<David Kaiser>http://web.mit.edu/dikaiser/www/#Publications

 

8-1-2007 I want to understand Feynman Diagrams. David Kaiser's book, "Drawing Theories Apart: The Dispersion of FEYNMAN DIAGRAMS in Postwar Physics" is enormously helpful. I would very much like to a technical explanation of how the physically motivated complex integrals involved in Feynman diagrams relate to the algebro-combinatorial exegesis provided by John Baez and his colleagues in categorified physics.

 

<cognitive science of mathematics>http://www.answers.com/main/ntquery;jsessionid=5r6tys2qb798?method=4&dsid=2222&dekey=Cognitive+science+of+mathematics&gwp=8&curtab=2222_1&sbid=lc01a

8-1-2007 "The cognitive science of mathematics is the study of mathematical ideas using the techniques of cognitive science. Specifically, it is the search for foundations of mathematics in human cognition."

<philosophy of real mathematics>http://www.dcorfield.pwp.blueyonder.co.uk/htdocs/2005_12_01_archive.html

 

8-1-2007 A gripping blog by David Corfield with strong links to other sites about mathematics, philosophy, and the philosophy of mathematics.

<A Brief History of the Universe>http://math.ucr.edu/home/baez/timeline.html

 

8-1-2007 For example, "50 million years from now - Africa collides with Europe, eliminating Mediterranean sea." This could possibly influence the religio-political problems in the Middle East.

 

<A MOMENT IN THE ZOOM OF A LENS: TOWARDS A DISCURSIVE PSYCHOLOGY OF MATHEMATICS TEACHING AND LEARNING>http://myweb.lsbu.ac.uk/~lermans/PMEpaper.html

8-1-2007 "Psychology has undergone considerable changes in the last decade or so, in line with developments in cultural studies, feminist research, postmodernism etc. In mathematics education research one can see the evidence of such developments, albeit as yet in a limited way. In publications over the last few years I have argued the case for a cultural psychology and for a move away from constructivism. In this paper I attempt to outline a research programme for mathematics education which is firmly based in cultural, discursive psychology. I discuss theoretical issues, refer to some of the important literature, and present some research in aspects of the programme."

 

<A Short Talk about Richard Feynman (2005)>http://www.stephenwolfram.com/publications/talks/feynman/

 

8-1-2007 One genius writes about another. What is a genius? A genius is someone who, first of all,  is prolific, and second, profoundly changes the history of a discipline. When I was growing up it somehow got into my head that a genius is merely someone who has had at least two great ideas. Only recently I came across a story that may account for my childish simplification. It seems that Paul Valery, the polymathic poet, essayist, and philosopher, conversed with Albert Einstein and asked him, "Master, you get up during the night to take note of your ideas on a small pad?" Einstein replied, "Ideas? Oh you know, you get two or three in your life!"

 

<Alexander Grothendieck>http://en.wikipedia.org/wiki/Grothendieck

8-1-2007 "Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century. He is also one of its most extreme scientific personalities, with achievements over a short span of years which are still astounding in their broad scope and sheer bulk, and a lifestyle later in his career which alienated even close followers. He made major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966, and was co-awarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter prize on ethical grounds in a letter to the media."

<Algebraic Topology - Andries Brouwer>http://www.win.tue.nl/~aeb/at/algtop.html#toc4

8-1-2007 "Some fragments of algebraic topology."

<Algebraic and Geometric Topology>http://www.msp.warwick.ac.uk/agt/2007/07/index.xhtml

 

8-1-2007 Many free downloads of mathematics papers in .PDF format.

 

<An Introduction to Galois Theory>http://nrich.maths.org/public/viewer.php?obj_id=1422

8-1-2007 "This is a short introduction to Galois theory. The level of this article is necessarily quite high compared to some Nrich articles, because Galois theory is a very difficult topic usually only introduced in the final year of an undergraduate mathematics degree. This article only skims the surface of Galois theory and should probably be accessible to a 17 or 18 year old school student with a strong interest in mathematics."

<Apollonian Gasket>http://mathworld.wolfram.com/ApollonianGasket.html

8-1-2007 "Consider three mutually tangent circles, and draw their inner Soddy circles. Then draw the inner Soddy circles of this circle with each pair of the original three, and continue iteratively. The steps in the process are illustrated above (Trott 2004, pp. 34-35)."

<Automated Reasoning>http://plato.stanford.edu/entries/reasoning-automated/

8-1-2007 "Reasoning is the ability to make inferences, and automated reasoning is concerned with the building of computing systems that automate this process. Although the overall goal is to mechanize different forms of reasoning, the term has largely been identified with valid deductive reasoning as practiced in mathematics and formal logic. In this respect, automated reasoning is akin to mechanical theorem proving."

<Beyond Confusion: An Assessment Glossary>http://www.aacu.org/peerreview/pr-sp02/pr-sp02reality.cfm

8-1-2007 "Higher education lacks a common vocabulary about assessment; and individuals use terms in mutating ways to refer to varying levels of analysis. Some interpretations imply external oversight or control, an unpleasant idea for most college faculty. Miscommunication and mistrust result from this confused language and are likely to interfere with developing the kind of useful value added assessment proposed by Benjamin, Hersh, and Klein."

<Can Jerry Seinfeld crack P vs NP?>http://weblog.fortnow.com/

8-1-2007 "The following is a quote from Comedian Jerry Seinfeld. The source is Seinfeld Universe: The Entire Domain by Greg Gattuso (Publisher of Nothing: The Newsletter for Seinfeld Fans, page 96.) I was great at Geometry. If I wanted to train someone as a comedian, I would make them do lots of proofs. That's what comedy is: a kind of bogus proof. You set up a fallacious premise and then prove it with rigorous logic. It just makes people laugh. You'll find that most of my stuff is based on that system ... You must think rationally on a completely absurd plane."

<Casselman's visual proof>http://www.ams.org/featurecolumn/archive/visual2.html

8-1-2007 "In this regular pentagon, call the side length s and the length of the diagonal d. Our job is to show that s and d are incommensurable."

<Category Theory>http://plato.stanford.edu/entries/category-theory/

8-1-2007 "Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical physics. Roughly, it is a general mathematical theory of structures and of systems of structures."

<Category theory, homological algebra>http://www.math.niu.edu/~rusin/known-math/index/18-XX.html

8-1-2007 "Category theory, a comparatively new field of mathematics, provides a universal framework for discussing fields of algebra and geometry. While the general theory and certain types of categories have attracted considerable interest, the area of homological algebra has proved most fruitful in areas of ring theory, group theory, and algebraic topology."

<Computers and Assessment in Science Education. ERIC Digest.>http://www.ericdigests.org/1997-1/science.html

8-1-2007 "Research and development efforts in alternative forms of assessment are on the rise in science education, along with growing interest in using computers in science assessment. A review of technological applications in science assessment was provided by Helgeson and Kumar (1993), recent developments in computer scorable, large scale tests were reported by Martinez and Bennett (1992), and a theme issue on computer-based science assessment was prepared by the Journal of Science Education and Technology (Vol. 4, Issue 1, 1995)."

<Dynamic Classroom>http://www.wolfram.com/solutions/precollege/videos/

8-4-2007 "Focused on creating a dynamic classroom, high school math teacher Abby Brown shares some of her work and experiences with Mathematica in videos that show how to incorporate Mathematica into your own teaching."

<Elementary Calculus>http://www.math.wisc.edu/~keisler/calc.html

 

8-1-2007 The magnificient free download of H. Jerome Keisler's textbook. I used this text in my high school calculus class for seniors, and we worked many problems from the early chapters. Student grades were based on class participation, quizzes (with eInstruction clickers), and on term papers that were required to be prepared as WORD documents using Geometer's SketchPad and MathType inserts.

 

<Enhancing Education Through Technology Conference>http://www.doe.mass.edu/edtech/teacher/workshop.html?section=schedule

8-1-2007 "Each spring the Massachusetts Department of Education holds a technology conference. The conference features presentations and exhibits by recipients of technology grants, as well as Department-sponsored technology projects. Details about the 2008 technology conference will be posted here when they are available."

<Galois Theory: Lectures Delivered at the University of Notre Dame>http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197041

 

8-1-2007 Free download in .PDF format of a classic work on Galois Theory. This is a book I studied when I was a graduate student.

 

<Higher-Dimensional Algebra>http://math.ucr.edu/home/baez/hda/

8-1-2007 "John Baez March 27, 2006 I'm just getting started on a project I've been wanting to do for a long time: a book higher-dimensional algebra and its applications to topology, representation theory, combinatorics, and quantum physics. Here's a tentative outline. Some of the sections have links to more detailed outlines of those sections. But, it's all incredibly unfinished."

<Intelligent Tutoring Systems>http://coe.sdsu.edu/eet/Articles/tutoringsystem/start.htm

8-1-2007 "THE NOTION OF INTELLIGENT MACHINES FOR TEACHING PURPOSES can be traced back to 1926 when Sidney L. Pressey built a machine with multiple choice questions and answers. This machine delivered questions and provided immediate feedback to the user. Educational Psychologists have since reported that carefully designed individualized tutoring produces the best learning for most people."

<Introduction to Non-Standard Analysis>http://www.gather.com/viewArticle.jsp?articleId=281474976755323

 

8-1-2007 I high point of my high school mathematics teaching career was teaching calculus to seniors based on Abraham Robinson's theory of infinitesimals. I was particularly happy with H. Jerome Keisler's free download of his magnificent textbook, "Elementary Calculus: An Approach Using Infinitesimals". Fred Kidd gives here a quick introduction.

 

<Joe Mitchell's Hotlist>http://www.ams.stonybrook.edu/~jsbm/hotlist.html

8-1-2007 Ginormous set of links.

<Journal for Research in Mathematics Education>http://my.nctm.org/eresources/journal_home.asp?journal_id=1

8-1-2007 "Promotes and disseminates disciplined scholarly inquiry into the teaching and learning of mathematics at all levels, including research reports, book reviews, and commentaries. Article downloads are free to individual members that subscribe to Journal for Research in Mathematics Education."

<Lindenbaum-Tarski algebra>http://planetmath.org/encyclopedia/LindenbaumAlgebra.html

8-1-2007 "Let L be a first order language."

<MITOPENCOURSEWARE>http://ocw.mit.edu/OcwWeb/index.htm

 

8-1-2007 Astounding resource for the auto-didact, and all other students.

 

<Mathematically Correct>http://www.mathematicallycorrect.com/

8-1-2007 "This web site is devoted to the concerns raised by parents and scientists about the invasion of our schools by the New-New Math and the need to restore basic skills to math education."

<Mathematicians come to life>http://www.oxfordtoday.ox.ac.uk/2005-06/v18n2/04.shtml

8-1-2007 "'One of the intriguing things about mathematics', says Professor Chapman, 'is the way in which equations that model one type of thing can often be adapted to model another. For example, it's the equations used by engineers to model the stresses and strains in buildings that are providing the springboard for some of my biological work on modelling human tissue.' Maini agrees: 'The power of mathematics is that similar mathematical equations can be applied to very, very different areas.'"

<Metacognition in Mathematical Problem Solving>http://www.aare.edu.au/98pap/yea98408.htm

8-1-2007 "The main purpose of the study is to understand the role of metacognition in mathematical problem solving to generate a set of recommendations for classroom instruction. The three specific aims are, firstly, to observe patterns of behaviours when students solve mathematical problems, secondly, to observe the types of metacognitive behaviours during mathematical problem solving, and thirdly, to understand the role of metacognition during problem solving. The thinking aloud method was used to generate data from a group of Year 7 students. Data analysis using a previously developed taxonomy of problem solving behaviours and Flavell's model of cognitive monitoring provide insights into metacognitive aspects of mathematical problem solving. This paper outlines the main research findings."

<Mirek's Cellebration>http://www.mirwoj.opus.chelm.pl/ca/

 

8-1-2007 Free download of a beautiful cellular automata simulator. My mathematics students delight in seeing the variety of examples that I project on the screen, leading for example to discussions about rules. I am particularly enthralled by questions about formalizing the relationship between the rules that generate the patterns, and the rules that the viewer might use to describe the emergent behavior of the patterns.

 

<National Council of Teachers of Mathematics>http://www.nctm.org/

 

8-1-2007 The major richly linked resource website for mathematics teachers.

 

<Popularization and Teaching>http://www.bangor.ac.uk/~mas010/publar.html

8-1-2007 Work of Ronnie Brown.

<Ronald Brown's Home Page>http://www.bangor.ac.uk/~mas010/

8-1-2007 "I have come to be involved in the analysis and development of underlying stuctures across several fields, and computation with these. One such area is the foundations of algebraic topology. This has led to an interest in multiple categories, and the applications of groupoids and multiple groupoids. The aim is make the algebra model better the geometry, so that more computations are possible. The computational side leads to a current interest in rewriting and Groebner bases."

<Standards-based mathematics>http://en.wikipedia.org/wiki/Standards-based_mathematics

8-1-2007 "Mathematics in this style have also been called "standards-based" instruction or "standards-based mathematics[1], or simply "reform mathematics"[2]. Less favorable terminology which have appeared in press and web articles include fuzzy math, "Where's the math"[3], "anti-math"[4], "math for dummies"[5], "no-math mathematics"[6], rainforest algebra [7], "Math for women and minorities, [8] and "new new math".[9] Traditional mathematics education has been called "Parrot Math" by critics. The direct instruction method has been criticized as "drill and kill"."

After three years of teaching algebra and geometry to high school students of whom one fifth are officially categorized as "special needs" students, my conclusion is that an entirely new approach to teaching mathematics is necessary. A series of "basic formal skills games" should be invented that bring about an implicit understanding of the following notions:

• value of numerical expression
• expression re-arrangement without changing value
• formation, transformation, and evaluation of numerical expressions
• distinction between numerical and algebraic expressions
• substitution of numbers for variables in algebraic expressions, thereby converting them to numerical expressions
• the distinction between expressions and statements
• formation, transformation, and evaluation of algebraic statements

Such games could involve all the usual manipulatives, enhanced by use of the T_Rig document camera system for educators and students to share their work and games.

 

<Teacher Resources/Mathematics>http://www.learner.org/resources/series66.html

8-1-2007 "A video instructional series on algebra for college and high school classrooms and adult learners; 26 half-hour video programs and coordinated books."

<The Fundamental Theorem of X>http://www.math.dartmouth.edu/~leejstem/mathematrix/funda.html

 

8-1-2007 My personal list of fundamental theorems of mathematics (idea for a book):

• of Algebra
• of Calculus
• of Galois Theory
• of Sheaf Theory
• of Enumerative Combinatorics
• of Analysis (the "e" theorem)
• of Finite Markov Chains
• of Algebraic Geometry (Nullstellenzatz)
• of Lie Groups
• of Finitely Generated Abelian Groups
• of Algebraic Topology (characteristic)
• of Differential Geometry (Gauss-Bonnet)

 

<The Inflationary Universe>http://csep10.phys.utk.edu/astr162/lect/cosmology/inflation.html

8-1-2007 "The Theory of Inflation: In the corresponding theory of inflation, the Universe, because of properties of elementary particles not accounted for in the standard big bang models, expands for a fleeting instant at its beginning at a much higher rate than that expected for the big bang. This period, which is called the inflationary epoch, is a consequence of the nuclear force breaking away from the weak and electromagnetic forces that it was unified with at higher temperatures in what is called a phase transition. (An example from everyday life of a phase transition is the conversion of ice to liquid water.) This phase transition is thought to have happened about 10-35 seconds after the creation of the Universe. It filled the Universe with a kind of energy called the vacuum energy, and as a consequence of this vacuum energy density (which plays the role of an effective cosmological constant), gravitation effectively became repulsive for a period of about 10-32 seconds. During this period the Universe expanded at an astonishing rate, increasing its size scale by about a factor of 10^50. Then, when the phase transition was complete the universe settled down into the big bang evolution that we have discussed prior to this point. This, for example, means that the entire volume of the Universe that we have been able to see so far (out to a distance of about 18 billion light years) expanded from a volume that was only a few centimeters across when inflation began!"

 

To me this kind of story beats any story of supernatural creation.

 

<The International Journal on Mathematics Education>http://www.emis.de/journals/ZDM/zdmp1.html

8-1-2007 "Zentralblatt für Didaktik der Mathematik (ZDM) is one of the oldest mathematics education research journals. The papers appearing in the six themed issues per year are strictly by invitation only followed by internal peer review by selected members from the editorial board and the international editorial advisory board. The journal exists to survey, discuss and extend current research-based and theoretical perspectives as well as to create a forum for critical analyses of issues within mathematics education. The audience is pre-dominantly mathematics education researchers around the world interested in current developments in the field."

<The Life of a Mathematician>http://www.fermentmagazine.org/rands/recoltes1.html

8-1-2007 "By December 2002, when I began this translation, 14 years had passed since the distribution, in 1988, of copies of Récoltes et Semailles by Alexander Grothendieck to mathematical colleagues around the world."

<The Pythagorean Theorem is Equivalent to the Parallel Postulate>http://www.cut-the-knot.org/triangle/pythpar/PTimpliesPP.shtml

8-1-2007 "The proofs of the Pythagorean Theorem seem to divide into three main types: proofs by shearing, which depend on theorems that the areas of parallelograms (or triangles) on equal bases with equal heights are equal, proofs by similarity, which depend on calculations of proportions of sides of similar triangles, and proofs by dissection, which depend on the observation that the acute angles of a right triangle are complementary. In each case, the argument can readily be traced back to consequences of the Parallel Postulate."

<The origins of Alexander Grothendieck's>http://www.bangor.ac.uk/~mas010/purs-stacks.html

8-1-2007 "In 1983 a new Manuscript by Alexander Grothendieck entitled `Pursuing Stacks' (referred to below as [PS]) began to be distributed from Bangor, UK, and it eventually ran to almost 600 pages. Many publications refer to [PS], which is now being edited by G. Maltsiniotis , for publication by the Société Mathématique de France. This is the story of how this Manuscript came about."

<Visual Explanations in Mathematics>http://www.ams.org/featurecolumn/archive/visual1.html

8-1-2007 "Perhaps the most famous and certainly one of the oldest visual explanations in mathematics is this visual proof of the Pythagorean theorem."

<Wolfram Demonstrations Project>http://demonstrations.wolfram.com/

 "An expanding collection of freely available, interactive Demonstrations in math, science, and many other areas - at all levels, from elementary education to front-line research."

8-2-2007 I have used this wonderful set of Mathematica demonstrations in my high school mathematics classes to stimulate discussions.

<correlation>http://en.wikipedia.org/wiki/Correlation

 

8-1-2007 I maintain student grades in a spreadsheet, and am often interested in having it calculate correlation between, say, student classroom performance and student test performance.

 

<intelligent assessment system>http://www.aleks.com/

8-1-2007 "ALEKS is a web-based, artificially intelligent assessment and learning system. ALEKS uses adaptive questioning to quickly and accurately determine exactly what a student knows and doesn't know in a course. ALEKS then instructs the student on the topics she is most ready to learn. As a student works through a course, ALEKS periodically reassesses the student to ensure that topics learned are also retained. ALEKS courses are very complete in their topic coverage and ALEKS avoids multiple choice questions. A student who shows a high level of mastery of an ALEKS course will be successful in the actual course she is taking."

 

 As a consequence of my work as a high school teacher of Algebra I and Algebra II, I developed the idea that the basic formal skills could be ranked in terms of which ones are prerequisites for others. For example, (at low resolution) basic arithmetic skills are a prerequisite for most algebra skills. Formally, this means basic formal skills can be considered the elements of a partially ordered set. Jean-Claude Falmagne at New York University and the University of California, Irvine, Jean-Paul Doignon at the University of Brussels, and several other scientists created ALEKS and provide at their website a free download of a paper, "The Assessment of Knowledge in Theory and in Practice" in which there are (high resolution) pictures of this partially ordered set.

<A New Kind of Science | ONLINE>http://www.wolframscience.com/nksonline/page-986?firstview=1

8-1-2007 The whole noodle.

<A group blog on math, physics and philosophy>http://golem.ph.utexas.edu/category/2006/09/ringoids.html

8-1-2007 "Ringoids - Posted by John Baez Just as a "group with many objects" is a groupoid, a "ring with many objects" is called a ringoid. Gregory Muller has emailed me a question about ringoids. I don’t want to get into the habit of posting emailed questions on this blog, because I already burnt myself out years ago helping moderate a newsgroup. But just this once, I will. (Famous last words...)"

 

My graduate supervisor and friend, Barry Mitchell, published "Rings with Several Objects" in Advances in Mathematics, Volume 8, Number 1, February 1972, pp. 1-161. I am not sure that John Baez is aware of that.

 

<Algebraic and Geometric Topology>http://www.msp.warwick.ac.uk/agt/2007/07/index.xhtml

8-1-2007 Many free downloads of mathematics papers in .PDF format.

<Andy Wuensche>http://www.cogs.susx.ac.uk/users/andywu/index.html

8-1-2007 Author of "The Global Dynamics of Cellular Automata".

<Barry Jay's Homepage>http://www-staff.it.uts.edu.au/~cbj/

8-1-2007 "pattern-matching calculi."

<Brown's representability theorem>http://www.answers.com/topic/brown-s-representability-theorem#copyright

8-1-2007 "In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions on a contravariant functor F on the homotopy category Hot of pointed CW complexes, to the category of sets Set, to be a representable functor. "

<Bulletin of the American Mathematical Society>http://www.ams.org/jourcgi/jrnl_toolbar_nav/bull_all

 

8-1-2007 Free downloads of mathematics papers in .PDF format.

 

<Centre for the Popularization of Mathematics>http://www.popmath.org.uk/centre/

8-1-2007 Symbolic sculpture, Sculpture Maths, Knots Exhibition, Raising Public Awareness of Mathematics.

<Classifying space of a discrete group>http://en.wikipedia.org/wiki/Classifying_space

8-1-2007 "For a discrete group G, BG is, roughly speaking, a path-connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial."

<Classifying spaces of categories>http://www.tac.mta.ca/tac/volumes/9/n5/9-05abs.html

8-1-2007 "In this paper we show how collapsing schemes can give us information on the homotopy type of the classifying space of a small category, when this category is presented by a complete rewrite system."

<Complexity Taxonomy>http://www.public.iastate.edu/~crb002/combData/compTax.html

8-1-2007 "One of my favorite websites is the Complexity Zoo. I had a little too much time on my hands and made a graphical taxonomy."

<Complexity Zoo>http://qwiki.caltech.edu/wiki/Complexity_Zoo

8-1-2007 "Welcome to the Complexity Zoo... There are now 462 classes and counting!"

<Covering space>http://en.wikipedia.org/wiki/Universal_cover

 

8-1-2007 A beautiful part of mathematics.

 

<Curry-Howard Correspondence>http://en.wikipedia.org/wiki/Curry-Howard_isomorphism

8-1-2007 "The Curry-Howard correspondence is the close relationship between computer programs and mathematical proofs; the correspondence is also known as the Curry-Howard isomorphism, or the formulae-as-types correspondence. A number of different formulations have been used, for a principle now identified as the independent discovery of the American mathematician Haskell Curry and logician William Alvin Howard."

<Dr. Christopher Francis Townsend>http://www.christophertownsend.org/

 

8-1-2007 Rich collection of category theory research papers available for download in .PDFs.

 

<F. William Lawvere>http://www.acsu.buffalo.edu/~wlawvere/

 

8-1-2007 Rich collection of free downloads of papers by a revered leader among category theorists. I was briefly a student of Professor Lawvere and can report that however much I ever wanted to invent "mathematical metaphysics," Professor Lawvere is the creator of what Professor Eilenberg told me (at a party around 1974 at Myles Tierney's loft in SOHO, New York City) is a better challenge, "metaphysical mathematics."

 

<Favorite Theorems: The Polynomial-Time Hierarchy>http://weblog.fortnow.com/2005/06/favorite-theorems-polynomial-time.html

8-1-2007 "The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space by Albert Meyer and Larry Stockmeyer, FOCS (then called SWAT) 1972. The title result of this paper gave an early example of a natural problem that provably does not have an efficient algorithm. But it is the second half of the paper that developed one of the most important concepts in computational complexity."

<Free Technical Books>http://www.freetechbooks.com/index.php

8-1-2007 "This site lists free online computer science, engineering and programming books, textbooks and lecture notes, all of which are legally and freely available over the Internet."

<Geometric Langlands Program>http://www2.math.northwestern.edu/langlands/index_moreinfo.htm

"This program is dedicated to the investigation of the geometric Langlands, its relationship to other areas of mathematics, and its relationship to physics." "The Langlands Program, launched by Robert Langlands in the late 60's, ties together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic functions. The Langlands conjecture predicts that there is a correspondence between n-dimensional representations of the Galois group of a number field and automorphic representations of the group GL(n) over the ring of adeles of this field. This conjecture has an analogue when the number field is replaced by the field of functions on a smooth projective curve defined over a finite field. In this setting, this conjecture has a geometric version, called the geometric Langlands correspondence. The role of the Galois group is now played by the fundamental group of the curve X, and the role of the automorphic representations is played by the so-called Hecke eigensheaves on the moduli spaces of G-bundles on X. This reformulation allows one to consider curves defined over arbitrary fields, such as the field of complex numbers."

<Higher-Dimensional Algebra: A Language for Quantum Spacetime>http://math.ucr.edu/home/baez/quantum_spacetime/

8-1-2007 "Category theory is a general language for describing things and processes - called "objects" and "morphisms". In this language, the counterintuitive features of quantum theory turn out to be properties that the category of Hilbert spaces shares with the category of cobordisms - in which objects are choices of "space", and morphisms are choices of "spacetime". The striking similarities between these categories suggests that "n-categories with duals" are a promising framework for a quantum theory of spacetime. We sketch the historical development of these ideas from Feynman diagrams, to string theory, topological quantum field theory, spin networks and spin foams, and especially recent work on open-closed string theory, quantum gravity coupled to point particles, and 4d BF theory coupled to strings."

<Higher-Gauge Theory and Elliptic Cohomology>http://golem.ph.utexas.edu/category/

8-1-2007 "After some fun in Greece, I’ve been holed up in Greenwich the last two days preparing my talk for the 2007 Abel Symposium. This is an annual get-together sponsored by the folks who put out the Abel prize, a belated attempt to create something like a Nobel prize for mathematicians."

 

One thing I get from reading this blog is a wonderful sense of how mathematical physics ideas are almost like jokes that come up in conversations between friends. Someone will say something that may or may not click, then someone else picks up on it and elaborates on it, and pretty soon people are rolling on the floor laughing.

 

<Joseph Goguen>http://www.cse.ucsd.edu/users/goguen/

 

8-1-2007 A most prolific computer scientist.

 

<List of conjectures>http://en.wikipedia.org/wiki/List_of_conjectures

8-1-2007 Are you looking for a problem to work on?

<Logical Methods in Computer Science>http://www.lmcs-online.org/index.php

 

8-1-2007 Free downloads of papers in .PDF format.

 

<Lost Causes in Theoretical Physics>http://www.mth.kcl.ac.uk/~streater/lostcauses.html

8-1-2007 "This page contains some remarks about research topics in physics which seem to me not to be suitable for students. Sometimes I form this view because the topic is too difficult, and sometimes because it has passed its do-by date. Some of the topics, for one reason or another, have not made any convincing progress."

<Mathematica Notation: Past and Future>http://www.stephenwolfram.com/publications/talks/mathml/mathml2.html

 

8-1-2007 Everyone knows that Stephen Wolfram is a genius. Here is a typically deep and thorough discussion of the history of mathematical notation, using Mathematica code to generate illustrations.

 

<Mathematics Preprint List 2006>http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/06/

 

8-1-2007 Free downloads of mathematics papers in .PDF format.

 

<Natural deduction>http://en.wikipedia.org/wiki/Natural_deduction

8-1-2007 "In mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it "naturally" occurs."

<Not Even Wrong>http://www.math.columbia.edu/~woit/wordpress/?p=352

 

8-1-2007 What an interesting blog this is.

 

<Papers of Paul Renteln>http://physics.csusb.edu/~prenteln/papers/papers.html

8-1-2007 "16. P. Renteln and A. Dundes, ``Foolproof: A Sampling of Mathematical Folk Humor", Notices of the AMS, 52 (2005) 24-34. pdf"

<The Theory of Multidimensional Persistence>http://www.cs.dartmouth.edu/~afra/papers.html

 

8-1-2007 Numerous interesting applications of algebraic topology.

 

<The work of Robert Langlands>http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/intro.html

8-1-2007 "Looking forward to eventual publication of his collected works, Professor Langlands is cooperating with us in providing on our site a large selection from his professional correspondence and previously unpublished work, as well as an almost complete collection of all of his published work." "My office in Ankara was next to that of Cahit Arf, and when I mentioned the question to him, he drew my attention to a paper of Hasse that had appeared in a journal not widely read, the Acta Salmanticensia of 1954. He fortunately had a reprint. So I could begin to think seriously about the matter. The critical idea came in April 1968 in a hotel room in Izmir, where I had gone to deliver a lecture. It was the understanding that all identities needed were consequences of four basic ones, formulated in the notes as the four main lemmas. Once this is understood and basic facts about Gauss sums are understood, as in the papers of Lamprecht and Davenport-Hasse, three of these four identities are not so difficult to establish. The second main lemma turned out, on the other hand, to be a major obstacle. Fortunately, as I discovered while leafing idly through journals in the library, either in Ankara or later in New Haven (I no longer remember), I came across Dwork's paper in which the first and the second main lemmas were proved."

<This Week's Finds in Mathematical Physics (Week 223)>http://math.ucr.edu/home/baez/week223.html

8-1-2007 "John Baez This week I'd like to talk about two aspects of higher gauge theory: p-form electromagnetism and nonabelian cohomology. Lurking behind both of these is the mathematics of n-categories, but I'll do my best to hide that until the end, to build up the suspense."

 

John Baez is one of the most prolific writers of intuitively helpful mathematics and physics.

 

<Universal Algebra and Diagrammatic Reasoning>http://math.ucr.edu/home/baez/universal/

8-1-2007 "Since the introduction of category theory, the old subject of "universal algebra" has diversified into a large collection of frameworks for describing algebraic structures. These include "monads" (formerly known as "triples"), the "algebraic theories" of Lawvere, and the "PROPs" of Adams, Mac Lane, Boardman and Vogt. We give an overview of these different frameworks, which are closely related, and explain how one can reason diagrammatically about algebraic structures defined using them. Our treatment of monads focuses on the abstract "bar construction". Our treatment of algebraic theories and PROPs explains how the latter are related to Feynman diagrams, and leads up to an adjunction between algebraic theories and PROPs which is analogous to the relation between classical and quantum physics. We conclude with some reflections on how features of our physical universe have influenced our notions of universal algebra."

<What is Formal Topology>http://www.3wftop.math.unipd.it/formal-topology.html

 

8-1-2007 It is almost as though there is a conspiracy to rid mathematics of points.

 

<arXiv.org>http://www.arxiv.org/

8-1-2007 "Open access to 432,006 e-prints in Physics, Mathematics, Computer Science, Quantitative Biology and Statistics"

<classifying space of a category>http://www.answers.com/classifying%20space%20of%20a%20category

 

8-1-2007 I still could not tell you the story of what a "classifying space" is, but this website has plenty of links for finding out.

 

<forcing>http://www.nationmaster.com/encyclopedia/Forcing_(mathematics)

 

8-1-2007 I always wanted to understand "forcing" and still cannot tell you the story.

 

<seeking the number of partial orders on a finite set>http://www.math.niu.edu/~rusin/known-math/99/part_ord

8-1-2007 "Sloane's sequences A000112 and A001035 (see http://akpublic.research.att.com/~njas/sequences/SA.html) give this information. Which one you want depends on whether you want to count them upto isomorphism or not. There are lots of references on enumeration of posets there as well (and also in the poset section of http://akpublic.research.att.com/~njas/sequences/classic.html, but these are probably the same references)."

<stuff, structure and properties>http://math.ucr.edu/home/baez/qg-spring2004/

8-1-2007 "The goal of this course was to understand the category-theoretic foundations of the combinatorics of Feynman diagrams. To keep things simple, we considered only Feynman diagrams for the perturbed quantum harmonic oscillator, instead of a full-fledged quantum field theory. But to handle even this simple case, we needed some interesting category theory."

<Advanced Quantum Mechanics II>http://www.phys.ualberta.ca/~gingrich/phys512/latex2html/phys512.html

 

8-1-2007 The whole book.

 

<Asymptotic expansion>http://en.wikipedia.org/wiki/Asymptotic_expansion

8-1-2007 "In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point."

<Asymptotic expansion of Feynman integrals near threshold>http://www.citebase.org/abstract?id=oai:arXiv.org:hep-ph/9711391

 

8-1-2007 Free download of paper in .PDF format.

 

<Books and Papers>http://math.ucr.edu/home/baez/papers.html

 

8-1-2007 Free downloads of papers by John Baez in .PDF format.

 

<Communications in Mathematical Physics>http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104288298

 

8-1-2007 Free downloads of papers in .PDF format.

 

<Feynman Diagrams>http://www2.slac.stanford.edu/vvc/theory/feynman.html

 

8-1-2007 Babytalk.

 

<Feynman diagram>http://encyclopedia.thefreedictionary.com/Feynman%20diagram

8-1-2007 "A Feynman diagram is a method for performing calculations in quantum field theory, invented by American physicist Richard Feynman."

<Finite-dimensional Feynman Diagrams>http://www.ams.org/featurecolumn/archive/feynman4.html

8-1-2007 "Calculating high-order derivatives of a function like ... can be very messy. A useful theorem reduces the calculation to combinatorics."

<From Finite Sets to Feynman Diagrams (2000)>http://citeseer.ist.psu.edu/baez00from.html

8-1-2007 "Abstract: `Categorification' is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with elementary arithmetic, where the category of finite sets serves as a categorified version of the set of natural numbers, with disjoint union and Cartesian product playing the role of addition and multiplication. We sketch how categorifying the integers leads naturally to the infinite loop space..."

<Hestenes Physics Education Research Papers>http://modeling.la.asu.edu/R&E/Research.html

 

8-1-2007 Free downloads of papers in .PDF format.

 

<John Baez's Stuff>http://math.ucr.edu/home/baez/

8-1-2007 "I'm a mathematical physicist. I specialize in quantum gravity and n-categories, but I'm interested in many other things too. I like talking about them here."

<Quotations by Mark Kac>http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Kac.html

8-1-2007 "There are surely worse things than being wrong, and being dull and pedantic are surely among them."

<Re: Feynman diagrams "unphysical">http://www.lns.cornell.edu/spr/2003-09/msg0053688.html

8-1-2007 "It seems to me that Feynman diagrams are merely a methematical trick,  to reduce an intractible integral to a tractable power series."

<Teaching Physics With the Physics Suite>http://www2.physics.umd.edu/~redish/Book/

8-1-2007 "These are close-to-final drafts of the text."

<The Franck-Hertz experiment>http://www.klingereducational.com/products/modern_physics_experiments/franck-hertz_experiments/franck-hertz_experiments.html

8-1-2007 "The Franck-Hertz experiment, (1913, Nobel Prize 1926), with the well defined periodic and equidistant minima and maxima of the collector electrode current, is undoubtedly one of the most impressive experiments to demonstrate and verify the quantum theory."

<The Nonrelativistic Propagator>http://www.phys.ualberta.ca/~gingrich/phys512/latex2html/node74.html

 

8-1-2007 In physics the three topics that I always come back to, because I do not understand them, are special relativity, Feynman diagrams, and thermodynamics.

 

<The Principle of Equivalence>http://csep10.phys.utk.edu/astr162/lect/cosmology/equivalence.html

 

8-1-2007 When I was thirteen years old (1955) living in Brooklyn, New York I had a circle of friends all interested in science. Among them was "Bobby" Reasenberg. One day he loaned me a book called "Giant Brains" by Edmund C. Berkeley. I was already interested in my mind/brain so this was a thrilling read. My mother observed my passion and actually located Mr. Berkeley and brought me to visit his office/workshop in Greenwich Village. He hired me as a laboratory assistant, and very kindly was a mentor to me. Thus began my career as a computer engineer. My family moved away from Brooklyn when I was fourteen, and the group of friends separated. Fast forward fifty years. While on a job search I came into contact with a person who, though not in that circle, was an epi-cycle upon it. He put me back in touch with Bob, who is a scientist at the Harvard-Smithsonian Center for Astrophysics. Our friendship has been renewed, and he is performing interesting experimental work on The Principle of Equivalence.

Principle    Of     Equivalence    Measurement   
 

<The Role of Physics Education Research in Reforming Undergraduate Education> http://www2.physics.umd.edu/~redish/Talks/Revitalize/sld001.htm

 

8-1-2007 Slideshow.

 

<Various Physics Education Links>http://www.physics.ucsb.edu/~circus/links.html

 

8-1-2007 A rich collection of links.

<Educational image display device>U.S. Patent 5863209

8-4-2007 "An image display device preferably used for educational applications is disclosed. In the image display device, an instructor gives a lecture while seeing an original document without being disturbed by any element of the device. In addition, a video camera, which is detachably attached to the camera mount, may be tilted to the left or right or tilted upward or downward so that it is not required to move the original document when the left, right, upper or lower side of the document has to be picked up by the camera. The above display device also includes a means used as either a whiteboard or an OHP-filmed image revealing panel. The device thus improves educational effect when it is used for educational applications. The device is not expensive thus being widely used for educational applications without forcing excessive costs on the owner."

The vertical movement of the camera in this device is by means of a rack-and-pinion inside the vertical column, unlike the T_Rig Camera Bracket, in which vertical movement is a friction slide (with adjustable level of friction). Also, unlike this device, the T_Rig Camera Bracket clamps to the edge of any desk, table, or rolling cart.