I found two interesting online resources on the Foundations of Mathematics:
- What is Mathematics: Gödel’s Theorem and Around by Karlis Podnieks
- Foundations of Mathematics by Alexander Sakharov
Other posts on “Math is hard”
I’d wish everyone a happy new year, but clearly the mathematically inclined all refuse to recognize arbitrary time divisions.
I’ve been tipped, via e-mail, to some other weblog discussions inspired by Math is hard:
Math is hard
Math is hard. Discuss.
Tim Chow
In the comments to this post, David MacIver provides an alternative, registration-free link to Tim Chow’s You Could Have Invented Spectral Sequences.
I poked around Tim Chow‘s site, and found two other interesting articles (in the form of old sci.math.research posts):
- Forcing for Dummies. Forcing is the technique invented by Paul Cohen to prove the independence of the Continuum Hypothesis.
- What is Class Field Theory? Class field theory describes the extensions of a field with abelian Galois group.
Baez Week 224
Week 224 of This Week’s Finds in Mathematical Physics is up.
Mathematical Reviews
The American Mathematical Society produces, at great expense, Mathematical Reveiws, which provides a capsule summary of every paper published in a mathematics journal. Here’s something that just occurred to me today: does anyone actually use these reviews? I’ve used MR as a bibliography of papers by a particular author, but I can’t say I’ve ever read one of the reviews out of anything but idle curiousity. Does anyone else rely on this feature of MR?
Update. Apparently, it’s just me. I’ve been informed via email that everyone uses MR reviews.
January Notices
The January Notices of the AMS is available. It features an article with the intriguing title You Could Have Invented Spectral Sequences.
Update. I’ve finally found the time to look at this article, and it is the simplest introduction to the subject I’ve ever seen. It acheives its simplicity by concentrating on the special case of the spectral sequence for filtered chain complexes.
Alg-top in CS
In a post below, I mentioned algebraic topology in computer science. A nice application of alg-top is for study of concurrency in distributed systems. For instance, one approach is to consider execution traces of a computational system being represented by time-directed paths through a space, and then to use alg-top methods to ask and answer questions about the structure of this space.
This approach leads naturally to a concept of homotopies of paths, equivalence classes of paths which may be transformed into one another via other paths in the space. What is different from traditional homotopy theory is that the paths are directed, and so these are referred to as directed homotopies or dihomotopies. Paths which are not dihomotopically equivalent represent execution traces on which there are events not reachable from one to another. For more on this, start with the GETCO conference pages.
+Plus 37
Plus Magazine is an e-zine produced in Britain and aimed at popularizing mathematics among school students. The latest issue (number 37) is now out.
New uses for Grothendieck topologies
I’ve been saying for a while that the big problems in computer science (eg, P vs NP; a theory of distributed systems; effective GO-playing machines; etc) need radical new methods, and I have suggested algebraic topology as a likely source of ideas. Joel Friedman has just applied some alg-top to boolean complexity, motivated by the P vs. NP problem, in Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity.