Well, the Carnival of Mathematics (by my count) is turning 21. I know it’s quite a party here, but I’m not allowed to put images of the blog’s secret lair on the internet, so you’ll have to imagine an appropriate level of debauchery.
To start things off, we’ll have a selection from everyone’s favorite math-flavored webcomic, xkcd:
Odd fact of the day: not only does Randall Munroe write an awesome webcomic, he also has a ballpit in his living room.
Lieven LeBruyn posts some preliminary but interesting looking stuff about a series of finite simple groups which calls Inguanodon series of simple groups (to see the guts of his construction, see the inguanodon dissected). The idea that the sporadic simple groups should fit into infinite series of something has been kicked around by various authors of this blog, but from a somewhat different perspective. LeBruyn seems to be working on the idea that there should be lots of different series of finite simple groups, which start overlapping with the classical series for large values, but include the sporadic groups in their small values.
Math for Mortals uses A problem courtesy of Shakespeare (which doesn’t match too well with my recollection of the Merchant of Venice, but let’s not quibble) as a jumping off point for a discussion of basic logic. However, I think the lesson that using logic can help you avoid execution is a good one.
In case you haven’t been keeping up with the latest on the arXivotubes, the curiously-named 11011110 obligingly catches us up on Six recent arXiv papers. I’ll let him do the summarizing, but will mention that a lot of wacky combinatorics was involved. I think if everyone would write a little summary of the arXiv papers they had read and understood for the benefit of humanity, the internet (and the arXiv) would be a better place. Too bad I’m too lazy to do so.
Math Less Travelled does a 4-part post on Perfect numbers, part I (and part II and part III, with a challenge in between). It’s a pretty elementary subject, but one that people have been unable unlock all the mysteries of centuries later. Funnily enough, one his suggested challenges isn’t finding the first odd perfect number.
JD2718 has another challenge, entitled McRib. I think it might be a bit easier than that odd perfect number bit.
Vlorbik pretends to be confused about why this post was his most popular in months. I think it might have something to do with the fact that it’s entirely about Legos. Of particular interest (though I bet a lot of you have seen it before) is Andrew Lipson’s Mathematical LEGO Sculptures. Yes, it is as awesome as it sounds.
360 gives a couple of more fun posts for the holiday season: one on the ties between Tryptophan and Game Theory (hint: if you were planning on challenging friends and family to a friendly round of Prisoner’s Dilemma, you may have missed your chance), and one on The Rubik’s Hypercube. As if the original wasn’t hard enough.
Blake Stacey complains a bit about Baggage, and the mathematical inconsistencies of airline baggage rules. I wish I could say that that was prime amongst MY travel worries.
Mikael Vejdemo Johansson gives a command (at my command, in fact!) performance at the Infinity Seminar explaining a bit about The Computation of -Structures. It was a helpful post (for me), but I still only like computing -structures when they’re trivial. At his own blog, he tells us about Wreath Products (which sort of sounds like it might be Christmas-related, but thankfully, is not).
Jeffrey Shalit has done something that I hope we’ll see quite a bit more of in the future: blogging on the work of one of his students. He gives a down to earth description of some of Narad Rampersad’s Work on Combinatorics on Words. I’ll admit, I don’t think I’ve ever thought much about the combinatorics of infinite words, but it looks pretty cool now.
Gary Laden ruminates for a bit on the question: Is There a Black Box in your Research Methodology? It’s not an issue for most of us, but it will probably start to loom as more mathematicians integrate computers into their research. I think it is just one more important reason to lean toward open source software, but on the other hand, I doubt any of us will be giving up Mathematica any time soon (at least until omath is actually functional).
Larry Ferlazzo points us to a website containing clips of math appearing in movies (loosely interpreted) in Math Movies And More. Never again will a group of mathematicians have to rent “It’s My Turn” just to watch the proof of the snake lemma in the first 5 minutes. Not that anyone I know has ever done that. The Monty Python clip of the international philosophy game (Greece vs. Germany) come highly recommended.
Reasonable Deviation has a very interesting, though pretty sketchy post on Smooth Sorting. Basically, someone has cooked up a system of ODEs which will sort numbers by just letting time run. If only someone would now write a post explaining why it works. Well, maybe next carnival. Anyways, definitely the winner of the coolest video award for this installment.
Mark Dominus reminds us in Lazy square roots of power series that it’s easy to say “just take the square root of this” and another thing entirely to convince a computer to do it for you. I can’t comment on the quality of his code, but I applaud the effort.
meeyauw treats us to a voluminous educational post on Boxes Without Topses and Pentominoes. It sounds like good activity to try with Tetris-loving schoolchildren (is there any other kind?).
Of course, what would a blog carnival be without the Fields medalist section? When looking for a good post from Terry Tao, of course, the carnivalist is spoiled for choice. One of the more interesting selections lately was On the property testing of hereditary graph and hypergraph properties. (As you may note, this post also played on my school spirit [institute spirit?] by name-checking by current intellectual home). As many of you know, there are a lot of interesting properties of graphs which is rather hard to check with 100% reliability, but it turns out, there some rather interesting things to say about more probabilistic tests.
For those of you looking for deep discussion of various innocuous seeming bits of linear algebra, look no further than Tim Gowers’s post on The exchange lemma and Gaussian elimination. Linear algebra may not sound like the most scintillating topic, but if Tim discovered facts about linear algebra that surprised him, it probably worth a look from the rest of us.
On his blog, Alain Connes gives (the first part of) his view of The heart of non-commutative geometry and explains what he means by “noncommutative spaces generate their own time.” Make sure you scroll down to the discussion between Connes and Urs Schreiber in the comment section. Whatever issues comments have, they do let us listen in on a lot of very interesting discussions.
John Armstrong (definitely not a Fields medalist) provides us a discussion of Faulhaber’s Famous Formula. Those of you who who haven’t encountered its fame, might learn a little something.
The n-Category Café marches on with blogging John Baez and Jim Dolan’s Geometric Representation Theory course. (The link is to lecture 12). This particular lecture is somewhat dear to my own heart because it involves the connection between braid groups and flag varieties, the multitudinous variations of which are among my favorite stories in mathematics. There’s a lot of exciting connections there they had no time for in this lecture, but I still recommend watching.
If that wasn’t enough about the symmetric group for you, have a look at Isabel’s post on Pattern Avoidance (and another) in the symmetric group. Any of you who like geometry better than combinatorics might still want to take a look. She doesn’t say anything about in the post, but in fact, the pattern avoidance of a permutation is intimately connected with the geometry of the corresponding Schubert varieties (see, for example, this paper and references therein).
Well, I hope you enjoyed the Carnival as much as I did. Start getting ready for the next installment at Wild about Math on December 14th, and keep on bloggin’.