Ars Mathematica

The investment bank Goldman Sachs is being sued by the SEC for allegedly selling an investment designed to lose money. The investment was built on a pool of mortgages that were likely to go into default. Initially, Erik Gerding at The Conglomerate (a legal blog) thought that the SEC would have difficulty winning the case, since Goldman had disclosed the contents of the pool. Then he had second thoughts, because of this paper, “Computational Complexity and Information Asymmetry in Financial Products”, by Arora, Barak, Brunnermeier, and Ge. The paper shows that even if you know the contents of the pool, detecting whether bad mortgages are hidden in the pool is an NP-complete problem, which is normally considered the hallmark of computational intractability.

In some disciplines, there is the notion of a Comment on a published article, which is what it sounds like: a short comment about the contents of the article (for example, that it’s wrong). Cat Dynamics links to an interesting account of physicist Rick Trebino’s (lengthy but ultimately successful) attempts to publish a Comment explaining why a published article is wrong.

I don’t think I’ve ever seen a Comment in a pure math journal. They’re common in statistics journals.

The New York Times has an online math blog by Steven Strogatz. Given the venue, it is written for an elementary audience. Here is a recent post on the method of exhaustion and how it allows you to approximate π.

The Insane Clown Posse has declared war on scientists. Check out these recent lyrics (warning, curse words ahead):

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When I said that I was about to eat a big bowl of non-blogging, I didn’t mean that I was going to turn off the website, but apparently that’s what the server thought I meant. I didn’t realize initially that the site was down because I was having unrelated DNS issues, and I thought that’s why I couldn’t access the site.

I discovered that the site was back up when I received my first notification that somebody had posted spam in the comment section.

This article by Henrik Lenstra has an intriguing quote:

The key notion underlying the second algorithm is that of “infrastructure”, a word coined by Shanks (see [11]) to describe a certain multiplicative structure that he detected within the period of the continued fraction expansion of √d. It was subsequently shown (see [7]) that this period can be “embedded” in a circle group of “circumference” Rd, the embedding preserving the cyclical structure. In the modern terminology of Arakelov theory, one may describe that circle group as the kernel of the natural map Pic0Z[√d] → PicZ[√d] from the group of “metrized line bundles of degree 0” on the “arithmetic curve” corresponding to Z[√d] to the usual class group of invertible ideals. By means of Gauss’s reduced binary quadratic forms one can do explicit computations in Pic0Z[√d] and in its “circle” subgroup.

I’m a sucker for anything that related elementary topics (like Pell’s equation) to advanced topics that I don’t understand (like Arakelov theory).

The 24-cell is a mysterious regular polytope in 4 dimensions, in that there’s no analogy to a Platonic solid. Since 2005, the campus at Penn State has featured a sculpture based on a 3-dimensional projection of this 4-dimensional object. The sculpture, known as the Octacube, was designed by Penn State mathematics professor Adrian Ocneanu. The sculpture was sponsored by Jill Grashof Anderson in honor of her husband, who was killed in the September 11th attacks.

Penn State has put out a shorter and longer article on the sculpture.

Mohammad Sal Moslehian has put together a website of counterexamples in functional analysis. (On my browser the HTML versions come out looking weird, but the PDFs look fine.)