Mathematics Seminar - 21 September 2012


Hancock Center Room 2023,  3:30 PM


James Callahan, Marist College '62
Professor Emeritus, Department of Mathematics and Statistics, Smith College 


Singularities of Maps: A  Brief Survey


One of the major threads of calculus ties the local behavior of a function to its derivative at a point. A familar example is the inverse function theorem: a function is invertible in a neighborhood of a point at which the derivative is nonzero. This covers most functions at most points. In the special, or singular case, where the derivative is zero, all other possible behaviors occur; the second derivative test begins their classification. For several functions of several variables—that is, for maps—the derivative is a linear map, and the singular case occurs when the derivative fails to have maximal rank. This talk surveys the singularities that appear in low dimensions; they are limited in number and can be directly visualized. The material is accessible to undergraduates familiar with linear algebra and multivariable calculus.