Online courses directory (2511)

This is the first semester of a two-semester sequence on Algebraic Geometry. The goal of the course is to introduce the basic notions and techniques of modern algebraic geometry. It covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. This course is an introduction to the language of schemes and properties of morphisms.

This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry.

The main aims of this seminar will be to go over the classification of surfaces (Enriques-Castelnuovo for characteristic zero, Bombieri-Mumford for characteristic p), while working out plenty of examples, and treating their geometry and arithmetic as far as possible.

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.

This course is devoted to the theory of Lie Groups with emphasis on its connections with Differential Geometry. The text for this class is Differential Geometry, Lie Groups and Symmetric Spaces by Sigurdur Helgason (American Mathematical Society, 2001).

Much of the course material is based on Chapter I (first half) and Chapter II of the text. The text however develops basic Riemannian Geometry, Complex Manifolds, as well as a detailed theory of Semisimple Lie Groups and Symmetric Spaces.

This course will give a detailed introduction to the theory of tensor categories and review some of its connections to other subjects (with a focus on representation-theoretic applications). In particular, we will discuss categorifications of such notions from ring theory as: module, morphism of modules, Morita equivalence of rings, commutative ring, the center of a ring, the centralizer of a subring, the double centralizer property, graded ring, etc.

This is the first semester of a one year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry.

The goal of this course is to describe some of the tools which enter into the proof of Sullivan's conjecture.

Geometry of Manifolds analyzes topics such as the differentiable manifolds and vector fields and forms. It also makes an introduction to Lie groups, the de Rham theorem, and Riemannian manifolds.

This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.

This course will focus on various aspects of mirror symmetry. It is aimed at students who already have some basic knowledge in symplectic and complex geometry (18.966, or equivalent). The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and rigor.

In this course, students take turns in giving lectures. For the most part, the lectures are based on Robert Osserman's classic book A Survey of Minimal Surfaces, Dover Phoenix Editions. New York: Dover Publications, May 1, 2002. ISBN: 0486495140.

This is an advanced topics course in model theory whose main theme is simple theories. We treat simple theories in the framework of compact abstract theories, which is more general than that of first order theories. We cover the basic properties of independence (i.e., non-dividing) in simple theories, the characterization of simple theories by the existence of a notion of independence, and hyperimaginary canonical bases.

In this graduate-level course, we will be covering advanced topics in combinatorial optimization. We will start with non-bipartite matchings and cover many results extending the fundamental results of matchings, flows and matroids. The emphasis is on the derivation of purely combinatorial results, including min-max relations, and not so much on the corresponding algorithmic questions of how to find such objects. The intended audience consists of Ph.D. students interested in optimization, combinatorics, or combinatorial algorithms.

This course provides an introduction to the mechanics of solids with applications to science and engineering. We emphasize the three essential features of all mechanics analyses, namely: (a) the geometry of the motion and/or deformation of the structure, and conditions of geometric fit, (b) the forces on and within structures and assemblages; and (c) the physical aspects of the structural system (including material properties) which quantify relations between the forces and motions/deformation.

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 measure

Mastery of these topics will be assessed via homework, quizzes/exams, and lab assignments.

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 course will survey the conditions of material life and changing social and economic conditions in medieval Europe with reference to the comparative context of contemporary Islamic, Chinese, and central Asian experiences. Subject covers the emergence and decline of feudal institutions, the transformation of peasant agriculture, living standards and the course of epidemic disease, and the ebb and flow of long-distance trade across the Eurasian system. Particular emphasis will be placed on the study of those factors, both institutional and technological, which have contributed to the emergence of capitalist organization and economic growth in Western Europe in contrast to the trajectories followed by the other major medieval economies.

This course is intended for first year graduate students and advanced undergraduates with an interest in design of ships or offshore structures. It requires a sufficient background in structural mechanics. Computer applications are utilized, with emphasis on the theory underlying the analysis. Hydrostatic loading, shear load and bending moment, and resulting primary hull primary stresses will be developed. Topics will include; ship structural design concepts, effect of superstructures and dissimilar materials on primary strength, transverse shear stresses in the hull girder, and torsional strength among others. Failure mechanisms and design limit states will be developed for plate bending, column and panel buckling, panel ultimate strength, and plastic analysis. Matrix stiffness, grillage, and finite element analysis will be introduced. Design of a ship structure will be analyzed by "hand" with desktop computer tools and a final design project using current applications for structural design of a section will be accomplished.

This course was originally offered in Course 13 (Department of Ocean Engineering) as 13.122. In 2005, ocean engineering subjects became part of Course 2 (Department of Mechanical Engineering), and this course was renumbered 2.082.   

This course introduces finite element methods for the analysis of solid, structural, fluid, field, and heat transfer problems. Steady-state, transient, and dynamic conditions are considered. Finite element methods and solution procedures for linear and nonlinear analyses are presented using largely physical arguments. The homework and a term project (for graduate students) involve use of the general purpose finite element analysis program ADINA. Applications include finite element analyses, modeling of problems, and interpretation of numerical results.