Content varies from year to year. An introduction to some of the major topics of present day combinatorics, in particular enumeration, partially ordered sets, and generating functions. This is a graduate-level course in combinatorial theory. The content varies year to year, according to the interests of the instructor and the students. The topic of this course is hyperplane arrangements, including background material from the theory of posets and matroids.

Content varies from year to year. An introduction to some of the major topics of present day combinatorics, in particular enumeration, partially ordered sets, and generating functions. This course serves as an introduction to major topics of modern enumerative and algebraic combinatorics with emphasis on partition identities, young tableaux bijections, spanning trees in graphs, and random generation of combinatorial objects. There is some discussion of various applications and connections to other fields.

This interactive activity for grades 8-12 features eight models that explore atomic arrangements for gases, solids, and liquids. Highlight an atom and view its trajectory to see how the motion differs in each of the three primary phases. As the lesson progresses, students observe and manipulate differences in attractions among atoms in each state and experiment with adding energy to produce state changes. More advanced students can explore models of latent heat and evaporative cooling. This item is part of the Concord Consortium, a nonprofit research and development organization dedicated to transforming education through technology.

This is the first semester of a two-semester sequence on Differential Analysis. Topics include fundamental solutions for elliptic; hyperbolic and parabolic differential operators; method of characteristics; review of Lebesgue integration; distributions; fourier transform; homogeneous distributions; asymptotic methods.

Fall: Fundamental solutions for elliptic, hyperbolic and parabolic differential operators. Method of characteristics. Review of Lebesgue integration. Distributions. Fourier transform. Homogeneous distributions. Asymptotic methods. Spring: Sobolev spaces. Fredholm alternative. Variable coefficient elliptic, parabolic and hyperbolic linear partial differential equations. Variational methods. Viscosity solutions of fully nonlinear partial differential equations. The main goal of this course is to give the students a solid foundation in the theory of elliptic and parabolic linear partial differential equations. It is the second semester of a two-semester, graduate-level sequence on Differential Analysis.

A useful essay that opens dialog between students and faculty about some of the struggles seen in teaching of mathematics.

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.

Compiled to support a course on teaching math and science in Early Childhood Education. Chapters include: Early Learning; Math & Science Connections; Theory & Teacher Effectiveness; Special Needs; Standards & Assessment; Lesson Planning; Centers & Environment; Teaching Science Content; Teaching Math Content; and Hands-on Learning.

Considers how institutions have been incorporated theoretically into explorations of growth and development. Four sets of institutions are examined in detail: the corporate sector, to study how ownership, strategy, and structure affect growth-related policies; financial institutions, to analyze how they condition savings and investment; labor market institutions, to investigate their impact on the determination of wage and production-related productivity; and the institutions associated with technology, such as universities, research laboratories, and corporate training centers, to consider how skill formulation is accomplished.

Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and mathematics who have completed calculus through partial differentiation. If your syllabus includes Chapter 10 (Linear Systems of Differential Equations), your students should have some preparation in linear algebra. In writing this book I have been guided by the these principles: • An elementary text should be written so the student can read it with comprehension without too much pain. I have tried to put myself in the student’s place, and have chosen to err on the side of too much detail rather than not enough. • An elementary text can’t be better than its exercises. This text includes 2041 numbered exercises, many with several parts. They range in difficulty from routine to very challenging. • An elementary text should be written in an informal but mathematically accurate way, illustrated by appropriate graphics. I have tried to formulate mathematical concepts succinctly in language that students can understand. I have minimized the number of explicitly stated theorems and defonitions, preferring to deal with concepts in a more conversational way, copiously illustrated by 299 completely worked out examples. Where appropriate, concepts and results are depicted in 188 figures

Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and mathematics who have completed calculus through partial differentiation. If your syllabus includes Chapter 10 (Linear Systems of Differential Equations), your students should have some preparation in linear algebra. In writing this book I have been guided by the these principles: • An elementary text should be written so the student can read it with comprehension without too much pain. I have tried to put myself in the student’s place, and have chosen to err on the side of too much detail rather than not enough. • An elementary text can’t be better than its exercises. This text includes 2041 numbered exercises, many with several parts. They range in difficulty from routine to very challenging. • An elementary text should be written in an informal but mathematically accurate way, illustrated by appropriate graphics. I have tried to formulate mathematical concepts succinctly in language that students can understand. I have minimized the number of explicitly stated theorems and defonitions, preferring to deal with concepts in a more conversational way, copiously illustrated by 299 completely worked out examples. Where appropriate, concepts and results are depicted in 188 figures

Continues 18.100. Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals.

As taught in 2006-2007 and 2007-2008.

Functional analysis begins with a marriage of linear algebra and metric topology. These work together in a highly effective way to elucidate problems arising from differential equations. Solutions are sought in an infinite dimensional space of functions.

This module paves the way by establishing the principal theorems (all due in part to the great Polish mathematician Stefan Banach) and exploring their diverse consequences. Topics to be covered will include:

– norm topology and topological isomorphism;
– boundedness of operators;
– compactness and finite dimensionality;
– extension of functionals;
– weak*-compactness;
– sequence spaces and duality;
– basic properties of Banach algebras.

Suitable for: Undergraduate students Level Four

Dr Joel F. Feinstein
School of Mathematical Sciences

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras.

Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.

Introduction to discrete and computational geometry. Topics covered: planar graphs, geometric graphs, the theory of crossings, extremal graph theory, arrangements of curves and points in the plane (mainly pseudolines and pseudocircles), problems involving distances, Gallai-Sylvester-type problems, Davenport-Schinzel sequences. Emphasis on teaching methods in combinatorial geometry. Many results presented are recent, and include open problems.

A rigorous introduction designed for mathematicians into perturbative quantum field theory, using the language of functional integrals. Basics of classical field theory. Free quantum theories. Feynman diagrams. Renormalization theory. Local operators. Operator product expansion. Renormalization group equation. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and string theory.

The full text of "Geospatial Analysis - A comprehensive guide" is provided on this website. It covers the full spectrum of analytical techniques that are provided within modern Geographic Information Systems (GIS) and related software products.Click here for a PDF extract (first 55 pages).Topics covered in detail include: * Geospatial analysis concepts * Core components of geospatial analysis, including distance and directional analysis, geometrical processing, map algebra, and grid models * Exploratory Spatial Data Analysis (ESDA) and spatial statistics, including spatial autocorrelation and spatial regression * Surface analysis, including surface form and flow analysis, gridding and interpolation methods, and visibility analysis * Network and locational analysis, including shortest path calculation, travelling salesman problems, facility location and arc routing Geocomputational methods, including agent-based modelling, artifical neural networks and evolutionary computing

Examines the development of computing techniques and technology in the nineteenth and twentieth centuries, particularly critical evaluation of how the very idea of "computer" changes and evolves over time. Emphasis is on technical innovation, industrial development, social context, and the role of government. Topics include Babbage, Hollerith, differential analyzers, control systems, ENIAC, radar, operations research, computers as scientific instruments, the rise of "computer science," artificial intelligence, personal computers, and networks. Includes class visits by members of the MIT community who have made important historical contributions. This course focuses on one particular aspect of the history of computing: the use of the computer as a scientific instrument. The electronic digital computer was invented to do science, and its applications range from physics to mathematics to biology to the humanities. What has been the impact of computing on the practice of science? Is the computer different from other scientific instruments? Is computer simulation a valid form of scientific experiment? Can computer models be viewed as surrogate theories? How does the computer change the way scientists approach the notions of proof, expertise, and discovery? No comprehensive history of scientific computing has yet been written. This seminar examines scientific articles, participants' memoirs, and works by historians, sociologists, and anthropologists of science to provide multiple perspectives on the use of computers in diverse fields of physical, biological, and social sciences and the humanities. We explore how the computer transformed scientific practice, and how the culture of computing was influenced, in turn, by scientific applications.

Selection of material from the following topics: calculus of variations (the first variation and the second variation); integral equations (Volterra equations; Fredholm equations, the Hilbert-Schmidt theorem); the Hilbert Problem and singular integral equations of Cauchy type; Wiener-Hopf Method and partial differential equations; Wiener-Hopf Method and integral equations; group theory.

" The fundamental concepts, and approaches of aerospace engineering, are highlighted through lectures on aeronautics, astronautics, and design. Active learning aerospace modules make use of information technology. Student teams are immersed in a hands-on, lighter-than-air (LTA) vehicle design project, where they design, build, and fly radio-controlled LTA vehicles. The connections between theory and practice are realized in the design exercises. Required design reviews precede the LTA race competition. The performance, weight, and principal characteristics of the LTA vehicles are estimated and illustrated using physics, mathematics, and chemistry known to freshmen, the emphasis being on the application of this knowledge to aerospace engineering and design rather than on exposure to new science and mathematics."

A general introduction to manifolds and Lie groups. The role of Lie groups in mathematics and physics. The exponential mapping. Correspondence with Lie algebras. Homogeneous spaces and transformation groups. Adjoint representation. Covering groups. Automorphism groups. Invariant differential forms and cohomology of Lie groups and homogeneous spaces. 18.101 recommended but not required. DThis 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.