June 30, 2009
Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture
Posted by John Baez
The ‘big three’ science publishers — Elsevier, Springer, and Wiley–Blackwell — like to argue that their high prices pay for high quality. Recent events cast a tinge of doubt on this. We all know about the case of El Naschie, and Elsevier’s fake medical journals. Now Springer has published a book that purports to contain elementary proofs of Fermat’s Last Theorem and Goldbach’s Conjecture!
Here it is:
- Nico F. Benschop, Associative Digital Network Theory: An Associative Algebra Approach to Logic, Arithmetic and State Machines, Springer Verlag, 2009. ISBN: 978-1-4020-9828-4.
June 26, 2009
This Book Needs a Title
Posted by John Baez
You may be familiar with Raymond Smullyan’s delightful books packed with puzzles and paradoxes. One of them — not my favorite — is called This Book Needs No Title.
Peter May and I are almost done editing a book that’s quite the opposite. It does need a title.
Background Essays Towards Higher Category Theory would be an accurate description, but it’s not very snappy. Towards Higher Category Theory is overly ambitious. Can you think of something better?
To help you dream up an appropriate title, here’s a draft of the preface. And if you spot mistakes in this preface, I’d like to hear about them. (The bibliography makes no pretensions to completeness, so surely many people will be offended by how we have neglected their work. If you’re one of those people, I apologize.)
Cohomology and Homotopy
Posted by David Corfield
In posts and this $n$Lab entry, Urs has been promoting his view of cohomology as about Hom spaces between objects in certain settings, where the unknown space is on the left. Similarly homotopy is where the unknown space is on the right. This got me thinking the following thoughts during some quiet moments in a conference this morning.
June 20, 2009
Hopf Algebras in Luxembourg
Posted by John Baez
Hopf algebras lie at a fascinating intersection of combinatorics, quantum physics, topology and category theory. Everyone should learn about them. Luckily, there is still some funding for students and young researchers to attend this conference and school on Hopf algebras:
- Hopf-in-Lux: School-Conference on Hopf algebras, July 13-17, 2009, University of Luxembourg, organized by Said Benayadi, Philippe Bonneau, Yael Fregier and Martin Schlichenmaier.
This Week’s Finds in Mathematical Physics (Week 276)
Posted by John Baez
In week276 of This Week’s Finds, hear the shocking news about this star:
Read about the Local Bubble, the Loop I Bubble, the cloudlets from Sco-Gen, and the “local fluff”. Come visit the $n$Lab! 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.
Kan Lifts
Posted by David Corfield
I’ve been thinking more about organising principles operating in mathematics. I remember Steenrod wrote a very illuminating sketch of algebraic topology in terms of extensions and lifts, which I can’t now retrieve. That got me wondering, if with Mac Lane we say
The notion of Kan extensions subsumes all the other fundamental concepts of category theory,
whatever happened to Kan lifts?
June 16, 2009
Accessible Even to a Philosopher
Posted by David Corfield
Edward Frenkel has a paper out today – Gauge Theory and Langlands Duality – which sets out from André’s Weil’s letter to his sister.
This is a remarkable document, in which Weil tries to explain, in fairly elementary terms (presumably, accessible even to a philosopher), the “big picture” of mathematics, the way he saw it. I think this sets a great example to follow for all of us.
Martin Krieger provided a translation of the letter. As for its accessibility, I can say that it did inspire chapter 4 of my book. Let’s hope Frenkel’s paper can also be inspirational. At first glance, however, it looks tough going.
June 15, 2009
2-Branes in 11 Dimensions
Posted by John Baez
I’m trying to learn a teeny bit more about supersymmetric membrane theories, and I’m so far behind that this old review article is proving helpful:
- Micheal J. Duff, Supermembranes: the first fifteen weeks, Classical and Quantum Gravity 5 (1988), 189–205.
I’m particularly fascinated by the classification of ‘fundamental super $p$-branes’ and its relation to normed division algebras:
- Reals: 2-branes in 4 dimensions, with 1 bosonic and 1 fermionic degree of freedom.
- Complexes: 3-branes in 6 dimensions, with 2 bosonic and 2-fermionic degrees of freedom.
- Quaternions: 5-branes in 10 dimensions, with 4 bosonic and 4 fermionic degrees of freedom.
- Octonions: 2-branes in 11 dimensions, with 8 bosonic and 8 fermionic degrees of freedom.
I talked about this ‘brane scan’ in incredibly elementary terms back in week118, but now I’d like to actually understand it. It’s supposed to follow from something like a classification of closed differential forms on super-Minkowski spacetimes, due to Achúcarro et al. I don’t know how it goes, but it seems potentially quite comprehensible.
If anyone can help me with this, I’d appreciate it a lot. But I really want to say a word about the 11-dimensional case — a brief explanation for complete novices such as myself — and then ask a question about that.
June 14, 2009
This Week’s Finds in Mathematical Physics (Week 275)
Posted by John Baez
In week275 of This Week’s Finds, read about progress towards proving the Cobordism Hypothesis.
June 10, 2009
Final Exams Again
Posted by John Baez
I’m busy grading final exams for my undergraduate number theory class. The class went quite well — perhaps because instead of proving quadratic reciprocity, I spent time teaching them about arithmetic functions, Dirichlet convolution, Möbius inversion and the like… topics which lead to lots of fun puzzles and computations.
Nonetheless, grading finals is always mind-numbing and dispiriting. I’m sure you’ve seen it — perfectly intelligent people grading finals, trading the most mean-spirited and witless of witticisms just to keep from going insane.
In that spirit, let me report three mildly amusing things I’ve seen so far. Don’t get your hopes up — they’re not nearly as funny as the proof of the infinitude of primes that I described last time I taught this class.
Indeed, I’m sure some of you have seen funnier final exams this year. If so, tell us about ‘em!
June 8, 2009
Strings, Fields, Topology in Oberwolfach
Posted by Urs Schreiber
This week’s workshop at MFO is on Strings, Fields, Topology. We started collecting notes and other material at
Oberwolfach Workshop, June 2009 – Strings, Fields Topology .
This includes today
- Christoph Schweigert and Ingo Runkel on [[CFT]] and algebra in modular tensor categories;
- Dan Freed on [[differential cohomology]] of [[string theory]] [[background fields]];
- Kevin Costello on quantum field theory in terms of [[factorization algebras]].
Algebraic Geometry for Category Theorists
Posted by John Baez
James Dolan and I have spent the last year or so talking about algebraic geometry, trying to learn the basics.
Algebraic geometry should be a lot of fun for category theorists — after all, this is the subject made Grothendieck invent topos theory! But alas, introductions to topos theory don’t seem to explain much about algebraic geometry, and introductions to algebraic geometry don’t seem to fully embrace topos theory. It seems that Grothendieck’s revolution never fully caught on. And that’s sort of sad.
June 6, 2009
Nonabelian Algebraic Topology
Posted by John Baez
Here it is — the magnum opus of cubical methods in algebraic topology:
- Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian algebraic topology: homotopy groupoids and filtered spaces.
It’s still just a preliminary version, but it’s 516 pages long, and packed with good stuff — and it’s free!
Grab a copy! I’m sure the authors will be grateful if you catch typos and other problems.
June 5, 2009
The Elusive Proteus
Posted by David Corfield
Cassirer writes in Determinism and Indeterminism in Modern Physics:
At first glance the universality of the action principle seems by no means beyond question. This universality could be attained only at the cost of a circumstance which, from the purely physical point of view, led again and again to difficulties and doubts. For the more universally the principle was conceived, the more difficult it became to specify clearly its proper concrete content. It becomes finally a kind of Proteus, displaying a new aspect on each new level of scientific knowledge. If we ask what precisely that “something” might be to which the property of a maximum or a minimum is ascribed, we receive no definite and unambiguous answer. (p. 50)
June 3, 2009
Mathematical Principles
Posted by David Corfield
I’ve been reading several works by Ernst Cassirer of late. In his Determinism and Indeterminism in Modern Physics, Yale University Press, 1956, (translation by O. Benfrey of ‘Determinismus und Indeterminismus in der modernen Physik’, 1936) he discusses the multi-levelled nature of physics: laws encompass measurements, and principles encompass laws.
Here in fact we find a basic methodological characteristic common to all genuine statements of principles. Principles do not stand on the same level as laws, for the latter are statements concerning specific concrete phenomena. Principles are not themselves laws, but rules for seeking and finding laws. This heuristic point of view applies to all principles. They set out from the presupposition of certain common determinations valid for all natural phenomena and ask whether in the specialized disciplines one finds something corresponding to these determinations, and how this “something” is to be defined in particular cases.
The power and value of physical principles consists in this capacity for “synopsis,” for a comprehensive view of whole domains of reality… Principles are invariably bold anticipations that justify themselves in what they accomplish by way of construction and inner organization of our total knowledge. They refer not directly to phenomena but to the form of the laws according to which we order these phenomena. A genuine principle, therefore, is not equivalent to a natural law. It is rather the birthplace of natural laws, a matrix as it were, out of which new natural laws may be born again and again. (pp. 52-53)
Journal Club — Geometric Infinity-Function Theory — Week 6
Posted by John Baez
guest post by Alex Hoffnung
Hi
This section is broken into two short parts. I will try to say a little about the first here and then head over to the $n$Lab to say more soon.
June 2, 2009
Treq Lila
Posted by John Baez
I should have been finishing a paper, but instead I finished an album.