A group blog on math, physics and philosophy

- Random Permutations (Part 13) (Jan 17, 2020)
- We can embed the groupoid of finite sets with permutation into a slightly larger groupoid with very nice properties.
- Random Permutations (Part 12) (Jan 11, 2020)
- I’ll describe the groupoid of ‘finite sets equipped with a permutation’ as a sum over Young diagrams. Taking the groupoid cardinality, this description gives a well-known formula for the probability that a random permutation belongs in any given conjugacy class.
*Compositionality*: First Issue (Dec 29, 2019)- There’s a new diamond open-access journal on applied category theory, and its first issue has been published!
- Schrödinger's Unified Field Theory (Dec 26, 2019)
- In 1947, Schrödinger got really excited about a certain unified field theory. But what did it actually amount to?
- Random Permutations (Part 11) (Dec 24, 2019)
- We can use groupoid cardinality to prove the Cycle Length Lemma, a fundamental result in the theory of random permutations.
- Applied Category Theory 2020 - Adjoint School (Dec 24, 2019)
- The deadline to apply for the next applied category theory school is January 15, 2020.
- Random Permutations (Part 10) (Dec 18, 2019)
- We can use groupoid cardinality to compute the moments of this random variable: the number of cycles of length k in a random permutation of an n-element set.
- Applied Category Theory Postdocs at NIST (Dec 13, 2019)
- The National Institute of Standards and Technology (NIST) is looking to expand its group of applied category theorists.
- Random Permutations (Part 9) (Dec 11, 2019)
- Treat the number of cycles of length k in a permutation of an n-element set as a random variable. What do the moments of this random variable approach as n approaches infinity?
- Random Permutations (Part 8) (Dec 8, 2019)
- Suppose raindrops are falling on your head, randomly and independently, at an average rate of one per minute. What’s the average of the cube of the number of raindrops that fall on your head in one minute? We can solve this with the help of groupoid cardinalities.
- Random Permutations (Part 7) (Dec 7, 2019)
- As n approaches infinity, the number of cycles of length k in a random permutation of an n-element set approaches a Poisson distribution with mean 1/k.
- Position at U. C. Riverside (Dec 5, 2019)
- U. C. Riverside is hiring.
- Topoi from Lawvere Theories (Dec 5, 2019)
- What’s the category of presheaves on a Lawvere theory like?
- Random Permutations (Part 6) (Dec 1, 2019)
- Now we’ll compute the expected number of cycles of length k in a random permutation of an n-element set. This will answer lots of questions about random permutations!
- Random Permutations (Part 5) (Nov 30, 2019)
- Here we’ll use bivariate generating functions to figure out the average number of cycles in a random permutation of an $n$-element set.
- Random Permutations (Part 4) (Nov 25, 2019)
- A random permutation of a large finite set has an almost 70% chance of having one cycle that contains at least half the elements! Let’s see why.
- Random Permutations (Part 3) (Nov 24, 2019)
- An attempt to get a better intuition for a random permutation of a large set.
- Random Permutations (Part 2) (Nov 24, 2019)
- In the limit as n approaches infinity, what is the probability that the shortest cycle in a randomly chosen permutation of an n-element set has length more than k?
- Random Permutations (Part 1) (Nov 24, 2019)
- What is the probability that a randomly chosen permutation of a large finite set has exactly some chosen number of fixed points?
- The Golomb-Dickman Constant (Nov 23, 2019)
- The Goldman–Dickman constant governs the average length of the longest cycle of a randomly chosen permutation.
- Total Maps of Turing Categories (Nov 18, 2019)
- guest post by Adam Ó Conghaile and Diego Roque We continue the 2019 Applied Category Theory School with a discussion of the paper Total maps of Turing categories by Cockett, Hofstra and Hrubeš. Thank you to Jonathan Gallagher for…
- Doubles for Monoidal Categories (Nov 15, 2019)
- As part of the ACT2019 schoo, Fosco Loregian and Bryce Clarke explain Pastro and Street’s paper on doubles for monoidal categories.
- Screw Theory (Oct 21, 2019)
- Screw theory is about the Euclidean group. A ‘screw’ is an element of the Lie algebra of this group: an infinitesimal translation together with an infinitesimal rotation.
- Diversity Workshop at UCR (Oct 14, 2019)
- We’re having a workshop on diversity in math on Friday 8 November 2019 at U. C. Riverside, right before the applied category theory meeting that weekend.
- Foundations of Math and Physics One Century After Hilbert (Oct 10, 2019)
- Review of
*Foundations of Mathematics and Physics One Century After Hilbert: New Perspectives* - Applied Category Theory Meeting at UCR (Part 2) (Sep 30, 2019)
- There will be a meeting on applied category theory on the weekend November 9–10 at U. C. Riverside. Here is the schedule.
- Partial Evaluations 2 (Sep 16, 2019)
- Carmen Constantin from the ACT 2019 Adjoint School talks about “partial evaluations”, weakly cartesian monads, and the bar construction.
- Stellenbosch is Hiring (Sep 12, 2019)
- Stellenbosch University is advertising two permanent faculty positions at the level of Senior Lecturer and Professor.
- The Riemann Hypothesis (Part 3) (Sep 11, 2019)
- How the Weil Conjectures lead us to the idea of ‘motives’.
- The Riemann Hypothesis (Part 2) (Sep 10, 2019)
- A baby version of the Riemann Hypothesis: Hasse’s theorem on elliptic curves.
- The Riemann Hypothesis (Part 1) (Sep 7, 2019)
- What does the Riemann Hypothesis actually mean?
- Homotopy Type Theory Electronic Seminar Talks (Sep 6, 2019)
- The Homotopy Type Theory Electronic Seminar Talks (HoTTEST) will be returning in Fall 2019.
- From Simplicial Sets to Categories (Aug 31, 2019)
- I need a reference to a proof of the fact that the left adjoint of the simplicial nerve preserves finite products.
- Turing Categories (Aug 27, 2019)
- Georgios Bakirtzis and Christian Williams discuss Hofstra and Cockett’s paper on Turing categories.
- Graphical Regular Logic (Aug 16, 2019)
- guest post by Sophie Libkind and David Jaz Myers This post continues the series from the Adjoint School of Applied Category Theory 2019….
- Evil Questions About Equalizers (Aug 16, 2019)
- Does the process of converting a category with finite colimits into a symmetric monoidal category give a functor from Rex to SMC that preserves equalizers?
- Even-Dimensional Balls (Aug 11, 2019)
- There’s a simple formula for the sum of the volumes of all even-dimensional balls of radius r. But what does it mean?
- The Conway 2-Groups (Aug 9, 2019)
- There are 24 inequivalent ways to extend the Conway group to a 2-group, i.e. a one-object weak 2-groupoid, where each 1-morphism has a circle’s worth of auto-2-morphisms.
- 2020 Category Theory Conferences (Aug 9, 2019)
- Learn the dates and locations for CT2019 and ACT2019.
- Applied Category Theory 2019 Talks (Jul 22, 2019)
- Read about the talks at Applied Category Theory 2019.
- The Riemann Hypothesis Says 5040 is the Last (Jul 6, 2019)
- Riemann’s Hypothesis is equivalent to a simple bound on the sum of divisors of a number.
- Structured Cospans (Jul 1, 2019)
- Learn about structured cospans, a new approach to open systems!
- Behavioral Mereology (Jun 29, 2019)
- Toby Smithe and Bruno Gavranovic discuss Fong, Myers, and Spivak’s paper on the concept of “part”.
- Meeting the Dialogue Challenge (Jun 25, 2019)
- Dan Shiebler and Alexis Toumi discuss Grammars as Parsers: Meeting the Dialogue Challenge by Matthew Purver, Ronnie Cann and Ruth Kempson.
- Katrina Honigs meets Grothendieck (Jun 20, 2019)
- A short true story about Grothendieck.
- Applied Category Theory Meeting at UCR (Part 1) (Jun 16, 2019)
- There’s an AMS special session on applied category theory at U.C. Riverside on the weekend of November 9–10, 2019.
- What's a One-Object Sesquicategory? (Jun 13, 2019)
- What’s a one-object sesquicategory called? A “semi-monoidal category”?
- Nonstandard Models of Arithmetic (Jun 5, 2019)
- Conversations on mathematical logic between Michael Weiss and John Baez, centered on nonstandard models of the Peano axioms of arithmetic.
- Why Category Theory Matters (Jun 3, 2019)
- In which I find that I’m leading a cabal of radical category theorists.
- A Question on Left Adjoints (May 28, 2019)
- Jade Master question about the existence of left adjoints for certain functors.
- Polyadic Boolean Algebras (May 23, 2019)
- How can we reformulate classical first-order logic more categorically?
- Young Diagrams and Schur Functors (May 20, 2019)
- Todd Trimble and I figured out a new way to think about Schur functors, which I explain briefly in these notes.
- Partial Evaluations 1 (May 16, 2019)
- A summary of some ideas from the paper “Monads, partial evaluations and rewriting” by Tobias Fritz and Paolo Perrone.
- Generalized Petri Nets (Apr 28, 2019)
- Various flavors of Petri net can be handled systematically using the theory of ‘generalized Petri nets’.
- Can 1+1 Have More Than Two Points? (Apr 19, 2019)
- Is there a cartesian closed category with finite coproducts such that there exist more than two morphisms from 1 to 1 + 1?
- The ZX-Calculus for Stabilizer Quantum Mechanics (Apr 14, 2019)
- Backens’ completeness proof for the ZX-calculus for stabilizer quantum mechanics.
- Normalising Quantum Circuits (Mar 24, 2019)
- Summary of “A finite presentation of CNOT-dihedral operators”, a paper on quantum circuits by Matthew Amy, Jianxin Chen and Neil J. Ross.
- Network Models from Petri Nets with Catalysts (Mar 24, 2019)
- We can build a network model from a Petri net with chosen “catalysts”!
- Sporadic SICs and Exceptional Lie Algebras III (Mar 14, 2019)
- There’s a relation between E8 and the problem of fitting as as many equiangular lines as possible into an 8-dimensional
*complex*Hilbert space. - Left Adjoints Between Categories of Posets (Mar 6, 2019)
- A lot of questions about various kinds of posets.
- Sporadic SICs and Exceptional Lie Algebras II (Feb 28, 2019)
- You can find nine equiangular lines in a 3-dimensional complex Hilbert space. They’re related to the Lie algebra E6!
- Tychonoffication (Feb 20, 2019)
- A Tychonoff space is one that’s homeomorphic to a subspace of a possibly infinite-dimensional cube.
- Sporadic SICs and Exceptional Lie Algebras I (Feb 20, 2019)
- Sporadic informationally complete systems of equiangular lines are connected to exceptional Lie algebras!
- Applied Category Theory 2019 (Feb 7, 2019)
- I hope to see you at this conference! Applied Category Theory 2019, July 15-19, 2019, Oxford, UK. Here’s some information about it, such as how to submit papers….
- Jacobi Manifolds (Feb 4, 2019)
- A bit about Jacobi manifolds.
- Symposium on Compositional Structures 3 (Jan 28, 2019)
- SYCO3 is taking place at the University of Oxford on 27-28 March, 2019.
- Applied Category Theory 2019 School (Jan 5, 2019)
- Apply for the Applied Category Theory 2019 school! Applications are due January 30th, 2019.
- HoTT 2019 (Dec 25, 2018)
- There will be an International Conference on Homotopy Type Theory, HoTT 2019, from August 12th to 17th, 2019 at Carnegie Mellon University in Pittsburgh, USA.
- Monads and Lawvere Theories (Dec 23, 2018)
- Jade Master has a question about morphisms of Lawvere theories and the corresponding morphisms of monads.
- 2-Groups in Condensed Matter Physics (Nov 1, 2018)
- 2-groups are starting to show up in theoretical condensed matter physics!
- Bar Constructions and Combinatorics of Polyhedra for n-Categories (Oct 29, 2018)
- Samuel Vidal has LaTeXed some notes by Todd Trimble.
- Analysis in Higher Gauge Theory (Oct 18, 2018)
- Andreas Gastel has a new paper ‘Canonical gauges in higher gauge theory’.
- Topoi of G-sets (Oct 14, 2018)
- Reinterpreting basic ideas of group theory in terms of topos theory.
- Applied Category Theory 2019 (Oct 2, 2018)
- Here is a pre-announcement for ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019.
- Applied Category Theory Course: Collaborative Design (Sep 26, 2018)
- Here are lectures on enriched profunctors and collaborative design, as part of a course on applied category theory.
- A Pattern That Eventually Fails (Sep 21, 2018)
- Some patterns go on a long time and then fail.
*p*-Local Group Theory (Sep 19, 2018)- What’s the relation between Sylow $p$-subgroups of finite groups, and $p$-completion in algebraic topology?
- What is Applied Category Theory? (Sep 18, 2018)
- Tai-Danae Bradley has a new booklet on applied category theory.
- Exceptional Quantum Geometry and Particle Physics (Aug 27, 2018)
- Michel Dubois-Violette and Ivan Todorov claim to get the Standard Model gauge group as the subgroup of automorphisms of the exceptional Jordan algebra that 1) preserves a copy of 10d Minkowski spacetime inside this Jordan algebra, and 2) also preserves a unit imaginary octonion—which lets us pick out a copy of 4d Minkowski spacetime inside 10d Minkowski spacetime!
*Compositionality*: Now Open For Submissions (Aug 24, 2018)- Our new journal is now open for submissions.
- Kan (Aug 22, 2018)
- Jake Bian has created a new add-on for Firefox called ‘Kan’, designed to make the nLab look nicer.
- The Duties of a Mathematician (Jul 18, 2018)
- What are the duties of a mathematician?
*Compositionality*: the Editorial Board (Jul 18, 2018)- The journal Compositionality has chosen an editorial board.
- Beyond Classical Bayesian Networks (Jul 7, 2018)
- A category-theoretic approach to Bayesian networks, explained by Pablo Andres-Martinez and Sophie Raynor.
- Symposium on Compositional Structures (Jul 4, 2018)
- The First Symposium on Compositional Structures will be taking place at the School of Computer Science of the University of Birmingham, 20-21 September, 2018.
- The Behavioral Approach to Systems Theory (Jun 15, 2018)
- As part of Applied Category 2018 school, Eliana Lorch and Joshua Tan discuss Jan Willems’ behavioral approach to open and interconnected systems.
- Fun for Everyone (Jun 13, 2018)
- Lorscheid’s paper is a great introduction to the mysterious world of ‘the field with one element’.
- Sets of Sets of Sets of Sets of Sets of Sets (Jun 9, 2018)
- There’s no way to make the square of the covariant powerset functor into a monad.
- Applied Category Theory: Resource Theories (Jun 4, 2018)
- An online course on applied category theory now turns to “resource theories”.
- Applied Category Theory: Ordered Sets (May 28, 2018)
- Here are 17 lectures on ordered sets, and exercises, and puzzles, and discussions that you can join.
- Laxification (May 25, 2018)
- Do you know a systematic way of turning lax monoidal functors into strict ones out of a bigger, “puffed-up” monoidal category?
- Tropical Algebra and Railway Optimization (May 24, 2018)
- You can use the tropical rig to help design railway time-tables.
- Circuits, Bond Graphs, and Signal-Flow Diagrams (May 19, 2018)
- An introduction to Brandon Coya’s thesis on category-theoretic aspects of engineering.
*Compositionality*(May 6, 2018)- There’s a new journal,
*Compositionality*, on the study of building big things from smaller things - for example, using category theory. - Props in Network Theory (Apr 27, 2018)
- Brandon Coya and I have a new paper on PROPs and electrical circuits. Here’s an overview.
- Dynamical Systems and Their Steady States (Apr 2, 2018)
- At part of the Applied Category Theory 2018 school, Maru Sarazola explains and compares two category-theoretic approaches to open dynamical systems.
- Cognition, Convexity, and Category Theory (Mar 10, 2018)
- Tai-Danae Bradley and Brad Theilman, students at the Applied Category Theory 2018 school, continue discussing category-theoretic approaches to linguistics.
- Coarse-Graining Open Markov Processes (Mar 4, 2018)
- Kenny Courser and I have a new paper on coarse-graining open Markov processes, which makes use of symmetric monoidal double categories. Let us know what you think!
- Hypergraph Categories of Cospans (Feb 28, 2018)
- As part of the Applied Category Theory school, Jonathan Lorand and Fabrizio Genovese explain Brendan Fong’s work on decorated cospan categories.
- Applied Category Theory at NIST (Feb 23, 2018)
- There will be a workshop on Applied Category Theory at the National Institute of Standards and Technology.
- Higher Algebra and Mathematical Physics (Feb 23, 2018)
- There will be a “double conference” on higher algebra, simultaneously at the Perimeter Institute in Waterloo and the Max Planck Institute in Bonn.
- Cartesian Bicategories (Feb 19, 2018)
- Read Daniel Cicala and Julian Hedges’ summary of the Carboni–Walters paper
*Cartesian Bicategories I*. - Homotopy Type Theory Electronic Seminar (Feb 8, 2018)
- Watch a series of online talks by the leading experts in homotopy type theory.
- Linguistics Using Category Theory (Feb 6, 2018)
- A summary of the paper “Mathematical Foundations for a Compositional Distributional Model of Meaning” Bob Coecke, Mehrnoosh Sadrzadeh, and Stephen Clark.
*m*Lab (Feb 5, 2018)- Check out the mLab - a friendly parody of the nLab.
- A Problem on Pushouts and Pullbacks (Feb 2, 2018)
- I’ve got a problem involving a cube in the category of finite sets. I could use some help.
- More Secrets of the Associahedra (Jan 25, 2018)
- How associahedra show up when you invert a formal power series.
- Statebox: A Universal Language of Distributed Systems (Jan 23, 2018)
- Statebox is a startup trying to combine categories, open games, dependent types, Petri nets, string diagrams, and blockchains into a universal language for distributed systems.
- A Categorical Semantics for Causal Structure (Jan 22, 2018)
- This is the first blog entry as part of the online reading seminar associated to Applied Category Theory 2018. Joseph Moeller and Dmitry Vagner explain some of Kissinger’s work on causality!
- An M5-Brane Model (Dec 23, 2017)
- The string Lie 2-algebra may be relevant to M5-branes.
- From the Icosahedron to E8 (Dec 11, 2017)
- Here are two ways to get $\mathrm{E}_8$ from the icosahedron. How are they related?
- Real Sets (Nov 23, 2017)
- How to construct a set of cardinality pi.
- Topology Puzzles (Nov 11, 2017)
- Let’s say the closed unit interval [0,1][0,1] maps onto a metric space XX if there is a continuous map from [0,1][0,1] onto XX. Similarly for the Cantor set. Puzzle 0. Does the Cantor set map onto the closed unit…
- Applied Category Theory Papers (Nov 3, 2017)
- Some papers on applied category theory.
- The Adjoint School (Oct 28, 2017)
- There’s still some room for people to attend this great school on applied category theory!
- Vladimir Voevodsky, June 4, 1966 - September 30, 2017 (Oct 1, 2017)
- Vladimir Voevodsky died on September 30, 2017.
- Applied Category Theory at UCR (Part 2) (Sep 22, 2017)
- There will be a bunch of talks on applied category theory on November 4th and 5th at U.C. Riverside!
- Applied Category Theory 2018 (Sep 12, 2017)
- Applied Category Theory (ACT 2018) is a conference in Leiden April 30th to May 4th, 2018 preceded by a summer school April 23rd to 27th.
- Postdoc in Applied Category Theory (Sep 8, 2017)
- guest post by Spencer Breiner One Year Postdoc Position at Carnegie Mellon/NIST We are seeking an early-career researcher with a background in category theory, functional programming and/or electrical engineering for a one-year post-doctoral position supported by an Early-concept Grant…
- A Puzzle on Multi-Sorted Lawvere Theories (Aug 27, 2017)
- I’m curious about when the inclusion into a coproduct is monic.
- Simplicial Sets vs. Simplicial Complexes (Aug 19, 2017)
- What’s the relation between geometrically realizing a simplicial complex directly and doing so indirectly by first turning it into a simplicial set?
- A Graphical Calculus for Proarrow Equipments (Aug 11, 2017)
- David Myers has been thinking about string diagrams for calculating with proarrow equipments. Check out the pictures!
- The Rise and Spread of Algebraic Topology (Aug 5, 2017)
- I’m giving a talk on the history of algebraic topology, and how it led a new conception of mathematics.
- A Compositional Framework for Reaction Networks (Jul 31, 2017)
- Blake Pollard and I have figured out how to describe open chemical reaction networks using the ideas of functorial semantics.
- A Bicategory of Decorated Cospans (Jul 8, 2017)
- Kenny Courser has a new paper called “A Bicategory of Decorated Cospans”.
- The Geometric McKay Correspondence (Part 2) (Jul 2, 2017)
- Learn how we resolve singularities using the concept of a ‘Hilbert scheme’.
- The Geometric McKay Correspondence (Part 1) (Jun 19, 2017)
- The icosahedron and E
_{8}are connected, with the E_{8}Dynkin diagram arising from a detailed study of how an icosahedron can shrink to zero size. - Eliminating Binders for Easier Operational Semantics (Jun 10, 2017)
- Bound variables cause a lot of complications. In this post Mike Stay explains how to simply avoid them.
- Enriched Lawvere Theories for Operational Semantics (Jun 7, 2017)
- Greg Meredith and Mike Stay are using multisorted Lawvere theories enriched over the category of graphs to do operational sematics!
- Applied Category Theory at UCR (Part 1) (Apr 5, 2017)
- The American Mathematical Society is having a meeting here at U. C. Riverside during the weekend of November 4th and 5th, 2017. I’m organizing a session on Applied Category Theory, and I’m looking for people to give talks.
- Jobs at U. C. Riverside (Apr 1, 2017)
- The Mathematics Department of the University of California at Riverside is trying to hire some visiting assistant professors. We plan to make decisions quite soon! The positions are open to applicants who have PhD or will have a PhD…
- Globular for Higher-Dimensional Knottings (Part 3) (Jan 4, 2017)
- See a 2-sphere embedded in 4-space!
- Globular for Higher-Dimensional Knottings (Part 2) (Dec 17, 2016)
- Higher-dimensional surfaces using
*Globular*. - Globular for Higher-Dimensional Knottings (Dec 16, 2016)
- guest post by Scott Carter About 7 months ago, Jamie Vicary contacted me with a Globular worksheet of which, initially, I could make neither heads nor tails. He patiently explained to me that what I was looking at was…
- Linear Algebraic Groups (Part 8) (Nov 29, 2016)
- Learn about algebraic sets and affine algebras!
- Linear Algebraic Groups (Part 7) (Nov 10, 2016)
- Parabolic subgroups, flag varieties, and q-factorials!
- Linear Algebraic Groups (Part 6) (Nov 7, 2016)
- The $q$-deformed version of Pascal’s triangle, and the decomposition of a Grassmannian into Bruhat cells.
- Linear Algebraic Groups (Part 5) (Nov 3, 2016)
- Grassmannians, maximal parabolic subgroups, and a bit of counting!
- Linear Algebraic Groups (Part 4) (Oct 30, 2016)
- Abstract projective planes, Pappus’s hexagon theorem and Klein’s Erlangen program.
*Higher Structures*Journal (Oct 26, 2016)- There’s a new journal called “Higher Structures”.
- Linear Algebraic Groups (Part 3) (Oct 26, 2016)
- Grassmannians, their Bruhat cells, and how to count their points.
- Linear Algebraic Groups (Part 2) (Oct 20, 2016)
- Lectures on algebraic groups: the idea of projective geometry.
- Linear Algebraic Groups (Part 1) (Oct 17, 2016)
- Lecture notes on linear algebraic group theory, starting with Euclid.
- Euclidean, Hyperbolic and Elliptic Geometry (Sep 26, 2016)
- What’s the best way to see Euclidean, hyperbolic and elliptic geometry as part of a 1-parameter family.
- Twitter (Sep 12, 2016)
- I’m now on Twitter. Watch me tweet.
- The Ultimate Question, and its Answer (Sep 9, 2016)
- What is the Ultimate Question, and what is its Answer?
- Topological Crystals (Part 4) (Aug 28, 2016)
- Finally, a gallery of examples of topological crystals!
- Compact Closed Bicategories (Aug 19, 2016)
- Mike Stay’s paper on compact closed symmetric monoidal bicategories is finally published!
- Two Miracles of Algebraic Geometry (Aug 10, 2016)
- Taking the Albanese variety defines an idempotent monad.
- Topological Crystals (Part 3) (Aug 6, 2016)
- How to build a topological crystal.
- Topological Crystals (Part 2) (Jul 27, 2016)
- Learn about the maximal abelian cover of a graph.
- Topological Crystals (Part 1) (Jul 23, 2016)
- There’s a way to get a “topological crystal” from a graph. Learn how it works!
- Good News (May 26, 2016)
- Good news about four former students.
- The Works of Charles Ehresmann (May 10, 2016)
- Charles Ehresmann’s complete works, over 3000 pages of mathematics, are now available for free online.
- Which Paths Are Orthogonal to All Cycles? (May 5, 2016)
- A question about the first homology group of a graph.
- Categorifying Lucas' Equation (May 4, 2016)
- Gavin Wraith asks: can you find a nice bijective proof that $1^2 + \cdots + 24^2 = 70^2$?
- Polygonal Decompositions of Surfaces (Apr 23, 2016)
- What are the most standard references for polygonal decompositions of compact smooth surfaces, so I can just cite those and use their terminology?
- Foundations of Mathematics (Mar 31, 2016)
- Different people have very different attitudes to “foundations”, and until we discuss those, we’ll be talking past each other.
- E
_{8}Is the Best (Mar 24, 2016) - We finally know that E
_{8}gives the densest lattice packing of spheres! - The Involute of a Cubical Parabola (Mar 23, 2016)
- Help me find the icosahedron hidden in the theory of wave propagation in 2 dimensions.
- Prime Numbers and the Riemann Hypothesis (Mar 21, 2016)
- A free book on the connection between prime numbers and the Riemann zeta function!
- The Most Common Prime Gaps (Mar 19, 2016)
- What are the most common gaps between primes?
- Weirdness in the Primes (Mar 15, 2016)
- A new discovery shakes up our idea of how random primes are.
- Category Theory Course Notes (Mar 10, 2016)
- Here are the notes for my Winter 2016 introductory course on category theory.
- Category Theory Seminar Notes (Mar 9, 2016)
- Here are the lecture notes for my Fall 2015 introduction to how category theory helps unify our view of mathematics.
- Hyperbolic Kac--Moody Groups (Mar 4, 2016)
- Is there a geometric construction of hyperbolic Kac–Moody groups?
- Integral Octonions (Part 12) (Jan 31, 2016)
- Tim Silverman explores the $\mathrm{E}_8$ lattice mod 2, and its connection with questions we’d studied earlier in this series.
- Thinking about Grothendieck (Jan 17, 2016)
- Barry Mazur has a new short essay on Grothendieck.
- Homotopy of Operads and Grothendieck-Teichmüller Groups (Jan 16, 2016)
- Benoit Fresse has finished a big two-value book on operads, which you can now see on his website.
- A Compositional Framework for Markov Processes (Jan 12, 2016)
- Open Markov processes — that is, Markov processes with specified ‘input’ and ‘output’ states — can be seen as morphisms in a category. This lets us take a big complicated Markov process and think of it as made of smaller open ones.
- Decorated Cospans (Jan 6, 2016)
- Brendan Fong has a paper on ‘decorated cospan categories’. They seem to play a big role in network theory!
- A Compositional Framework for Passive Linear Networks (Dec 28, 2015)
- Yes, you can find categories everywhere — even in electrical engineering! The reason is that we live in a world of networks, and networks tend to be morphisms in categories.
- Operads and Phylogenetic Trees (Dec 22, 2015)
- What do phylogenetic trees have to do with operads? A lot!
- Globular (Dec 5, 2015)
- Jamie Vicary writes about some new software for calculating in $n$-categories.
- 42 (Dec 3, 2015)
- In
*The Hitchhiker’s Guide to the Galaxy*, the number 42 was revealed to be the Answer to the Ultimate Question of Life, the Universe, and Everything. But we never learned what the question was! - Burritos for Category Theorists (Nov 11, 2015)
- At last, burritos have been explained in a clear, conceptual way.
- Cakes, Custard, Categories and Colbert (Nov 3, 2015)
- Eugenia Cheng will be on TV this Wednesday, November 4th.
- Four Tribes of Mathematicians (Oct 20, 2015)
- Read about different motivations for doing mathematics.
- Configurations of Lines and Models of Lie Algebras (Part 2) (Oct 5, 2015)
- Triality, its action on the 24-cell, and the ‘tetrality’ description of $\mathfrak{so}(8)$.
- Configurations of Lines and Models of Lie Algebras (Part 1) (Oct 4, 2015)
- Learn a bit about the 27 lines on a smooth cubic surface.
- The Free Modular Lattice on 3 Generators (Sep 19, 2015)
- How is the free modular lattice on 3 generators related to 8-dimensional space?
- A Wrinkle in the Mathematical Universe (Aug 17, 2015)
- The outer automorphism of $S_6$ is an exceptional thing.
- Two Cryptomorphic Puzzles (Aug 9, 2015)
- Two puzzles whose solutions are related.
- Category Theory 2015 (Jul 19, 2015)
- You can see slides from Category Theory 2015.
- Semigroup Puzzles (Jun 9, 2015)
- Two puzzles that involve nothing but playing with equations.
- A 2-Categorical Approach to the Pi Calculus (May 26, 2015)
- Greg Meredith and Mike Stay have a new paper on modeling the pi calculus with 2-categories.
- PROPs for Linear Systems (May 22, 2015)
- PROPs were developed in topology, along with operads, to describe spaces with lots of operations on them. But now some of us are using them to think about ‘signal-flow diagrams’ in control theory—an important branch of engineering. I talked…
- The Origin of the Word "Quandle" (May 21, 2015)
- How did ‘quandles’ get their name?
- Categories in Control (Apr 28, 2015)
- To understand ecosystems, ultimately will be to understand networks. - B. C. Patten and M. Witkamp A while back I decided one way to apply my math skills to help save the planet was to start pushing toward green…
- Information and Entropy in Biological Systems (Apr 7, 2015)
- You can join a workshop on entropy and information in biological systems, virtually, by watching the streaming video.
- Split Octonions and the Rolling Ball (Mar 14, 2015)
- Watch a rolling ball illustrate an exceptional incidence geometry!
- Concepts of Sameness (Part 4) (Feb 27, 2015)
- Three approaches to defining equality.
- Concepts of Sameness (Part 3) (Feb 25, 2015)
- What good is the equation x = x?
- Concepts of Sameness (Part 2) (Feb 23, 2015)
- Some notes on the Gongsun Long’s ‘white horse paradox’.
- Concepts of Sameness (Part 1) (Feb 23, 2015)
- Here I’ll start drafting a paper on ‘concepts of sameness’, which will be a chapter in Elaine Landry’s
*Categories for the Working Philosopher*. - Quantum Physics and Logic at Oxford (Feb 18, 2015)
- The 12th International Workshop on Quantum Physics and Logic is being held in Oxford from July 13th to 17th, 2015.
- Can a Computer Solve Lebesgue's Universal Covering Problem? (Feb 12, 2015)
- Let’s try to prove that the solution to Lebesgue’s universal covering problem is computable.
- Higher-Dimensional Rewriting in Warsaw (Feb 9, 2015)
- There will be a conference on Higher-Dimensional Rewriting and Applications, 28-29 June 2015, in Warsaw.
- Lebesgue's Universal Covering Problem (Feb 4, 2015)
- Read about some progress on Lebesgue’s universal covering problem.
- Braided 2-Groups from Lattices (Jan 1, 2015)
- There are some easy ways to get braided 2-groups from lattices.
- Can One Explain Schemes to a Biologist? (Dec 31, 2014)
- Mumford’s obituary of Grothendieck was rejected by
*Nature*. - Integral Octonions (Part 11) (Dec 11, 2014)
- Why there are 270 rescaled copies of the first shell of the E
_{8}lattice sitting in the second shell. - Integral Octonions (Part 10) (Dec 1, 2014)
- How is the Leech lattice related to the exceptional Jordan algebra?
- Integral Octonions (Part 9) (Nov 23, 2014)
- Learn about using integral octonions and the E
_{8}lattice to construct the Leech lattice. - Integral Octonions (Part 8) (Nov 15, 2014)
- With a suitable Lorentzian metric, the lattice of self-adjoint 3 x 3 octonionic matrices with
*integral*octonions as entries becomes isometric to a 27-dimensional lattice that may play a role in bosonic string theory. - Integral Octonions (Part 7) (Nov 5, 2014)
- Greg Egan just showed that $\mathrm{E}_{10}$: is the lattice of $2 \times 2$ self-adjoint matrices with integral octonions as entries! Here’s the proof.
- M-theory, Octonions and Tricategories (Oct 5, 2014)
- John Huerta has completed the fourth of a series of papers on supersymmetry and division algebras.
- A Packing Pessimization Problem (Sep 21, 2014)
- What convex centrally symmetric shape is worst at tiling the plane with high density?
- Uncountably Categorical Theories (Aug 31, 2014)
- Boris Zilber’s thoughts on ‘logically perfect’ structures.
- The Ten-Fold Way (Part 4) (Aug 12, 2014)
- Todd Trimble’s note on ‘the super Brauer group and super division algebras’, reprinted.
- The Ten-Fold Way (Part 3) (Aug 7, 2014)
- A detailed introduction to the Brauer–Wall monoid, which gives an algebraic explanation of the ten-fold way.
- The Ten-Fold Way (Part 2) (Jul 22, 2014)
- Algebraic thoughts on the threefold way and ten-fold way.
- The Ten-Fold Way (Part 1) (Jul 19, 2014)
- There are 10 associative real super-division algebras, 10 classical families of compact symmetric spaces, 10 ways that Hamiltonians can get along with time reversal and charge conjugation symmetry… and even 10 dimensions of spacetime in string theory! Are these related? Yes!
- Describing PROPs Using Generators and Relations (Jul 10, 2014)
- What’s a good reference for ‘presentations’ of symmetric monoidal category?
- Spans and the Categorified Heisenberg Algebra (Part 2) (May 25, 2014)
- Watch a video of a talk about Jeffrey Morton and Jamie Vicary’s work on the categorified Heisenberg algebra.
- Finite Products Theories (Apr 20, 2014)
- I need a slight generalization of a classic result on algebraic theories. You may know who proved it!
- Operads and Trees (Mar 31, 2014)
- It would be nice if someone had already proved these conjectured results relating trees and operads.
- Translating Grothendieck's Biography into English (Mar 18, 2014)
- Leila Schneps is raising funds for an English translation of volume 3 of a biography of Grothendieck.
- Relative Entropy (Feb 13, 2014)
- John Baez and Tobias Fritz have a new category-theoretic characterization of relative entropy.
- Network Theory Talks at Oxford (Feb 10, 2014)
- There will be 4 talks on “Network Theory” and one on “Operads and the Tree of Life” at the University of Oxford.
- Tensor Categories and TQFT in Erlangen (Jan 30, 2014)
- There will be a meeting on tensor categories and topological field theory in Erlangen on March 4-6, 2014.
- Wormholes and Entanglement (Jan 19, 2014)
- The formation of a wormhole and the creation of a particle-antiparticle pair are closely related processes in a 3d TQFT. This sheds some light on how wormholes and entanglement are similar.
- Categories for the Working Philosopher (Nov 18, 2013)
- Elaine Landry is editing a book called
*Categories for the Working Philosopher*. - Clay Mathematics Conference (Sep 24, 2013)
- There is a Clay Research Conference from September 29 to October 4, 2013 at Oxford University.
- Integral Octonions (Part 6) (Sep 17, 2013)
- Learn about the Cayley integers, a lattice in the octonions that’s closed under multiplication.
- Integral Octonions (Part 5) (Sep 3, 2013)
- Let’s count the number of vertices, edges, triangles, tetrahedra and so on in the root polytope of E
_{8}! - Integral Octonions (Part 4) (Aug 3, 2013)
- More on semiregular polytopes and honeycombs related to the Cayley integer octonions.
- Spans and the Categorified Heisenberg Algebra (Aug 1, 2013)
- Some talks on spans of groupoids, symmetric monoidal bicategories and how these structures shed new light on the combinatorics of quantum mechanics.
- Integral Octonions (Part 3) (Jul 29, 2013)
- Let’s take a look at the E
_{8}lattice and the Lie algebra by the same name. - Integral Octonions (Part 2) (Jul 25, 2013)
- Belinskii, Khalatnikov and Lifshitz described homogeneous cosmologies that become chaotic as you go back toward the Big Bang. What does this have to do with octonionic integers?
- Integral Octonions (Part 1) (Jul 23, 2013)
- Learn how integral octonions are connected to the Weyl group of the hyperbolic Kac–Moody algebra E
_{10}. - Quasicrystals and the Riemann Hypothesis (Jun 14, 2013)
- Freeman Dyson asserted a relation between quasicrystals and the Riemann Hypothesis. What does it amount to, actually?
- The Selected Papers Network (Jun 14, 2013)
- Given that social networks already exist, all we need for truly open scientific communication is a convention on a consistent set of tags and IDs for discussing papers. Christopher Lee has developed software that makes this work.
- Semantics of Proofs in Paris (May 17, 2013)
- There’s a program on “Semantics of Proofs and Certified Mathematics” in Paris from April 22nd to July 11tth, 2014.
- How Does Applied Math Impact Foundations? (Apr 30, 2013)
- This weekend I’m giving a talk on “The Foundations of Applied Mathematics”. It’s mostly about how the widespread use of diagrams in engineering, biology, and the like may force us to think about math in new ways, at least…
- Category-Theoretic Foundations in Irvine (Apr 5, 2013)
- You can now see the list of talks at Category-Theoretic Foundations of Mathematics Workshop at the Department of Logic and Philosophy of Science of U.C. Irvine, May 4-5, 2013.
- Iterating the Free Monoid Construction (Apr 3, 2013)
- If you cube the free monoid monad, do you get back where you started… in some weak sense?
- The Hipparchus Operad (Apr 1, 2013)
- It seems the Schröder–Hipparchus numbers are the numbers of operations in an operad that was first discovered around 100 BC.
*M*_{13}(Feb 10, 2013)- You can use groupoids to give a nice description of the Mathieu group $M_{12}$!
- Category-Theoretic Foundations in Irvine (Dec 18, 2012)
- There will be a conference on Category-Theoretic Foundations of Mathematics at the Department of Logic and Philosophy of Science, U.C. Irvine on May 4-5, 2013.
- Morton and Vicary on the Categorified Heisenberg Algebra (Jul 10, 2012)
- Jeffrey Morton and Jamie Vicary have a new paper on the categorification of quantum mechanics using spans of groupoids.
- Compact Closed Bicategories (Jun 6, 2012)
- Mike Stay has a new paper on compact closed bicategories and their many applications.
- Quivering with Excitement (May 4, 2012)
- Any projective variety is some Grassmannian of some quiver representation.
- Teleparallel Gravity as a Higher Gauge Theory (Apr 20, 2012)
- How to see Einstein’s old theory of ‘teleparallel gravity’ as a theory involving a 2-connection.
- Math 2.0 (Feb 16, 2012)
- Welcome to Math 2.0: a forum for discussing an upgrade of the mathematical environment.
- The Cost of Knowledge (Feb 9, 2012)
- An explanation of the Elsevier boycott, from the mathematician’s point of view.
- Good News (Feb 7, 2012)
- Good news about category theory in Macquarie, Jeffrey Morton’s new position, and his work with Jamie Vicary on categorified quantum mechanics.
- Banning Elsevier (Jan 26, 2012)
- Join the ban of Elsevier.
- The Café Will Be Closed 2 Days in January (Dec 30, 2011)
- The Café will be closed for 48 hours starting at 6 pm CST on January 5th, 2012.
- Categorifying Fractional Euler Characteristics (Dec 16, 2011)
- Frenkel, Stroppel and Sussan have a way to define things with fractional Euler characteristics.
- What Might Be Done About High Prices of Journals? (Dec 12, 2011)
- Do you have suggestions for how to lower the prices of mathematical journals?
- TeX in Gmail (Dec 5, 2011)
- There’s a plugin for TeX in Gmail.
- The (∞,1)-Category of (∞,n)-Categories (Nov 30, 2011)
- There’s a new paper by Barwick and Schommer-Pries on the (∞,1)-category of (∞n)-categories.
- A Math Puzzle Coming From Chemistry (Oct 23, 2011)
- Try this graph theory puzzle coming from chemistry.
- Weak Systems of Arithmetic (Oct 11, 2011)
- There are lots of interesting axiom systems for arithmetic weaker than Peano arithmetic.
- The Inconsistency of Arithmetic (Sep 27, 2011)
- Edward Nelson has claimed to prove the inconsistency of Peano Arithmetic.
- Division Algebras and Supersymmetry III (Sep 22, 2011)
- John Huerta explains his new paper, which constructs some Lie 2-supergroups that are important in superstring theory.
- Mathematics of Planet Earth at Banff (Sep 8, 2011)
- Let’s have a math conference in 2013 at Banff on the theme
*The Mathematics of Planet Earth*. - Math and Physics on Google+ (Sep 8, 2011)
- Fun math and physics items at Google+.
- The Genius In My Basement (Aug 30, 2011)
- There’s a new book about the mathematician Simon Norton.
- Klein 2-Geometry XII (Aug 17, 2011)
- An attempt to see teleparallel gravity as a higher gauge theory.
- The Strangest Numbers in String Theory (Aug 15, 2011)
- A
*Scientific American*article on normed division algebras, especially the octonions, and their role in string theory. - Geometries: Diffeomorphism Classes vs Quilts (Aug 13, 2011)
- What follows is a guest post by Greg Weeks. If your memory extends back before the formation of this blog to the glory days of sci.physics.research, you should remember Greg….
- Operads and the Tree of Life (Jul 7, 2011)
- Tomorrow I’m giving a talk about an operad that shows up in biology. I wrote my lecture notes in the form of a blog entry: John Baez, Operads and the tree of life, Combinatorics Seminar, Université du Québec à…
- Christopher Walker on Hall Algebras (Jun 11, 2011)
- Christopher Walker has successfully defended his thesis,
*A Categorification of Hall Algebras*. - Category-Theoretic Characterizations of Entropy III (Jun 5, 2011)
- Now we’re trying to generalize our characterization of entropy from the classical to the quantum case.
- Category-Theoretic Characterizations of Entropy II (May 8, 2011)
- More work on trying to understand entropy in category-theoretic terms.
- Category-Theoretic Characterizations of Entropy (May 1, 2011)
- Tobias Fritz, Tom Leinster and John Baez are pondering category-theoretic characterizations of entropy.
- Higher Categories for Concurrency (Apr 29, 2011)
- Mike Stay on a bicategorical model of Boudol’s ‘blue calculus’.
- Higher Structures in China II (Apr 21, 2011)
- There is a conference Higher structures in China II taking place at Jilin University on August 8th to 10th, 2011.
- Quantum Theory and Gravitation in Zurich (Apr 17, 2011)
- There’s a conference on Quantum Theory and Gravitation at the ETH in Zurich, June 14–24, 2011.
- The Three-Fold Way (Part 6) (Apr 13, 2011)
- See how the Threefold Way solves two ‘problems’ with real and quaternionic quantum theory.
- Higher Gauge Theory, Division Algebras and Superstrings (Mar 22, 2011)
- See the slides for a talk explaining John Huerta’s work on higher gauge theory, division algebras and superstrings.
- A Categorified Supergroup for String Theory (Mar 11, 2011)
- John Huerta has used the octonions to construct a Lie 3-supergroup that describes the parallel transport of 2-branes in 11 dimensions.
- Liang Kong on Levin-Wen Models (Mar 7, 2011)
- Liang Kong is using tricategories in condensed matter physics!
- Heron's Formula (Feb 15, 2011)
- See a 4-dimensional proof of Heron’s formula for the area of a triangle.
- Rényi Entropy and Free Energy (Feb 11, 2011)
- Up to a temperature-dependent factor, Rényi entropy is just free energy!
- The Three-Fold Way (Part 5) (Feb 8, 2011)
- Learn about the functors going between the categories of real, complex and quaternionic Hilbert spaces.
- The Three-Fold Way (Part 4) (Jan 29, 2011)
- In the fourth episode of The Three-Fold Way, we’ll see how turning a particle around 180 degrees is related to making it go backwards in time.
- Math Journal Workshop at MSRI (Jan 25, 2011)
- There is a workshop on math journals at MSRI from February 14th to 16th, 2011.
- Doodling in Math Class (Dec 17, 2010)
- Vi Hart has some cool math videos.
- A Global Glance on Categories in Logic (Dec 15, 2010)
- Freely dowloadable papers on logic… for a limited time.
- Higher Gauge Theory, TQFT and Quantum Gravity in Lisbon (Dec 14, 2010)
- There’s a workshop and school on Higher Gauge Theory, TQFT and Quantum Gravity in Lisbon during February 7-13, 2011.
- The Three-Fold Way (Part 3) (Dec 13, 2010)
- In the third episode of The Three-Fold Way, we’ll see that the “q” in “qubit” stands for
*quaternion*— at least when you think of them as representations of SU(2). - The Library: Three Jeremiads (Dec 9, 2010)
- Those of you interested in journal prices and the like will enjoy this article pointed out by David Roberts: Robert Darnton, The library: three jeremiads, The New York Review of Books, December, 2010. The author is a historian at Harvard…
- Quantum Foundations Mailing List (Dec 9, 2010)
- Bob Coecke and Jamie Vicary have set up a mailing list on “quantum foundations”.
- The Three-Fold Way (Part 2) (Dec 7, 2010)
- The ‘three-fold way’ first manifests itself as a classification of unitary group representations — let’s see how that works.
- The Three-Fold Way (Part 1) (Dec 3, 2010)
- In the first of a series of posts on real, complex and quaternionic quantum theory, we describe some problems with the real and complex theories.
- Homological Algebra Puzzle (Dec 2, 2010)
- A homological algebra puzzle about kernels and cokernels.
- Category Theory Puzzle (Dec 2, 2010)
- A puzzle about categories enriched over the category of abelian groups.
- Solèr's Theorem (Dec 1, 2010)
- In 1995 Maria Pia Solèr proved an amazing theorem that characterizes Hilbert spaces over the real numbers, complex numbers and quaternions.
- State-Observable Duality (Part 3) (Nov 29, 2010)
- The third and last of a short series of posts on the foundations of quantum theory: the Koecher–Vinberg classification of self-dual homogeneous convex cones, and its relation to observable-state duality.
- State-Observable Duality (Part 2) (Nov 27, 2010)
- The second of a short series of posts on the foundations of quantum theory: the theorem by Jordan, von Neumann and Wigner classifying ‘finite-dimensional formally real Jordan algebras’.
- State-Observable Duality (Part 1) (Nov 25, 2010)
- The first of a short series of posts on the foundations of quantum theory: a review of normed division algebras.
- The Art of Math (Oct 22, 2010)
- Sophie Hebden has written a nice gentle introduction to the work of John Baez’s students that was funded by a grant from the Foundational Questions Institute.
- Vanity and Ambition in Mathematics (Oct 14, 2010)
- Some essays on mathematics by Jonah Sinick.
- Pictures of Modular Curves (I) (Oct 6, 2010)
- The first of a series of posts by Tim Silverman, with lots of pretty pictures.
- A Hopf Algebra Structure on Hall Algebras (Oct 4, 2010)
- Christopher Walker has a new paper, “A Hopf algebra structure on Hall algebras”.
- Democracy and Mathematics (Oct 4, 2010)
- A message from André Joyal on reforming mathematics.
- Jacob Biamonte on Tensor Networks (Sep 29, 2010)
- Jacob Biamonte wants your comments on his paper about string diagrams in quantum computation and condensed matter physics, which are called “tensor networks”.
- Categories and Information in Oxford (Sep 11, 2010)
- There’s a conference on “The categorical flow of information in quantum physics and linguistics” in Oxford from October 29th to 31st, 2010.
- Grothendieck's "Tohoku" paper (Sep 7, 2010)
- Michael Barr has made available an English translation of Grothendieck’s “Tohoku” paper.
- Bimonoids from Biproducts (Sep 1, 2010)
- Every object in a category with finite biproducts is a bimonoid in a unique way.
- The Philosophy of the Logic of Sheaves (Sep 1, 2010)
- On October 19 - 21, 2010, there’s a conference in Columbia on the philosophical aspects of sheaf logic.
- Categorification in Portugal (Aug 31, 2010)
- The latest news from Jeffrey Morton.
- This Week's Finds (Week 301) (Aug 27, 2010)
- The new
*This Week’s Finds*can be found over on Azimuth. - This Week's Finds in Mathematical Physics (Week 300) (Aug 11, 2010)
- In “week300”, learn how to categorify the Riemann zeta function and get a groupoid whose cardinality is $\pi^2/6$.
- Thermodynamics and Wick Rotation (Aug 6, 2010)
- Help Mike Stay with some questions about the analogy between quantum mechanics and statistical mechanics!
- Azimuth (Jul 30, 2010)
- John Baez has a new blog, called Azimuth, which is about ecological and technological issues.
- Zeta Functions: Dedekind Versus Hasse-Weil (Jul 28, 2010)
- Some help on Hasse-Weil zeta functions, anyone?
- Commutative Separable Algebras II (Jul 22, 2010)
- More questions about commutative separable algebras, especially over the integers.
- Commutative Separable Algebras (Jul 16, 2010)
- What are some commutative semisimple algebras that aren’t separable?
- Homological Algebra Puzzle (Jul 1, 2010)
- What is the walking epimorphism?
- 2010 Octoberfest (Jun 20, 2010)
- There’s a category theory workshop on October 23-24, 2010 at Dalhousie University in Halifax, Canada.
- This Week's Finds in Mathematical Physics (Week 299) (Jun 13, 2010)
- In week299 of This Week’s Finds, hear about the school on Quantum Information and Computer Science that was recently held in Oxford, and the subsequent workshop on Quantum Physics and Logic. Then, find out what Bill Schmitt told me about combinatorics.
- Boycotting the Nature Publishing Group (Jun 9, 2010)
- Nature Publishing Group is massively hiking its prices on many journals… and the University of California is considering ways to fight back.
- Categories, Logic and Physics 7 at Birmingham (Jun 2, 2010)
- The 7th Categories in Logic and Physics workshop will be in Birmingham, England on September 21st, 2010.
- The Quantum Whisky Club (May 28, 2010)
- Fun at Oxford.
- This Week's Finds in Mathematical Physics (Week 298) (May 15, 2010)
- In “week298” of This Week’s Finds, learn about finite subgroups of the unit quaternions, finite subloops of the unit octonions, Lie n-superalgebras built using division algebras, and Duff and Ferrara’s ideas connecting exceptional groups to quantum information theory. And learn what a "hyperdeterminant" is!
- This Week's Finds in Mathematical Physics (Week 297) (May 10, 2010)
- In “week297” of This Week’s Finds, see knot sculptures and learn about special relativity in finance, lazulinos, some peculiar infinite sums, and a marvelous fact about the number 12. Then: Dirichlet forms and electrical circuits!
- This Week's Finds in Mathematical Physics (Week 296) (Apr 26, 2010)
- In week296 of This Week’s Finds, get a free copy of Jerry Shurman’s book
*The Geometry of the Quintic*, and read my attempt to construct a compact dagger-category whose morphisms are electrical circuits made of resistors. - This Week's Finds in Mathematical Physics (Week 295) (Apr 17, 2010)
- In “week295” of This Week’s Finds, learn about the principle of least power, Poincaré duality for electrical circuits — and a curious generalization of Hamiltonian mechanics that involves entropy as well as energy.
- Alex Hoffnung Heading to Ottawa (Mar 25, 2010)
- Alex Hoffnung got a job at the University of Ottawa!
- This Week's Finds in Mathematical Physics (Week 294) (Mar 22, 2010)
- In “week294”, hear an account of Gelfand’s famous math seminar in Moscow. Read what Jan Willems thinks about control theory and bond graphs. Learn the proof of Tellegen’s theorem. Meet some categories where the morphisms are circuits, learn why a category object in Vect is just a 2-term chain complex.. and gaze at Saturn’s rings edge-on.
- An Invitation to Higher Gauge Theory (Again) (Mar 17, 2010)
- You can now see ‘An invitation to higher gauge theory’ in near-final form.
- Division Algebras and Supersymmetry II (Mar 13, 2010)
- The real numbers, complex numbers, quaternions and octonions give Lie 2-superalgebras that describe the parallel transport of superstrings, and Lie 3-superalgebras that describe the parallel transport of 2-branes!
- Postdoc Position in Lisbon (Mar 5, 2010)
- There’s a postdoc position in Lisbon for someone working on categorification or gerbes.
- Categorification in Faro (Mar 5, 2010)
- The XIXth Oporto Meeting on Geometry, Topology and Physics will take place on July 19 – 23, 2010 in Faro, Portugal.
- 2-Branes and Supergravity Theories (Mar 4, 2010)
- What about supergravity theories in dimensions 4, 5, 7 and 11?
- An Invitation to Higher Gauge Theory (Mar 1, 2010)
- Read the draft of an expository paper on higher gauge theory by John Huerta and John Baez.
- Algorithmic Thermodynamics (Feb 14, 2010)
- Read a new paper about the relation between thermodynamics and algorithmic information theory.
- Grothendieck Said: "Stop" (Feb 10, 2010)
- People were trying to republish Grothendieck’s
*magnum opus*. But then… - This Week's Finds in Mathematical Physics (Week 293) (Feb 6, 2010)
- In week293, catch up on recent papers and books about $n$-categories. Hear about last weekend’s Conference on the Mathematics of Environmental Sustainability and Green Technology at Harvey Mudd College. And learn how to think of networks of resistors as chain complexes which are also morphisms in a category.
- Quantum Physics and Logic at Oxford (Feb 3, 2010)
- There will be a workshop on Quantum Physics and Logic on Saturday and Sunday May 29-30, 2010 at Oxford.
- This Week's Finds in Mathematical Physics (Week 292) (Jan 30, 2010)
- In “week292”, learn about Henry Paynter’s “bond graphs” — a branch of applied category theory still waiting to be understood.
- This Week's Finds in Mathematical Physics (Week 291) (Jan 23, 2010)
- In week291, listen to a crab canon on a Möbius strip, see how the Mandelbrot set mimics infinitely many Julia sets, and learn about 2-ports, 3-ports, and Poincaré duality for electrical circuits.
- This Week's Finds in Mathematical Physics (Week 290) (Jan 16, 2010)
- In “week290” of This Week’s Finds, read about categorification in analysis. Ponder a number puzzle. Meet the five most popular 1-ports: resistances, capacitances, inertances, effort sources and flow sources. And learn a bit more rational homotopy theory!
- Going to Singapore (Jan 14, 2010)
- John’s leave of absence was approved.
- This Week's Finds in Mathematical Physics (Week 289) (Jan 8, 2010)
- In “week289” of This Week’s Finds, learn how E8 shows up in condensed matter physics. Continuing exploring the grand analogy between different physical systems! And learn about differential graded Lie algebras in rational homotopy theory.
- The Math of Environmental Sustainability at Harvey Mudd (Jan 6, 2010)
- Ever wonder what mathematicians can do to help solve the environmental problems facing all of us? I do. Maybe this will help: 2010 Harvey Mudd College Mathematics Conference on the Mathematics of Environmental Sustainability and Green Technology, Harvey Mudd,…
- Coenergy (Jan 4, 2010)
- What’s coenergy? Is ‘kinetic coenergy’ just kinetic energy regarded as a function of velocity rather than momentum?
- This Week's Finds in Mathematical Physics (Week 288) (Jan 1, 2010)
- In “week288”, start exploring a huge set of analogies linking many branches of physics… and keep learning about rational homotopy theory. This time we’ll tackle differential graded Lie algebras.
- This Week's Finds in Mathematical Physics (Week 287) (Dec 20, 2009)
- In “week287” of This Week’s Finds, hear about the history of categorical logic, and continue learning about rational homotopy theory - especially Sullivan’s approach based on differential forms.
- Can Five-Year-Olds Compute Coproducts? (Dec 13, 2009)
- Some cognitive scientists are using category theory to understand the development of reasoning ability in children.
- This Week's Finds in Mathematical Physics (Week 286) (Dec 9, 2009)
- In week286 of This Week’s Finds, begin learning about rational homotopy! More will follow in Weeks to come…
- This Week's Finds in Mathematical Physics (Week 285) (Dec 6, 2009)
- In “week285” of This Week’s Finds, discover the beauty of roots. Then, hear what happened on the second day of the session on homotopy theory and higher algebraic structures in the AMS conference at U.C. Riverside!
- Dangerous Knowledge (Dec 1, 2009)
- There’s a BBC documentary called “Dangerous Knowledge”, about “four brilliant mathematicians — Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing — whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide”.
- Dold--Kan Question (Nov 27, 2009)
- Why is the normalized singular cochain complex of a space a differential graded coalgebra?
- The 50 Best Physics Blogs (Nov 26, 2009)
- There’s a list of the 50 best physics blogs… and we’re on it!…
- This Week's Finds in Mathematical Physics (Week 284) (Nov 25, 2009)
- In “week284” of This Week’s Finds, learn about the pentagon-decagon-hexagon identity, golden triangles, categorified quantum groups, the Halperin-Carlsson conjecture, real Johnson-Wilson theories, Picard 2-stacks, quasicategories, motivic cohomology theory, and toric varieties!
- The 1000th Post on the n-Category Cafe (Nov 13, 2009)
- Meet our new hosts: Alex Hoffnung, Tom Leinster, Mike Shulman and Simon Willerton!
- This Week's Finds in Mathematical Physics (Week 283) (Nov 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!
- Courant Algebroids From Categorified Symplectic Geometry (Nov 10, 2009)
- Read a draft of a new paper that gets Courant algebroids from 2-plectic manifolds.
- Who Discovered the Icosahedron? (Nov 4, 2009)
- Was it Theaetetus or the Scots?
- Interview with Manin (Nov 2, 2009)
- Read a new interview with Yuri Manin.
- This Week's Finds in Mathematical Physics (Week 282) (Oct 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…
- Math: Folk Wisdom in an Electronic Age (Oct 26, 2009)
- People are starting to debate the best ways to use the internet to gather, store and access ‘folk wisdom’ about mathematics.
- Cheaper Online Textbooks? (Oct 22, 2009)
- The U.S. Department of Education is spending $300,000 to develop online textbooks. But how about free online textbooks?
- This Week's Finds in Mathematical Physics (Week 281) (Oct 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.
- Girih by Egan (Oct 16, 2009)
- If you like quasicrystals, check out this new applet by Greg Egan.
- Borisov on Shapes of Cells in
*n*-Categories (Oct 6, 2009) - Can you comment on a paper about comparing definitions of $n$-categories?
- Israel Gelfand, September 2, 1913 - October 5, 2009 (Oct 6, 2009)
- Israel Gelfand died on Monday October 5th, at the age of 96.
- Sex Advice From Mathematicians (Oct 1, 2009)
- The most amusing email of the day.
- What Do Mathematicians Need to Know About Blogging? II (Sep 30, 2009)
- Help me write an opinion piece on math blogging for the
*AMS Notices*! - This Week's Finds in Mathematical Physics (Week 280) (Sep 28, 2009)
- Hear about the courses Ashtekar and Rovelli taught at the quantum gravity summer school at Corfu.
- Questions on
*n*-Curvature (Sep 26, 2009) - Some questions for Urs.
- Homotopy Theory and Higher Algebraic Structures at UC Riverside (Sep 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!
- Movie Trivia Question (Sep 24, 2009)
- A trivia question involving Star Trek and two Bogart movies.
- Towards a Computer-Aided System for Real Mathematics (Sep 10, 2009)
- Arnold Neumaier is seeking useful comments on his proposal for a Formal Mathematical Language for doing and communicating mathematics on the computer.
- The Pi Calculus II (Sep 8, 2009)
- Mike Stay tries to explain the pi calculus to John Baez.
- This Week's Finds in Mathematical Physics (Week 279) (Sep 6, 2009)
- In week279, read about division algebras and supersymmetry - and a bit about tuning systems and infinitely groupoidified logic.
- Division Algebras and Supersymmetry (Sep 1, 2009)
- Learn how special properties of the reals, complexes, quaternions and octonions make supersymmetry tick in dimensions 3, 4, 6, and 10.
- SSE Composite Index Bubble II (Sep 1, 2009)
- Was Didier Sornette right in predicting a plunge in the Shanghai Stock Exchange (SSE) Composite Index?
- This Week's Finds in Mathematical Physics (Week 278) (Aug 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!
- Higher-Dimensional Algebra VII: Groupoidification (Aug 23, 2009)
- Check out our new paper “Higher-dimensional algebra VII: groupoidification”.
- The Pi Calculus (Aug 15, 2009)
- The pi calculus is a process calculus. But what does
*that*mean? - Asymptotics of the Magnitude of Metric Spaces (Aug 13, 2009)
- Learn about Tom Leinster and Simon Willerton’s new paper ‘On the asymptotic magnitude of subsets of Euclidean space’.
- Dan Ghica on Software vs Hardware (Aug 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.
- This Week's Finds in Mathematical Physics (Week 277) (Aug 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.
- What Do Mathematicians Need to Know About Blogging? (Aug 9, 2009)
- You can help me write an opinion piece on math blogging for the AMS
*Notices*. - Computation and the Periodic Table II (Aug 8, 2009)
- I’m giving a talk at LICS 2009 – this year’s annual conference on Logic in Computer Science.
- Groupoidfest 09 (Aug 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.
- Question About Exotic Smooth Structures (Jul 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?
- Generalizing "One-to-One" and "Onto" (Jul 28, 2009)
- Jonas Frey has settled some questions about the 2-categorical duals of the notions of ‘full’ and ‘fully faithful’.
- Arithmetic Geometry at the Newton Institute (Jul 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.
- Thomas Noll's Talks at Chicago on Mathematical Music Theory (Jul 23, 2009)
- Thomas Fiore has written a guest post about Thomas Noll’s talks on mathematical music theory.
- SSE Composite Index Bubble? (Jul 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.
- The Monads Hurt My Head --- But Not Anymore (Jul 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?
- A Puzzle From Gavin Wraith (Jul 16, 2009)
- A puzzle by Gavin Wraith, who found a wonderfully high-powered solution.
- Searching for a Video Proof of "Seven Trees in One" (Jul 16, 2009)
- James Propp is searching for a video proof that he saw once.
- Ben Zvi's Lectures on Topological Field Theory II (Jul 14, 2009)
- This is the second set of notes on David Ben-Zvi’s lectures on topological field theory at Northwestern University.
- Categorifying Nicomachus of Gerasa's Equation (Jul 13, 2009)
- Can you find a bijective proof of a formula for the sum of cubes of the first $n$ natural numbers?
- Ben-Zvi's Lectures on Topological Field Theory I (Jul 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.
- A Prehistory of n-Categorical Physics II (Jul 7, 2009)
- I’m back to writing that Prehistory of
*n*-Categorical Physics. I want to finish it before I die. - Course on Topological Quantum Field Theory in Almería (Jul 5, 2009)
- There will be a school on TQFTs in Almería, Spain from October 19th to 23rd, 2009.
- Open Access to Taxpayer-Funded Research (Jul 2, 2009)
- nn-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…
- Elsevier Pays for Favorable Book Reviews (Jul 2, 2009)
- Elsevier offered gift certificates to academics to write 5-star reviews of a book on Amazon and Barnes & Noble.
- Springer Verlag Publishes 'Proof' of Goldbach's Conjecture (Jun 30, 2009)
- Springer Verlag has published a book purporting to give elementary proofs of Fermat’s Last Theorem and Goldbach’s Conjecture.
- This Book Needs a Title (Jun 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?
- Hopf Algebras in Luxembourg (Jun 20, 2009)
- On July 13th to 17th 2009 there will be a conference and school on Hopf algebras in Luxembourg.
- This Week's Finds in Mathematical Physics (Week 276) (Jun 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.
- 2-Branes in 11 Dimensions (Jun 15, 2009)
- A question about 2-branes in 11 dimensions.
- This Week's Finds in Mathematical Physics (Week 275) (Jun 14, 2009)
- Read about progress on proving the Cobordism Hypothesis.
- Final Exams Again (Jun 10, 2009)
- What’s the funniest thing you’ve seen on a final exam this year?
- Algebraic Geometry for Category Theorists (Jun 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.
- Nonabelian Algebraic Topology (Jun 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. - Journal Club --- Geometric Infinity-Function Theory --- Week 6 (Jun 3, 2009)
- Alex Hoffnung on topological field theories.
- Treq Lila (Jun 2, 2009)
- I made an album
*Treq Lila*, in which all the melodies were created by cellular automata. - 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.
- 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.
- 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?
- 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’.
- 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. - 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!
- 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.
- Odd Currency Puzzle (May 7, 2009)
- A puzzle concerning odd American currency.
- The Foibles of Science Publishing (May 4, 2009)
- News about Elsevier’s fake medical journal and problems at
*Scientific American*. - A Riddle (Apr 29, 2009)
- See if you can solve this riddle.
- The Earth - For Physicists (Apr 28, 2009)
- A history of the Earth in 3000 words or less, for physicists.
- 'Kervaire Invariant One Problem' Solved (Apr 23, 2009)
- It seems Michael Hopkins has solved a famous old topology problem due to Milnor and Kervaire.
- Comparative Smootheology, IV (Apr 23, 2009)
- Take a look at Martin Laubinger’s thesis on smooth spaces.
- Kamnitzer on Categorifying Tangle Invariants (Apr 14, 2009)
- A brief summary of a talk by Joel Kamnitzer at the workshop on Categorification and Geometrisation from Representation Theory.
- Commutative Diagrams in Toulouse (Apr 8, 2009)
- There will be a workshop on Computer Algebra Methods and Commutativity of Algebraic Diagrams in Toulouse on October 16th and 17th, 2009.
- Categorification and Topology (Apr 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. - Categories, Quanta and Concepts at the Perimeter Institute (Mar 31, 2009)
- There will be a workshop called Categories, Quanta, Concepts at the Perimeter Institute from June 1st to 5th, 2009.
- The Algebra of Grand Unified Theories III (Mar 25, 2009)
- Is the SO(10) grand unified theory of particle physics still experimentally viable? A plea for help from phenomenologists.
- Unitary Representations of the Poincaré Group (Mar 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!
- Open Access at MIT (Mar 22, 2009)
- MIT has passed an open-access mandate!
- The Algebra of Grand Unified Theories II (Mar 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!
- The Okubo Algebra (Mar 13, 2009)
- Okubo invented a mysterious 8-dimensional division algebra while pondering quarks. What’s it all about?
- This Week's Finds in Mathematical Physics (Week 274) (Mar 12, 2009)
- Learn about infinite-dimensional 2-vector spaces and representations of 2-groups on these.
- Banning Open Access II (Mar 8, 2009)
- Conyers defends his bill which would prohibit the National Institute of Health from mandating open access to the research they fund.
- Super-Yang-Mills Theory (Mar 5, 2009)
- Why is supersymmetic Yang–Mills theory really supersymmetric? A plea for help.
- New Structures for Physics III (Feb 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.
- Monoidal Closed Categories And Their Deviant Relatives (Feb 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.
- Homotopical Algebraic Geometry in Seville (Feb 21, 2009)
- There will be a school on Homotopy Theory and Algebraic Geometry in Seville from September 7th to 13th, 2009.
- Satanist Rocketeer Puzzle (Feb 19, 2009)
- Two puzzles.
- Banning Open Access (Feb 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.
- Baković and Jurčo on Classifying Topoi for Topological Bicategories (Feb 14, 2009)
- Igor Bakoviċ and Branislaw Jurčo have a new paper out.
- Lurie on TQFT and the Cobordism Hypothesis (Feb 13, 2009)
- Watch videos of four talks by Jacob Lurie at the University of Texas at Austin.
- The Algebra of Grand Unified Theories (Feb 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.
- Link Homology in Paris (Feb 10, 2009)
- In 2009 there will be a summer school on link homology in Paris.
- The Cocktail Party Version (Feb 4, 2009)
- Jeffrey Morton explains what he’s working on.
- New York City Category Seminar (Jan 29, 2009)
- There’s a new seminar on categories, organized by Noson Yanofsky, meeting regularly at the CUNY Grad Center in Manhattan.
- The Third Time is the Charm (Jan 29, 2009)
- A puzzle about math and physics.
- Can Particle Physicists Regain Control of Their Journals? (Jan 28, 2009)
- Learn about the Sponsoring Consortium for Open Access Publishing in Particle Physics.
- Bruce Bartlett's Thesis (Jan 27, 2009)
- Bruce Bartlett’s thesis on topological quantum field theory is on the arXiv now.
- Categorified Symplectic Geometry and the String Lie 2-Algebra (Jan 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.
- Limits in the 2-Category of 2-Hilbert Spaces (Jan 22, 2009)
- What limits are present in the 2-category of finite-dimensional 2-Hilbert spaces?
- Petit Topos, Gros Topos (Jan 22, 2009)
- What’s the relation between the petit topos and the gros topos of a scheme?
- Hopf Algebras from Posets (Jan 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.
- The Toric Variety Associated to the Weyl Chambers (Dec 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?
- Joint Math Meetings in Washington DC (Dec 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?
- Groupoidification Made Easy (Dec 26, 2008)
- Now you can learn about groupoidification in a paper that has proofs of all the most basic results!
- Tilings (Dec 24, 2008)
- Here are some pretty pictures of tilings, suitable for use as computer wallpaper.
- Infinite-Dimensional Representations of 2-Groups (Dec 24, 2008)
- Here’s a new paper on infinite-dimensional representations of 2-groups — representations on ‘measurable categories’.
- Science and the Environment (Dec 24, 2008)
- Scientists may soon be running science policy in the US. What should we do?
- Higher Structures in Göttingen - Part II (Dec 22, 2008)
- On February 5-7 2009 there will be a workshop in Göttingen on Higher Structures in Topology and Geometry.
- Super Version of 2-Plectic Geometry for Classical Superstrings? (Dec 16, 2008)
- Here are some basic questions about describing the classical superstring using the ‘super’ analogue of multisymplectic geometry, if such a thing exists.
- This Week's Finds in Mathematical Physics (Week 273) (Dec 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.
- Science Citation Index (Dec 9, 2008)
- The Science Citation Index is increasingly being used to evaluate research. There are lots of reasons why this is a bad idea.
- A Quick Algebra Quiz (Dec 9, 2008)
- Here’s a quick test of your mathematical reflexes.
- Smooth Structures in Ottawa (Dec 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.
- This Week's Finds in Mathematical Physics (Week 272) (Nov 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.
- Groupoidfest 08 - A Brief Report (Nov 28, 2008)
- A brief report on the 2008 Groupoidfest held at UCR, with an emphasis on
*groupoidification*. - Categorification in Glasgow (Nov 24, 2008)
- There will be a workshop on categorification in Glasgow from April 13th to April 18th, 2009.
- Locally Compact Hausdorff Abelian Groups (Nov 12, 2008)
- Learn the Principal Structure Theorem for locally compact Hausdorff abelian groups — the kind of groups that show up in Pontryagin duality.
- Variations on Pontryagin Duality (Nov 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. - Google Books --- More Open Access? (Oct 28, 2008)
- A settlement has been reached which may give us more free access to Google Books.
- Lie Theory Through Examples 4 (Oct 28, 2008)
- Classifying representations using weights.
- This Week's Finds in Mathematical Physics (Week 271) (Oct 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!
- Open Access at the University of California (Oct 24, 2008)
- News about open access at the University of California.
- Groupoidfest 08 (Oct 24, 2008)
- Some talks at the Groupoidfest being held in Riverside on November 22nd-23rd, 2008.
- What is Categorification? (Oct 22, 2008)
- The meaning of the word ‘categorification’, and some key examples.
- Lie Theory Through Examples 3 (Oct 21, 2008)
- Finally the concept of ‘weight lattice’ rears its ugly head.
- Categorification in New Scientist (Oct 20, 2008)
- Richard Elwes at
*New Scientist*has written an article on knot theory that mentions categorification. - Morton on 2-Vector Spaces and Groupoids (Oct 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.
- The Nature of Time (Oct 13, 2008)
- The Foundational Questions Institute is having an essay contest on the nature of time.
- This Week's Finds in Mathematical Physics (Week 270) (Oct 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.
- Lie Theory Through Examples 2 (Oct 7, 2008)
- Moving on up a dimension, now let’s look at the A
_{3}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. - Quantum Theory and Analysis (Sep 30, 2008)
- Some lecture notes for a class on Hilbert spaces, self-adjoint operators, and their applications to quantum theory.
- Lie Theory Through Examples 1 (Sep 30, 2008)
- The beginning of a course on Lie theory through examples.
- OctoberFest 08 (Sep 27, 2008)
- This October, the annual Canadian conference on categories will be at Concordia University in Montreal.
- 5 (Sep 12, 2008)
- A talk on special properties of the number 5.
- 24 (Sep 12, 2008)
- A talk on my favorite number of all: 24.
- New Structures for Physics II (Sep 10, 2008)
- You can help review some more papers for Bob Coecke’s
*New Structures for Physics*. - 8 (Sep 7, 2008)
- A talk on one of my favorite numbers: 8.
- Military Research (Sep 6, 2008)
- What sort of research is the US military funding today?
- Comparative Smootheology, III (Sep 3, 2008)
- The third episode in our continuing comparison of various frameworks for differential geometry.
- This Week's Finds in Mathematical Physics (Week 269) (Aug 31, 2008)
- See more of Jupiter’s moon Io — and learn about honeycombs, the work of Kelvin, the Weaire-Phelan structure, and gas clathrates.
- New Structures for Physics I (Aug 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.
- Henri Cartan, July 8, 1904 - August 13, 2008 (Aug 22, 2008)
- Henri Cartan, a founding member of Bourbaki, died August 13th at the age of 104.
- The Weaire--Phelan Structure (Aug 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.
- Polish Spaces (Aug 21, 2008)
- Are second-countable locally compact Hausdorff spaces always homeomorphic to complete separable metric spaces?
- Elvis Zap does the Calculus Rap (Aug 20, 2008)
- Professor Elvis Zap explains the rules of differential calculus.
- Spivak on Derived Manifolds (Aug 19, 2008)
- David Spivak has an interesting thesis on ‘derived differential geometry’.
- Bimodules Versus Spans (Aug 14, 2008)
- Some puzzles about bimodules in the opposite of the category of sets.
- This Week's Finds in Mathematical Physics (Week 268) (Aug 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!
- Visiting Grothendieck (Aug 6, 2008)
- The story of someone who tried to visit Alexander Grothendieck.
- Categorified Symplectic Geometry and the Classical String (Aug 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.
- Getting Started Early (Jul 31, 2008)
- The man who had Ph.D. students before he was born.
- Light Mills (Jul 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?
- Causality in Discrete Models of Spacetime (Jul 27, 2008)
- Gavin Wraith has some questions about causality in discrete spacetimes described as simplicial complexes.
- Categories, Logic and Foundations of Physics in Oxford (Jul 25, 2008)
- On August 23rd and 24th, there will be a conference on Categories, Logic and Foundations of Physics at Oxford.
- This Week's Finds in Mathematical Physics (Week 267) (Jul 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.
- Theorems Into Coffee IV (Jul 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.
- This Week's Finds in Mathematical Physics (Week 266) (Jun 20, 2008)
- Read about Jupiter’s moon Io, the Pythagorean tuning system, and the categorical groups conference now taking place in Barcelona.
- Fundamental 2-Groups and 2-Covering Spaces (Jun 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’.
- Computation and the Periodic Table (Jun 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.
- Categories, Logic and Foundations of Physics: The Videos (Jun 9, 2008)
- Now videos are available for the 2nd conference on Categories, Logic and Foundations of Physics.
- 2-Groups in Barcelona (Jun 8, 2008)
- There’s a workshop on categorical groups in Barcelona during June 16-20, 2008.
- Farey Sequences and the Stern-Brocot Tree (Jun 7, 2008)
- An approach to modular curves based on Farey sequences and the Stern-Brocot tree.
- Classical String Theory and Categorified Symplectic Geometry (Jun 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!
*Cahiers*Free Online! (May 27, 2008)- The Ehresmann’s famous journal is now free online.
- Double Categories --- Warm and Cuddly? (May 27, 2008)
- Abstract of a talk on double categories by Ross Street.
- Number Theory on YouTube? (May 26, 2008)
- Number theory abstracts on YouTube?
- 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.
- This Week's Finds in Mathematical Physics (Week 264) (May 19, 2008)
- Learn about Phobos, homotopy groups of spheres, and theta functions.
- 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.
- 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!
- 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!
- Theorems Into Coffee (May 4, 2008)
- Prove this theorem, and you’ll get some coffee!
- Comparative Smootheology, II (Apr 17, 2008)
- Comparing Chen’s ‘differentiable spaces’ and Souriau’s ‘diffeological spaces’.
- This Week's Finds in Mathematical Physics (Week 263) (Apr 6, 2008)
- John Thompson and Jacques Tits won the 2008 Abel Prize for their work on group theory. Find out what they did.
- A Puzzle About Blueshifts (Apr 1, 2008)
- A puzzle about blueshifts for rotating black holes, posed by Greg Egan.
- This Week's Finds in Mathematical Physics (Week 262) (Mar 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 e
_{8}, and a categorification of sl(2). - Test Your Singlish (Mar 29, 2008)
- Guess the meaning of some words in Singlish.
- Groupoidfest in Riverside (Mar 21, 2008)
- The next Groupoidfest is at UCR, November 22-23 2008.
- This Week's Finds in Mathematical Physics (Week 261) (Mar 20, 2008)
- Learn about the Engraved Hourglass Nebula… and read an ode to the number 3. Also: try your hand at some trefoil puzzles.
- Following Singapore's Lead (Mar 12, 2008)
- A Los Angeles elementary school drastically improves its math teaching… by following the lead of Singapore.
- Physics, Topology, Logic and Computation: a Rosetta Stone (Mar 11, 2008)
- Toward a general theory of systems and processes.
- Learning to Love Topos Theory (Mar 9, 2008)
- Steve Vickers on topos theory.
- Some Puzzles (Mar 8, 2008)
- Three puzzles to keep you entertained.
- Kim on Fundamental Groups in Number Theory (Mar 1, 2008)
- Minhyong Kim on Diophantine geometry and the fundamental group.
- Computer Scientists Needed Now (Mar 1, 2008)
- John Baez begs for help on the computation section of his ‘Rosetta Stone’ paper connecting physics, topology, logic and computation.
- New Hire at UCR (Feb 27, 2008)
- Julia Bergner has accepted a tenure-track position at UCR. She works on the homotopy theory of homotopy theories.
- Logicians Needed Now (Feb 21, 2008)
- Mike Stay and I are writing a paper… and we need help from logicians!
- Kostant on E
_{8}(Feb 20, 2008) - Videos and lecture notes of Bertram Kostant’s talk on E
_{8}and the algebra of the Standard Model. - Harvard Research Free Online (Feb 18, 2008)
- Harvard arts and sciences faculty vote to make their research free online!
- Verity on ∞-Categories From Topology (Feb 12, 2008)
- Dominic Verity talks about a weak ∞-category of cobordisms.
- Geometric Representation Theory (Lecture 25) (Feb 12, 2008)
- Groupoidifying the commutation relations between annihilation and creation operators in quantum mechanics. An in-class experiment demonstrating these relations.
- Metric Spaces (Feb 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.
- Modular Forms (Feb 1, 2008)
- What are modular forms like beyond level 1? How about level 2, for starters?
- Geometric Representation Theory (Lecture 24) (Jan 31, 2008)
- How to groupoidify the harmonic oscillator.
- Geometric Representation Theory (Lecture 23) (Jan 29, 2008)
- James Dolan on groupoidifying the Hall algebra of the A
_{2}quiver. - The Yoneda Embedding as a Reflection (Jan 25, 2008)
- A pictorial way to think about the Yoneda embedding (or ‘continuation passing transform’).
- Classifying Spaces for 2-Groups (Jan 24, 2008)
- Just as principal bundles are classified by maps into a ‘classifying space’, the same is true for principal 2-bundles.
- Geometric Representation Theory (Lecture 22) (Jan 18, 2008)
- James Dolan groupoidifies the “Hall algebra” of a quiver.
- Strong NDR Pairs --- A Technical Question (Jan 16, 2008)
- Some niggling technical questions about topology.
- Geometric Representation Theory (Lecture 21) (Jan 15, 2008)
- The winter quarter of the Geometric Representation Theory seminar begins to tackle examples. Groupoidifying and
*q*-deforming the harmonic oscillator. - The Continuation Passing Transform and the Yoneda Embedding (Jan 10, 2008)
- The Yoneda embedding is familiar in category theory. The continuation passing transform is familiar in computer programming. They’re secretly the same!
- Geometric Representation Theory (Lecture 20) (Jan 10, 2008)
- At last: the Fundamental Theorem of Hecke Operators!
- A Tiny Taste of the History of Mechanics (Jan 7, 2008)
- Some incredibly sketchy lecture notes on the history of mechanics from Aristotle to Newton.
- How I Learned to Love the Nerve Construction (Jan 6, 2008)
- Guest post by Tom Leinster on the work of Mark Weber.
- Geometric Representation Theory (Lecture 19) (Jan 5, 2008)
- James Dolan lays the final piece of groundwork for the Fundamental Theorem of Hecke Operators.
- Geometric Representation Theory (Lecture 18) (Jan 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’.
- Non-Mathematician Rediscovers R
^{n}, C^{n}(Dec 30, 2007) - Based on the philosophy of Ayn Rand, a nonmathematician has reinvented some familiar algebras — and issued a press release about it.
- The QG-TQFT Blues (Dec 29, 2007)
- See Scott Carter sing the quantum gravity, topological quantum field theory blues.
- Geometric Representation Theory (Lecture 17) (Dec 26, 2007)
- James Dolan on degroupoidification and ‘transfer’ for the zeroth homology of groupoids.
- This Week's Finds in Mathematical Physics (Week 260) (Dec 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
- Roytenberg on Weak Lie 2-Algebras (Dec 21, 2007)
- Dmitry Roytenberg has proposed a fully general notion of categorified Lie algebra.
- Geometric Representation Theory (Lecture 16) (Dec 21, 2007)
- Turning a group acting on a set into a groupoid — the ‘weak quotient’ or ‘action groupoid’.
- Geometric Representation Theory (Lecture 15) (Dec 13, 2007)
- James Dolan on various forms of decategorification.
- This Week's Finds in Mathematical Physics (Week 259) (Dec 10, 2007)
- Hear what may be lurking in the Egg Nebula, and learn about a mathematical phantom called the “field with one element”.
- One Geometry (Math Raps) (Dec 7, 2007)
- A rappin’ proof of the Bolzano–Weierstrass theorem, by Steve Sawin.
- Look Around You (Dec 3, 2007)
- Did you know that the largest number is 45 billion?
- An Unlikely Result (Dec 3, 2007)
- A. N. Other puts ‘An unlikely result’ on the arXiv.
- Astronomers Destroy Universe (Dec 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? - Geometric Representation Theory (Lecture 14) (Nov 29, 2007)
- Matrix mechanics and Hecke operators. A failed attempt to state the Fundamental Theorem of Hecke Operators.
- Rejecta Mathematica (Nov 27, 2007)
- You can now publish your rejected papers in
*Rejecta Mathematica*. - This Week's Finds in Mathematical Physics (Week 258) (Nov 26, 2007)
- Read about the flow of grainy substances, polycyclic aromatic hydrocarbons in outer space, and Deligne’s conjecture on Hochschild cohomology.
- Poncelet's Porism (Nov 26, 2007)
- A question from Gavin Wraith, requiring help from someone who knows elliptic curves and also old-fashioned synthetic geometry.
- Categories, Logic and Physics in London (Nov 26, 2007)
- Bob Coecke and Andreas Döring are running a workshop on ‘Categories, Logic and Foundations of Physics’ at Imperial College.
- Geometric Representation Theory (Lecture 13) (Nov 22, 2007)
- James Dolan on using braids to describe Hecke operators between flag representations.
- Geometric Representation Theory (Lecture 12) (Nov 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.
- Geometric Representation Theory (Lecture 11) (Nov 16, 2007)
- How to describe Hecke operators between flag representations using matrices.
- The Dangers of Complex Analysis (Nov 13, 2007)
- Mathematicians like to redefine everyday words and make them into technical terms. This can be dangerous.
- Geometric Representation Theory (Lecture 10) (Nov 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. - Pfeiffer on Modular Tensor Categories (Nov 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.
- Geometric Representation Theory (Lecture 9) (Nov 9, 2007)
- Getting irreducible representations of symmetric groups from flag representations. Using ‘crackpot matrices’ to describe Hecke operators between flag representations.
- Geometric Representation Theory (Lecture 8) (Nov 6, 2007)
- Categorifying and
*q*-deforming Pascal’s triangle. - Geometric Representation Theory (Lecture 7) (Nov 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*! - Comet Holmes (Oct 30, 2007)
- Comet Holmes suddenly got a million times brighter last week, and now it’s visible with the naked eye.
- Higher Clifford Algebras (Oct 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!
- Fundamental Physics: Where We Stand Today (Oct 29, 2007)
- Slides from an easy talk on the open problems in fundamental physics.
- Steve Fever (Oct 27, 2007)
- A new story by Greg Egan — free online!
- Geometric Representation Theory (Lecture 6) (Oct 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. - Geometric Representation Theory (Lecture 5) (Oct 22, 2007)
- James Dolan on how to get intertwining operators called “Hecke operators” from relations between types of geometrical figures.
- Geometric Representation Theory (Lecture 3) (Oct 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!
- Geometric Representation Theory (Lecture 2) (Oct 17, 2007)
- James Dolan on geometric representation theory: transformation groups, logic and the orbi-simplex.
- This Week's Finds in Mathematical Physics (Week 257) (Oct 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.
- Geometric Representation Theory (Lecture 4) (Oct 12, 2007)
- Categorifying and
*q*-deforming the theory of binomial coefficients — and multinomial coefficients! — using the analogy between projective geometry and set theory. - Geometric Representation Theory (Lecture 1) (Oct 7, 2007)
- Video of the first lecture in the new seminar on Geometric Representation Theory at U. C. Riverside.
- Spans in Quantum Theory (Oct 1, 2007)
- You can use spans to understand why quantum processes act so much like pieces of spacetime.
- Rotations in the 7th Dimension (Sep 26, 2007)
- Is there an exceptional isomorphism involving the Spin group in 7 dimensions?
- The Catsters Strike Again: "String Diagrams" (Sep 24, 2007)
- Check out the
*other*Catster. - The Virtues of
*American Scientist*(Sep 24, 2007) - While some science magazines have declined, the
*American Scientist*is still great. - The Catsters' Latest Hit: "Adjunctions" (Sep 20, 2007)
- Learn about adjunctions on YouTube!
- Deep Beauty: Understanding the Quantum World (Sep 19, 2007)
- A symposium honoring the 75th anniversary of von Neumann’s book on quantum mechanics.
- The Catsters on YouTube: "Monads" (Sep 16, 2007)
- Watch some videos from the new group out of Sheffield — the Catsters!
- Toward a Higher-Dimensional Wiki (Sep 2, 2007)
- Let’s talk about setting up a wiki for
*n*-category theory and other aspects of higher-dimensional algebra! - Axis of Evil? Or Axis of Opportunity? (Aug 31, 2007)
- A paper by Michael Longo argues that the axes of elliptic galaxies don’t point in random directions.
- Journal Publishers Hire the "Pit Bull of PR" (Aug 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.
- This Week's Finds in Mathematical Physics (Week 256) (Aug 27, 2007)
- Lie
*n*-algebroids and Lie*n*-groupoids in classical mechanics… and the return of the Tale of Groupoidification - Gerbes in
*The Guardian*(Aug 22, 2007) - A
*Guardian*article about how hard it is to understand the concept of ‘gerbe’. - Higher Gauge Theory and the String Group (Aug 17, 2007)
- A talk leading up to the concept of ‘characteristic classes for 2-bundles’, and a nice problem involving them.
- Math Journal Wiki (Aug 12, 2007)
- A wiki to help us keep an eye on math journals, their prices, their publishers, and their policies.
- This Week's Finds in Mathematical Physics (Week 255) (Aug 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…
- Questions about Modules (Aug 5, 2007)
- What kinds of categories are categories of modules of some ring.
- Astronomical London (Aug 1, 2007)
- The Royal Observatory, Greenwich.
- Web Spamming by Academic Publishers (Jul 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.
- Higher Gauge Theory and Elliptic Cohomology (Jul 31, 2007)
- A talk on higher gauge theory and its relation to the work of Stolz and Teichner.
- Homotopy Theory and Higher Categories in Barcelona (Jul 26, 2007)
- During the 2007-2008 academic year there will be a program on Homotopy Theory and Higher Categories in Barcelona.
- Question About Representations of Finite Groups (Jul 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?
- Astronomical Paris (Jul 18, 2007)
- A tour of L’Observatoire de Paris.
- This Week's Finds in Mathematical Physics (Week 254) (Jul 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.
- Derek Wise on Cartan Geometry and MacDowell--Mansouri Gravity (Jul 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.
- Mathematical Paris (Jul 7, 2007)
- In Paris, you can get your name on a street sign just by being good in math.
*Really*good. - This Week's Finds in Mathematical Physics (Week 253) (Jun 28, 2007)
- From the Standard Model to SU(5), SO(10), E
_{6}… and maybe even on to E_{8}, with a friendly tip of the hat to symmetric spaces like the complexified octonionic plane. - Cohomology and Computation (Week 27) (Jun 19, 2007)
- Defining the cohomology of algebraic gadgets using the bar construction.
- Cohomology and Computation (Week 26) (Jun 18, 2007)
- An example of the bar construction: puffing up a point to the free contractible
*G*-space*EG*, important in group cohomology. - More Mysteries of the Number 24 (Jun 17, 2007)
- Kummer’s 24 hypergeometric functions: what are they?
- Why Math Teachers Get Grumpy (Jun 13, 2007)
- Grading final exams — a test of ones soul.
- Quantization and Cohomology (Week 27) (Jun 8, 2007)
- Review of what we’ve done in this course; prospectus of what’s still to be done.
- Cohomology and Computation (Week 25) (Jun 6, 2007)
- Getting monads, comonads and simplicial objects from adjoint functors.
- Quadratic Reciprocity (Jun 4, 2007)
- Quadratic Reciprocity is one of the gems of elementary number theory. Why does this famous proof actually work?
- Quantization and Cohomology (Week 26) (Jun 1, 2007)
- Cech cohomology in terms of anafunctors and ananatural transformations.
- Quantization and Cohomology (Week 25) (Jun 1, 2007)
- How describing bundles in terms of Cech cohomology secretly amounts to describing them in terms of smooth anafunctors.
- Quantization and Cohomology (Week 24) (Jun 1, 2007)
- Three approaches to connections and gauge transformations, leading up to one based on Toby Bartels’ notion of “smooth anafunctor”.
- 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.
- Link Homology and Categorification in Kyoto (May 26, 2007)
- Link Homology and Categorification — a conference in Kyoto, May 10-25, 2007.
- Congratulations to David! (May 25, 2007)
- David Corfield found a permanent position at the University of Kent at Canterbury!
- Cohomology and Computation (Week 24) (May 24, 2007)
- What makes the bar construction tick?
- Questions on n-Categories and Topology (May 21, 2007)
- Questions by Bruce Westbury on $n$-categories and topology
- 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.
- 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?
- Puzzle #12 (May 12, 2007)
- What’s the size of New Jersey, shaped like a dog bone, and named after a famous woman?
- Cohomology and Computation (Week 23) (May 11, 2007)
- The bar construction: a way to get simplicial sets from algebraic gadgets.
- Quantization and Cohomology (Week 23) (May 10, 2007)
- Connections on principal bundles — a modern treatment.
- Cohomology and Computation (Week 22) (May 8, 2007)
- How to count holes.
- 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.
- 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! - This Week's Finds in Mathematical Physics (Week 251) (May 6, 2007)
- Foundations of quantum mechanics — the news from Les Treilles.
- This Week's Finds in Mathematical Physics (Week 250) (Apr 26, 2007)
- The Popescu-Rohrlich game, double cosets, atomic invariant relations and spans of groupoids.
- Cohomology and Computation (Week 21) (Apr 20, 2007)
- Why mathematicians like to take algebraic gadgets and topological spaces and turn them into simplicial sets.
- Quantization and Cohomology (Week 21) (Apr 20, 2007)
- Understanding the ‘action’ in classical mechanics as a smooth functor — the case of a path groupoid.
- Cohomology and Computation (Week 20) (Apr 20, 2007)
- Why do mathematicians like simplices so much? Why are they better than other shapes?
*Incandescence*(Apr 14, 2007)- Greg Egan’s next novel is coming out in 2008 — but you can already read a story set in the same universe.
- Topos Theory in the
*New Scientist*(Apr 14, 2007) - Robert Matthews on Chris Isham’s work on topos theory and physics.
- Schur Functors (Apr 12, 2007)
- What’s the right way to think about Schur functors?
- Quantization and Cohomology (Week 20) (Apr 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.
- This Week's Finds in Mathematical Physics (Week 249) (Apr 9, 2007)
- Klein’s Erlangen program, now available in English! And: the Tale of Groupoidification, continued.
- Why Mathematics Is Boring (Apr 6, 2007)
- Storytellers have developed many strategies for luring in readers and keeping them interested. Mathematicians systematically avoid these.
- Cohomology and Computation (Week 19) (Apr 6, 2007)
- The origins of cohomology in the study of ‘syzygies’, or relations between relations.
- Quantization and Cohomology (Week 19) (Apr 4, 2007)
- Finding critical points of the action in a general context: a smooth category equipped with a smooth ‘action’ functor.
- This Week's Finds in Mathematical Physics (Week 248) (Mar 29, 2007)
- Watch movies of coronal mass ejections, auroras, and tornados on the Sun! Then: the Tale of Groupoidification continues.
- Relativity on the World-Wide Web (Mar 28, 2007)
- Chris Hillman is back with a new improved guide to learning relativity!
- This Week's Finds in Mathematical Physics (Week 247) (Mar 23, 2007)
- Symmetry: from the ancient Greeks through medieval Islam to E
_{8}and groupoidification. - News about E
_{8}(Mar 19, 2007) - A huge new calculation involving the exceptional Lie group E
_{8}. - Quantum Logic, Topology and Categories at Oxford (Mar 16, 2007)
- Conferences on categories, logic, computation and topology at Oxford this summer.
- Classical vs Quantum Computation (Week 18) (Mar 15, 2007)
- 2-categories of computation, and an intriguing relation to catastrophe theory.
- Quantization and Cohomology (Week 18) (Mar 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.
- Snowglobe Models (Mar 10, 2007)
- Constructing strange ‘nonstandard models’ of typed λ-calculi.
- We're Under Attack! (Mar 9, 2007)
- We’re experiencing problems due to a trackback spambot.
- Gurski on Tricategories (Mar 9, 2007)
- If your work takes you into the land of weak 3-categories, you’ll need to read this.
- Classical vs Quantum Computation (Week 17) (Mar 9, 2007)
- String worldsheets illustrating processes of computation!
- Quantization and Cohomology (Week 17) (Mar 7, 2007)
- Getting Hilbert spaces and operator algebras from categories.
- A Topos Foundation for Theories of Physics (Mar 3, 2007)
- A major new paper by Isham and Döring.
- Classical vs Quantum Computation (Week 16) (Mar 2, 2007)
- The λ-calculus for… high school calculus!
- Quantization and Cohomology (Week 16) (Feb 27, 2007)
- More examples of path-integral quantization: the particle in a potential.
- This Week's Finds in Mathematical Physics (Week 246) (Feb 25, 2007)
- The trouble with fundamental physics today.
- How to Write Mathematics Badly (Feb 23, 2007)
- A hilarious video.
- Classical vs Quantum Computation (Week 15) (Feb 23, 2007)
- The λ-theory of commutative rings.
- An Introduction to Algebraic Topology (Feb 21, 2007)
- Want to learn a little algebraic topology?
- Quantization and Cohomology (Week 15) (Feb 21, 2007)
- An example of path integral quantization: the free particle on a line (continued).
- Classical vs Quantum Computation (Week 14) (Feb 16, 2007)
- From typed lambda-calculus to cartesian closed categories and back.
- Quantization and Cohomology (Week 14) (Feb 15, 2007)
- An example of path integral quantization: the free particle on a line.
- This Week's Finds in Mathematical Physics (Week 245) (Feb 12, 2007)
- Read about Fields Institute workshop on Higher Categories and Their Applications.
- Classical vs Quantum Computation (Week 13) (Feb 9, 2007)
- Lambek and Scott’s definition of a typed lambda-calculus.
- Infinitely Categorified Calculus (Feb 9, 2007)
- In noncommutative geometry, we must infinitely categorify the calculus!
- Quantization and Cohomology (Week 13) (Feb 6, 2007)
- Statistical mechanics and temperature-dependent mathematics.
- This Week's Finds in Mathematical Physics (Week 244) (Feb 3, 2007)
- How to measure the size of a category.
- Classical vs Quantum Computation (Week 12) (Feb 1, 2007)
- How to see computation as a process: rewrite rules.
- In Memory of Max Kelly (Feb 1, 2007)
- Letters in appreciation of Max Kelly.
- Quantization and Cohomology (Week 12) (Jan 31, 2007)
- Classical mechanics, quantum mechanics and statistical mechanics as ‘matrix mechanics’ over various rigs (ri
**n**gs without**n**negatives). - Classical vs Quantum Computation (Week 11) (Jan 25, 2007)
- Lots of examples of 2-categories!
- Classical vs Quantum Computation (Week 10) (Jan 24, 2007)
- Categorifying the concept of ‘category’!
- Quantization and Cohomology (Week 11) (Jan 24, 2007)
- What’s really going on with quantization?
- Higher Categories and Their Applications (Jan 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. - Animals That Are Also Verbs (Jan 21, 2007)
- A fly can fly. A duck can duck. And a dog can dog you.
- Traces in Ottawa (Jan 19, 2007)
- A workshop on applications of traces to algebra, analysis and categorical logic.
- Knot Homology in Faro (Jan 19, 2007)
- This year’s Oporto Meeting Geometry, Topology and Physics will actually be held in Faro, The theme is knot homology!
- Multiplicative Intuitionistic Linear Logic (Jan 18, 2007)
- Translating from how logicians think about mulitplicative intuitionistic linear logic to how category theorists might think about it.
- Quantization and Cohomology (Week 10) (Jan 17, 2007)
- Classical versus quantum mechanics: the Lagrangian and Hamiltonian approaches
- Classical vs Quantum Computation (Week 9) (Jan 5, 2007)
- Now we’ll categorify the concept of ‘category’ to see computation as a process!
- Puzzle #11 (Dec 31, 2006)
- When was the toothpick invented?
- This Week's Finds in Mathematical Physics (Week 243) (Dec 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…
- The Earliest Stars? (Dec 25, 2006)
- Have people caught sight of the earliest stars in the Universe?
- arXiv Policy Statement? (Dec 24, 2006)
- Where is the arXiv’s official policy statement about guaranteed open access?
- Puzzle #9 (Dec 24, 2006)
- Which philosopher is also known as the RaMBaM?
- Lectures on Classical Mechanics (Dec 21, 2006)
- Lectures notes on classical mechanics - now available in printed form!
- Quantization and Cohomology (Week 9) (Dec 21, 2006)
- A glimpse of geometric quantization.
- This Week's Finds in Mathematical Physics (Week 242) (Dec 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.
- Higher Categories at the Fields Institute (Dec 14, 2006)
- Schedule of talks at the Fields Institute workshop on “Higher Categories and Their Applications”
- New Light Particle(s) Discovered? (Dec 7, 2006)
- Hints of particles with masses near 7 and 19 MeV?
- Guiraud on Higher-Dimensional Rewrite Rules (Dec 6, 2006)
- The three dimensions of proof.
- Classical vs Quantum Computation (Week 8) (Dec 1, 2006)
- How Curry’s Fixed Point Theorem is related to Cantor’s diagonal argument.
- Quantization and Cohomology (Week 8) (Dec 1, 2006)
- From particles to membranes, continued.
- Puzzle #8 (Nov 30, 2006)
- Which mathematician was rumored in 1999 to be secretly in charge of one of the world’s largest countries?
- Puzzle #7 (Nov 25, 2006)
- Which bird can sleep with half its brain while the other half stays awake?
- A Third Model of the String Lie 2-Algebra (Nov 23, 2006)
- Wagemann gives a third construction of the one-parameter family of Lie 2-algebras associated to any simple Lie algebra.
- Classical vs Quantum Computation (Week 7) (Nov 21, 2006)
- Building a computer using the lambda calculus: the Fixed Point Theorem.
- This Week's Finds in Mathematical Physics (Week 241) (Nov 20, 2006)
- Gravitational wave detection in Louisiana, higher gauge theory and the dodecahedron.
- Classical vs Quantum Computation (Week 6) (Nov 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).
- Quantization and Cohomology (Week 7) (Nov 15, 2006)
- Generalizing classical mechanics from particles to strings and higher-dimensional membranes.
- Higher Gauge Theory (Nov 15, 2006)
- Transparencies for a talk on higher gauge theory.
- Breen on Gerbes and 2-Gerbes (Nov 13, 2006)
- A new expository paper.
- Quantization and Cohomology (Week 6) (Nov 13, 2006)
- The canonical 1-form, extended phase space and Rovelli’s covariant formulation of classical mechanics.
- Quantization and Cohomology (Week 5) (Nov 13, 2006)
- The canonical 1-form on the cotangent bundle of a manifold, and what it does for classical mechanics.
- Puzzle #6 (Nov 13, 2006)
- Which famous buildings were named after a form of food - or was it the other way around?
- Tales of the Dodecahedron (Nov 12, 2006)
- Slides from a talk on the history of the dodecahedron, from Pythagoras through Plato to Poincaré.
- Quantization and Cohomology (Week 4) (Nov 8, 2006)
- Hamiltonian dynamics and symplectic geometry.
- Quantization and Cohomology (Week 3) (Nov 8, 2006)
- From Lagrangian to Hamiltonian dynamics.
- Puzzle #5 (Nov 5, 2006)
- When was the Roman empire sold, and who bought it?
- Classical vs Quantum Computation (Week 5) (Nov 2, 2006)
- The naturality of currying - and a new "bubble" notation for currying and uncurrying.
- A Categorical Manifesto (Nov 2, 2006)
- Why are categories important in computer science?
- Puzzle #4 (Oct 30, 2006)
- Such a brivla, built from the rafsi for the component gismu and cmavo, is called a lujvo.
- Classical vs Quantum Computation (Week 4) (Oct 26, 2006)
- Currying and uncurrying, evaluation and coevaluation: basic aspects of the "quantum lambda-calculus".
- Lauda on Topological Field Theory and Tangle Homology (Oct 25, 2006)
- Aaron Lauda’s thesis on topological field theory and tangle homology.
- Puzzle #3 (Oct 25, 2006)
- A special law to prevent votes from being counted.
- This Week's Finds in Mathematical Physics (Week 240) (Oct 23, 2006)
- Read about Dolan and Trimble’s work on cartesian closed categories and
*holodeck games!* - Classical vs Quantum Computation (Week 3) (Oct 20, 2006)
- Guest lecture by James Dolan: Holodeck strategies and cartesian closed categories.
- A Course on Classical Mechanics (Oct 19, 2006)
- A course on classical mechanics from a mathematical viewpoint.
- Australian Category Theory (Oct 18, 2006)
- What’s happening at the Australian Category Seminar.
- Classical vs Quantum Computation (Week 2) (Oct 17, 2006)
- An intro to closed monoidal categories.
- Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection (Oct 11, 2006)
- A question from Bruce Bartlett
- Quantization and Cohomology (Week 2) (Oct 11, 2006)
- The Lagrangian approach to classical mechanics
- Zooming Out in Time (Oct 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.
- Classical vs Quantum Computation (Week 1) (Oct 6, 2006)
- Types and operations; categories as theories; monoidal categories versus categories with finite products.
- Quantization and Cohomology (Week 1) (Oct 5, 2006)
- How the dynamics of p-branes resembles the statics of (p+1)-branes.
*New Scientist*Reacts! (Oct 5, 2006)*New Scientist*now has a blog entry defending their article on Shawyer’s electromagnetic drive.- Trimble's Definition of Tetracategory (Oct 5, 2006)
- Todd Trimble has redrawn his legendary lost definition of tetracategories.
- λ-Calculus in Paris (Oct 4, 2006)
- Typed Lambda Calculi and Applications - a conference in Paris, June 26-28, 2007
- Classical vs Quantum Computation (Week 1) (Oct 3, 2006)
- Computation, the lambda calculus and cartesian closed categories: an overview.
- Voevodsky on the Homotopy λ-Calculus (Oct 2, 2006)
- Voevodsky has released a “very short note on the homotopy λ-calculus”.
- Topos Theory at Chicago (Oct 2, 2006)
- Lectures on topos theory at Chicago
- Puzzle #2 (Oct 1, 2006)
- A spooky link between Tycho Brahe and Shakespeare.
- Dimensional Analysis and Coordinate Systems (Sep 30, 2006)
- More gnarly issues: dimensional analysis and coordinate systems.
- Dimensional Analysis (Sep 21, 2006)
- Let’s amass a collection of wisdom on gnarly issues in physics - starting with dimensional analysis.
- A Plea to Save
*New Scientist*(Sep 19, 2006) - The SF writer Greg Egan has issued a plea to save the magazine New Scientist.
- Higher Categories and their Applications (Sep 13, 2006)
- In January 2007, the Fields Institute is having a workshop on Higher Categories and their Applications.
- Groupoids and Stacks in Physics and Geometry (Sep 13, 2006)
- In 2007, the Institut Henri Poincaré will be running a program on Groupoids and Stacks in Physics and Geometry.
- This Week's Finds in Mathematical Physics (Week 239) (Sep 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! - Doctrines (Sep 3, 2006)
- In 1969 Lawvere invented “doctrines”. A doctrine is a roughly a kind of category, thought of as a kind of logical theory.
- TeXnical Issues (Sep 2, 2006)
- Having trouble reading the equations on this blog, or posting comments? Try this.
- Ringoids (Sep 2, 2006)
- A group with many objects is a groupoid. A ring with many objects is a ringoid!
- Letter from Grothendieck (Aug 31, 2006)
- Newly released letter from Alexander Grothendieck.
- Quantum Computation and Symmetric Monoidal Categories (Aug 28, 2006)
- Ordinary computer scientists like cartesian closed categories. Quantum computer scientists like symmetric monoidal categories with duals.
- Categorifying CCCs: Computation as a Process (Aug 28, 2006)
- “Cartesian closed n-categories” may help us see computation as a process.
- CCCs and the λ-calculus (Aug 28, 2006)
- Let’s discuss cartesian closed categories and the λ-calculus!
- Categories and Computation (Aug 24, 2006)
- What’s the relation between the lambda-calculus, computation and CCCs?
- Categorifying the Dijkgraaf-Witten model (Aug 24, 2006)
- Martins and Porter categorify the Dijkgraaf-Witten model.
- Puzzle #1 (Aug 23, 2006)
- What was Uncle Sam’s last name?
- Lectures on n-Categories and Cohomology (Aug 22, 2006)
- Lectures on n-Categories and Cohomology, by John Baez and Michael Shulman
- Dark Matter in the Bullet Cluster (Aug 17, 2006)
- Hard evidence for dark matter in the Bullet Cluster

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