The Unreasonable Effectiveness of Derivatives
Professor Thomas Hunter
May 20, 2003
Bailey Hall 100
Dinner with Professor Hunter: Contact Bill Zwicker (firstname.lastname@example.org) by Monday Evening if you'd like to join us for dinner after the talk- free for students.
A substantial cluster of the great ideas of twentieth century mathematics were generated by Grothendieck's grand unification of number theory and geometry. Among other things, these ideas provided the foundation for the results which provide the jumping off point for Andrew Wiles' leap to a proof of Fermat's Last Theorem. Unfortunately, the details of this theory--especially when built from the ground up and in full generality--are more abstract, more convoluted, and more difficult than most students (and many working mathematicians) are willing to tolerate. In this lecture I will aim to give an intuitive and elementary peek at a tiny part of these ideas. Taking the simplest nontrivial case, I hope to show how one can provide a notion of infinitesimal that is good enough to explain the unreasonable effectiveness of derivatives in situations which are very far from their place of theoretical origin. Prerequisites for the talk are a familiarity with the language of linear algebra and facility with the arithmetic of polynomials.
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