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Undergraduate mini-course on the Calculus of Analytic Critical Points
October 11, 2007 (02:00 PM PDT - 05:00 PM PDT)
Brendan Hassett (Rice University)
Every calculus student learns the second-derivative test for critical points: Depending on the sign of the second derivative, we can decide whether the critical point is a maximum or minimum. However, the test is inconclusive if the second derivative is zero. For functions in two variables, there is a second-derivative test for critical points using the matrix of second-order partial derivatives. This works well when the determinant of this matrix is nonzero; these are called nondegenerate critical points.
Over the last three years, undergraduate students at Rice University have been doing research on critical points of analytic functions in two variables, classifying degenerate critical points using invariants generalizing those from multivariable calculus. Their work revolves around developing computational techniques for evaluating these invariants efficiently. We will present these techniques, as well as the mathematical principles underlying them.
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