The March Notices of the AMS features a retrospective on George Dantzig, who died in 2005, and a description of his simplex method to solve linear programming problems.

Linear programming is the solution of constrained optimization problems with linear objective function and linear constraints. Since the set of allowable input values is a convex polytope, the solution must (usually) be one of the vertices of the polytope. One sure-fire method for finding the solution is to try every vertex. For small problems this is an eminently practical procedure, but for large problems this takes time exponential in the number of vertices. The simplex method is the first algorithm that in practice efficiently solves linear programming problems by only trying a subset of the vertices. In the worst case, the simplex algorithm can still visit every vertex, but for the vast majority of practical problems the algorithm only takes polynomial time. (There are more recent algorithms that have better worst-case performance. I have a upcoming post planned on the subject.)

This month’s What is… is János Kollár on What is… a minimal model. Minimal models are the current target of research in birational algebraic geometry — the current hope is that most (for a well-defined meaning of most) birational equivalence classes of algebraic varieties have a minimal model. Minimal models are not unique, but two birationally equivalent minimal models are closely related. Most introductions to the subject I have seen take an algebraic approach. Kollár instead answers the question in complex analytic terms.

(I had more for this post, but WordPress just ate it. More later.)

Mock theta function are a mysterious family of functions defined by Srinivasa Ramanujan. Ramanujan defined them in a letter to G. H. Hardy. Only part of his letter survives, so the actual definition of what makes a function a mock theta function has been lost. The surviving remnant contains examples which make the resemblace to theta functions clear.

I ran across this press release put out by the University of Wisconsin-Madison about a breakthrough in mock theta functions, and I had no idea what they were actually announcing. The best I can piece together is that Sander Zwegers, in his dissertation gave the first general definition of mock theta functions that included most of Ramanujan’s examples, and that this work was extended by Kathrin Bringmann and Ken Ono. Some details can be found in this preprint.

Lieven Le Bruyn has an intriguing post about the Mathieu groups. For a century the Mathieu groups were the only sporadic simple groups known. Lieven describes how to construct the groups geometrically.

Just under the wire…

The February Notices of the AMS features an article on the Four Vertex Theorem. The Four Vertex Theorem states that any simple closed curve in the plane (other than the circle) must have four extrema in its curvature.

Alexander Yong does the honors in this month’s entry in the What is… series, What is… a Young tableau. Young tableau arise in many contexts in combinatorics, most dramatically in parametrizing the irreducible representations of the symmetric group.

There are two articles about Mikio Sato, one a summary of his research contributions, and the other an interview. Sato is the inventor hyperfunctions,

We live in a golden age for popular accounts of advanced mathematical research. Robert Griess reviews Symmetry and the Monster, One of the Greatest Quests of Mathematics, by Mark Ronan. The book documents the history of the classification of finite simple groups. (In his review, Griess sounds like a skeptic on the question of whether the classification can truly be said to be done.)

Alain Connes has joined the web as a member of the group weblog, Noncommutative Geometry. From a quick scan of the posts, it looks as if the other members are Arup, Masoud Khalkhali, and David Goss. (David Corfield mentioned the new site last week. The comment thread there features an interesting discussion of how mathematicians would rather be torn apart by wild dogs than be identified as members of someone else’s “school”.)

Terry Tao has started a solo weblog, What’s new. I welcome everyone concerned to the world of weblogs, and congratulate them for making the wise choice of getting no real work done ever again.

The second Carnival of Mathematics is out.

I have a fever, so posting from me will be light. Before I go back to my sick-bed, though, I wanted to draw your attention to this web site that documents bias in Wikipedia. Most of the examples listed are not math-related, but item number 16 is:

Wikipedia has many entries on mathematical concepts, but lacks any entry on the basic concept of an elementary proof. Elementary proofs require a rigor lacking in many mathematical claims promoted on Wikipedia.

Further perusing of the site reveals that they mean elementary proofs in number theory, i.e. proofs don’t use complex analysis. Apparently belief in the complex numbers is a form of bias.

Via Good Math, Bad Math.

New York Magazine has an interesting article on the best way to use praise in education. Experiments show that praise can be an effective teaching technique, but interestingly, only praise for someone’s work ethic. Praise of someone’s intelligence can actually hurt performance.

The inaugural edition of the Carnival of Mathematics is up.