The n-Category Café

A group blog on math, physics and philosophy
David Corfield: Mathematical Emotion (November 19, 2009)
Does mathematical writing express emotion?
Alexander Hoffnung: Cobordism and Topological Field Theories Week 3 (November 18, 2009)
Week 3 of the cobordism and TFT seminar introduces symmetric monoidal categories and TFT’s.
Mike Shulman: Combinatorial-Game Categories (November 16, 2009)
Joyal’s category of Conway games is the initial “combinatorial-game category.” What is the terminal one?
David Corfield: A Rose by Any Other Name (November 16, 2009)
Getting Ola Bratteli’s name right
John Baez: The 1000th Post on the n-Category Cafe (November 13, 2009)
Meet our new hosts: Alex Hoffnung, Tom Leinster, Mike Shulman and Simon Willerton!
John Baez: This Week’s Finds in Mathematical Physics (Week 283) (November 11, 2009)
See galaxies in visible, infrared and ultraviolet light. Read about who first discovered the regular icosahedron: Theatetus or the ancient Scots. And learn more about the geometry of the icosahedron!
John Baez: Courant Algebroids From Categorified Symplectic Geometry (November 10, 2009)
Read a draft of a new paper that gets Courant algebroids from 2-plectic manifolds.
David Corfield: Fraïssé Limits (November 6, 2009)
How should we understand a Fraïssé Limit category theoretically?
John Baez: Who Discovered the Icosahedron? (November 4, 2009)
Was it Theaetetus or the Scots?
Urs Schreiber: Notions of Space (November 4, 2009)
A survey of Jacob Lurie’s “Structured Spaces”.
Tom Leinster: An Adventure in Analysis (November 4, 2009)
How a problem about magnitude of metric spaces was solved.
David Corfield: The Arrow of Time in Cat (November 3, 2009)
Does the same broken symmetry that affects Set affect Cat?
John Baez: Interview with Manin (November 2, 2009)
Read a new interview with Yuri Manin.
Mike Shulman: Generalized Operads in Classical Algebraic Topology (October 30, 2009)
Two obscure-seeming kinds of generalized operad help make categorical sense out of two constructions in classical algebraic topology.
John Baez: This Week’s Finds in Mathematical Physics (Week 282) (October 29, 2009)
In week282 of This Week’s Finds, visit Mercury: Learn how this planet’s powerful magnetic field interacts with the solar wind to produce flux transfer events and plasmoids. Then read about the web of connections between associative, commutative, Lie and…
John Baez: Math: Folk Wisdom in an Electronic Age (October 26, 2009)
People are starting to debate the best ways to use the internet to gather, store and access ‘folk wisdom’ about mathematics.
Urs Schreiber: Mathematical Foundations of Quantum Field and Perturbative String Theory (October 24, 2009)
A Call for Comments on a Call for Papers on an issue on Mathematical Foundations of Quantum Field and Perturbative String Theory.
John Baez: Cheaper Online Textbooks? (October 22, 2009)
The U.S. Department of Education is spending $300,000 to develop online textbooks. But how about free online textbooks?
Alexander Hoffnung: Cobordism and Topological Field Theories Week 2 (October 22, 2009)
Week 2 of the cobordism and TFT seminar introduces oriented manifold and cobordisms in a more formal way.
Mike Shulman: Aesthetics of Commutative Diagrams (October 21, 2009)
How to lay out a large commutative diagram aesthetically?
John Baez: This Week’s Finds in Mathematical Physics (Week 281) (October 20, 2009)
In “week281” of This Week’s Finds, learn about the newly discovered ring of Saturn, tilings with 5-fold and 10-fold quasisymmetry, and the latest news on quantum gravity from Corfu.
Mike Shulman: Syntax, Semantics, and Structuralism, I (October 19, 2009)
A brief summary of syntax and semantics of type theory and logic, as a basis for continuing discussions about structural and material foundations.
Simon Willerton: Halmos on Writing Mathematics (October 19, 2009)
Over in a discussion at Math Overflow I was reminded about Halmos’ great article on writing mathematics, which I highly recommend to all graduate students (or anyone else, for that matter). P. R. Halmos, How to write mathematics, L’Enseignement…
David Corfield: Math Overflow (October 19, 2009)
The new math question and answer site
John Baez: Girih by Egan (October 16, 2009)
If you like quasicrystals, check out this new applet by Greg Egan.
Alexander Hoffnung: Seminar on Cobordism and Topological Field Theories at UCR (October 15, 2009)
Lecture 1 of the Seminar on Cobordism and Topological Field Theories at UCR.
David Corfield: A Dual for Set? (October 14, 2009)
Question about structural set theory
Urs Schreiber: Path-Structured Smooth (∞,1)-Toposes (October 13, 2009)
A talk on a context for higher synthetic differential geometry.
Alexander Hoffnung: MSRI Workshops on Knot Homology Theories (October 11, 2009)
Announcement of MSRI workshops on knot homology theories.
Simon Willerton: More Magnitude of Metric Spaces and Problems with Penguins (October 10, 2009)
Learn about the tenuous link between emperor penguins and the magnitude of metric spaces.
Mike Shulman: Associativity Data in an (∞,1)-Category (October 8, 2009)
Guest post by Emily Riehl A popular slogan is that (∞,1)-categories (also called quasi-categories or ∞-categories) sit somewhere between categories and spaces, combining some of the features of both. The analogy with spaces is fairly clear, at least to…
John Baez: Borisov on Shapes of Cells in n-Categories (October 6, 2009)
Can you comment on a paper about comparing definitions of $n$-categories?
John Baez: Israel Gelfand, Sept. 2nd 1913 – Oct. 5th 2009 (October 6, 2009)
Israel Gelfand died on Monday October 5th, at the age of 96.
David Corfield: Shell What? (October 5, 2009)
My ol blog pages
Alexander Hoffnung: Special Issue on “Categorification in Representation Theory” (October 2, 2009)
Announcement of Special Issue on “Categorification in Representation Theory”
David Corfield: Steve Lack Wins Medal (October 2, 2009)
Steve Lack wins Australian Mathematics Society medal.
John Baez: Sex Advice From Mathematicians (October 1, 2009)
The most amusing email of the day.
Urs Schreiber: D’Auria-Fré-Formulation of Supergravity (September 30, 2009)
An nLab entry on the D’Auria-Fre formulation of supergravity and its interpretation as a higher gauge theory.
John Baez: What Do Mathematicians Need to Know About Blogging? II (September 30, 2009)
Help me write an opinion piece on math blogging for the AMS Notices!
John Baez: This Week’s Finds in Mathematical Physics (Week 280) (September 28, 2009)
Hear about the courses Ashtekar and Rovelli taught at the quantum gravity summer school at Corfu.
David Corfield: The Mathematical Vocation (September 28, 2009)
Collingwood on the value of the pure mathematician.
John Baez: Questions on n-Curvature (September 26, 2009)
Some questions for Urs.
John Baez: Homotopy Theory and Higher Algebraic Structures at UC Riverside (September 24, 2009)
We’re having a special session on homotopy theory and higher algebraic structures here at UCR on November 7th and 8th. I hope you can come by!
John Baez: Movie Trivia Question (September 24, 2009)
A trivia question involving Star Trek and two Bogart movies.
Urs Schreiber: A Seminar on a Geometric Model for TMF (September 21, 2009)
A seminar on geometric models for tmf cohomology theory by Stephan Stolz and Peter Teichner.
David Corfield: Proof by Coinduction (September 18, 2009)
How similar is coinduction to induction?
Urs Schreiber: A Seminar on Gromov-Witten Invariants (September 17, 2009)
A mathematician’s seminar on Gromov-Witten theory.
Urs Schreiber: nLab - Latest Changes (September 17, 2009)
Trying to install better software support for logging latest changes to the nLab.
Urs Schreiber: Inbound Citations (September 15, 2009)
A list of citations of entries at the nCafé blog and/or at the nLab wiki.
John Baez: Towards a Computer-Aided System for Real Mathematics (September 10, 2009)
Arnold Neumaier is seeking useful comments on his proposal for a Formal Mathematical Language for doing and communicating mathematics on the computer.
Urs Schreiber: Interactive Science Symposium (September 8, 2009)
A sympositum on the use of web technology for communication among scientists.
John Baez: The Pi Calculus II (September 8, 2009)
Mike Stay tries to explain the pi calculus to John Baez.
David Corfield: Coalgebraic Modal Logic (September 7, 2009)
Putting together ideas of modal logic and coalgebra with ways of modelling first-order theories.
Urs Schreiber: A Seminar on a Survey of Elliptic Cohomology (September 6, 2009)
An nLab entry accompanying a seminar on elliptic cohomology.
John Baez: This Week’s Finds in Mathematical Physics (Week 279) (September 6, 2009)
In week279, read about division algebras and supersymmetry - and a bit about tuning systems and infinitely groupoidified logic.
Urs Schreiber: Traces in Bicategories (September 4, 2009)
An article by Ponto and Shulman on traces in bicategories.
David Corfield: Where Have All the Solitons Gone? (September 2, 2009)
A request for examples of applied soliton theory.
John Baez: Division Algebras and Supersymmetry (September 1, 2009)
Learn how special properties of the reals, complexes, quaternions and octonions make supersymmetry tick in dimensions 3, 4, 6, and 10.
Urs Schreiber: nLab Migration Done (September 1, 2009)
The nLab wiki has successfully migrated to a better server.
John Baez: SSE Composite Index Bubble II (September 1, 2009)
Was Didier Sornette right in predicting a plunge in the Shanghai Stock Exchange (SSE) Composite Index?
Alexander Hoffnung: Ben-Zvi’s Lectures on Topological Field Theory III (September 1, 2009)
This is the third set of notes on David Ben-Zvi’s lectures on topological field theory at Northwestern University.
John Baez: This Week’s Finds in Mathematical Physics (Week 278) (August 30, 2009)
In week278, hear the latest about Betelgeuse. Read how red supergiants spew out dust which eventually forms planets like ours. Watch a hypervelocity collision in a distant solar system. Learn the new way to make graphene, and read my history of the Earth - for physicists. And when you’re ready: dive into groupoidification!
David Corfield: Feynman the Babylonian (August 27, 2009)
Feynman’s concerns about an axiomatic approach
John Baez: Higher-Dimensional Algebra VII: Groupoidification (August 23, 2009)
Check out our new paper “Higher-dimensional algebra VII: groupoidification”.
Urs Schreiber: Notes on Differential Nonabelian Cohomology (August 21, 2009)
Notes on differential nonabelian cohomology.
Urs Schreiber: nLab Migration (August 19, 2009)
Request for help with testing the migrated nLab wiki installation.
David Corfield: Chasing around the Triangle (August 17, 2009)
Steve Awodey compares foundational frameworks: sets, types and categories.
John Baez: The Pi Calculus (August 15, 2009)
The pi calculus is a process calculus. But what does that mean?
John Baez: Asymptotics of the Magnitude of Metric Spaces (August 13, 2009)
Learn about Tom Leinster and Simon Willerton’s new paper ‘On the asymptotic magnitude of subsets of Euclidean space’.
John Baez: Dan Ghica on Software vs Hardware (August 13, 2009)
A few notes on Dan Ghica’s ideas about ‘function interface models for hardware compilation’ — a potentially very practical application of symmetric monoidal closed categories.
John Baez: This Week’s Finds in Mathematical Physics (Week 277) (August 12, 2009)
In “week277”, find out what’s a million times thinner than paper, stronger than diamond, a better conductor than copper, and absorbs exactly pi times the fine structure constant of the light you shine through it.
Urs Schreiber: Question about ∞-Colimits (August 10, 2009)
What can one say about hocolimits over domains that are themselves hocolimits?
John Baez: What Do Mathematicians Need to Know About Blogging? (August 9, 2009)
You can help me write an opinion piece on math blogging for the AMS Notices.
John Baez: Computation and the Periodic Table II (August 8, 2009)
I’m giving a talk at LICS 2009 – this year’s annual conference on Logic in Computer Science.
Urs Schreiber: Question on Synthetic Differential Forms (August 3, 2009)
On a realization of differential forms as cosimplicial generalized smooth algebras.
John Baez: Groupoidfest 09 (August 2, 2009)
Groupoidfest 09 is being held on October 24–25, 2009 at the Department of Mathematics of the University of Colorado, Boulder, and it’s being organized by Arlan Ramsay.
John Baez: Question About Exotic Smooth Structures (July 30, 2009)
I’ve heard an interesting story about using Khovanov homology to help prove the existence of exotic smooth structures in 4 dimensions. Could you help check to see if it’s correct?
Urs Schreiber: Question on Models for (∞,1)-Functor Categories (July 29, 2009)
How and under which conditions are (∞,1)–functor categories modeled my simplicial model structures on functor categories.
John Baez: Generalizing “One-to-One” and “Onto” (July 28, 2009)
Jonas Frey has settled some questions about the 2-categorical duals of the notions of ‘full’ and ‘fully faithful’.
John Baez: Arithmetic Geometry at the Newton Institute (July 24, 2009)
There will be an introductory workshop on non-abelian fundamental groups in arithmetic geometry at the Newton Institute from July 27th to 31st, 2009.
Urs Schreiber: Verity on Descent for Strict ω-Groupoid Valued Presheaves (July 23, 2009)
Dominc Verity characterizes the descent condition for infinity-groupoid valued presheaves that happen to take values in strict infinity-groupoids.
Urs Schreiber: nLab – How to get started (July 23, 2009)
Quick easy information on how to make use of the nLab.n
John Baez: Thomas Noll’s Talks at Chicago on Mathematical Music Theory (July 23, 2009)
Thomas Fiore has written a guest post about Thomas Noll’s talks on mathematical music theory.
John Baez: SSE Composite Index Bubble? (July 21, 2009)
The geophysicist Didier Sornette and his coauthors predict that a bubble in the SSE will burst between July 17th and July 27th, with 60% probability.
John Baez: The Monads Hurt My Head — But Not Anymore (July 19, 2009)
I’m ready to learn some more advanced topics in the theory of monads. Here’s a list of them. Can you help me out?
David Corfield: Being Tentative on nLab (July 17, 2009)
Is nLab the best place to discuss tentative ideas?
John Baez: A Puzzle From Gavin Wraith (July 16, 2009)
A puzzle by Gavin Wraith, who found a wonderfully high-powered solution.
John Baez: Searching for a Video Proof of “Seven Trees in One” (July 16, 2009)
James Propp is searching for a video proof that he saw once.
John Baez: Ben Zvi’s Lectures on Topological Field Theory II (July 14, 2009)
This is the second set of notes on David Ben-Zvi’s lectures on topological field theory at Northwestern University.
John Baez: Categorifying Nicomachus of Gerasa’s Equation (July 13, 2009)
Can you find a bijective proof of a formula for the sum of cubes of the first $n$ natural numbers?
John Baez: Ben-Zvi’s Lectures on Topological Field Theory I (July 8, 2009)
These are the first of some notes of David Ben-Zvi’s lectures at a workshop on topological field theories at Northwestern University, held in May 2009.
John Baez: A Prehistory of n-Categorical Physics II (July 7, 2009)
I’m back to writing that Prehistory of n-Categorical Physics. I want to finish it before I die.
David Corfield: Generalized Homotopy Theory (July 6, 2009)
On doing the parallel generalization on homotopy as has been done on cohomology.
John Baez: Course on Topological Quantum Field Theory in Almería (July 5, 2009)
There will be a school on TQFTs in Almería, Spain from October 19th to 23rd, 2009.
John Baez: Open Access to Taxpayer-Funded Research (July 2, 2009)
n-Café regulars will know about Representative Conyer’s bill that would repeal the National Institute of Health’s public access policy and forbid other US funding agencies from mandating open access to research papers written with the help of federal grant…
John Baez: Elsevier Pays for Favorable Book Reviews (July 2, 2009)
Elsevier offered gift certificates to academics to write 5-star reviews of a book on Amazon and Barnes & Noble.
Urs Schreiber: Laubinger on Lie Algebras for Frölicher Groups (July 1, 2009)
Martin Laubinger studies Lie algebras of groups with generalized smooth manifold structure: Froelicher groups.
John Baez: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture (June 30, 2009)
Springer Verlag has published a book purporting to give elementary proofs of Fermat’s Last Theorem and Goldbach’s Conjecture.
John Baez: This Book Needs a Title (June 26, 2009)
Peter May and I are almost done editing a volume of survey articles that should help people get going on higher categories. What should we call it?
David Corfield: Cohomology and Homotopy (June 26, 2009)
On the duality between cohomology and homotopy
John Baez: Hopf Algebras in Luxembourg (June 20, 2009)
On July 13th to 17th 2009 there will be a conference and school on Hopf algebras in Luxembourg.
John Baez: This Week’s Finds in Mathematical Physics (Week 276) (June 20, 2009)
In “week276”, read about Betelgeuse, the Local Bubble, the Loop I Bubble, the cloudlets from Sco-Gen, and the “local fluff”. Get to know the nLab. And learn how Paul-André Mélliès and Nicolas Tabareau have taken some classic results of Lawvere on algebraic theories and generalized them to other kinds of theories, like PROPs.
David Corfield: Kan Lifts (June 20, 2009)
Where is the dual of thenotion of Kan extensions?
David Corfield: Accessible Even to a Philosopher (June 16, 2009)
Frenkel’s new expository paper
John Baez: 2-Branes in 11 Dimensions (June 15, 2009)
A question about 2-branes in 11 dimensions.
John Baez: This Week’s Finds in Mathematical Physics (Week 275) (June 14, 2009)
Read about progress on proving the Cobordism Hypothesis.
John Baez: Final Exams Again (June 10, 2009)
What’s the funniest thing you’ve seen on a final exam this year?
Urs Schreiber: Strings, Fields, Topology in Oberwolfach (June 8, 2009)
Notes and material on Oberwolfach workshop on Strings, Fields and Topology
John Baez: Algebraic Geometry for Category Theorists (June 8, 2009)
You can now see the introduction to a paper James Dolan is writing about algebraic geometry for category theorists. You can also see 5 lectures he gave on this topic.
John Baez: Nonabelian Algebraic Topology (June 6, 2009)
Ronald Brown, Philip J. Higgins and Rafael Sivera have come out with a book called Nonabelian algebraic topology: homotopy groupoids and filtered spaces. You can download it for free.
David Corfield: The Elusive Proteus (June 5, 2009)
Continuing the thought that mathematics and physics share in having difficult to articulate general principles.
David Corfield: Mathematical Principles (June 3, 2009)
Cassirer describes certain physical principles. Are there corresponding ones in mathematics?
John Baez: Journal Club — Geometric Infinity-Function Theory — Week 6 (June 3, 2009)
Alex Hoffnung on topological field theories.
John Baez: Treq Lila (June 2, 2009)
I made an album Treq Lila, in which all the melodies were created by cellular automata.
John Baez: The Mathematics of Music at Chicago (May 31, 2009)
Thomas Noll will give some talks on the mathematics of music at the University of Chicago.
John Baez: Quantum Gravity and Quantum Geometry in Corfu (May 28, 2009)
There will be a summer school in Corfu from September 13th to September 20th, with courses on quantum gravity, renormalization and higher gauge theory.
David Corfield: Metric Coinduction (May 28, 2009)
When does coinduction differ from induction?
Urs Schreiber: Alm on Quantization as a Kan Extension (May 26, 2009)
An observation on a relation between Kan extensions and path integral quantization – by Johan Alm.
John Baez: Elsevier Journal Prices (May 24, 2009)
Can you help Ted Jacobson and I find information about the price of Elsevier journals compared to journals put out by other publishers?
John Baez: Charles Wells’ Blog (May 22, 2009)
Charles Wells has a new blog. I especially like the entries on ‘sketches’ and on ‘how “math is logic” ruined mathematics for a generation’.
David Corfield: Where is the Philosophy of Physics? (May 19, 2009)
Should we devote more time to the philosophy of physics?
Urs Schreiber: TFT at Northwestern (May 19, 2009)
Reports from a conference on extended topological quantum field theory at Northwestern University.
John Baez: A Prehistory of n-Categorical Physics (May 18, 2009)
Help John Baez write a paper on the prehistory of n-categorical physics — the prehistory of a history that hasn’t quite started yet.
John Baez: Higher Structures in Göttingen III (May 18, 2009)
There’s another workshop on higher structures at the Courant Research Centre in Göttingen. It sounds great!
Urs Schreiber: Journal Club – Geometric Infinity-Function Theory – Week 4 (May 18, 2009)
Chris Brav reviews technical details about tensor products and integral transforms of quasi-coherent sheaves on perfect stacks.
David Corfield: The Relevance of Predicativity (May 15, 2009)
Whether the predicative/impredicative split is noticeable in mainstream mathematics.
Urs Schreiber: Journal Club – Geometric Infinity-Function Theory – Week 3 (May 11, 2009)
This week in our Journal Club on [[geometric ∞-function theory]] Bruce Bartlett talks about section 3 of “Integral Transforms”: perfect stacks. So far we had Week 1: Alex Hoffnung on Introduction Week 2, myself on Preliminaries See here for…
John Baez: Smooth Structures in Ottawa II (May 9, 2009)
A summary of some talks at the Fields Workshop on Smooth Structures in Logic, Category Theory and Physics.
David Corfield: In Search of Terminal Coalgebras (May 8, 2009)
Trying to understand a talk on terminal coalgebras
John Baez: Odd Currency Puzzle (May 7, 2009)
A puzzle concerning odd American currency.
David Corfield: nLab - More General Discussion (May 6, 2009)
Discussing nLab
David Corfield: Gerundives (May 5, 2009)
Naming ‘things to be categorified’
Urs Schreiber: Journal Club – Geometric Infinity-Function Theory – Week 2 (May 5, 2009)
Preliminaries for the discussion of geometric infinity-function theory: higher categories, higher sheaves, higher algebra, higher traces and what it all means.
John Baez: The Foibles of Science Publishing (May 4, 2009)
News about Elsevier’s fake medical journal and problems at Scientific American.
David Corfield: Taming the Boundless (April 30, 2009)
How in the modern period the boundless is divided between the infinite and the indefinite, and the former approached mathematically.
John Baez: A Riddle (April 29, 2009)
See if you can solve this riddle.
John Baez: The Earth - For Physicists (April 28, 2009)
A history of the Earth in 3000 words or less, for physicists.
Urs Schreiber: Journal Club – Geometric Infinity-Function Theory – Week 1 (April 27, 2009)
Journal Club on Geometric oo-Function theory part I: Introduction to integral transforms.
John Baez: ‘Kervaire Invariant One Problem’ Solved (April 23, 2009)
It seems Michael Hopkins has solved a famous old topology problem due to Milnor and Kervaire.
David Corfield: Afternoon Fishing (April 23, 2009)
A question on a picture of Spec(Q).
John Baez: Comparative Smootheology, IV (April 23, 2009)
Take a look at Martin Laubinger’s thesis on smooth spaces.
Urs Schreiber: Master in Mathematical Physics at Hamburg University (April 20, 2009)
A new master program in mathematical physics at University of Hamburg.
John Baez: Kamnitzer on Categorifying Tangle Invariants (April 14, 2009)
A brief summary of a talk by Joel Kamnitzer at the workshop on Categorification and Geometrisation from Representation Theory.
Urs Schreiber: Report on 88th Peripatetic Seminar on Sheaves and Logic (April 11, 2009)
Mike Shulman reports on the 88th Peripatetic Seminar on Sheaves and Logic
Urs Schreiber: Journal Club – Geometric Infinity-Function Theory (April 10, 2009)
A place to discuss and learn about the work by Ben-Zvi/Francis/Nadler on geometric infinity-function theory and its application in infinity-quantum field theory.
John Baez: Commutative Diagrams in Toulouse (April 8, 2009)
There will be a workshop on Computer Algebra Methods and Commutativity of Algebraic Diagrams in Toulouse on October 16th and 17th, 2009.
Urs Schreiber: Graphical Category Theory Demonstrations (April 7, 2009)
On computer aided diagrammatic reasoning.
John Baez: Categorification and Topology (April 7, 2009)
Here are the slides for a talk explaining some hypotheses relating n-categories and topology, and Jacob Lurie’s new work on these hypotheses.
Urs Schreiber: Cohomology (April 4, 2009)
On the general (infinity,1)-categorical notion of cohomology.
David Corfield: A River and a Trickle (April 3, 2009)
On the brief appearance of Lakatos in a Handbook of philosophy of mathematics
David Corfield: Convex Spaces (April 1, 2009)
Fritz on convex spaces as algebras for the finite Giry monad
John Baez: Categories, Quanta and Concepts at the Perimeter Institute (March 31, 2009)
There will be a workshop called Categories, Quanta, Concepts at the Perimeter Institute from June 1st to 5th, 2009.
David Corfield: First-Order Logical Duality (March 27, 2009)
Announcing a recent PhD thesis which looks to extend Stone duality to a duality between syntax and semantic for first-order theories.
John Baez: The Algebra of Grand Unified Theories III (March 25, 2009)
Is the SO(10) grand unified theory of particle physics still experimentally viable? A plea for help from phenomenologists.
David Corfield: The Borromean Link Configuration (March 23, 2009)
Looking for a general formulation of situations where 3 entities are pairwise unconnected, yet connected as a triple.
John Baez: Unitary Representations of the Poincaré Group (March 22, 2009)
This blog entry is supposed to be a forum for learning and discussing Wigner’s classification of the representations of the Poincaré group. Ask and answer questions about this subject here!
John Baez: Open Access at MIT (March 22, 2009)
MIT has passed an open-access mandate!
Urs Schreiber: Twisted Differential String- and Fivebrane-Structures (March 20, 2009)
An article on twisted differential nonabelian cohomology and its application to anomaly cancellation in string theory.
John Baez: The Algebra of Grand Unified Theories II (March 16, 2009)
If you know a bit of group representation theory and you’ve always wanted to understand some particle physics, now is your chance: read a gentle expository account of the algebraic patterns lurking behind three famous Grand Unified Theories!
David Corfield: Geometric Help Wanted (March 13, 2009)
What does a result of Felix Klein mean?
John Baez: The Okubo Algebra (March 13, 2009)
Okubo invented a mysterious 8-dimensional division algebra while pondering quarks. What’s it all about?
John Baez: This Week’s Finds in Mathematical Physics (Week 274) (March 12, 2009)
Learn about infinite-dimensional 2-vector spaces and representations of 2-groups on these.
David Corfield: Synergism (March 10, 2009)
Are we maximising the collaborative power of our blog?
John Baez: Banning Open Access II (March 8, 2009)
Conyers defends his bill which would prohibit the National Institute of Health from mandating open access to the research they fund.
John Baez: Super-Yang-Mills Theory (March 5, 2009)
Why is supersymmetic Yang–Mills theory really supersymmetric? A plea for help.
Urs Schreiber: Question on Homotopical Structure on SimpSet (March 5, 2009)
A question on (directed) path objects in Kan complexes and weak Kan complexes.
David Corfield: The Stabilizer of a Subcategory (March 2, 2009)
Extending John Baez’s definition of the stabilizer of an object in a category to general subcategories
David Corfield: Two Streams in the Philosophy of Mathematics (March 2, 2009)
Announcing a philosophy of mathematics conference
Urs Schreiber: Question on Geometric Function Theory (February 27, 2009)
On the choice of “nice generalized geometric functions” in a general context of oo-stacks.
John Baez: New Structures for Physics III (February 27, 2009)
Mike Stay and I have finished what we hope is the final version of our paper for Bob Coecke’s book on New Structures for Physics. Peter Selinger’s paper for this book is also done.
David Corfield: Lakatos as Dialectical Realist (February 26, 2009)
Considering Lakatos’s idea of mathematicians aiming at the real through dialogue
John Baez: Monoidal Closed Categories And Their Deviant Relatives (February 22, 2009)
I’m curious to know whether a deviant definition of ‘monoidal closed category’ is equivalent to the usual one, or see counterexamples if it’s not.
John Baez: Homotopical Algebraic Geometry in Seville (February 21, 2009)
There will be a school on Homotopy Theory and Algebraic Geometry in Seville from September 7th to 13th, 2009.
John Baez: Satanist Rocketeer Puzzle (February 19, 2009)
Two puzzles.
David Corfield: Can -oids save Group Theory 101? (February 17, 2009)
Intriguing questions from Lieven le Bruyn.
David Corfield: Dialectical Realism in Utrecht (February 16, 2009)
My talk on Lautman in Utrecht
John Baez: Banning Open Access (February 16, 2009)
There’s a bill in the US House of Representatives that would repeal the NIH’s public access policy… and ban similar policies by other federal agencies.
John Baez: Baković and Jurčo on Classifying Topoi for Topological Bicategories (February 14, 2009)
Igor Bakoviċ and Branislaw Jurčo have a new paper out.
John Baez: Lurie on TQFT and the Cobordism Hypothesis (February 13, 2009)
Watch videos of four talks by Jacob Lurie at the University of Texas at Austin.
John Baez: The Algebra of Grand Unified Theories (February 10, 2009)
John Huerta is taking his oral exam soon. He’ll give a talk on the group representation theory underlying three famous grand unified theories, and how they fit together in a larger pattern. You can see the slides now.
John Baez: Link Homology in Paris (February 10, 2009)
In 2009 there will be a summer school on link homology in Paris.
Urs Schreiber: Moerdijk on Infinity-Operads (February 6, 2009)
Ieke Moerdijk on (infinity,1)-operads.
Urs Schreiber: Dendroidal Sets and Infinity-Operads (February 5, 2009)
Notes from a talk by Ieke Moerdijk on dendroidal sets, with a few remarks on presheaves on the category of posets.
John Baez: The Cocktail Party Version (February 4, 2009)
Jeffrey Morton explains what he’s working on.
David Corfield: Last Person Standing (February 2, 2009)
On the effects of nobody being left who understands the totality of the huge proof of a result.
John Baez: New York City Category Seminar (January 29, 2009)
There’s a new seminar on categories, organized by Noson Yanofsky, meeting regularly at the CUNY Grad Center in Manhattan.
John Baez: The Third Time is the Charm (January 29, 2009)
A puzzle about math and physics.
John Baez: Can Particle Physicists Regain Control of Their Journals? (January 28, 2009)
Learn about the Sponsoring Consortium for Open Access Publishing in Particle Physics.
John Baez: Bruce Bartlett’s Thesis (January 27, 2009)
Bruce Bartlett’s thesis on topological quantum field theory is on the arXiv now.
John Baez: Categorified Symplectic Geometry and the String Lie 2-Algebra (January 26, 2009)
A new paper shows how to build the string Lie 2-algebra by taking a compact Lie group with its canonical closed 3-form and then using ideas from multisymplectic geometry.
David Corfield: Abstract Stone Duality (January 26, 2009)
Paul Taylor’s programme to provide a computable foundation for topology
John Baez: Limits in the 2-Category of 2-Hilbert Spaces (January 22, 2009)
What limits are present in the 2-category of finite-dimensional 2-Hilbert spaces?
John Baez: Petit Topos, Gros Topos (January 22, 2009)
What’s the relation between the petit topos and the gros topos of a scheme?
John Baez: Hopf Algebras from Posets (January 14, 2009)
From a sufficiently nice category, we can get a Hopf algebra. But what, exactly, does ‘sufficiently nice’ mean? Maybe you can help me out.
David Corfield: The Space of Robustness (January 14, 2009)
A discussion of a blog post by Tim Gowers on the structure of mathematics.
David Corfield: Truth as Value and Duty (January 12, 2009)
Manin on mathematical wisdom
Urs Schreiber: Ben-Zvi on Geometric Function Theory (January 8, 2009)
An exposition by David Ben-Zvi on central ideas of geometric function theory, its categorification and relation to groupoidification.
David Corfield: Categories, Logic and Physics in London (January 8, 2009)
Reporting on the 4th CLP workshop
Urs Schreiber: nLab – General Discussion (January 4, 2009)
A place for general discussion concerning the nLab.
John Baez: The Toric Variety Associated to the Weyl Chambers (December 31, 2008)
The Weyl chambers of a simple Lie algebra form a ‘fan’, and this fan gives a toric variety. But what’s that toric variety like? What does it mean?
David Corfield: Organizing the Pages at nLab (December 31, 2008)
How to organise pages at nLab
John Baez: Joint Math Meetings in Washington DC (December 31, 2008)
I’m going to the Joint Mathematics Meetings in Washington DC from January 5th to January 8th, 2009, and giving talks on 2-groups and groupoidification. Who else will be in town?
Urs Schreiber: Groupoidification from sigma-Models? (December 30, 2008)
On constructing sigma-models from differential nonabelian cocycles and how this gives rise to methods of groupoidification.
John Baez: Groupoidification Made Easy (December 26, 2008)
Now you can learn about groupoidification in a paper that has proofs of all the most basic results!
John Baez: Tilings (December 24, 2008)
Here are some pretty pictures of tilings, suitable for use as computer wallpaper.
John Baez: Infinite-Dimensional Representations of 2-Groups (December 24, 2008)
Here’s a new paper on infinite-dimensional representations of 2-groups — representations on ‘measurable categories’.
John Baez: Science and the Environment (December 24, 2008)
Scientists may soon be running science policy in the US. What should we do?
David Corfield: Bridge Building (December 23, 2008)
Might the cohomology of dynamical systems provide a meeting ground for researchers on the ‘combinatorics’ side of mathematics, and those on the ‘theory-building’ side?
John Baez: Higher Structures in Göttingen - Part II (December 22, 2008)
On February 5-7 2009 there will be a workshop in Göttingen on Higher Structures in Topology and Geometry.
David Corfield: The Microcosm Principle (December 19, 2008)
The microcosm principle finds aspects of the small reflected in the large.
John Baez: Super Version of 2-Plectic Geometry for Classical Superstrings? (December 16, 2008)
Here are some basic questions about describing the classical superstring using the ‘super’ analogue of multisymplectic geometry, if such a thing exists.
David Corfield: What to Make of Mathematical Difficulties (December 16, 2008)
How to make the most of obstructions to a theory
John Baez: This Week’s Finds in Mathematical Physics (Week 273) (December 15, 2008)
Read more about the geysers of Enceladus. Hear the history of mineral evolution, from chondrites to the Big Splat, the Late Heavy Bombardment, the Great Oxidation Event, and Snowball Earth… to now. Then, learn about Pontryagin duality.
David Corfield: The Status of Coalgebra (December 9, 2008)
Is coalgebra just as important as algebra?
John Baez: Science Citation Index (December 9, 2008)
The Science Citation Index is increasingly being used to evaluate research. There are lots of reasons why this is a bad idea.
John Baez: A Quick Algebra Quiz (December 9, 2008)
Here’s a quick test of your mathematical reflexes.
John Baez: Smooth Structures in Ottawa (December 7, 2008)
There will be a conference on Smooth Structures in Logic, Category Theory and Physics at the University of Ottawa on May 1-3, 2009.
Urs Schreiber: Question on Infinity-Yoneda (December 4, 2008)
What is known about the Yoneda lemma in a higher categorical context?
Urs Schreiber: Zhu on Lie’s Second Theorem for Lie Groupoids (December 3, 2008)
On Lie integration of Lie algebroids.
David Corfield: Tom Leinster in The Reasoner (November 30, 2008)
Tom Leinster interviewed for The Reasoner
John Baez: This Week’s Finds in Mathematical Physics (Week 272) (November 29, 2008)
See what the Cassini probe saw as it shot through the Enceladus plumes, see what happens when the Io flux tube hits Jupiter, learn two new properties of the number 6, and discover the wonders of standard Borel spaces, commutative von Neumann algebras, and Polish groups.
John Baez: Groupoidfest 08 - A Brief Report (November 28, 2008)
A brief report on the 2008 Groupoidfest held at UCR, with an emphasis on groupoidification.
Urs Schreiber: nLab (November 28, 2008)
We have set up a Wiki accompanying the n-Category Cafe.
David Corfield: Coalgebraically Thinking (November 27, 2008)
How coalgebra differs from algebra.
Urs Schreiber: Mathematics, Physics and Philosophy (November 26, 2008)
An essay, or the like, on motivations for interdisciplinary research at the boundary of math, physics and philosophy.
Urs Schreiber: Bär on Fiber Integration in Differential Cohomology (November 26, 2008)
On fiber integration in differential cohomology and the notion of generalized smooth spaces used for that.
David Corfield: Beyond the Blog (November 25, 2008)
How to make best use of blog material.
John Baez: Categorification in Glasgow (November 24, 2008)
There will be a workshop on categorification in Glasgow from April 13th to April 18th, 2009.
David Corfield: Mathematical Robustness (November 20, 2008)
Does the ‘robust’ appearance of mathematical entities indicate reality?
Urs Schreiber: Higher and Graded Geometric Structures in Göttingen (November 19, 2008)
A workshop on higher bundles, gerbes, groupoids and supermanifolds.
David Corfield: Fourth Categories, Logic and Physics Workshop (November 18, 2008)
Categories, Logic and Physics workshop announcement.
Urs Schreiber: Local Nets and Co-Sheaves (November 14, 2008)
Co-sheaf condition (codescent) for Haag-Kastler nets of local quantum observables?
Urs Schreiber: Frobenius Algebras and the BV Formalism (November 14, 2008)
Bruce Bartlett is looking at the latest article by Cattaneo and Mnev on BV-quantization of Chern-Simons theory.
Urs Schreiber: Baković on Bigroupoid 2-Torsors (November 13, 2008)
Bakovic on G-principal bundles for G a bigroupoid.
Urs Schreiber: Talk: Local Nets from Parallel Transport 2-Functors (November 12, 2008)
A talk on the relation between AQFT and extended functorial QFT.
John Baez: Locally Compact Hausdorff Abelian Groups (November 12, 2008)
Learn the Principal Structure Theorem for locally compact Hausdorff abelian groups — the kind of groups that show up in Pontryagin duality.
John Baez: Variations on Pontryagin Duality (November 8, 2008)
Here are some basic questions about the process of taking a locally compact abelian group A to the group of homomorphisms from A to the invertible complex numbers.
David Corfield: Entropy, Diversity and Cardinality (Part 2) (November 7, 2008)
Continuation of a discussion of entropy and cardinality and its relevance to ecology
Urs Schreiber: Lie 3-Algebras on the Membrane (?) (November 6, 2008)
Recently a trinary bracket appears in the study of the supermembrane which is sometimes addressed as a homotopy algebraic structure.
David Corfield: The Internal Language of a 2-Topos (November 6, 2008)
Extracting the internal language from a 2-topos
David Corfield: Maximum Entropy Distributions (November 2, 2008)
Why are familiar distributions so often maximum shannon entropy distributions for their class?
Urs Schreiber: Higher Structures in Math and Physics in Lausanne (November 2, 2008)
A workshop in Lausanne on Higher Structures in Mathematics and Physics
Urs Schreiber: Twisted Differential Nonabelian Cohomology (October 30, 2008)
Work on theory and applications of twisted nonabelian differential cohomology.
John Baez: Google Books — More Open Access? (October 28, 2008)
A settlement has been reached which may give us more free access to Google Books.
John Baez: Lie Theory Through Examples 4 (October 28, 2008)
Classifying representations using weights.
John Baez: This Week’s Finds in Mathematical Physics (Week 271) (October 26, 2008)
Read about massive volcanic eruptions on Jupiter’s moon Io, allotropes of sulfur, quasicrystals in various dimensions, Jeffrey Morton’s extension of the “groupoidification” program, and Stephen Summers’ review of new work on constructive quantum field theory!
John Baez: Open Access at the University of California (October 24, 2008)
News about open access at the University of California.
John Baez: Groupoidfest 08 (October 24, 2008)
Some talks at the Groupoidfest being held in Riverside on November 22nd-23rd, 2008.
Urs Schreiber: Hopf Algebraic Renormalization (October 23, 2008)
The starting point and basic idea of combinatorial Hopf algebraic techniques in perturbative quantum field theory.
David Corfield: Reviewing Ruelle’s Book (October 23, 2008)
Announcing my review of Ruelle’s ‘The Mathematician’s Brain’.
John Baez: What is Categorification? (October 22, 2008)
The meaning of the word ‘categorification’, and some key examples.
Urs Schreiber: Codescent and the van Kampen Theorem (October 21, 2008)
On codescent, infinity-co-stacks, fundamental infinity-groupoids, natural differential geometry and the van Kampen theorem
David Corfield: John McKay Visits Kent (October 21, 2008)
What I took from John McKay’s talk on finite simple groups.
John Baez: Lie Theory Through Examples 3 (October 21, 2008)
Finally the concept of ‘weight lattice’ rears its ugly head.
Urs Schreiber: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack (October 20, 2008)
Hepworth on the idea of volume forms on a stack in terms of sections of 2-vector bundles.
John Baez: Categorification in New Scientist (October 20, 2008)
Richard Elwes at New Scientist has written an article on knot theory that mentions categorification.
Urs Schreiber: Talk in Göttingen: Second Nonabelian Differential Cohomology (October 19, 2008)
A talk on second nonabelian differential cohomology.
John Baez: Morton on 2-Vector Spaces and Groupoids (October 19, 2008)
Jeffrey Morton has a paper that turns groupoids into 2-vector spaces — a key step toward constructing the Dijkgraaf–Witten model as an extended TQFT.
David Corfield: Entropy, Diversity and Cardinality (Part 1) (October 16, 2008)
Entropy in relation to biodiversity and categories
Urs Schreiber: String- and Fivebrane-Structures (October 14, 2008)
A new article on String- and Fivebrane structures and some previous articles on Fivebrane structures.
Urs Schreiber: Freely Generated ω-Categories (October 14, 2008)
The notion of strict infinity-categories which are “freely generated”.
John Baez: The Nature of Time (October 13, 2008)
The Foundational Questions Institute is having an essay contest on the nature of time.
John Baez: This Week’s Finds in Mathematical Physics (Week 270) (October 12, 2008)
See lava on Jupiter’s moon Io. Hear about Greg Egan’s new novel. And then, learn about some little-known interactions between the numbers 5, 8, 12, and 24.
Urs Schreiber: Semistrict Infinity-Categories and ω-Semi-Categories (October 8, 2008)
On strict infinity categories with weak identities.
David Corfield: New Directions in the Philosophy of Mathematics (October 8, 2008)
To celebrate the founding of MIMS, the mathematics department of the recently unified Manchester University, it was proposed that various workshops named ‘New Directions in…’ be run. They kindly agreed to allow Alexandre Borovik and me to organise one…
Urs Schreiber: Yet Another Model ω-Question (October 7, 2008)
On the relation between the model category structure on smooth infinity-groupoids and on differential algebras.
John Baez: Lie Theory Through Examples 2 (October 7, 2008)
Moving on up a dimension, now let’s look at the A3 lattice. This arises naturally from the group SU(4), but you’ve also seen it in grocery stores if you ever paid attention to stacks of oranges.
David Corfield: The Blog of Fun (October 5, 2008)
The ‘field with one element’ has been honoured by a great accolade. As announced here, it has been awarded a blog all to itself. Not bad for an entity with dubious existence credentials….
Urs Schreiber: More Model ω-Questions (October 2, 2008)
More questions on model category theory in general and the model structure on omega-categories in particular.
David Corfield: Mathematical Reality (October 1, 2008)
Some thoughts on mathematical reality as pieces of mathematics inevitably encountered in different ways.
Urs Schreiber: (ω+ω)-Categories (?) (September 30, 2008)
A tentative suggestion for weak oo-categories.
John Baez: Quantum Theory and Analysis (September 30, 2008)
Some lecture notes for a class on Hilbert spaces, self-adjoint operators, and their applications to quantum theory.
John Baez: Lie Theory Through Examples 1 (September 30, 2008)
The beginning of a course on Lie theory through examples.
John Baez: OctoberFest 08 (September 27, 2008)
This October, the annual Canadian conference on categories will be at Concordia University in Montreal.
Urs Schreiber: Some Model ω-Questions (September 27, 2008)
The model structure on omega-categories should induce a model structure on omega-category valued presheaves. What can be said about this?
David Corfield: Mathematical Miniatures (September 26, 2008)
Short essays by Beno Eckmann
Urs Schreiber: Group Cocycles and Simplices (September 22, 2008)
Question on simplicial group cocycles.
Urs Schreiber: Back and Catching Up (September 21, 2008)
A list of things to read.
David Corfield: The Volume of a Differentiable Stack (September 15, 2008)
Weinstein’s generalisation of the Baez-Dolan groupoid cardinality concept to stacks.
John Baez: 5 (September 12, 2008)
A talk on special properties of the number 5.
John Baez: 24 (September 12, 2008)
A talk on my favorite number of all: 24.
Urs Schreiber: Planar Algebras, TFTs with Defects (September 11, 2008)
On planar algebras interpreted as 2D TFTs with defect lines.
John Baez: New Structures for Physics II (September 10, 2008)
You can help review some more papers for Bob Coecke’s New Structures for Physics.
David Corfield: Who are Various? (September 10, 2008)
Workshop announcement and an enquiry about a reference made by Saunders Mac Lane
David Corfield: Reliability (September 9, 2008)
A response to an article on the unreliability of the mathematical literature.
Urs Schreiber: Smooth Differential Graded Algebra? (September 9, 2008)
What is the right definition of generalized smooth differential graded-commutative algebras?
Urs Schreiber: Equivariant deRham Cohomology and Action Lie Algebroids (September 7, 2008)
The relation between equivariant deRham cohomology and the Weil algebra of the corresponding action Lie algebroid.
John Baez: 8 (September 7, 2008)
A talk on one of my favorite numbers: 8.
John Baez: Military Research (September 6, 2008)
What sort of research is the US military funding today?
Urs Schreiber: Waldorf on Transport Functors and Connections on Gerbes (September 6, 2008)
A talk on parallel 2-transport.
John Baez: Comparative Smootheology, III (September 3, 2008)
The third episode in our continuing comparison of various frameworks for differential geometry.
John Baez: This Week’s Finds in Mathematical Physics (Week 269) (August 31, 2008)
See more of Jupiter’s moon Io — and learn about honeycombs, the work of Kelvin, the Weaire-Phelan structure, and gas clathrates.
David Corfield: Imperfections, Ambiguities and Physics (August 30, 2008)
Can Lautman’s concept of the dialectical relationship between perfection and imperfection be realised in physics?
John Baez: New Structures for Physics I (August 29, 2008)
Please comment on two chapters of a forthcoming book edited by Bob Coecke: ‘Introduction to categories and categorical logic’ by Abramsky and Tzevelekos, and ‘Categories for the practicing physicist’ by Coecke and Paquette.
John Baez: Henri Cartan, July 8, 1904 - August 13, 2008 (August 22, 2008)
Henri Cartan, a founding member of Bourbaki, died August 13th at the age of 104.
David Corfield: Melliès on Functorial Boxes (August 22, 2008)
Paul-André Melliès has a tutorial on incorporating functors between categories into string diagrams… and applications to logic.
John Baez: The Weaire–Phelan Structure (August 21, 2008)
The best known solution to an old math problem posed by Kelvin now graces the National Aquatic Center at the Olympic games in Beijing.
Urs Schreiber: Homotopical Algebra with Applications to Mathematical Physics (August 21, 2008)
A workshop on homotopical methods in mathematical physics.
John Baez: Polish Spaces (August 21, 2008)
Are second-countable locally compact Hausdorff spaces always homeomorphic to complete separable metric spaces?
David Corfield: Freed on Chern-Simons (August 21, 2008)
Daniel Freed has come out with an expository paper on Chern-Simons theory.
John Baez: Elvis Zap does the Calculus Rap (August 20, 2008)
Professor Elvis Zap explains the rules of differential calculus.
John Baez: Spivak on Derived Manifolds (August 19, 2008)
David Spivak has an interesting thesis on ‘derived differential geometry’.
Urs Schreiber: Bergman on Infinity-Vector Bundles Coupled to Topological Strings (August 17, 2008)
topological string, B-model, derived category, B-brane, infinity-vector space, infinity connection
Urs Schreiber: Differential Graded Clifford Algebra (August 17, 2008)
On Clifford-defomrations of Chevalley-Eilenberg and Weil algebras.
Urs Schreiber: Connections on Nonabelian Gerbes and their Holonomy (August 15, 2008)
An article on transport 2-functors.
David Corfield: Lautman on Reciprocity (August 14, 2008)
Albert Lautman has a paper on the use of analysis in number theory — with a special emphasis on “reciprocity”.
John Baez: Bimodules Versus Spans (August 14, 2008)
Some puzzles about bimodules in the opposite of the category of sets.
John Baez: This Week’s Finds in Mathematical Physics (Week 268) (August 6, 2008)
See a marvelous view of Io, and then learn more about Frobenius algebras than you probably wanted to know — and a bit about modular tensor categories and the mathematics of music, too!
John Baez: Visiting Grothendieck (August 6, 2008)
The story of someone who tried to visit Alexander Grothendieck.
John Baez: Categorified Symplectic Geometry and the Classical String (August 2, 2008)
In this new version of our paper, we systematically explain how n-dimensional field theories give n-plectic manifolds. We also say how a B field affects the 2-plectic structure for a string.
John Baez: Getting Started Early (July 31, 2008)
The man who had Ph.D. students before he was born.
Urs Schreiber: Pre- and Postdictions of the NCG Standard Model (July 30, 2008)
Some predictions and postdictions derived from the NCG Kaluza-Klein standard model.
David Corfield: Category Theory and Model Theory (July 29, 2008)
Category theory and model theory
John Baez: Light Mills (July 28, 2008)
The Crookes radiometer is also known as a ‘light mill’ — a little glass bulb with a windmill in it, with vanes black on one side and white on the other. It puzzled Reynolds, Maxwell and even Einstein. Do we really understand it yet?
John Baez: Causality in Discrete Models of Spacetime (July 27, 2008)
Gavin Wraith has some questions about causality in discrete spacetimes described as simplicial complexes.
John Baez: Categories, Logic and Foundations of Physics in Oxford (July 25, 2008)
On August 23rd and 24th, there will be a conference on Categories, Logic and Foundations of Physics at Oxford.
David Corfield: Real versus Complex Numbers (July 24, 2008)
Comparing the reals and the complex numbers
John Baez: This Week’s Finds in Mathematical Physics (Week 267) (July 23, 2008)
In “week267” see the tilings of the Alhambra and learn about wallpaper groups, 17 wallpaper groups, their corresponding 2d orbifolds, the role of 2-groups as symmetries of orbifolds, the work of Carrasco and Cegarra on hypercrossed complexes, and the work of João Faria Martins on the fundamental 2-group of a 2-knot.
David Corfield: Girard on the Limitations of Categories (July 23, 2008)
Girard on categories
David Corfield: Hierarchy and Emergence (July 18, 2008)
Emergence in hierarchies
Urs Schreiber: Some ω-Questions (July 18, 2008)
Questions on omega-categorical constructions in the context of descent and cohomology.
Urs Schreiber: News on Measures on Groupoids? (July 17, 2008)
Benjamin Bahr apparently thought about measures on groupoids of connections.
John Baez: Theorems Into Coffee IV (July 12, 2008)
The first Theorems into Coffee prize is awarded. Read about Steve Lack’s work on PROPs, and try your hand at the latest Theorems into Coffee challenge.
Urs Schreiber: Talk on AQFT from FQFT and Applications (July 10, 2008)
On local nets constructed from transport 2-functors and examples relating to lattice models, Hopf spin chains, asymptotic inclusion of subfactors. And some remarks on the relation between conformal nets and vertex operator algebras.
Urs Schreiber: Basics of Poisson Reduction and BV, I (July 7, 2008)
Basics of symplectic reduction and Poisson reduction.
David Corfield: Sphere Eversion (July 7, 2008)
Everting the sphere
David Corfield: A Small Observation (July 3, 2008)
A duality in 2-vector spaces
Urs Schreiber: The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism) (July 1, 2008)
Walter Suijlekom explains a relation between Connes-Kreimer Hopf algebras of Feynman diagrams with the master equation in BV-formalism.
Urs Schreiber: Block on L-∞ Module Categories (June 30, 2008)
On Jonathan Block’s concept of modules over differential graded algebras.
Urs Schreiber: The Manifold Geometries of QFT, I (June 30, 2008)
Some notes on the first day at “The manifold geometries of QFT” at the MPI in Bonn.
David Corfield: Lerman on Orbifolds (June 30, 2008)
Lerman on orbifolds
David Corfield: Michael Polanyi and Personal Knowledge (June 28, 2008)
Polanyi on personal knowledge
Urs Schreiber: AQFT from Lattice Models (?) (June 27, 2008)
On lattice AQFT models such as Hopf spin chains and their derivation of 2-C*-algebra valued 2-functors.
Urs Schreiber: Eli Hawkins on Geometric Quantization, II (June 27, 2008)
Eli Hawkins explains his method of getting a quantum algebra from the convolution algebra of sections on a symplectic groupoid.
Urs Schreiber: Category Algebras (June 26, 2008)
On an interpretation of the category algebra of a category in terms of pull-push of finite sets through correspondence spaces.
David Corfield: Tim Porter on Formal Homotopy Quantum Field Theories and 2-Groups (June 24, 2008)
Tim Porter on HQFT
Urs Schreiber: Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups (June 23, 2008)
A talk by Behrang Noohi.
Urs Schreiber: Aldrovandi on Non-Abelian Gerbes and 2-Bundles (June 23, 2008)
A talk by Aldrovandi on different perspectives on gerbes and 2-bundles.
Urs Schreiber: Landsman on Quantization of Poisson Algebras Associated to Lie Algebroids (June 22, 2008)
A quick review of Landsman’s result on strict deformation quantization of Poisson manifolds dual to Lie algebroids: the quantum algebra is nothing but the groupoid algebra of the Lie groupoid integrating the Lie algebroid.
Urs Schreiber: Eli Hawkins on Geometric Quantization, I (June 20, 2008)
Some basics and some aspects of geometric quantization. With an emphasis on the geometric quantization of duals of Lie algebras and duals of Lie algebroids.
John Baez: This Week’s Finds in Mathematical Physics (Week 266) (June 20, 2008)
Read about Jupiter’s moon Io, the Pythagorean tuning system, and the categorical groups conference now taking place in Barcelona.
John Baez: Fundamental 2-Groups and 2-Covering Spaces (June 19, 2008)
You can see the slides of David Roberts’ talk on the fundamental 2-group of a topological space, and the corresponding notion of ‘2-covering space’.
Urs Schreiber: Schommer-Pries on Classification of 2-Dimensional Extended TFT (June 18, 2008)
Chris Schommer-Pries classifies 2-functors from a 2-category of 2-dimensional cobordisms.
Urs Schreiber: Teleman on Topological Construction of Chern-Simons Theory (June 17, 2008)
A talk by Constant Teleman on extended Chern-Simons QFT and what to assign to the point.
David Corfield: The Mathematical Sublime (June 17, 2008)
The sublime
John Baez: Computation and the Periodic Table (June 14, 2008)
The analogies between physics, topology, logic and computer science, visible so clearly with the help of symmetric monoidal closed categories, are just the tip of a larger iceberg involving $n$-categories. The Periodic Table seems to be a useful guide here.
Urs Schreiber: An Exercise in Groupoidification: The Path Integral (June 13, 2008)
A remark on the path integral in view of groupoidification and Sigma-model quantization.
Urs Schreiber: A Groupoid Approach to Quantization (June 12, 2008)
On Eli Hawkins’ groupoid version of geometric quantization.
David Corfield: Klein 2-Geometry XI (June 12, 2008)
categorified geometry
John Baez: Categories, Logic and Foundations of Physics: The Videos (June 9, 2008)
Now videos are available for the 2nd conference on Categories, Logic and Foundations of Physics.
John Baez: 2-Groups in Barcelona (June 8, 2008)
There’s a workshop on categorical groups in Barcelona during June 16-20, 2008.
John Baez: Farey Sequences and the Stern-Brocot Tree (June 7, 2008)
An approach to modular curves based on Farey sequences and the Stern-Brocot tree.
Urs Schreiber: Urban Myths in Contemporary Cosmology (June 6, 2008)
On urban-myth-like memes in the contemporary theoretical cosmology community.
David Corfield: Dumbing Down (June 4, 2008)
Mathematics exams for 16 year olds are getting easier, it is claimed. It’s fairly easy to check for yourself. Take a look at the Arithmetic, Algebra and Geometry papers from 1959 and compare with a contemporary specimen GCSE paper. Even…
John Baez: Classical String Theory and Categorified Symplectic Geometry (June 2, 2008)
Just as any symplectic manifold gives a Lie algebra of observables, any 2-plectic manifold gives a Lie 2-algebra of observables. This shows up in string theory!
Urs Schreiber: HIM Trimester on Geometry and Physics, Week 4 (May 29, 2008)
Talk in Stanford on nonabelian differential cohomology.
David Corfield: Manin on Foundations (May 28, 2008)
Manin on mathematical foundations
John Baez: Cahiers Free Online! (May 27, 2008)
The Ehresmann’s famous journal is now free online.
John Baez: Double Categories — Warm and Cuddly? (May 27, 2008)
Abstract of a talk on double categories by Ross Street.
John Baez: Number Theory on YouTube? (May 26, 2008)
Number theory abstracts on YouTube?
John Baez: This Week’s Finds in Mathematical Physics (Week 265) (May 26, 2008)
Read about Europa, the Pythagorean pentagram, Bill Schmitt’s work on Hopf algebras in combinatorics, the magnum opus of Aguiar and Mahajan, and quaternionic analysis.
Urs Schreiber: Workshop: Non-Commutative Constructions in Arithmetic and Geometry (May 22, 2008)
Conference in London on non-commutative constructions in arithmetic and geometry
Urs Schreiber: Hopkins-Lurie on Baez-Dolan (May 20, 2008)
Hopkins and Lurie give a proof of the Baez-Dolan hypothesis on tangles and extended TQFT.
Urs Schreiber: Relation between AQFT and Extended Functorial QFT (May 20, 2008)
On how to obtain from any extended d-dimensional QFT a local net of observables.
David Corfield: Ambiguity Theory (May 19, 2008)
Galois theory as the study of ambiguity
Urs Schreiber: Integrals and Valuations using Geometric Logic (May 19, 2008)
Bas Spitters on the theory of integration in a topos.
John Baez: This Week’s Finds in Mathematical Physics (Week 264) (May 19, 2008)
Learn about Phobos, homotopy groups of spheres, and theta functions.
Urs Schreiber: Harrison’s Geometric Calculus (May 18, 2008)
On Harrison’s generalized formalism for differential calculus and differential geometry.
Urs Schreiber: Electric-Magnetic-Duality and Hodge Duality Extended to Differental Cocycles (May 17, 2008)
On the electric-magnetic dual formulation of higher abelian Yang-Mills theory.
John Baez: Convenient Categories of Smooth Spaces (May 17, 2008)
Chen spaces and Souriau’s diffeological spaces are two great contexts for differential geometry. Alex Hoffnung and his thesis advisor just wrote a paper studying these in detail.
John Baez: Theorems Into Coffee III (May 15, 2008)
What is the symmetric monoidal category of finite sets and partially defined functions the PROP for? Prove your answer and win some coffee!
Urs Schreiber: HIM Trimester Geometry and Physics, Week 1 (May 14, 2008)
On vertex operator algebras, operads and Segals QFT axioms. On integration over supermanifolds, supergroupoids and the two notions of supercategories.
Urs Schreiber: E8 Quillen Superconnection (May 10, 2008)
On Quillen superconnections on E8-bundles and on the mathematical interpretation of the connection appearing in an article by A. Lisi.
Urs Schreiber: On Lie oo-Theory (May 8, 2008)
Talk on the Lie infinity-algebroid perspective on the BRST complex and its relation to the integrated picture of action infinity-groups.
David Corfield: Pernicious Symbolization (May 8, 2008)
Ryle against Carnap
Urs Schreiber: Integrability of Lie Brackets (May 7, 2008)
Crainic and Fernandes on integration of Lie algebroids to Lie groupoids.
Urs Schreiber: Ambimorphic? (May 6, 2008)
On adjunctions between spaces and n-groupoids induced by homming out of the fundamental n-groupoid.
David Corfield: Quaternionic Analysis (May 6, 2008)
Quaternionic analysis
John Baez: Theorems Into Coffee II (May 6, 2008)
Another coffee challenge: prove that matrices of integers are morphisms in the PROP for commutative and cocommutative Hopf algebras!
John Baez: Theorems Into Coffee (May 4, 2008)
Prove this theorem, and you’ll get some coffee!
Urs Schreiber: FQXi (May 4, 2008)
The Foundational Questions Institute runs its second grant competition.
David Corfield: Questions on 2-covers (April 30, 2008)
2-covers
David Corfield: Returning to Lautman (April 29, 2008)
I mentioned in an earlier post that Albert Lautman had a considerable influence on my decision to turn to philosophy. I recently found out that his writings have been gathered together and republished as Les mathématiques, les idées et le…
Urs Schreiber: Dual Formulation of String Theory and Fivebrane Structures (April 28, 2008)
An article which discusses lifts through the 7-fold connected cover of the structure group of the tangent bundle in the context of electric-magnetic duality in string theory.
Urs Schreiber: Charges and Twisted Bundles, IV: Anomaly Cancellation (April 27, 2008)
How the fermionic anomaly may cancel against the charge anomaly in higher gauge theory: the Green-Schwarz mechanism
Urs Schreiber: Charges and Twisted Bundles, III: Anomalies (April 25, 2008)
On the anomalies arising in (higher) gauge theories in the presence of electric and magnetic charges.
Urs Schreiber: Thoughts (Mostly on Super ∞-Things) (April 22, 2008)
Thoughts while travelling and talking.
Urs Schreiber: Higher Hochschild Cohomology and Differential Forms on Mapping Spaces (April 18, 2008)
On “brane topology”, the higher order generalization of Hochschild cohomology. And on Chen iterated integrals.
John Baez: Comparative Smootheology, II (April 17, 2008)
Comparing Chen’s ‘differentiable spaces’ and Souriau’s ‘diffeological spaces’.
Urs Schreiber: A Topos for Algebraic Quantum Theory (revised) (April 15, 2008)
Bas Spitters on topos-theory of noncommutative C*-algebras and quantum physics.
David Corfield: Klein 2-Geometry X (April 15, 2008)
Acting on groups
Urs Schreiber: Sigma-Models and Nonabelian Differential Cohomology (April 11, 2008)
Notes on nonabelian differential cohomology and its application to classical and qantum parallel transport.
David Corfield: Categorical Sheaves (April 9, 2008)
Toën and vezzosi on categorical sheaves
David Corfield: Petitition to Save USQ Mathematics (April 9, 2008)
Terry Tao on saving USQ Maths
John Baez: This Week’s Finds in Mathematical Physics (Week 263) (April 6, 2008)
John Thompson and Jacques Tits won the 2008 Abel Prize for their work on group theory. Find out what they did.
David Corfield: 2-Structure Types (April 2, 2008)
2-structure types
John Baez: A Puzzle About Blueshifts (April 1, 2008)
A puzzle about blueshifts for rotating black holes, posed by Greg Egan.
Urs Schreiber: Limits and Push-Forward (March 31, 2008)
Question on the relation between push-forward of functors and (indexed) limits.
John Baez: This Week’s Finds in Mathematical Physics (Week 262) (March 30, 2008)
Learn about the Southern Ring Nebula, the frosty dunes of Mars, quantum technology in Singapore, atom chips, graphene transistors, nitrogen-vacancy pairs in diamonds, a new construction of e8, and a categorification of sl(2).
John Baez: Test Your Singlish (March 29, 2008)
Guess the meaning of some words in Singlish.
David Corfield: Categorified Quantum Groups (March 27, 2008)
categorifying quantum groups
Urs Schreiber: What Has Happened So Far (March 27, 2008)
A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
Urs Schreiber: Nonabelian Differential Cohomology in Street’s Descent Theory (March 22, 2008)
A discussion of differential nonabelian cocycles classifying higher bundles with connection in the context of the general theory of descent and cohomology with coefficients in infnity-category valued presheaves as formalized by Ross Street.
John Baez: Groupoidfest in Riverside (March 21, 2008)
The next Groupoidfest is at UCR, November 22-23 2008.
Urs Schreiber: Crossed Menagerie (March 21, 2008)
Detailed notes by Tim Porter on “crossed gadgetry and cohomology in algebra and topology”.
John Baez: This Week’s Finds in Mathematical Physics (Week 261) (March 20, 2008)
Learn about the Engraved Hourglass Nebula… and read an ode to the number 3. Also: try your hand at some trefoil puzzles.
David Corfield: The World of L (March 17, 2008)
Breaking news in number theory
Urs Schreiber: Slides: On Nonabelian Differential Cohomology (March 13, 2008)
On the notion of nonabelian differential cohomology.
Urs Schreiber: Chern-Simons Actions for (Super)-Gravities (March 12, 2008)
On Chern-Simons actions for (super-)gravity.
John Baez: Following Singapore’s Lead (March 12, 2008)
A Los Angeles elementary school drastically improves its math teaching… by following the lead of Singapore.
David Corfield: A Strange Link (March 11, 2008)
Tim Porter on global actions
John Baez: Physics, Topology, Logic and Computation: a Rosetta Stone (March 11, 2008)
Toward a general theory of systems and processes.
John Baez: Learning to Love Topos Theory (March 9, 2008)
Steve Vickers on topos theory.
John Baez: Some Puzzles (March 8, 2008)
Three puzzles to keep you entertained.
Urs Schreiber: Space and Quantity (March 6, 2008)
Notes on spaces and smooth spaces, function algebras and smooth function algebras.
David Corfield: Worrying About 2-Logic (March 6, 2008)
More on 2-logic — some possible problems.
Urs Schreiber: Sections of Bundles and Question on Inner Homs in Comma Categories (March 4, 2008)
On inner homs in comma categories, motivated from a description of spaces of sections of bundles in terms of such.
Urs Schreiber: Charges and Twisted n-Bundles, II (March 4, 2008)
Rephrasing Freed’s action functional for differential cohomology in terms of L-oo connections in a simple toy example.
Urs Schreiber: Infinity-Groups with Specified Composition (March 3, 2008)
On infinity-groups and infinity-categories with specified composition, and their closedness.
David Corfield: A Deep Sense of Miserable Ignorance (March 3, 2008)
Peirce on pedagogy
John Baez: Kim on Fundamental Groups in Number Theory (March 1, 2008)
Minhyong Kim on Diophantine geometry and the fundamental group.
John Baez: Computer Scientists Needed Now (March 1, 2008)
John Baez begs for help on the computation section of his ‘Rosetta Stone’ paper connecting physics, topology, logic and computation.
Urs Schreiber: Charges and Twisted n-Bundles, I (February 29, 2008)
Generalized charges are very well understood using generalized differential cohomology. Here I relate that to the nonabelian differential cohomology of n-bundles with connection.
David Corfield: Peirce on Mathematics (February 28, 2008)
Charles Peirce on mathematics.
Urs Schreiber: (Generalized) Differential Cohomology and Lie Infinity-Connections (February 27, 2008)
On generalized differential cohomology and its relation to infinity-parallel transport and Lie-infinity connections.
John Baez: New Hire at UCR (February 27, 2008)
Julia Bergner has accepted a tenure-track position at UCR. She works on the homotopy theory of homotopy theories.
Urs Schreiber: Impressions on Infinity-Lie Theory (February 26, 2008)
Thoughts on infinity Lie theory.
Urs Schreiber: What I learned from Urs (February 26, 2008)
Bruce Bartlett talks about some aspects of the program of systematically understanding the quantization of Sigma-models in terms of sending parallel transport n-functors to the cobordism representations which encode the quantum field theory of the n-particles charged under them.
David Corfield: A Question or Two (February 25, 2008)
Is there a categorified spectrum?
Urs Schreiber: Lurie on Extended TQFT (February 21, 2008)
Jacob Lurie on extended TQFT and the Baez-Dolan Cobordism Hypothesis
John Baez: Logicians Needed Now (February 21, 2008)
Mike Stay and I are writing a paper… and we need help from logicians!
John Baez: Kostant on E8 (February 20, 2008)
Videos and lecture notes of Bertram Kostant’s talk on E8 and the algebra of the Standard Model.
Urs Schreiber: Categories, Logic and Physics in London, II (February 19, 2008)
Second workshop on “Categories, Logic and Foundations of Physics” in London.
John Baez: Harvard Research Free Online (February 18, 2008)
Harvard arts and sciences faculty vote to make their research free online!
Urs Schreiber: Slides: L-Infinity Connections and Applications (February 15, 2008)
Talks on L-infinity algebra connections.
David Corfield: 2-Galois and 2-Logic (February 13, 2008)
Categorifying logic
John Baez: Verity on ∞-Categories From Topology (February 12, 2008)
Dominic Verity talks about a weak ∞-category of cobordisms.
David Corfield: Lautman Conference (February 12, 2008)
Lautman conference
John Baez: Geometric Representation Theory (Lecture 25) (February 12, 2008)
Groupoidifying the commutation relations between annihilation and creation operators in quantum mechanics. An in-class experiment demonstrating these relations.
Urs Schreiber: Construction of Cocycles for Chern-Simons 3-Bundles (February 12, 2008)
On how to interpret the geometric construction by Brylinksi and McLaughlin of Cech cocycles classified by Pontrjagin classes as obstructions to lifts of G-bundles to String(G)-2-bundles.
Urs Schreiber: Question on Smooth Functions (February 11, 2008)
On morphisms of algebras of smooth functions coming from pullback along smooth maps.
John Baez: Metric Spaces (February 9, 2008)
Tom Leinster on the cardinality of a metric space, thought of as an enriched category — and an interesting relation to geometric measure theory.
Urs Schreiber: Smooth 2-Functors and Differential Forms (February 6, 2008)
An article on the relation between smooth 2-functors with values in strict 2-groups, and an outline of the big picture that this sits in.
Urs Schreiber: Chern-Simons States from L-infinity Bundles, III: States over the Circle (February 4, 2008)
On computing the states of Chern-Simons theory over the circle from the L-infinity algebraic model of the Chern-Simons 3-bundle over BG.
David Corfield: Albert Lautman (February 4, 2008)
The philosophy of Albert Lautman
David Corfield: Virtually Real or Really Virtual (February 2, 2008)
If NASA can have a presence in Second Life with their CoLab project, maybe we at the Café should be thinking about the next step….
John Baez: Modular Forms (February 1, 2008)
What are modular forms like beyond level 1? How about level 2, for starters?
Urs Schreiber: States of Chern-Simons Theory (February 1, 2008)
A list of selected literature discussing Chern-Simons theory and its space of states.
John Baez: Geometric Representation Theory (Lecture 24) (January 31, 2008)
How to groupoidify the harmonic oscillator.
Urs Schreiber: L-infinity Associated Bundles, Sections and Covariant Derivatives (January 30, 2008)
Associated L-infinity structures are obtained from Lie action infinity-algebroids, leading to a concept of sections and covariant derivatives in this context.
John Baez: Geometric Representation Theory (Lecture 23) (January 29, 2008)
James Dolan on groupoidifying the Hall algebra of the A2 quiver.
Urs Schreiber: Differential Forms and Smooth Spaces (January 28, 2008)
On turning differential graded-commutative algebras into smooth spaces, and interpreting these as classifying spaces.
John Baez: The Yoneda Embedding as a Reflection (January 25, 2008)
A pictorial way to think about the Yoneda embedding (or ‘continuation passing transform’).
Urs Schreiber: Integration Without Integration (January 24, 2008)
On how integration and transgression of differential forms is realized in terms of inner homs applied to transport n-functors and their corresponding Lie oo-algebraic connection data.
John Baez: Classifying Spaces for 2-Groups (January 24, 2008)
Just as principal bundles are classified by maps into a ‘classifying space’, the same is true for principal 2-bundles.
John Baez: Geometric Representation Theory (Lecture 22) (January 18, 2008)
James Dolan groupoidifies the “Hall algebra” of a quiver.
Urs Schreiber: Spaceoids (January 17, 2008)
Paolo Bertozzini, Roberto Conti and Wicharn Lewkeeratiyutkul on categorified spaces and the many-object version of C-star algebras.
David Corfield: 101 things to do with a 2-classifier (January 17, 2008)
What to do with a 2-classifier.
Urs Schreiber: Slides: On the BV-Formalism (BV Part XI) (January 16, 2008)
A discussion of BV-formalism from a Lie oo-algebraic perspective.
John Baez: Strong NDR Pairs — A Technical Question (January 16, 2008)
Some niggling technical questions about topology.
John Baez: Geometric Representation Theory (Lecture 21) (January 15, 2008)
The winter quarter of the Geometric Representation Theory seminar begins to tackle examples. Groupoidifying and q-deforming the harmonic oscillator.
Urs Schreiber: BV-Formalism, Part X: Symplectic Structures (January 14, 2008)
A Lie oo-algebraic perspective on the antibracket.
David Corfield: 2-Toposes (January 12, 2008)
What happens when you categorify topos theory. You get 2-topos theory… but what’s that like?
Urs Schreiber: Ginot and Stiénon on Characteristic Classes of 2-Bundles (January 11, 2008)
Stinot and Stienon on 2-bundles and their characteristic classes.
John Baez: The Continuation Passing Transform and the Yoneda Embedding (January 10, 2008)
The Yoneda embedding is familiar in category theory. The continuation passing transform is familiar in computer programming. They’re secretly the same!
John Baez: Geometric Representation Theory (Lecture 20) (January 10, 2008)
At last: the Fundamental Theorem of Hecke Operators!
Urs Schreiber: The Concept of a Space of States, and the Space of States of the Charged n-Particle (January 9, 2008)
On the notion of topos-theoretic quantum state objects, the proposed definition by Isham and Doering and a proposal for a simplified modification for the class of theories given by charged n-particle sigma-models.
John Baez: A Tiny Taste of the History of Mechanics (January 7, 2008)
Some incredibly sketchy lecture notes on the history of mechanics from Aristotle to Newton.
John Baez: How I Learned to Love the Nerve Construction (January 6, 2008)
Guest post by Tom Leinster on the work of Mark Weber.
John Baez: Geometric Representation Theory (Lecture 19) (January 5, 2008)
James Dolan lays the final piece of groundwork for the Fundamental Theorem of Hecke Operators.
Urs Schreiber: Dijkgraaf-Witten and its Categorification by Martins and Porter (January 4, 2008)
On Dijkgraaf-Witten theory as a sigma model, and its categorification by Martins and Porter.
Urs Schreiber: On BV Quantization, Part IX: Antibracket and BV-Laplacian (January 4, 2008)
The meaning of antibracket, BV-Laplacian and master equation in BV-formalism: Clifford structures on generalized smooth spaces.
Urs Schreiber: Comparative Smootheology (January 3, 2008)
A survey and comparison of the various notions of generalized smooth spaces, by Andrew Stacey.
David Corfield: Two Cultures in the Philosophy of Mathematics? (January 2, 2008)
Should we have a conference bringing together the two kinds of philosophers of mathematics?
John Baez: Geometric Representation Theory (Lecture 18) (January 1, 2008)
An intro to degroupoidification: the process of turning groupoids into vector spaces, and spans of groupoids into linear operators. A key prerequisite: ‘groupoid cardinality’.
John Baez: Non-Mathematician Rediscovers Rn, Cn (December 30, 2007)
Based on the philosophy of Ayn Rand, a nonmathematician has reinvented some familiar algebras — and issued a press release about it.
Urs Schreiber: Transgression of n-Transport and n-Connections (December 30, 2007)
On the general idea of transgression of n-connections and on the underlying machinery of generalized smooth spaces and their differential graded-commutative algebras of differential forms.
Urs Schreiber: BF-Theory as a Higher Gauge Theory (December 29, 2007)
On interpreting BF-theory as a higher gauge theory.
John Baez: The QG-TQFT Blues (December 29, 2007)
See Scott Carter sing the quantum gravity, topological quantum field theory blues.
David Corfield: Challenges for the Future (December 28, 2007)
Benjamin Mann’s 23 challenges.
John Baez: Geometric Representation Theory (Lecture 17) (December 26, 2007)
James Dolan on degroupoidification and ‘transfer’ for the zeroth homology of groupoids.
Urs Schreiber: Lie oo-Connections and their Application to String- and Chern-Simons n-Transport (December 25, 2007)
A discussion of connections for general L-infinity algebras and their application to String- and Chern-Simons n-transport.
John Baez: This Week’s Finds in Mathematical Physics (Week 260) (December 25, 2007)
The Retina and Ring Nebulae, my Christmas eve guide to free math and physics books online, and the “exceptional series” of Lie algebras
John Baez: Roytenberg on Weak Lie 2-Algebras (December 21, 2007)
Dmitry Roytenberg has proposed a fully general notion of categorified Lie algebra.
David Corfield: Progic VI (December 21, 2007)
Graphical models
Urs Schreiber: Non-Commutative Structures in Arithmetic and Geometry (December 21, 2007)
A conference on non-commutative structures.
John Baez: Geometric Representation Theory (Lecture 16) (December 21, 2007)
Turning a group acting on a set into a groupoid — the ‘weak quotient’ or ‘action groupoid’.
Urs Schreiber: Basic Bundle Theory and K-Cohomology Invariants (December 17, 2007)
A book on bundle theory and K-cohomology.
David Corfield: Is Mathematics Special? (December 17, 2007)
A conference to be held in Vienna in May 2008 tackles the question “Is Mathematics Special?”
David Corfield: A Dialogue on Infinity (December 15, 2007)
The application Alexandre Borovik and I submitted to the John Templeton Foundation as part of their funding of the core theme of infinity was successful. We intend to discuss and disseminate ideas via a blog – A Dialogue on Infinity….
John Baez: Geometric Representation Theory (Lecture 15) (December 13, 2007)
James Dolan on various forms of decategorification.
Urs Schreiber: Program Geometry and Physics at HIM (December 11, 2007)
A program on geometry and physics takes place summer 2008 at the HIM on Bonn.
John Baez: This Week’s Finds in Mathematical Physics (Week 259) (December 10, 2007)
Hear what may be lurking in the Egg Nebula, and learn about a mathematical phantom called the “field with one element”.
Urs Schreiber: Local Nets from 2-Transport (December 9, 2007)
How to obtain a local net of observables from an extended functorial QFT.
John Baez: One Geometry (Math Raps) (December 7, 2007)
A rappin’ proof of the Bolzano–Weierstrass theorem, by Steve Sawin.
Urs Schreiber: A Topos for Algebraic Quantum Theory (December 6, 2007)
A summary of some key points of Chris Heunen’s and Bas Spitters’ article.
David Corfield: Progic V (December 6, 2007)
I’ve come across something promissing for the Progic project. Apparently there is a way to complete the analogy: propositional logic : predicate logic :: Bayesian networks: ? The answer, it is claimed, is ‘probabilistic relational models’. Now before we…
Urs Schreiber: The Principle of General Tovariance (December 5, 2007)
Landsmann proposes that physical laws should be formulated such that they may be internalized into any topos.
John Baez: Look Around You (December 3, 2007)
Did you know that the largest number is 45 billion?
John Baez: An Unlikely Result (December 3, 2007)
A. N. Other puts ‘An unlikely result’ on the arXiv.
John Baez: Astronomers Destroy Universe (December 1, 2007)
Analysis of a media kerfuffle: can looking at the Universe actually hasten its demise? Is that what Krauss and Dent’s paper really said? What did it really say?
Urs Schreiber: On BV Quantization, Part VIII (November 29, 2007)
Towards understading BV by computing the charged n-particle internal to Z-categories, secretly following AKSZ.
John Baez: Geometric Representation Theory (Lecture 14) (November 29, 2007)
Matrix mechanics and Hecke operators. A failed attempt to state the Fundamental Theorem of Hecke Operators.
John Baez: Rejecta Mathematica (November 27, 2007)
You can now publish your rejected papers in Rejecta Mathematica.
John Baez: This Week’s Finds in Mathematical Physics (Week 258) (November 26, 2007)
Read about the flow of grainy substances, polycyclic aromatic hydrocarbons in outer space, and Deligne’s conjecture on Hochschild cohomology.
John Baez: Poncelet’s Porism (November 26, 2007)
A question from Gavin Wraith, requiring help from someone who knows elliptic curves and also old-fashioned synthetic geometry.
John Baez: Categories, Logic and Physics in London (November 26, 2007)
Bob Coecke and Andreas Döring are running a workshop on ‘Categories, Logic and Foundations of Physics’ at Imperial College.
Urs Schreiber: Concordance (November 23, 2007)
On the notion of concordance of 2-bundles and, more generally, on a notion of omega-anafunctor and a possible closed structure on the category of omega-categories with omega-anafunctors between them.
David Corfield: MA in Reasoning (November 23, 2007)
I would like to announce that we in the Centre for Reasoning here in Canterbury are launching a new MA course for September 2008. As you can see, this offers the chance to select from four core modules: Logical reasoning,…
John Baez: Geometric Representation Theory (Lecture 13) (November 22, 2007)
James Dolan on using braids to describe Hecke operators between flag representations.
Urs Schreiber: Something Like Lie-Rinehart ∞-pairs and the BV-complex (BV, part VII) (November 20, 2007)
Notes on something like Lie ∞-algebroids in the light of the BV complex.
John Baez: Geometric Representation Theory (Lecture 12) (November 18, 2007)
Heisenberg’s matrix mechanics and its many generalizations, such as the category of relations and the weak 2-category of spans. Understanding Hecke operators in terms of spans.
John Baez: Geometric Representation Theory (Lecture 11) (November 16, 2007)
How to describe Hecke operators between flag representations using matrices.
David Corfield: Category Theory and Biology (November 15, 2007)
Does category theory have anything to say to biology?
Urs Schreiber: Modules for Lie infinity-Algebras (November 13, 2007)
On modules for Lie infinity algebras in general and the definition given by Lars Kjeseth in particular.
John Baez: The Dangers of Complex Analysis (November 13, 2007)
Mathematicians like to redefine everyday words and make them into technical terms. This can be dangerous.
John Baez: Geometric Representation Theory (Lecture 10) (November 12, 2007)
Simultaneously categorifying and q-deforming Pascal’s triangle will lead us to a categorified quantum group. Here we take the first steps in that direction.
John Baez: Pfeiffer on Modular Tensor Categories (November 12, 2007)
Hendryk Pfeiffer describes the sort of gadget whose representations form a modular tensor category… and shows how to reconstruct this gadget from its modular tensor category of representations.
John Baez: Geometric Representation Theory (Lecture 9) (November 9, 2007)
Getting irreducible representations of symmetric groups from flag representations. Using ‘crackpot matrices’ to describe Hecke operators between flag representations.
John Baez: Geometric Representation Theory (Lecture 8) (November 6, 2007)
Categorifying and q-deforming Pascal’s triangle.
Urs Schreiber: 2-Vectors in Trondheim (November 5, 2007)
On line 2-bundles.
John Baez: Geometric Representation Theory (Lecture 7) (November 4, 2007)
James Dolan on two applications of Hecke operators: showing that any doubly transitive permutation representation is the direct sum of two irreducible representations, and getting ahold of the irreducible representations of n!
Urs Schreiber: n-Bundle Obstructions for Bruce (November 4, 2007)
On the global description of n-bundles obstructing lifts through shifted central extensions.
Guest: Concrete Groups and Axiomatic Theories II (November 3, 2007)
More by Todd Trimble on the duality between symmetry and structure: that is, between groups of transformations of finite sets and complete axiomatic theories.
Urs Schreiber: On Noether’s Second (BV, Part VI) (October 31, 2007)
On Noether’s second theorem and ghost/antighost pairing.
Urs Schreiber: BV for Dummies (Part V) (October 30, 2007)
Some elements of BV formalism, or rather of the Koszul-Tate-Chevalley-Eilenberg resolution, in a simple setup with ideosyncratic remarks on higher vector spaces.
John Baez: Comet Holmes (October 30, 2007)
Comet Holmes suddenly got a million times brighter last week, and now it’s visible with the naked eye.
John Baez: Higher Clifford Algebras (October 30, 2007)
A talk by Chris Douglas reporting on his work with Arthur Bartels and André Henriques on “higher Clifford algebras”. They’re related to elliptic cohomology and they form a 3-category!
John Baez: Fundamental Physics: Where We Stand Today (October 29, 2007)
Slides from an easy talk on the open problems in fundamental physics.
John Baez: Steve Fever (October 27, 2007)
A new story by Greg Egan — free online!
Guest: Concrete Groups and Axiomatic Theories I (October 26, 2007)
Todd Trimble on the duality between symmetry and structure: that is, between groups of transformations of finite sets and complete axiomatic theories.
John Baez: Geometric Representation Theory (Lecture 6) (October 26, 2007)
Categorifying and q-deforming the binomial coefficients. Why are the q-binomial coefficients polynomials with natural number coefficients? And, why are they “palindromic” polynomials? Bruhat classes and Schubert cells.
Urs Schreiber: On String- and Chern-Simons n-Transport (October 23, 2007)
Slides on String- and Chern-Simons n-Transport.
John Baez: Geometric Representation Theory (Lecture 5) (October 22, 2007)
James Dolan on how to get intertwining operators called “Hecke operators” from relations between types of geometrical figures.
Urs Schreiber: On Lie N-tegration and Rational Homotopy Theory (October 20, 2007)
On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan’s old article on this issue in particular.
Urs Schreiber: Advanced Course on Simplicial Methods in Higher Categories (October 19, 2007)
Lectures on simplicial methods in higher category theory.
John Baez: Geometric Representation Theory (Lecture 3) (October 19, 2007)
When you have any structure on a set, it has a group of symmetries. Here James Dolan shows how to work backwards: given the symmetries, how read off an axiom system describing the structure those symmetries preserve!
David Corfield: Klein 2-Geometry IX (October 18, 2007)
More thoughts on 2-geometry
Urs Schreiber: On Weak Cokernels for 2-Groups (October 17, 2007)
On weak cokernels of 2-groups.
John Baez: Geometric Representation Theory (Lecture 2) (October 17, 2007)
James Dolan on geometric representation theory: transformation groups, logic and the orbi-simplex.
Urs Schreiber: n-Curvature, Part III (October 16, 2007)
Curvature is the obstruction to flatness. Believe it or not.
Urs Schreiber: Loday and Pirashvili on Lie 2-Algebras (secretly) (October 16, 2007)
On the non-standard monoidal structure on 2-term chain complexes induced from the monoidal structure on Baez-Crans 2.vector spaces.
Urs Schreiber: What is the Fiber? (October 15, 2007)
On bundle gerbes and whether or not they “are” 2-bundles.
John Baez: This Week’s Finds in Mathematical Physics (Week 257) (October 15, 2007)
In “week257”, learn about astrophysics, number theory, topos theory in physics, distributive laws for monads, and hear what’s happening to the Tale of Groupoidification.
John Baez: Geometric Representation Theory (Lecture 4) (October 12, 2007)
Categorifying and q-deforming the theory of binomial coefficients — and multinomial coefficients! — using the analogy between projective geometry and set theory.
Urs Schreiber: Obstructions to n-Bundle Lifts Part II: The BIG Diagram (October 12, 2007)
A big diagram illustrating the situation of lifting a bundle through a String-like extension.
Urs Schreiber: BV-Formalism, Part IV (October 11, 2007)
Lie algebroids of action groupoids and their relation to BRST formalism.
Urs Schreiber: Question on Weak Pullbacks along Sequences (October 11, 2007)
How to weakly pull a functor along a short sequence of groupoids?
Urs Schreiber: Categorified Clifford Algebra and weak Lie n-Algebras (October 9, 2007)
On weak Lie n-algebras, differential graded Clifford algebra and Roytenberg’s work on weak Lie 2-algebras.
David Corfield: Progic IV (October 9, 2007)
More on unity probability theory and logic
John Baez: Geometric Representation Theory (Lecture 1) (October 7, 2007)
Video of the first lecture in the new seminar on Geometric Representation Theory at U. C. Riverside.
Urs Schreiber: Report on AIM Workshop: Towards Relative Symplectic Field Theory (October 5, 2007)
James Stasheff reports on the AIM workshop Towards Relative Symplectic Field Theory
Urs Schreiber: Cohomology of the String Lie 2-Algebra (October 2, 2007)
On the Lie 2-algebra cohomlogy of the String Lie 2-algebra and its relation to twisted K-theory.
John Baez: Spans in Quantum Theory (October 1, 2007)
You can use spans to understand why quantum processes act so much like pieces of spacetime.
David Corfield: New Blog (September 28, 2007)
Jeff Morton has a new blog!
David Corfield: Progic III (September 28, 2007)
More about reconciling logic and probability theory
Urs Schreiber: Detecting Higher Order Necklaces (September 27, 2007)
Nils Baas on higher order structures, Enrico Vitale on weak cokernels and a speculation on weak Lie n-algebras triggered by discussion with Pavol Severa.
John Baez: Rotations in the 7th Dimension (September 26, 2007)
Is there an exceptional isomorphism involving the Spin group in 7 dimensions?
Urs Schreiber: An Invitation to Higher Dimensional Mathematics and Physics (September 25, 2007)
A public talk supposed to illustrate some of the basic reasoning underlying higher dimensional algebra.
David Corfield: Progic II (September 25, 2007)
More on merging probability theory and logic.
John Baez: The Catsters Strike Again: “String Diagrams” (September 24, 2007)
Check out the other Catster.
Urs Schreiber: Obstructions, Tangent Categories and Lie N-tegration (September 24, 2007)
Thoughts on n-bundle theory in terms of Lie n-algebras.
John Baez: The Virtues of American Scientist (September 24, 2007)
While some science magazines have declined, the American Scientist is still great.
John Baez: The Catsters’ Latest Hit: “Adjunctions” (September 20, 2007)
Learn about adjunctions on YouTube!
John Baez: Deep Beauty: Understanding the Quantum World (September 19, 2007)
A symposium honoring the 75th anniversary of von Neumann’s book on quantum mechanics.
David Corfield: Progic (September 18, 2007)
My colleague here in Canterbury Jon Williamson is part of an international research group, progicnet, whose aim is to find a good integration of probability theory and first-order logic. For one reason or another, some technical projects get counted…
John Baez: The Catsters on YouTube: “Monads” (September 16, 2007)
Watch some videos from the new group out of Sheffield — the Catsters!
Urs Schreiber: Groupoidfest 07 (September 14, 2007)
A conference on groupoids.
Urs Schreiber: Obstructions for n-Bundle Lifts (September 12, 2007)
On obstructions to lifting the structure n-group of n-bundles.
David Corfield: Tim Gowers Joins the Blogosophere (September 11, 2007)
Gowers’ new blog
Urs Schreiber: Notes by Jurčo on generalized Bundle Gerbes (September 10, 2007)
Branislav Jurco has some notes on nonabelian bundle 2-gerbes and on associated bundle 1-gerbes.
Urs Schreiber: Lie n-Algebra Cohomology (September 7, 2007)
On characteristic classes of n-bundles.
David Corfield: Category Theory in Machine Learning (September 5, 2007)
Does category theory have a future in machine learning?
Urs Schreiber: Arrow-Theoretic Differential Theory IV: Cotangents (September 3, 2007)
Cotangents and morphisms of Lie n-algebroids from arrow-theoretic differential theory.
John Baez: Toward a Higher-Dimensional Wiki (September 2, 2007)
Let’s talk about setting up a wiki for n-category theory and other aspects of higher-dimensional algebra!
John Baez: Axis of Evil? Or Axis of Opportunity? (August 31, 2007)
A paper by Michael Longo argues that the axes of elliptic galaxies don’t point in random directions.
Urs Schreiber: On Hess and Lack on Bundles of Categories (August 30, 2007)
On Hess and Lack’s notion of “bundles of categories”.
Urs Schreiber: Question about von Neumann Algebras (August 30, 2007)
On von Neumann algebras and their bimodules.
John Baez: Journal Publishers Hire the “Pit Bull of PR” (August 29, 2007)
A publisher’s group called PRISM lobbying against open access journals has hired Eric Dezenhall, the “pit bull of PR”. Learn his strategies.
Urs Schreiber: Arrow-Theoretic Differential Theory III: Higher Morphisms (August 28, 2007)
Using arrow-theoretic differential theory to understand higher morphisms of Lie n-algebras.
John Baez: This Week’s Finds in Mathematical Physics (Week 256) (August 27, 2007)
Lie n-algebroids and Lie n-groupoids in classical mechanics… and the return of the Tale of Groupoidification
David Corfield: The Reasoner (August 26, 2007)
Contributing to The Reasoner.
Urs Schreiber: The G and the B (August 25, 2007)
How to get the bundle governing Generalized Complex Geometry from abstract nonsense and arrow-theoretic differential theory.
Urs Schreiber: That Shift in Dimension (August 24, 2007)
What makes the Kontsevich-Cattaneo-Felder theorem tick? How can it be that an n-dimensional quantum field theory is encoded in an (n+1)-dimensional one?
Urs Schreiber: Wilson Loop Defects on the String (August 23, 2007)
On Alekseev and Monnier’s work on quantizing Wilson loop observables for the WZW model.
Urs Schreiber: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III) (August 22, 2007)
On Lyakhovich and Sharapov’s insights into mechanics, holography and BV formalism.
John Baez: Gerbes in The Guardian (August 22, 2007)
A Guardian article about how hard it is to understand the concept of ‘gerbe’.
David Corfield: Justificatory Narratives (August 22, 2007)
What’s special about special functions.
Urs Schreiber: More on Tangent Categories (August 21, 2007)
More comments on the nature of tangent categories and their relation to the notion of shifted tangent bundles to differential graded spaces.
Urs Schreiber: On BV Quantization, Part II (August 18, 2007)
A review of elements of the Batalin-Vilkovisky formalism, with an eye towards my claim that this describes configuration spaces which are Lie n-algebroids.
Urs Schreiber: On BV Quantization. Part I. (August 17, 2007)
On BV-formalism applied to Chern-Simons theory and its apparent relation to 3-functorial extentended QFT.
John Baez: Higher Gauge Theory and the String Group (August 17, 2007)
A talk leading up to the concept of ‘characteristic classes for 2-bundles’, and a nice problem involving them.
David Corfield: Future Gazing (August 17, 2007)
What to expect in the next year
Urs Schreiber: On Roberts and Ruzzi’s Connections over Posets (August 16, 2007)
On J. Roberts and G. Ruzzi’s concept of G-bundles with connection over posets and its relation to analogous notions discussed at length at the n-Cafe.
David Corfield: Math Teach Wiki (August 15, 2007)
Back from Tuscany, I find two e-mails requests awaiting me. First, and I’m now very late on this story, Alexandre Borovik asked me to draw attention to the plight of a Mathematical Summer School held in Turkey. Second, Tim Porter…
Urs Schreiber: The Canonical 1-Particle, Part II (August 14, 2007)
More on the canonical quantization of the charged n-particle for the case of a 1-particle propagating on a lattice.
Urs Schreiber: Lazaroiu on G-Flows on Categories (August 14, 2007)
Lazaroiu’s concept of a “graded category wiuth shifts” is the same as that of a category “with G-flow”.
Urs Schreiber: Question about Tensor Categories (August 13, 2007)
Hendryk Pfeiffer asked me to forward the following question to the Café. Dear n-category people, I have a question about tensor categories on which I would appreciate comments and references. As probably several people are interested in this, I…
John Baez: Math Journal Wiki (August 12, 2007)
A wiki to help us keep an eye on math journals, their prices, their publishers, and their policies.
John Baez: This Week’s Finds in Mathematical Physics (Week 255) (August 12, 2007)
In week255, hear what happened at the 2007 Abel Symposium in Oslo. Read explanations of Jacob Lurie and Ulrike Tillman’s talks on cobordism n-categories, Dennis Sullivan and Ralph Cohen’s talks on string topology, Stephan Stolz’s talk on cohomology and…
Urs Schreiber: String and Chern-Simons Lie 3-Algebras (August 10, 2007)
A talk on Chern-Simons Lie n-algebras.
Urs Schreiber: Arrow-Theoretic Differential Theory, Part II (August 8, 2007)
A remark on maps of categorical vector fields, inner derivations and higher homotopies of L-infinity algebras.
Urs Schreiber: Adinkras (August 7, 2007)
The concept of an “Adinkra” - a graph used to describe representations of N-extended d=1 supersymmetry algebras - remarkably resembles some categorical structures which appear in the context of supersymmetry.
John Baez: Questions about Modules (August 5, 2007)
What kinds of categories are categories of modules of some ring.
Urs Schreiber: Gauge Tranformations of n-Bundles and (n-1)-Gerbes (August 4, 2007)
The notion of gauge transformation of n-bundles and (n-1)-gerbes.
Urs Schreiber: QFT of Charged n-Particle: Towards 2-Functorial CFT (August 3, 2007)
Towards a 2-functorial description of 2-dimensional conformal field theory. A project description.
Urs Schreiber: QFT of Charged n-Particle: Extended Worldvolumes (August 2, 2007)
Passing from locally to globally refined extended QFTs by means of the adjointness property of the Gray tensor product of the n-particle with the timeline.
Urs Schreiber: Solving Transformations for One of the Two n-Functors (August 1, 2007)
What is the weakest possible condition on a morphism of n-functors to allow solving for one of the two n-functors?
John Baez: Astronomical London (August 1, 2007)
The Royal Observatory, Greenwich.
John Baez: Web Spamming by Academic Publishers (July 31, 2007)
Have you ever thought you were getting a PDF file of a journal article, only to hit a webpage from a publisher demanding money for it? Then you’ve been web spammed.
John Baez: Higher Gauge Theory and Elliptic Cohomology (July 31, 2007)
A talk on higher gauge theory and its relation to the work of Stolz and Teichner.
David Corfield: Algebra 1 versus Algebra 2 (July 28, 2007)
Rota on two algebras
Urs Schreiber: Arrow-Theoretic Differential Theory (July 27, 2007)
We propose and study a notion of a tangent (n+1)-bundle to an arbitrary n-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
John Baez: Homotopy Theory and Higher Categories in Barcelona (July 26, 2007)
During the 2007-2008 academic year there will be a program on Homotopy Theory and Higher Categories in Barcelona.
David Corfield: Delphic Inspiration (July 25, 2007)
Mathematics and Narrative meeting
John Baez: Question About Representations of Finite Groups (July 25, 2007)
If you have a representation of a finite group, can you pick a basis such that all the matrices you get have entries lying in some cyclotomic field?
Urs Schreiber: Physical Systems as Topoi, Part III (July 23, 2007)
The third part of the talk.
Urs Schreiber: Physical Systems as Topoi, Part II (July 22, 2007)
Second part of the talk.
Urs Schreiber: Physical Systems as Topoi, Part I (July 22, 2007)
Doering on his work with Isham on topos theoretic models of quantum theory and general theories of physics.
Urs Schreiber: Making AdS/CFT Precise (July 22, 2007)
The last session of Recent Developments in QFT in Leipzig was general discussion, which happened to be quite interesting for various reasons….
Urs Schreiber: Deformation Quantization of Surjective Submersions (July 22, 2007)
Waldman on noncommutative deformations of total spaces of fiber bundles.
Urs Schreiber: Tangent Categories (July 21, 2007)
An arrow-theoretic formulation of tangency and its relation to inner automorphisms.
John Baez: Astronomical Paris (July 18, 2007)
A tour of L’Observatoire de Paris.
David Corfield: Mathematical Imperatives (July 16, 2007)
Mathematical ‘must’
David Corfield: George Mackey (July 14, 2007)
Who will collect the letters of George Mackey?
John Baez: This Week’s Finds in Mathematical Physics (Week 254) (July 13, 2007)
Connes and Marcolli’s new book, Witten’s new paper, exceptional Lie superalgebras and the Standard Model… and the Tale of Groupoidification, continued.
David Corfield: Breaking Out of the Box (July 13, 2007)
Try n-categories in spheres!
David Corfield: Return of the Euler Characteristic of a Category (July 9, 2007)
Leinster’s second paper on the Euler characteristic.
John Baez: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity (July 7, 2007)
Derek Wise’s thesis: how the geometry of the MacDowell–Mansouri and Chern–Simons approaches to gravity become clearer if we use Cartan connections.
John Baez: Mathematical Paris (July 7, 2007)
In Paris, you can get your name on a street sign just by being good in math. Really good.
David Corfield: Kernels in Machine Learning III (July 6, 2007)
More about kernels, including heat kernels, in machine learning.
Urs Schreiber: Multiplicative Structure of Transgressed n-Bundles (July 5, 2007)
On how to think of multiplicative structures on n-bundles with connection.
Urs Schreiber: Supercategories (July 4, 2007)
A proposal for a definition of “supercategory” and an arrow-theoretic understanding of some elements of supersymmetry.
Urs Schreiber: The Inner Automorphism 3-Group of a Strict 2-Group (July 4, 2007)
On the definition and construction of the inner automorphism 3-group of any strict 2-group, and how it plays the role of the universal 2-bundle.
David Corfield: Kernels in Machine Learning II (June 28, 2007)
More about kernels in machine learning.
Urs Schreiber: Conference on Categories in Geometry and Physics (June 28, 2007)
A conference on categories and their application in geometry and physics, taking place in Split, Croatia.
John Baez: This Week’s Finds in Mathematical Physics (Week 253) (June 28, 2007)
From the Standard Model to SU(5), SO(10), E6… and maybe even on to E8, with a friendly tip of the hat to symmetric spaces like the complexified octonionic plane.
Urs Schreiber: Colorings of Graphs (June 26, 2007)
Dmitry Kozlov on understanding the graph coloring problem by looking at simplicial Hom-complexes of graphs.
Urs Schreiber: Why Theoretical Physics is Hard… (June 25, 2007)
A remark on the holographic principle, on behalf of Witten’s paper on 3-dimensional gravity.
David Corfield: Kernels in Machine Learning I (June 25, 2007)
The use of kernels in machine learning
Urs Schreiber: Some Recreational Thoughts on Super-Riemannian Cobordisms (June 23, 2007)
On super-Riemannian structure on cobordisms in terms of super-Poincaré connections.
David Corfield: Faith and Reason (June 21, 2007)
Polanyi on faith and reason
Urs Schreiber: Curvature, the Atiyah Sequence and Inner Automorphisms (June 20, 2007)
On the notion of curvature 2-functor in light of morphisms from the path sequence of the base to the Atiyah sequence of the bundle.
David Corfield: Degeneracy (June 19, 2007)
Papers on degenerate n-categories
John Baez: Cohomology and Computation (Week 27) (June 19, 2007)
Defining the cohomology of algebraic gadgets using the bar construction.
Urs Schreiber: Waldorf on Parallel Transport Functors (June 18, 2007)
Review of parallel transport functors by Konrad Waldorf.
Urs Schreiber: Generalized Geometric Langlands is False (June 18, 2007)
C. Teleman on a counter example to the generalised geometric Langlands conjecture.
John Baez: Cohomology and Computation (Week 26) (June 18, 2007)
An example of the bar construction: puffing up a point to the free contractible G-space EG, important in group cohomology.
John Baez: More Mysteries of the Number 24 (June 17, 2007)
Kummer’s 24 hypergeometric functions: what are they?
David Corfield: Opetopes as Trees (June 16, 2007)
A new paper on opetopes.
Urs Schreiber: Polyvector Super-Poincaré Algebras (June 14, 2007)
Superextension of Poincare algebras and how these give rise to brane charges.
John Baez: Why Math Teachers Get Grumpy (June 13, 2007)
Grading final exams — a test of ones soul.
Urs Schreiber: Spectral Triples and Graph Field Theory (June 12, 2007)
Yan Soibelman is thinking about spectral stringy geometry.
David Corfield: Two ArXiv Papers (June 12, 2007)
Two papers on bicategories.
David Corfield: More Mathematical Blogging (June 12, 2007)
A new mathematics blog.
Urs Schreiber: Extended QFT and Cohomology II: Sections, States, Twists and Holography (June 10, 2007)
How transformations of extended d-dimensional quantum field theories are related to (d-1)-dimensional quantum field theories. How this is known either as twisting or as, in fact, holography.
John Baez: Quantization and Cohomology (Week 27) (June 8, 2007)
Review of what we’ve done in this course; prospectus of what’s still to be done.
Urs Schreiber: Extended Quantum Field Theory and Cohomology, I (June 8, 2007)
On understanding extended quantum field theory and generalized cohomology.
Urs Schreiber: Large Smooth Categories (June 8, 2007)
Stacks versus categories internal to sheaves.
Urs Schreiber: The n-Café Quantum Conjecture (June 8, 2007)
Why it seems that quantum mechanics ought to be the de-refinement of a refined theory which lives in one categorical degree higher than usual.
David Corfield: The Curious Incident of the Dog in the Night-time (June 8, 2007)
The cyclic category.
David Corfield: Categorifying Quantum Mechanics (June 7, 2007)
Two approaches to categorifying quantum mechanics — compare them!
John Baez: Cohomology and Computation (Week 25) (June 6, 2007)
Getting monads, comonads and simplicial objects from adjoint functors.
David Corfield: Whose 2-Vector Spaces? (June 6, 2007)
2-vector spaces for elliptic cohomology.
David Corfield: June Events (June 5, 2007)
June conferences
John Baez: Quadratic Reciprocity (June 4, 2007)
Quadratic Reciprocity is one of the gems of elementary number theory. Why does this famous proof actually work?
Urs Schreiber: Connections on String-2-Bundles (June 3, 2007)
On connections on String 2-bundles.
John Baez: Quantization and Cohomology (Week 26) (June 1, 2007)
Cech cohomology in terms of anafunctors and ananatural transformations.
David Corfield: The Woodstock of the Mind (June 1, 2007)
The Hay Festival
John Baez: Quantization and Cohomology (Week 25) (June 1, 2007)
How describing bundles in terms of Cech cohomology secretly amounts to describing them in terms of smooth anafunctors.
John Baez: Quantization and Cohomology (Week 24) (June 1, 2007)
Three approaches to connections and gauge transformations, leading up to one based on Toby Bartels’ notion of “smooth anafunctor”.
Urs Schreiber: On the Bar Construction (May 31, 2007)
Todd Trimble on the bar construction.
Urs Schreiber: The Second Edge of the Cube (May 31, 2007)
Differentiating parallel transport anafunctors to Cartan-Ehresmann connections.
Urs Schreiber: Kock on Higher Connections (May 31, 2007)
Anders Kock gives a synthetic differential description of parallel n-transport using strict n-fold categories.
Urs Schreiber: What is a Lie Derivative, really? (May 31, 2007)
On the arrow-theory behind Lie derivatives.
John Baez: This Week’s Finds in Mathematical Physics (Week 252) (May 28, 2007)
The long-range weather report on Neptune, hot Neptunes in other solar systems, the electromagnetic snake at the center of our galaxy, and Hecke operators.
John Baez: Link Homology and Categorification in Kyoto (May 26, 2007)
Link Homology and Categorification — a conference in Kyoto, May 10-25, 2007.
John Baez: Congratulations to David! (May 25, 2007)
David Corfield found a permanent position at the University of Kent at Canterbury!
Urs Schreiber: Derivation Lie 1-Algebras of Lie n-Algebras (May 25, 2007)
On ordinary Lie algebras “of derivations” of Lie n-algebras.
David Corfield: Going Hi-Tech (May 25, 2007)
Slides on n-categories
John Baez: Cohomology and Computation (Week 24) (May 24, 2007)
What makes the bar construction tick?
David Corfield: The Two Cultures of Mathematics Revisited (May 22, 2007)
Revisiting the idea of two cultures in mathematics.
John Baez: Questions on n-Categories and Topology (May 21, 2007)
Questions by Bruce Westbury on $n$-categories and topology
Urs Schreiber: Chern Lie (2n+1)-Algebras (May 19, 2007)
Generalizing Baez-Crans Lie 2n-algebras to Chern and to Chern-Simons Lie (2n+1)-algebras.
Urs Schreiber: Calculations Inside Semisimple Categories (May 18, 2007)
Bruce Bartlett on computations in semisimple categories.
David Corfield: Linear Algebra Done Right (May 18, 2007)
Doing linear algebra right! — the trouble with determinants.
John Baez: Penrose on Angular Momentum: An Approach to Combinatorial Space-Time (May 18, 2007)
Roger Penrose’s paper Angular Momentum: an Approach to Combinatorial Space-Time is now available in electronic form.
David Corfield: Two Physics Papers Involving Categorification (May 16, 2007)
Physics papers dealing with categorification
John Baez: People Who May or May Not Actually Exist (May 12, 2007)
A lot of famous people may or may not actually exist. What are the odds?
John Baez: Puzzle #12 (May 12, 2007)
What’s the size of New Jersey, shaped like a dog bone, and named after a famous woman?
John Baez: Cohomology and Computation (Week 23) (May 11, 2007)
The bar construction: a way to get simplicial sets from algebraic gadgets.
Urs Schreiber: Zoo of Lie n-Algebras (May 10, 2007)
A menagerie of examples of Lie n-algebras and of connections taking values in these, including the String 2-connection and the Chern-Simons 3-connection.
John Baez: Quantization and Cohomology (Week 23) (May 10, 2007)
Connections on principal bundles — a modern treatment.
Urs Schreiber: Workshop on Elliptic Cohomology in Hamburg (May 9, 2007)
A little workshop on elliptic cohomology, mainly the functorial QFT approach by Stolz and Teichner, in Hamburg, June 2007.
Urs Schreiber: Report from “Workshop on Higher Gauge Theory” (May 9, 2007)
Report-back on a little symposium titled “Higher Gauge Theory” (but concerned just with abelian gerbes) that took place at the AEI in Golm.
John Baez: Cohomology and Computation (Week 22) (May 8, 2007)
How to count holes.
John Baez: Quantization and Cohomology (Week 22) (May 8, 2007)
Smooth functors and beyond: why we need nontrivial principal U(1) bundles to describe phases in geometric quantization.
John Baez: Support the Banff Protocol! (May 8, 2007)
There’s a new AMS Notices article on how the board of Topology resigned to protest Elsevier’s high prices. Support the Banff Protocol — avoid publishing in highly expensive journals!
David Corfield: Klein 2-Geometry VIIIS (May 7, 2007)
A few additional Klein 2-geometry thoughts.
John Baez: This Week’s Finds in Mathematical Physics (Week 251) (May 6, 2007)
Foundations of quantum mechanics — the news from Les Treilles.
Urs Schreiber: The First Edge of the Cube (May 4, 2007)
The notion of smooth local i-trivialization of transport n-functors for n=1.
Urs Schreiber: Grandis on Collared Cobordisms and TQFT (May 2, 2007)
Marco Grandis discusses the arrow-theory of collared cobordisms and topological quantum field theory.
Urs Schreiber: Symposium on “Higher Gauge Theory” in Golm (May 2, 2007)
A workshop on abelian gerbes, Deligne cohomology and strings.
David Corfield: Probability in Amsterdam (May 1, 2007)
Foundations of the Formal Sciences conference in Amsterdam.
John Baez: This Week’s Finds in Mathematical Physics (Week 250) (April 26, 2007)
The Popescu-Rohrlich game, double cosets, atomic invariant relations and spans of groupoids.
Urs Schreiber: n-Curvature (April 25, 2007)
The n-curvature canonically associated with a transport n-functor.
David Corfield: Learning from Our Ancestors (April 25, 2007)
What can we learn from earlier mathematicians?
David Corfield: Who’s on the Right Track? (April 23, 2007)
Connes on the wrong track for quantum gravity.
Urs Schreiber: Report-Back on BMC (April 22, 2007)
Bruce Bartlett reports from the British Mathematics Colloquium 2007.
John Baez: Cohomology and Computation (Week 21) (April 20, 2007)
Why mathematicians like to take algebraic gadgets and topological spaces and turn them into simplicial sets.
John Baez: Quantization and Cohomology (Week 21) (April 20, 2007)
Understanding the ‘action’ in classical mechanics as a smooth functor — the case of a path groupoid.
John Baez: Cohomology and Computation (Week 20) (April 20, 2007)
Why do mathematicians like simplices so much? Why are they better than other shapes?
Urs Schreiber: Some Conferences (April 19, 2007)
A conference on bundles and gerbes, another one on topology, and comments on associated 2-vector bundles and String connections.
David Corfield: The Field With One Element (April 17, 2007)
For some grand theory building and an answer to the question ‘What is the field with one element?’, see Nikolai Durov’s New Approach to Arakelov Geometry.
David Corfield: The Two Cultures of Mathematics (April 17, 2007)
‘Theory-builders’ and ‘problem-solvers’.
John Baez: Incandescence (April 14, 2007)
Greg Egan’s next novel is coming out in 2008 — but you can already read a story set in the same universe.
John Baez: Topos Theory in the New Scientist (April 14, 2007)
Robert Matthews on Chris Isham’s work on topos theory and physics.
John Baez: Schur Functors (April 12, 2007)
What’s the right way to think about Schur functors?
David Corfield: Structure and Pseudorandomness (April 12, 2007)
Terence Tao has written three delightful posts detailing his views on the relationship between structure and pseudorandomness in mathematics.
David Corfield: Category Theory as Esperanto (April 11, 2007)
If category theory is the Esperanto of mathematics, how many mathematicians speak it fluently?
John Baez: Quantization and Cohomology (Week 20) (April 11, 2007)
Generalizing smooth manifolds to smooth spaces, so we can define smooth categories and study the principal of least action in a very general setting.
John Baez: This Week’s Finds in Mathematical Physics (Week 249) (April 9, 2007)
Klein’s Erlangen program, now available in English! And: the Tale of Groupoidification, continued.
John Baez: Why Mathematics Is Boring (April 6, 2007)
Storytellers have developed many strategies for luring in readers and keeping them interested. Mathematicians systematically avoid these.
David Corfield: Whatever Happened to the Categories? (April 6, 2007)
Are we doing our job as broadcasters well? Max Tegmark has a new paper out on the physical universe as an abstract mathematical structure. Not a whiff of categories, let alone n-categories. Tegmark has read some of philosophy of…
John Baez: Cohomology and Computation (Week 19) (April 6, 2007)
The origins of cohomology in the study of ‘syzygies’, or relations between relations.
David Corfield: Automated Theorem Proving (April 5, 2007)
In Brussels, we heard from Koen Vervloesem about attempts towards better automated theorem provers. Readers of my book will know that I devoted its second chapter to automated theorem provers, to provide a relief against which to consider ‘real…
John Baez: Quantization and Cohomology (Week 19) (April 4, 2007)
Finding critical points of the action in a general context: a smooth category equipped with a smooth ‘action’ functor.
Urs Schreiber: Oberwolfach CFT, Tuesday Morning (April 3, 2007)
On Q-systems, on the Drinfeld Double and its modular tensor representation category, and on John Roberts ideas on nonabelian cohomology and QFT.
Urs Schreiber: Oberwolfach CFT, Monday Evening (April 3, 2007)
Some random notes concerning, bundles, functorial quantum field theory and algebraic QFT.
David Corfield: Rota on Combinatorics (April 2, 2007)
From an interview with Gian-Carlo Rota and David Sharp: Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that…
Urs Schreiber: Oberwolfach CFT, Monday Morning (April 2, 2007)
Some basic concepts and facts concerning local representations of nets of local algebras on the real line.
Urs Schreiber: Oberwolfach CFT, Arrival Night (April 1, 2007)
Some musings on the relation of AQFT to functorial QFT.
David Corfield: Bernard Williams on Scientism (April 1, 2007)
In Brussels, Brendan Larvor took us through a range of options for those of us who want our philosophy of mathematics to take serious notice of the history of mathematics. A distinction he relied upon was one Bernard Williams introduced…
Urs Schreiber: Some Notes on Local QFT (March 31, 2007)
Some aspects of the AQFT description of 2d CFT.
John Baez: This Week’s Finds in Mathematical Physics (Week 248) (March 29, 2007)
Watch movies of coronal mass ejections, auroras, and tornados on the Sun! Then: the Tale of Groupoidification continues.
John Baez: Relativity on the World-Wide Web (March 28, 2007)
Chris Hillman is back with a new improved guide to learning relativity!
David Corfield: Philosophising in Brussels (March 24, 2007)
I’m off to Brussels tomorrow to speak at a conference called Perspectives on Mathematical Practices 2007. You can see the notes for my talk.
John Baez: This Week’s Finds in Mathematical Physics (Week 247) (March 23, 2007)
Symmetry: from the ancient Greeks through medieval Islam to E8 and groupoidification.
Urs Schreiber: History of Understanding Bundles with Connection using Parallel Transport around Loops (March 23, 2007)
A list of some papers involved in the historical development of the idea of expressing bundles with connection in terms of their parallel transport around loops.
Urs Schreiber: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps (March 22, 2007)
In which sense D-branes generalize to “bi-branes”.
Urs Schreiber: Recent Developments in QFT in Leipzig (March 21, 2007)
A conference on new developments in Quantum Field Theory at the Max Planck Institute in Leipzig.
David Corfield: Generalising Hopf Algebras (March 20, 2007)
Gizem Karaali’s paper ‘On Hopf algebras and their generalizations’, in which she describes Hopf algebras and five attempts to generalise them.
Urs Schreiber: QFT of Charged n-Particle: The Canonical 1-Particle (March 19, 2007)
On the category of paths whose canonical Leinster measure reproduces the path integral measure appearing in the quantization of the charged particle.
John Baez: News about E8 (March 19, 2007)
A huge new calculation involving the exceptional Lie group E8.
John Baez: Quantum Logic, Topology and Categories at Oxford (March 16, 2007)
Conferences on categories, logic, computation and topology at Oxford this summer.
Urs Schreiber: Poisson-Sigma Models, Lie Algebroids, Deformations and Higher Analogues in Vienna (March 16, 2007)
Announcement of ESI Workshop on Lie Algebroids in Summer 2007.
John Baez: Classical vs Quantum Computation (Week 18) (March 15, 2007)
2-categories of computation, and an intriguing relation to catastrophe theory.
Urs Schreiber: Star-Structures and Daggers (March 15, 2007)
Question on the compatibility of daggers and stars in monoidal 2-categories with duals.
David Corfield: Manin on Mathematics (March 15, 2007)
On the ArXiv today, Yuri Manin has one of those wide-ranging overviews of the life of mathematics: Mathematical knowledge: internal, social and cultural aspects. One comment - When Poincaré said that there are no solved problems, there are only problems…
Urs Schreiber: Differentiating Lie Groupoids to Lie Algebroids (March 14, 2007)
On an attempt to find a nice way to formulate the definition of the Lie algebroid obtained from a Lie groupoid.
John Baez: Quantization and Cohomology (Week 18) (March 14, 2007)
From particles to strings. First: building a Hilbert space from a category C equipped with an "amplitude" functor A: C → U(1). Then: building a 2-Hilbert space from a 2-category C equipped with a 2-functor A: C → U(1)Tor.
John Baez: Snowglobe Models (March 10, 2007)
Constructing strange ‘nonstandard models’ of typed λ-calculi.
John Baez: We’re Under Attack! (March 9, 2007)
We’re experiencing problems due to a trackback spambot.
John Baez: Gurski on Tricategories (March 9, 2007)
If your work takes you into the land of weak 3-categories, you’ll need to read this.
John Baez: Classical vs Quantum Computation (Week 17) (March 9, 2007)
String worldsheets illustrating processes of computation!
Urs Schreiber: Canonical Measures on Configuration Spaces (March 8, 2007)
On how the Leinster weighting on a category might provide path integral measures in physics.
John Baez: Quantization and Cohomology (Week 17) (March 7, 2007)
Getting Hilbert spaces and operator algebras from categories.
Urs Schreiber: QFT of Charged n-Particle: Sheaves of Observables (March 6, 2007)
On the concepts of sheaves and nets of algebras of observables in quantum field theory.
David Corfield: Computer Science and Physics (March 5, 2007)
Samson Abramsky’s paper: What are the fundamental structures of concurrency? We still don’t know!
Urs Schreiber: QFT of Charged n-Particle: Disk Path Integral for String in trivial KR Field (March 5, 2007)
The arrow-theoretic perspective on the path integral for the disk diagram of the open string.
John Baez: A Topos Foundation for Theories of Physics (March 3, 2007)
A major new paper by Isham and Döring.
David Corfield: Dynamics of Mathematical Reason (March 2, 2007)
I’m having a spot of bother getting a paper published. It’s about the philosopher Michael Friedman’s treatment of mathematics in his Dynamics of Reason. I’d be grateful for any comments from the Café clientele.
John Baez: Classical vs Quantum Computation (Week 16) (March 2, 2007)
The λ-calculus for… high school calculus!
Urs Schreiber: QFT of Charged n-Particle: Gauge Theory Kinematics (February 28, 2007)
A review of Fleischhack’s discussion of a Weyl algebra on spaces of generalized connections.
Urs Schreiber: QFT of Charged n-Particle: Algebra of Observables (February 28, 2007)
The algebra of observables as certain endomorphisms of the n-category of sections.
John Baez: Quantization and Cohomology (Week 16) (February 27, 2007)
More examples of path-integral quantization: the particle in a potential.
John Baez: This Week’s Finds in Mathematical Physics (Week 246) (February 25, 2007)
The trouble with fundamental physics today.
Urs Schreiber: Amplimorphisms and Quantum Symmetry, II (February 25, 2007)
A remark on the appearance of iterated module n-categories in quantum field theory.
David Corfield: The Health Book (February 23, 2007)
The book — Why Do People Get Ill? — is finally out there in the shops.
John Baez: How to Write Mathematics Badly (February 23, 2007)
A hilarious video.
John Baez: Classical vs Quantum Computation (Week 15) (February 23, 2007)
The λ-theory of commutative rings.
David Corfield: Noncommutative Geometry Blog (February 21, 2007)
A new blog on noncommutative geometry.
David Corfield: Cake Talk (February 21, 2007)
A talk on Bayesianism, information geometry, and nonparametric statistics.
John Baez: An Introduction to Algebraic Topology (February 21, 2007)
Want to learn a little algebraic topology?
John Baez: Quantization and Cohomology (Week 15) (February 21, 2007)
An example of path integral quantization: the free particle on a line (continued).
David Corfield: Congratulations! (February 17, 2007)
It’s John’s wedding day today!
John Baez: Classical vs Quantum Computation (Week 14) (February 16, 2007)
From typed lambda-calculus to cartesian closed categories and back.
Urs Schreiber: QFT of Charged n-Particle: T-Duality (February 16, 2007)
Topological T-duality as a pull-push transformation of sections of the 2-particle.
John Baez: Quantization and Cohomology (Week 14) (February 15, 2007)
An example of path integral quantization: the free particle on a line.
Urs Schreiber: Higher Morphisms of Lie n-Algebras and L-infinity Algebras (February 15, 2007)
On higher morphisms of Lie n-algebras, and higher homotopies between L_oo algebras.
David Corfield: Category Theoretic Probability Theory II (February 13, 2007)
More on what category theory has to say about probability theory.
Urs Schreiber: Slides for Freed’s Andrejewski Lecture (February 13, 2007)
Lecture by Freed on Freed-Hopkins-Teleman.
Urs Schreiber: QFT of Charged n-Particle: Dynamics (February 12, 2007)
Definition of the dynamics of the charged quantum particle by pull-push along correspondences of path spaces.
John Baez: This Week’s Finds in Mathematical Physics (Week 245) (February 12, 2007)
Read about Fields Institute workshop on Higher Categories and Their Applications.
David Corfield: Why Do I Bother? (February 10, 2007)
Terence Tao’s paper ‘What is good mathematics?’, and what the philosophy of mathematics might say about this.
John Baez: Classical vs Quantum Computation (Week 13) (February 9, 2007)
Lambek and Scott’s definition of a typed lambda-calculus.
Urs Schreiber: Day on RCFTs (February 9, 2007)
Brian Day on a relation between fusion rings, braided tensor categories and “realisation” in enriched category theory.
John Baez: Infinitely Categorified Calculus (February 9, 2007)
In noncommutative geometry, we must infinitely categorify the calculus!
Urs Schreiber: Isham on Arrow Fields (February 8, 2007)
On Chris Isham’s notion of an “arrow field” on a category.
Urs Schreiber: QFT of Charged n-particle: Chan-Paton Bundles (February 7, 2007)
Chan-Paton bundles from the pull-push quantization of the open 2-particle.
David Corfield: Category Theoretic Probability Theory (February 7, 2007)
Having noticed (e.g., here and here) that what I do in my day job (statistical learning theory) has much to do with my hobby (things discussed here), I ought to be thinking about probability theory in category theoretic terms….
John Baez: Quantization and Cohomology (Week 13) (February 6, 2007)
Statistical mechanics and temperature-dependent mathematics.
Urs Schreiber: Amplimorphisms and Quantum Symmetry, I (February 6, 2007)
Localized endomorphisms of quantum observables in arrow-theoretic terms.
Urs Schreiber: Amplimorphisms (February 5, 2007)
A certain notion of twisted morphisms of algebras – and how to think about it.
David Corfield: In the Footsteps of Rudolf Carnap II (February 5, 2007)
The next day I set off East to Jena, following the path taken by Carnap, and by my host, David Green, a British mathematician who works on the cohomology of finite groups. While in Wuppertal, David had become interested…
David Corfield: In the Footsteps of Rudolf Carnap I (February 5, 2007)
Last week I gave a couple of talks in Germany. Thursday saw me in the town of Wuppertal, famous for its Schwebebahn, a railway built above the river Wupper, which snakes its way through the middle of the town. As…
John Baez: This Week’s Finds in Mathematical Physics (Week 244) (February 3, 2007)
How to measure the size of a category.
Urs Schreiber: Higher Structure in Geometry and Physics – and in Paris (February 2, 2007)
Slides of talks for a conference on higher structures in geometry and physics
Urs Schreiber: Huisken on Uniformization, II (February 2, 2007)
Something about Ricci flow and the proof of the Poincare conjecture.
Urs Schreiber: Huisken on Uniformization, I (February 2, 2007)
Some ideas behind the proof of the Poincare conjecture by means of Ricci flow.
John Baez: Classical vs Quantum Computation (Week 12) (February 1, 2007)
How to see computation as a process: rewrite rules.
Urs Schreiber: Towards the FFRS Description of 2dCFT (B) (February 1, 2007)
“2-processes”: 2-morphisms, adjunctions, the exchange law and the Frobenius property
John Baez: In Memory of Max Kelly (February 1, 2007)
Letters in appreciation of Max Kelly.
John Baez: Quantization and Cohomology (Week 12) (January 31, 2007)
Classical mechanics, quantum mechanics and statistical mechanics as ‘matrix mechanics’ over various rigs (rings without nnegatives).
David Corfield: No Need to Apologise (January 29, 2007)
Café regular John Armstrong has a blog. It goes by the name of The Unapologetic Mathematician. A subtle allusion to Hardy’s A Mathematician’s Apology, playing cleverly on the two meanings of apology?…
Urs Schreiber: CFT in Oberwolfach (January 29, 2007)
An Oberwolfach meeting on conformal field theory.
David Corfield: Another Interview (January 28, 2007)
It’s worth taking a look at an interview Mikio Sato gave to Emmanuel Andronikof in 1990, published in February’s Notices of the American Mathematical Society. Sato is famous for algebraic analysis, D-modules, and the like, about which I know next…
David Corfield: Peering Through the Veil (January 27, 2007)
Twice in recent days I have confronted the possibility of experiencing a kind of alienation due to interviews. First, my co-author Darian Leader and I were interviewed by the New Scientist about our book Why Do People Get Ill?. A…
Urs Schreiber: Globular Extended QFT of the Charged n-Particle: String on BG (January 26, 2007)
The string on the classifying space of a strict 2-group.
John Baez: Classical vs Quantum Computation (Week 11) (January 25, 2007)
Lots of examples of 2-categories!
John Baez: Classical vs Quantum Computation (Week 10) (January 24, 2007)
Categorifying the concept of ‘category’!
Urs Schreiber: The Globular Extended QFT of the Charged n-Particle: Definition (January 24, 2007)
Turning a classical parallel transport functor on target space into a quantum propagation functor on parameter space.
Urs Schreiber: Cocycle Category (January 24, 2007)
Bruce Bartlett reports on Rick Jardine’s concept of ‘cocycle categories’ and their relation to anafunctors.
John Baez: Quantization and Cohomology (Week 11) (January 24, 2007)
What’s really going on with quantization?
David Corfield: Two Café Owners Interviewed (January 23, 2007)
As neither John nor Urs has announced it, readers might like to find out about their motivations for starting and running this blog in an interview they gave to Bruce Bartlett, available in written form and also as an MP3…
John Baez: Higher Categories and Their Applications (January 23, 2007)
The Fields Institute workshop on n-categories was a lot of fun. If you couldn’t make it, you can still see what it was like.
Urs Schreiber: Fusion and String Field Star Product (January 21, 2007)
A description of the string field star product and of fusion of loop group representations from the point of view of functorial transport.
John Baez: Animals That Are Also Verbs (January 21, 2007)
A fly can fly. A duck can duck. And a dog can dog you.
David Corfield: More on Duality (January 19, 2007)
Continuing our earlier discussion about duality, it’s worth noting a distinction that Lawvere and Rosebrugh introduce in chapter 7 of their Sets for Mathematics between ‘formal’ and ‘concrete’ duality. Formal duality concerns mere arrow reversal in the relevant diagrams,…
John Baez: Traces in Ottawa (January 19, 2007)
A workshop on applications of traces to algebra, analysis and categorical logic.
John Baez: Knot Homology in Faro (January 19, 2007)
This year’s Oporto Meeting Geometry, Topology and Physics will actually be held in Faro, The theme is knot homology!
John Baez: Multiplicative Intuitionistic Linear Logic (January 18, 2007)
Translating from how logicians think about mulitplicative intuitionistic linear logic to how category theorists might think about it.
David Corfield: Duality between Probability and Optimization (January 18, 2007)
One of the reasons I have an interest in what we find out about mechanics in different rigs is that many machine learning algorithms are expressible in thermodynamic form, as the tutorial, Energy-Based Models: Structured Learning Beyond Likelihoods, by…
Urs Schreiber: D-Branes from Tin Cans, III: Homs of Homs (January 18, 2007)
Sections of sections, their pairing and n-disk correlators.
John Baez: Quantization and Cohomology (Week 10) (January 17, 2007)
Classical versus quantum mechanics: the Lagrangian and Hamiltonian approaches
Urs Schreiber: Towards the FFRS Description of 2dCFT (A) (January 16, 2007)
2-dimensional cobordisms, topological quantum field theory and Frobenius algebras
Urs Schreiber: Aaronson on the Nature of Quantum Mechanics (January 16, 2007)
Scott Aaronson on the nature of quantum mechanics.
Urs Schreiber: Khovanov Homology (January 14, 2007)
Khovanov homology and its generalization from links to tangles.
Urs Schreiber: The First Part of the Story of Quantizing by Pushing to a Point… (January 13, 2007)
Taking sections of transport functors by pushing them to a point.
David Corfield: Ubiquitous Duality (January 11, 2007)
I’m in one of those phases where everywhere I look I see the same thing. It’s Fourier duality and its cousins, a family which crops up here with amazing regularity. Back in August, John wrote: So, amazingly enough, Fourier duality…
David Corfield: Universality in Particularity (January 8, 2007)
To keep me from brooding on the pleasure I’m missing out on by not being with my Café co-hosts in Toronto, let me try out a blog post. In just about every academic endeavour to which I’ve applied myself, I…
John Baez: Classical vs Quantum Computation (Week 9) (January 5, 2007)
Now we’ll categorify the concept of ‘category’ to see computation as a process!
Urs Schreiber: FFRS on Uniqueness of CFT: Sewing as Natural Transformation (January 4, 2007)
On FFRS’s formulation of sewing constraints in terms of natural transformations.
Urs Schreiber: FFRS on Uniqueness of CFT: Morphisms into Transport Functors (January 3, 2007)
On morphisms into functors and states and sections in the FFRS description of conformal field theory.
Urs Schreiber: FFRS on Uniqueness of Conformal Field Theory (January 3, 2007)
A strengthening of the FFRS theorem on 2-dimensional rational conformal field theory.
David Corfield: Research Proposals (January 3, 2007)
There’s nothing quite like a research proposal to give you a sense of some of the big stories out there. Try Geometry and Quantum Theory for what’s happening in Holland of relevance to the Café. From a couple of years…
John Baez: Puzzle #11 (December 31, 2006)
When was the toothpick invented?
John Baez: This Week’s Finds in Mathematical Physics (Week 243) (December 26, 2006)
In week243 of This Week’s Finds, hear about Claude Shannon, his sidekick Kelly, and how they used information theory to make money at casinos and the stock market. Hear about the new book Fearless Symmetry, which explains fancy number…
John Baez: The Earliest Stars? (December 25, 2006)
Have people caught sight of the earliest stars in the Universe?
John Baez: arXiv Policy Statement? (December 24, 2006)
Where is the arXiv’s official policy statement about guaranteed open access?
John Baez: Puzzle #9 (December 24, 2006)
Which philosopher is also known as the RaMBaM?
Urs Schreiber: Conference: Lie Algebroids and Lie Groupoids in Differential Geometry (December 21, 2006)
Conference in Sheffield on Lie algebroids and Lie groupoids in Differential Geometry
John Baez: Lectures on Classical Mechanics (December 21, 2006)
Lectures notes on classical mechanics - now available in printed form!
Urs Schreiber: A Little Bit of Geometric Langlands: Relation to Integrable Systems (December 21, 2006)
On Hitchin integrable systems and their role in geometric Langlands duality.
David Corfield: Common Applications (December 21, 2006)
Why does the same piece of mathematics find many applications?
John Baez: Quantization and Cohomology (Week 9) (December 21, 2006)
A glimpse of geometric quantization.
Urs Schreiber: Excellent Math in Bonn: Opening Colloquium in January (December 20, 2006)
Opening colloquium of the new “cluster of excellence” of mathematics in Bonn.
Urs Schreiber: Research Blogging (December 20, 2006)
On research blogging.
Urs Schreiber: Universal Gerbes (December 18, 2006)
What is known about universal gerbes?
John Baez: This Week’s Finds in Mathematical Physics (Week 242) (December 18, 2006)
Photos of Saturn, its ring and moons. Unmanned NASA missions versus sending canned primates to Mars. Jeffrey Mortons’ work on topological quantum field theory.
Urs Schreiber: Frobenius Algebroids with Invertible Products (December 15, 2006)
Thinking of algebroids with invertible products as Frobenius algebroids.
John Baez: Higher Categories at the Fields Institute (December 14, 2006)
Schedule of talks at the Fields Institute workshop on “Higher Categories and Their Applications”
David Corfield: Back from NIPS 2006 (December 13, 2006)
Background knowledge in machine learning
David Corfield: Bicat is Not Triequivalent to Gray (December 13, 2006)
Steve Lack’s new paper.
Urs Schreiber: Communicating Thoughts on the Web (December 13, 2006)
Thinking, communicating and having math on the web.
Urs Schreiber: Navigating Geometric Langlands by Analogies (December 12, 2006)
Some basic analogies that help us navigate the geometric Langlands duality.
Urs Schreiber: Local Transition of Transport, Anafunctors and Descent of n-Functors (December 8, 2006)
Concepts and examples of what would be called transition data or descent data for n-functors.
John Baez: New Light Particle(s) Discovered? (December 7, 2006)
Hints of particles with masses near 7 and 19 MeV?
John Baez: Guiraud on Higher-Dimensional Rewrite Rules (December 6, 2006)
The three dimensions of proof.
Urs Schreiber: Talks at “Higher Categories and their Applications” (December 5, 2006)
Talks on 2-functorial TFT and CFT.
Urs Schreiber: Postdoctoral Position at the Interface of Algebra, Conformal Field Theory and String Theory (December 5, 2006)
Postdoc position in algebra, CFT and strings.
Urs Schreiber: What Does the Classifying Space of a 2-Category Classify? (December 4, 2006)
Notes by Baas, Boekstedt and Kro on classifying spaces for 2-vector bundles.
David Corfield: Klein 2-Geometry VIII (December 1, 2006)
Continuing Klein 2-geometry
John Baez: Classical vs Quantum Computation (Week 8) (December 1, 2006)
How Curry’s Fixed Point Theorem is related to Cantor’s diagonal argument.
John Baez: Quantization and Cohomology (Week 8) (December 1, 2006)
From particles to membranes, continued.
John Baez: Puzzle #8 (November 30, 2006)
Which mathematician was rumored in 1999 to be secretly in charge of one of the world’s largest countries?
Urs Schreiber: Nicolai on E10 and Supergravity (November 29, 2006)
H. Nicolai on further progress in checking the hypothesis that the dynamics of supergravity is encoded in geodesic motion on a Kac-Moody group coset.
Urs Schreiber: D-Branes from Tin Cans, II (November 28, 2006)
Gerbe modules from 2-sections.
David Corfield: NIPS 2006 (November 27, 2006)
NIPS 2006 conference, Vancouver
John Baez: Puzzle #7 (November 25, 2006)
Which bird can sleep with half its brain while the other half stays awake?
Urs Schreiber: 2-Monoid of Observables on String-G (November 24, 2006)
Rep(L G) from 2-sections.
Urs Schreiber: The Baby Version of Freed-Hopkins-Teleman (November 23, 2006)
The Freed-Hopkins-Teleman result and its baby version for finite groups, as explained by Simon Willerton.
Urs Schreiber: The 1-Dimensional 3-Vector Space (November 23, 2006)
Alg(C) as the 3-category of canonical 1-dimensional 3-vector spaces.
John Baez: A Third Model of the String Lie 2-Algebra (November 23, 2006)
Wagemann gives a third construction of the one-parameter family of Lie 2-algebras associated to any simple Lie algebra.
John Baez: Classical vs Quantum Computation (Week 7) (November 21, 2006)
Building a computer using the lambda calculus: the Fixed Point Theorem.
Urs Schreiber: Basic Question on Homs in 2-Cat (November 21, 2006)
A basic question on internal homs in 2-categories.
David Corfield: Philosophy as Stance (November 21, 2006)
Philosophy and taking a stance.
Urs Schreiber: Derived Categories in Utah (November 20, 2006)
A workshop on derived categories and their application in string theory.
John Baez: This Week’s Finds in Mathematical Physics (Week 241) (November 20, 2006)
Gravitational wave detection in Louisiana, higher gauge theory and the dodecahedron.
Urs Schreiber: Categorical Trace and Sections of 2-Transport (November 17, 2006)
A general concept of extended QFT and its relation to the Kapranov-Ganter 2-character.
David Corfield: MacIntyre on Rational Judgment (November 16, 2006)
How explicit must rational reasoning be?
John Baez: Classical vs Quantum Computation (Week 6) (November 15, 2006)
From lambda-terms to string diagrams. Building a computer inside a cartesian closed category with an object X with X ≅ hom(X,X).
John Baez: Quantization and Cohomology (Week 7) (November 15, 2006)
Generalizing classical mechanics from particles to strings and higher-dimensional membranes.
John Baez: Higher Gauge Theory (November 15, 2006)
Transparencies for a talk on higher gauge theory.
John Baez: Breen on Gerbes and 2-Gerbes (November 13, 2006)
A new expository paper.
John Baez: Quantization and Cohomology (Week 6) (November 13, 2006)
The canonical 1-form, extended phase space and Rovelli’s covariant formulation of classical mechanics.
John Baez: Quantization and Cohomology (Week 5) (November 13, 2006)
The canonical 1-form on the cotangent bundle of a manifold, and what it does for classical mechanics.
John Baez: Puzzle #6 (November 13, 2006)
Which famous buildings were named after a form of food - or was it the other way around?
John Baez: Tales of the Dodecahedron (November 12, 2006)
Slides from a talk on the history of the dodecahedron, from Pythagoras through Plato to Poincaré.
David Corfield: The Tasks of Philosophy (November 10, 2006)
What should philosophy study?
David Corfield: Mathematics Under the Microscope (November 9, 2006)
Book about mathematics
Urs Schreiber: Flat Sections and Twisted Groupoid Reps (November 8, 2006)
A comment on Willerton’s explanation of twisted groupoid reps in terms of flat sections of n-bundles.
John Baez: Quantization and Cohomology (Week 4) (November 8, 2006)
Hamiltonian dynamics and symplectic geometry.
John Baez: Quantization and Cohomology (Week 3) (November 8, 2006)
From Lagrangian to Hamiltonian dynamics.
Urs Schreiber: Chern-Simons Lie-3-Algebra Inside Derivations of String Lie-2-Algebra (November 7, 2006)
The Chern-Simons Lie 3-algebra sits inside that of inner derivations of the string Lie 2-algebra.
Urs Schreiber: Dijkgraaf-Witten Theory and its Structure 3-Group (November 6, 2006)
The idea of Dijkgraaf-Witten theory and its reformulation in terms of parallel volume transport with respect to a structure 3-group.
David Corfield: Infinite-Dimensional Exponential Families (November 6, 2006)
Information geometry for kernel methods in machine learning.
John Baez: Puzzle #5 (November 5, 2006)
When was the Roman empire sold, and who bought it?
Urs Schreiber: A 3-Category of Twisted Bimodules (November 3, 2006)
A 3-category of twisted bimodules.
John Baez: Classical vs Quantum Computation (Week 5) (November 2, 2006)
The naturality of currying - and a new "bubble" notation for currying and uncurrying.
John Baez: A Categorical Manifesto (November 2, 2006)
Why are categories important in computer science?
David Corfield: Academic Commons (November 2, 2006)
Enclosure of Academic Commons
David Corfield: Klein 2-Geometry VII (November 1, 2006)
Continuing to categorify the Erlanger Program
Urs Schreiber: Bakovic on 2-Groupoid 2-Torsors (October 31, 2006)
Igor Bakovic defines and classifies categorified groupoid bundles.
Urs Schreiber: Markl on Natural Differential Operators (October 31, 2006)
M. Markl explains how to think about “naturality” of differential operators.
David Corfield: Foundations (October 30, 2006)
What are foundations?
John Baez: Puzzle #4 (October 30, 2006)
Such a brivla, built from the rafsi for the component gismu and cmavo, is called a lujvo.
Urs Schreiber: WZW as Transition 1-Gerbe of Chern-Simons 2-Gerbe (October 29, 2006)
How the WZW 1-gerbe arises as the transition 1-gerbe of the Chern-Simons 2-gerbe.
Urs Schreiber: Hopkins Lecture on TFT: Chern-Simons (October 27, 2006)
Basic and advanced concepts in Chern-Simons theory.
John Baez: Classical vs Quantum Computation (Week 4) (October 26, 2006)
Currying and uncurrying, evaluation and coevaluation: basic aspects of the "quantum lambda-calculus".
John Baez: Lauda on Topological Field Theory and Tangle Homology (October 25, 2006)
Aaron Lauda’s thesis on topological field theory and tangle homology.
Urs Schreiber: Hopkins Lecture on TFT: Infinity-Category Definition (October 25, 2006)
A suggestion for a definition of (extended) TFT in terms of morphisms of omega-categories.
Urs Schreiber: Hopkins Lecture on TFT: Introduction and Outlook (October 25, 2006)
Introductory lecture by M. Hopkins on topological field theory.
John Baez: Puzzle #3 (October 25, 2006)
A special law to prevent votes from being counted.
David Corfield: Knowledge of the Reasoned Fact (October 23, 2006)
In a comment I raised the question of what to make of our expectation that behind different manifestations of an entity there is one base account, of which these manifestations are consequences. If I point out to you three…
John Baez: This Week’s Finds in Mathematical Physics (Week 240) (October 23, 2006)
Read about Dolan and Trimble’s work on cartesian closed categories and holodeck games!
John Baez: Classical vs Quantum Computation (Week 3) (October 20, 2006)
Guest lecture by James Dolan: Holodeck strategies and cartesian closed categories.
Urs Schreiber: Algebras as 2-Categories and its Effect on Algebraic Geometry (October 19, 2006)
A question by Bruce Bartlett about categories of algebras, algebras as categories and the possible implications for non-commutative algebraic geometry.
John Baez: A Course on Classical Mechanics (October 19, 2006)
A course on classical mechanics from a mathematical viewpoint.
John Baez: Australian Category Theory (October 18, 2006)
What’s happening at the Australian Category Seminar.
Urs Schreiber: Topology in Trondheim and Kro, Baas & Bökstedt on 2-Vector Bundles (October 18, 2006)
Baas, Dundas and Kro with new insights into 2-vector bundles.
Urs Schreiber: D-Branes from Tin Cans: Arrow Theory of Disks (October 18, 2006)
On disk holonomy and boundary conditions.
John Baez: Classical vs Quantum Computation (Week 2) (October 17, 2006)
An intro to closed monoidal categories.
Guest: Crossed Products and Not All That (October 16, 2006)
A. Rivero on crossed products of C*-algebras, with an eye on Connes’ spectral description of the standard model.
Urs Schreiber: Adjunctions and String Composition (October 16, 2006)
A remark on describing string composition and string field products in terms of adjunctions in categories of paths.
David Corfield: The Encyclopaedists’ Dream (October 15, 2006)
What is epistemology?
Urs Schreiber: Bundle Gerbes: Connections and Surface Transport (October 13, 2006)
Definition of connection on a bundle gerbe and of the surface transport computed from it.
Urs Schreiber: Bundle Gerbes: General Idea and Definition (October 13, 2006)
The general idea of bundle gerbes and the basic definition.
Urs Schreiber: What is the Categorified Gelfand-Naimark Theorem? (October 11, 2006)
A lightning review of the categorified Gelfand-Naimark theorem.
David Corfield: Wittgenstein and Thurston on Understanding (October 11, 2006)
What Wittgenstein might have said about the famous Jaffe-Quinn debate - and what Thurston did say.
David Corfield: Euler Characteristic of a Category (October 11, 2006)
Baez and Dolan defined the cardinality of a groupoid. Now Tom Leinster has defined a more general concept: the Euler characteristic of a category!
John Baez: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection (October 11, 2006)
A question from Bruce Bartlett
John Baez: Quantization and Cohomology (Week 2) (October 11, 2006)
The Lagrangian approach to classical mechanics
Urs Schreiber: Talks by Hopkins and Willerton (October 10, 2006)
Talk by Hopkins of TFT and by Willerton on Dijkgraaf-Witten.
Urs Schreiber: Kosmann-Schwarzbach & Weinstein on Lie Algebroid Classes (October 10, 2006)
Kosmann-Schwarzbach on classes of Lie algebroids generalizing the character of a Lie algebra.
Urs Schreiber: On n-Transport: Descent of the Universal Transition (October 9, 2006)
How the universal transition mediates descent from a surjection down to the original space.
David Corfield: Philosophy of Physics (October 9, 2006)
Publication of ‘Philosophy of Physics’
John Baez: Zooming Out in Time (October 8, 2006)
To understand what a few degrees of global warming can do, we must study the Earth’s climate over the last 65 million years.
John Baez: Classical vs Quantum Computation (Week 1) (October 6, 2006)
Types and operations; categories as theories; monoidal categories versus categories with finite products.
Urs Schreiber: On n-Transport: Universal Transition (October 6, 2006)
Paths in categories from universal transitions.
David Corfield: So Irigaray was Right? (October 6, 2006)
Navier-Stokes solution
John Baez: Quantization and Cohomology (Week 1) (October 5, 2006)
How the dynamics of p-branes resembles the statics of (p+1)-branes.
John Baez: New Scientist Reacts! (October 5, 2006)
New Scientist now has a blog entry defending their article on Shawyer’s electromagnetic drive.
John Baez: Trimble’s Definition of Tetracategory (October 5, 2006)
Todd Trimble has redrawn his legendary lost definition of tetracategories.
John Baez: λ-Calculus in Paris (October 4, 2006)
Typed Lambda Calculi and Applications - a conference in Paris, June 26-28, 2007
David Corfield: Emulating Aristotle (October 4, 2006)
How to emulate Aristotle
Guest: Wang at AustMS 2006 (October 4, 2006)
The following “guest post” is by David Roberts: At the Australian Mathematical Society’s annual meeting in Sydney, Brian Wang gave the talk Gerbes, D-branes and loop group representations. This is of particular interest to at least one blogger here, so…
John Baez: Classical vs Quantum Computation (Week 1) (October 3, 2006)
Computation, the lambda calculus and cartesian closed categories: an overview.
John Baez: Voevodsky on the Homotopy λ-Calculus (October 2, 2006)
Voevodsky has released a “very short note on the homotopy λ-calculus”.
John Baez: Topos Theory at Chicago (October 2, 2006)
Lectures on topos theory at Chicago
David Corfield: The Consolation of n-Categories (October 1, 2006)
What has mathematics meant to people through the ages?
David Corfield: Klein 2-Geometry VI (October 1, 2006)
Resuming the categorification of Kleinian Geometry
John Baez: Puzzle #2 (October 1, 2006)
A spooky link between Tycho Brahe and Shakespeare.
John Baez: Dimensional Analysis and Coordinate Systems (September 30, 2006)
More gnarly issues: dimensional analysis and coordinate systems.
Urs Schreiber: Puzzle Pieces Falling Into Place (September 28, 2006)
On the 3-group which should be underlying Chern-Simons theory.
Urs Schreiber: 2-Groups and Algebras (September 28, 2006)
Bundles of algebras from principal 2-group transitions.
David Corfield: Toleration (September 27, 2006)
Toleration and the ends of conversation
Urs Schreiber: Bulk Fields and Induced Bimodules (September 27, 2006)
Bulk field insertions in 2D CFT in terms of 2-transport: endomorphisms of 2-monoids.
Urs Schreiber: Fahrenberg and Raussen on Continuous Paths (September 26, 2006)
A theory of continuous n-paths modulo continuous reparameterization.
David Corfield: Our Raison D’être (September 26, 2006)
Marni Sheppeard reports from the AustMS2006 conference, which, as anyone who knows about Australian mathematics might expect, is holding a category theory session. Dominic Verity is giving one of the talks, in which he considers the raison d’être for higher…
David Corfield: Mathematical Kinds (September 23, 2006)
Do mathematical entities belong to kinds?
John Baez: Dimensional Analysis (September 21, 2006)
Let’s amass a collection of wisdom on gnarly issues in physics - starting with dimensional analysis.
David Corfield: Categorification in Uppsala (September 21, 2006)
Earlier this month the Mathematics Institute at Uppsala University hosted a conference called Categorification in Algebra and Topology, clearly a theme close to our collective heart. As yet there are only a handful of participants’ notes available (Scott Morrison’s are…
David Corfield: The Why and Wherefore of History (September 21, 2006)
Here are some notes for my talk at the Berlin workshop. Fortunately I was upgraded to a 45-minute talk. Even so, I didn’t manage to reach the last part where I discuss David Carr’s ideas. I would be interested in…
Urs Schreiber: Differential n-Geometry (September 20, 2006)
A quest for arrow-theoretic differential geometry.
John Baez: A Plea to Save New Scientist (September 19, 2006)
The SF writer Greg Egan has issued a plea to save the magazine New Scientist.
David Corfield: Searching for a New Epistemology in Berlin (September 18, 2006)
Reporting from a workshop in Berlin.
Urs Schreiber: Quantum n-Transport (September 14, 2006)
An attempt to understand the path integral for an n-dimensional field theory as a coproduct operation over transport n-functors.
John Baez: Higher Categories and their Applications (September 13, 2006)
In January 2007, the Fields Institute is having a workshop on Higher Categories and their Applications.
John Baez: Groupoids and Stacks in Physics and Geometry (September 13, 2006)
In 2007, the Institut Henri Poincaré will be running a program on Groupoids and Stacks in Physics and Geometry.
Urs Schreiber: Freed on Higher Structures in QFT, I (September 12, 2006)
Freed’s old observation that n-dimensional QFT assigns (n-d)-Hilbert spaces to d-dimensional volumes.
Urs Schreiber: Wirth and Stasheff on Homotopy Transition Cocycles (September 11, 2006)
Stasheff recalls an old result by Wirth on passing between fibrations and their homotopy transition cocycles.
David Corfield: Mathematics and Virtual Reality (September 9, 2006)
Is Mathematics a Massively-Multiplayer Online Role-Playing Game?
Urs Schreiber: Kock on 1-Transport (September 8, 2006)
A new preprint by Anders Kock on the synthetic formulation of the notion of parallel transport.
Urs Schreiber: Connes on Spectral Geometry of the Standard Model, IV (September 8, 2006)
The details of the spectral triple which describes the standard model.
John Baez: This Week’s Finds in Mathematical Physics (Week 239) (September 8, 2006)
Read about the open access movement, Freeman Dyson’s 1951 lecture notes, the origins of mathematics in little clay figures called “tokens”, and Koszul duality for L∞-algebras!
David Corfield: Category Theory and Philosophy (September 8, 2006)
The place of category theory in philosophy
Urs Schreiber: Connes on Spectral Geometry of the Standard Model, III (September 7, 2006)
First technical details for spectral action theory.
Urs Schreiber: On n-Transport: 2-Vector Transport and Line Bundle Gerbes (September 7, 2006)
Associated 2-transport, 2-representations and bundle gerbes with connection.
David Corfield: Mathematical Circuit Components (September 7, 2006)
What mathematics and radio circuits might have in common.
Urs Schreiber: Connes on Spectral Geometry of the Standard Model, II (September 6, 2006)
On the nature and implication of Connes’ identification of the spectral triple of the standard model coupled to gravity.
Urs Schreiber: Connes on Spectral Geometry of the Standard Model, I (September 6, 2006)
Connes is getting closer to the spectral triple encoding the standard model coupled to gravity. Part I: some background material.
Urs Schreiber: n-Transport and Higher Schreier Theory (September 5, 2006)
Understanding n-transport in terms of Schreier theory for groupoids.
David Corfield: The History of n-Categories (September 4, 2006)
How to write the history of mathematics.
David Corfield: Klein 2-Geometry V (September 4, 2006)
We resume the attempt to categorify Kleinian geometry
John Baez: Doctrines (September 3, 2006)
In 1969 Lawvere invented “doctrines”. A doctrine is a roughly a kind of category, thought of as a kind of logical theory.
John Baez: TeXnical Issues (September 2, 2006)
Having trouble reading the equations on this blog, or posting comments? Try this.
John Baez: Ringoids (September 2, 2006)
A group with many objects is a groupoid. A ring with many objects is a ringoid!
John Baez: Letter from Grothendieck (August 31, 2006)
Newly released letter from Alexander Grothendieck.
Urs Schreiber: 10D SuGra 2-Connection (August 28, 2006)
On the Lie 2-algebra governing 10-dimensional supergravity.
John Baez: Quantum Computation and Symmetric Monoidal Categories (August 28, 2006)
Ordinary computer scientists like cartesian closed categories. Quantum computer scientists like symmetric monoidal categories with duals.
John Baez: Categorifying CCCs: Computation as a Process (August 28, 2006)
“Cartesian closed n-categories” may help us see computation as a process.
John Baez: CCCs and the λ-calculus (August 28, 2006)
Let’s discuss cartesian closed categories and the λ-calculus!
Urs Schreiber: Picturing Morphisms of 3-Functors (August 25, 2006)
Diagrams governing morphisms of 3-functors.
John Baez: Categories and Computation (August 24, 2006)
What’s the relation between the lambda-calculus, computation and CCCs?
John Baez: Categorifying the Dijkgraaf-Witten model (August 24, 2006)
Martins and Porter categorify the Dijkgraaf-Witten model.
John Baez: Puzzle #1 (August 23, 2006)
What was Uncle Sam’s last name?
John Baez: Lectures on n-Categories and Cohomology (August 22, 2006)
Lectures on n-Categories and Cohomology, by John Baez and Michael Shulman
Urs Schreiber: n-Curvature (August 21, 2006)
Definition of n-curvature and the nature of “fake curvature”.
Urs Schreiber: On n-Transport, Part II (August 21, 2006)
Smooth transport, differentials of transport, and nonabelian differential cocycles.
Urs Schreiber: On n-Transport, Part I (August 18, 2006)
The concept of n-transport and its local trivialization and transition.
Urs Schreiber: SuGra 3-Connection Reloaded (August 17, 2006)
Discussion of n-category description of supergravity continued.
John Baez: Dark Matter in the Bullet Cluster (August 17, 2006)
Hard evidence for dark matter in the Bullet Cluster
Distler: Inaugural Post (August 16, 2006)
Brand new blog.

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