Ars Mathematica
Dedicated to the mathematical arts.
Rigid Analytic Geometry
Wikipedia’s article on rigid analytic geometry links to an interesting survey paper by Brian Conrad on the subject. Rigid analytic geometry is the attempt to translate the theory of complex analytic geometry to the p-adics. The theory is surprisin...
~ published: Wednesday at 23:40 ~ permalink
Groups of groups
You know, mathematical terminology cannot be parodied. Mathematicians have invented groups, semigroups, quasigroups, pseudogroups, and two mostly-unrelated concepts both known as groupoids. They have invented both formal groups and quantum groups, neith...
~ published: 08/22 at 23:19 ~ permalink
Bombers Do What Euler Could Not
Continuing the architectural theme, Isabel at God Plays Dice has a post on the ultimate fate of the real world Königsberg bridge problem. Königsberg had seven bridges, and in 1736 Euler proved it was impossible to find a path that allowed you to cross ...
~ published: 08/18 at 00:51 ~ permalink
Falkirk Wheel
Now that this is primarily an architecture blog, here’s Falkirk Wheel, a science-fiction-looking rotating boat elevator in Scotland.Actual math content soon....
~ published: 08/14 at 15:22 ~ permalink
Perfect Groups Viewed Topologically
A. J. Berrick has an interesting paper explaining how a topologist thinks about group theory. Topology and group theory are connected throught the fundamental group. For every group, topologists can construct a space with that group as its fundamental g...
~ published: 08/04 at 20:17 ~ permalink
Hibert
I have just learned from multiple Google searches that a) apparently I think Hilbert is spelled Hibert (with no l), but b) Google is smart enough to correct me. Next year I’m letting Google do my taxes....
~ published: 07/30 at 10:22 ~ permalink
Out of Print Math Books
There is a new site, outofprintmath, that is collecting information on which out-of-print books the mathematical reading public would like to see brought back in print. At the moment there are 67 books listed, many of them are surprisingly well-known. I...
~ published: 07/26 at 23:34 ~ permalink
Representations of GL(n)
David Speyer gives a nice introduction to the representations of GL(n) at the Secret Blogging Seminar....
~ published: 07/24 at 22:54 ~ permalink
Elementary Proof of Hairy Ball Theorem
Todd at Topological Musings has posted an elementary proof of the Hairy Ball Theorem: the theorem that all vector fields on a even-dimensional sphere must vanish somewhere. The elementary proof is by Milnor....
~ published: 07/23 at 12:49 ~ permalink
Topologically Non-Trivial Highway
The intersection of two interstate highways, I-95 and I-695 near Baltimore, is topologically non-trivial; it features a non-trivial braiding. Unfortunately, the interchange is scheduled to be redesigned.Via Low-Dimensional Topology....
~ published: 07/17 at 13:23 ~ permalink
Clay Mathematics Institute Library
The Clay Mathematics Institute has placed their library of publications online. Their most high-profile publication (other than the Millennium Problems) is Morgan and Tian’s write-up of the proof of the Poincare Conjecture. They have an interestin...
~ published: 07/14 at 03:50 ~ permalink
Intute
A group of UK universities have put together a database of links to online resources in various academic subjects, called Intute. Their mathematics section is particularly impressive. (They’ve already linked to almost every online math book I can ...
~ published: 07/13 at 21:42 ~ permalink
36th Carnival of Mathematics
Rigorous Trivialities is hosting the 36th Carnival of Mathematics.The blog also has a long running series expounding the basics of algebraic geometry. The latest post covers blowing up....
~ published: 07/11 at 13:44 ~ permalink
Wanted: Theorem about Cocomplete Categories
I’m pretty sure that a certain theorem about cocomplete categories must be true, and I’m even pretty sure that I know how to write down a proof. (Famous last words, I know.) But I have the feeling that the result is already known, and I just...
~ published: 07/05 at 15:38 ~ permalink
Li’s Preprint
Yesterday, everyone was all atwitter over a new preprint by Xian-Jin Li containing a purported proof the Riemann Hypothesis. The optics of it looked good (Li is clearly not a crank), but Terry Tao has identified an apparent error.More at Not Even Wrong....
~ published: 07/03 at 12:04 ~ permalink
Wells on Mathematical Language
Charles Wells, author (with Michael Barr) of Toposes, Triples, and Theories, now has a blog, gyre and gimble, devoted to how mathematicians use language.He notes that the idea of completed infinity, which mathematicians take for granted, is still not well...
~ published: 06/27 at 21:11 ~ permalink
The Kingston University Scandal
I’d rather post about math, but gossip is so much easier for a novice WordPress user! And this one is a doozy.The scandal in question is still growing and has already received much notoriety, with stories appearing in the BBC, The Times, and even ...
~ published: 06/25 at 16:16 ~ permalink
Nilpotent Infinitesimals II
This is a follow-up to this post.Nilpotent infinitesimals allow you to define objects like the “double point”, which is the solution set of x2 = 0 on the line. Intuitively, the double point is the point x = 0, plus another point infinitesimal...
~ published: 06/25 at 01:15 ~ permalink
Complex Cobordism and the Stable Homotopy of Spheres
Doug Ravenal has made the latest version of his book Complex Cobordism and the Stable Homotopy of Spheres available online....
~ published: 06/24 at 19:27 ~ permalink
Non-standard analysis in economics
I see, via Yet Another Sheep, that nonstandard analysis has spread to mathematical economics. Robert Anderson has a book manuscript available, Infinitesimal Methods in Mathematical Economics which explains how to apply nonstandard analysis to approximate...
~ published: 06/23 at 23:59 ~ permalink
Dimensions movie
Jos Leys emailed me to let me know about Dimensions, a computer-generated movie illustrating dimensions 2, 3, and 4. The trailer is here. The whole movie can be downloaded for free at dimensions-math.org. (It is also available on DVD.)The movie was ren...
~ published: 06/20 at 18:58 ~ permalink
Brouwer Fixed Point Theorem
One idiosyncratic interest of mine is mathematical economics. I was looking through Volume 2 of the Handbook of Mathematical Economics when I spotted a paper by Scarf called “The Computation of Equilibrium Prices: An Exposition”. The real su...
~ published: 06/17 at 21:01 ~ permalink
Universal Family
If you’re trying to look up the term universal family, my recommendation is to not to use the search string “universal family” in Google.Update. In the comments, Matt Heath points out that this post is already in the top ten for “...
~ published: 06/14 at 23:26 ~ permalink
Field With One Element
Lieven LeBruyn has a series of posts about the “field with one element”, here, here, and here. The field with one element does not exist, of course, but Tits pointed out a long time ago that you can think of the symmetric groups as Lie groups...
~ published: 06/12 at 22:34 ~ permalink
Chemistry Video
This video explaining the basics of chemical reactions may be the definitive educational document of the YouTube Era....
~ published: 06/09 at 10:21 ~ permalink
Shoup on Algebra
Victor Shoup has written an introductory (abstract) algebra textbook that emphasizes the algorithmic aspects. The best part? You can download it here.Via God Plays Dice....
~ published: 06/02 at 02:29 ~ permalink
Spam Must Go On. It Will Go On.
I happened to check the spam filter, and discovered that our spam is getting depressed, but has determined to persevere. Today’s spam included this comment:I have become my own version of an optimist. If I can’t make it through one door, I’ll g...
~ published: 06/01 at 17:03 ~ permalink
Nilpotent Infinitesimals I
I’ve been writing a post explaining the practical difference between the synthetic and nonstandard notions of infinitesimals. It was getting a bit long, so I’m splitting it into two posts, of which this is the first.Synthetic differential geo...
~ published: 05/19 at 23:58 ~ permalink
Stolz-Cesaro Theorem
Topological Musings has a post up about a cute little result, the Stolz-Cesaro theorem. The result is a discrete analogue of l’Hôpital’s rule.Via God Plays Dice....
~ published: 05/15 at 22:35 ~ permalink
Perspective
I was looking at the stats for Ars Math, when I saw that we’ve had 525 posts. I thought “Wow, that’s a lot of posts.” Then my eye happened to glance at the count of the number of comments the Akismet plug-in has deleted as spam: ...
~ published: 05/13 at 21:14 ~ permalink
Selling Infinitesimals
J. L. Bell’s A Primer of Infinitesimal Analysis (an intro to synthetic differential geometry) begins with a series of quotes to motivate why we should think of the reals as containing infinitesimals. The quotes all involve the idea that philosophic...
~ published: 05/06 at 22:26 ~ permalink
Drexelmath
I deny the fact that I just went three weeks without posting. Nothing on the Internet can be trusted, even timestamps on blog posts.The math librarian at Drexel University, Peggy Dominy, has a blog. Most of the posts are about acquisitions by DrexelR...
~ published: 04/30 at 20:47 ~ permalink
Groups of Order Sixteen
When I first took abstract algebra, I loved theorems classifying all of the groups of a certain order. Here is a paper I would have loved, The Groups of Order Sixteen Made Easy. Normally, the classification of groups of order 16 is described in terms of...
~ published: 04/08 at 21:40 ~ permalink
Sierpinski carpet
This is a test post to see what’s involved in uploading images.This is of course the Sierpinski carpet. What’s interesting to me is that many objects that, in a previous age dominated by a picture of the physical world as a continuum, seemed ...
~ published: 04/04 at 00:16 ~ permalink
2008 Abel Prize
The 2008 Abel Prize has been announced. This year’s winners are John Thompson and Jacques Tits.Thompson is most famous for his work on the Feit-Thompson theorem, that every group of odd order is solvable. Solvable groups resemble upper triangular ...
~ published: 03/28 at 22:24 ~ permalink
Proof Style
What makes a well-written proof? Who writes proofs well? Discussions of mathematical exposition usually revolve around larger-scale questions, such as how to organize the material, what kinds of examples to use, or how much background is necessary; by a...
~ published: 03/22 at 23:37 ~ permalink
Looting the Library
I promised a while back to write a post describing why so many statistics have a central limit theorem. I went to the library to look up the result I had in mind, to refresh my memory as to the details. The book I wanted was checked out. I thought abou...
~ published: 03/18 at 22:19 ~ permalink
Schwartz-Christoffel Formula for Multiply-Connected Domains
A reader sent me two news articles (here and here) announcing a generalization of the Schwarz-Christoffel mapping in complex analysis. The paper itself is not freely available, but I found this summary from SIAM news that fills out many of the details.Th...
~ published: 03/14 at 22:32 ~ permalink
Representative from Fermilab
Sorry for the light posting; life has been interfering with my blogging schedule.When Scoop Jackson was in Congress, a running joke was that he was the Senator from Boeing, abbreviated Jackson (D-Boeing). Now, Congress has an honest-to-God Representative...
~ published: 03/10 at 23:26 ~ permalink
Cauchy-Schwartz Theorem
While I was driving in my car today, I thought of a proof of the Cauchy-Schwartz theorem. I’m sure that it is completely unoriginal, but it has the advantages of both being longer and requiring more background than the usual proof (which you can fi...
~ published: 02/28 at 21:25 ~ permalink
Your Weakness, Revealed
Peter Woit quotes from a reminiscence by Peter Goddard from a physics conference in 1971:With great technical mastery, he was covering the board with special functions, doing manipulations that I knew from my studies with Alan White (who was also at the S...
~ published: 02/28 at 00:53 ~ permalink
Carleson’s Theorem
I’m intrigued by the beginning of a new series of posts at the Everything Seminar about harmonic analysis. This particular post talks about the relationship of singular integral operators and Carleson’s Theorem. Carleson’s Theorem (tha...
~ published: 02/23 at 22:19 ~ permalink
Reforming the Calculus Class, Permanently
Calculus has been the subject of immense amounts of educational material, ranging from textbooks to blog posts. Unfortunately, that is now all obsolete. The definitive presentation of calculus is here: Calculus, the Musical....
~ published: 02/19 at 23:51 ~ permalink
Dressing to Impress Mathematicians
Brad de Long, an economist, has a post up about the significance of how he dresses for specific audiences. In particular, the consequences of wearing ties:With math-oriented students, however, a tie tells them that I spend too little time thinking about ...
~ published: 02/15 at 21:07 ~ permalink
Noether and Molien’s Theorems
Charles at Rigorous Trivialities has written a post outlining the proof of two pretty theorems from the invariant theory of finite groups: Noether’s theorem that the ring of invariants is finitely generated, and Molien’s formula for the number...
~ published: 02/13 at 21:52 ~ permalink
Statistics Not Sadistic
Not only is John Armstrong a failed crackpot, he is wrong about statistics. Statistics is, from the mathematical point of view, a perfectly interesting subject; this fact is carefully concealed from us by statisticians. For example, most mathematicians ...
~ published: 02/06 at 23:00 ~ permalink
Dummit and Foote panned
Isabel at God Plays Dice finds the definitive review of Dummit and Foote, here at adequacy.org.(If you have any questions as to the objectivity of adequacy.org, I suggest checking out their Wikipedia page.)...
~ published: 02/02 at 21:27 ~ permalink
Notre Dame Mathematical Lectures
The Notre Dame Mathematical Lectures series is now freely available online. It was a fairly small series of lecture notes on various topics. The most famous was probably Emil Artin’s lectures on Galois Theory....
~ published: 01/28 at 20:34 ~ permalink
Four Color Theorem and Lie Algebras
Thanks to Greg Muller, I’m looking at this paper by Dror Bar-Natan that reduces the Four Color Theorem to a plausible statement about Lie algebras. Now we just have to hope this new conjecture does not not require hundreds of pages of computer gene...
~ published: 01/26 at 22:51 ~ permalink
25th Carnival of Mathematics
The 25th Carnival of Mathematics is up at Walking Randomly....
~ published: 01/25 at 20:09 ~ permalink