Treatment of the laws of thermodynamics and their applications to equilibrium and the properties of materials. Provides a foundation to treat general phenomena in materials science and engineering, including chemical reactions, magnetism, polarizability, and elasticity. Develops relations pertaining to multiphase equilibria as determined by a treatment of solution thermodynamics. Develops graphical constructions that are essential for the interpretation of phase diagrams. Treatment includes electrochemical equilibria and surface thermodynamics. Introduces aspects of statistical thermodynamics as they relate to macroscopic equilibrium phenomena.

"This class will be constructed as a lecture-discussion, the purpose being to engage important theoretical issues while simultaneously studying their continuing historical significance. To enhance discussion, three debates will be held in class. Each student will be required to participate in one of these debates. Each student will also be required to write three short papers. Class participation is essential and will be factored into the final grade.The course will portray the history of theory neither as the history of architectural theory exclusively, nor as a series of prepackaged static pronouncements, but as part of a broader set of issues with an active history that must be continually probed and queried. The sequence of topics will not be absolutely predetermined, but some of the primary issues that will be addressed are: pedagogy, professionalism, nature, modernity and the Enlightenment. Classroom discussions and debates are intended to demonstrate differences of opinion and enhance awareness of the consequences that these differences had in specific historical contexts."

The course consists of a sampling of topics from algebraic combinatorics. The topics include the matrix-tree theorem and other applications of linear algebra, applications of commutative and exterior algebra to counting faces of simplicial complexes, and applications of algebra to tilings.

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.

Topics vary from year to year. Fall Term: Numerical properties and vanish theorems for ample, nef, and big line bundles and vector bundles; multiplier ideals and their applications

Survey and special topics designed for graduate students in the brain and cognitive sciences. Emphasizes ethological studies of natural behavior patterns and their analysis in laboratory work, with contributions from field biology (mammology, primatology), sociobiology, and comparative psychology. Stresses human behavior but also includes major contributions from studies of other vertebrates and of invertebrates.

Students carry out independent experimental study under the direction of a member of the Biology Department faculty. Subject allows students with a strong interest in independent research to fulfill the project laboratory requirement for the Biology Department Program in the context of a research laboratory at MIT. Written and oral presentation of the research results is required. The permission of the faculty supervisor and the Biology Undergraduate Office must be obtained in advance. Instruction and practice in written and oral communication provided.

This is an introductory (i.e. first year graduate students are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry.

" 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."

" 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."

Geometry of pseudoconvex domains, the Monge-Ampere equation, Hodge theory on Kaehler manifolds, the theory of toric varieties and (time permitting) some applications to combinatorics.

This graduate-level course focuses on one-dimensional nonparametric statistics developed mainly from around 1945 and deals with order statistics and ranks, allowing very general distributions. For multidimensional nonparametric statistics, an early approach was to choose a fixed coordinate system and work with order statistics and ranks in each coordinate. A more modern method, to be followed in this course, is to look for rotationally or affine invariant procedures. These can be based on empirical processes as in computer learning theory. Robustness, which developed mainly from around 1964, provides methods that are resistant to errors or outliers in the data, which can be arbitrarily large. Nonparametric methods tend to be robust.

The main goal of this course is to study the generalization ability of a number of popular machine learning algorithms such as boosting, support vector machines and neural networks. Topics include Vapnik-Chervonenkis theory, concentration inequalities in product spaces, and other elements of empirical process theory.

" This course covers a collection of geometric techniques that apply broadly in modern algorithm design."

We will discuss numerous research problems that are related to the internet. Sample topics include: routing algorithms such as BGP, communication protocols such as TCP, algorithms for intelligently selecting a resource in the face of uncertainty, bandwidth sensing tools, load balancing algorithms, streaming protocols, determining the structure of the internet, cost optimization, DNS-related problems, visualization, and large-scale data processing. The seminar is intended for students who are ready to work on challenging research problems. Each lecture will discuss: methods used today issues and problems formulation of concrete problems potential new lines of research A modest amount of background information will be provided so that the importance and context of the problems can be understood. No previous study of the internet is required, but experience with algorithms and/or theoretical computer science at the graduate/research level is needed.

The emergence of Western science: the systematization of natural knowledge in the ancient world, the transmission of the classical legacy to the Latin West, and the revolt from classical thought during the scientific revolution. Examines scientific concepts in light of their cultural and historical contexts.

This course introduces students to the technique of instrumental neutron activation analysis. This is a non-destructive analytical technique for the determination of elemental abundances at very low levels in a wide variety of samples, geological to biological.

The emphasis of this course is to use Trace Element Geochemistry to understand the origin and evolution of igneous rocks. The approach is to discuss the parameters that control partitioning of trace elements between phases and to develop models for the partitioning of trace elements between phases in igneous systems, especially between minerals and melt. Subsequently, published papers that are examples of utilizing Trace Element Geochemistry are read and discussed.

Solid-state diffusion, homogeneous and heterogeneous chemical reactions, and spinodal decomposition. Heat conduction in solids, convective and radiative heat transfer boundary conditions. Fluid dynamics, 1-D solutions to the Navier-Stokes equations, boundary layer theory, turbulent flow, and coupling with heat conduction and diffusion in fluids to calculate heat and mass transfer coefficients.

" 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."