Tag Archives: galois

Duel at Dawn

Speaking of Galois, my review of Amir Alexander’s Duel at Dawn is up at BN Review today.  The book draws an interesting connection between the Romantic literary area and the invention of the “romantic” mathematical hero, of whom Galois is obviously the sterling example.  But Alexander commendably reaches past the endlessly-repeated Galois story to cover a lot of material less familiar to readers of pop math; I learned a lot about Abel, Bolyai, D’Alembert, and Cauchy (who was constantly getting rebuked by his deans for teaching epsilons and deltas in first-year calculus!)

The uncollected and very worthwhile David Foster Wallace essay “Rhetoric and the Math Melodrama,” which I mention towards the end of the piece, can be found in .pdf here.

Also, writing this review gave me the opportunity to use the word “emo” in print for the first time.  I hope my younger readers will let me know whether my usage is roughly correct.

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The entropy of Frobenius

Since Thurston, we know that among the diffeomorphisms of surfaces the most interesting ones are the pseudo-Anosov diffeomorphisms; these preserve two transverse folations on the surface, stretching one and contracting the other by the same factor.  The factor, usually denoted \lambda, is called the dilatation of the diffeomorphism and its logarithm is called the entropy. It turns out that \lambda, which is evidently a real number greater than 1, is in fact an algebraic integer, the largest eigenvalue of a matrix that in some sense keeps combinatorial track of the action of the diffeomorphism on the surface.  You might think of it as a kind of measure of the “complexity” of the diffeomorphism.  A recent preprint by my colleague Jean-Luc Thiffeault says much about how to compute these dilatations in practice, and especially how to hunt for diffeomorphisms whose dilatation is as small as possible.

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