. Three-dimensional particle kinematics. Force-momentum formulation for systems of particles and for rigid bodies (direct method). Newton-Euler equations. Work-enery (variational) formulation for systems particles and for rigid bodies (indirect method). Virtual displacements and work. Lagrange's equations for systems of particles and for rigid bodies. Linearization of equations of motion. Linear stability analysis of mechanical systems. Free and forced vibration of linear damped lumped parameter multi-degree of freedom models of mechanical systems. Application to the design of ocean and civil engineering structures such as tension leg platforms.

An introduction to theoretical studies of systems of many interacting components, the individual dynamics of which may be simple, but the collective dynamics of which are often nonlinear and analytically intractable. Topics vary from year to year. Format includes both pedagogical lectures and round-table reviews of current literature. Subjects of interest include: problems in natural science (e.g., geology, ecology, and biology) where quantitative theory is still in development; problems in physics, such as turbulence, that demonstrate powerful concepts such as scaling and universality; and modern computational methods for the simulation and study of such problems. Discussions in context of contemporary experimental or observational data.

An introduction to theoretical studies of systems of many interacting components, the individual dynamics of which may be simple, but the collective dynamics of which are often nonlinear and analytically intractable. Topics vary from year to year. Format includes both pedagogical lectures and round-table reviews of current literature. Subjects of interest include: problems in natural science (e.g., geology, ecology, and biology) where quantitative theory is still in development; problems in physics, such as turbulence, that demonstrate powerful concepts such as scaling and universality; and modern computational methods for the simulation and study of such problems. Discussions in context of contemporary experimental or observational data.

An introduction to theoretical studies of systems of many interacting components, the individual dynamics of which may be simple, but the collective dynamics of which are often nonlinear and analytically intractable. Topics vary from year to year. Format includes both pedagogical lectures and round-table reviews of current literature. Subjects of interest include: problems in natural science (e.g., geology, ecology, and biology) where quantitative theory is still in development; problems in physics, such as turbulence, that demonstrate powerful concepts such as scaling and universality; and modern computational methods for the simulation and study of such problems. Discussions in context of contemporary experimental or observational data.

Introduction to nonlinear deterministic dynamical systems. Nonlinear ordinary differential equations. Planar autonomous systems. Fundamental theory: Picard iteration, contraction mapping theorem, and Bellman-Gronwall lemma. Stability of equilibria by Lyapunov's first and second methods. Feedback linearization. Application to nonlinear circuits and control systems. Alternate years. Description from course website: This course provides an introduction to nonlinear deterministic dynamical systems. Topics covered include: nonlinear ordinary differential equations; planar autonomous systems; fundamental theory: Picard iteration, contraction mapping theorem, and Bellman-Gronwall lemma; stability of equilibria by Lyapunov's first and second methods; feedback linearization; and application to nonlinear circuits and control systems.