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Fall 2009 Abstracts |
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(or how to destroy children's illusions) Franklin Vera Pacheco Friday, November 20 15:10-16:00 BA6183 Math is about thinking about how to stop having to think. In this talk we will try to sum the first n terms of any sequence without thinking. Yevgeniy Liokumovich Friday, November 6 15:10-16:00 BA6183 Mathematician or not, everyone can imagine a sphere stretched in some way and frozen. But is it really as intuitive as it seems to be? We may explore this by asking "simple" questions about Riemannian 2-spheres. Question 1. Given a Riemannian 2-sphere of diameter 1 and a closed curve of length 1 on it, is it always possible to contract the curve to a point without stretching it too much? If that fails, maybe we should ask for the area to be bounded? Question 2. How long is a shortest non-trivial closed geodesic on a Riemannian 2-sphere? For all the familiar spheres, like ellipsoids, it is less or equal to twice the diameter (the maximum distance between two points on a sphere). Is it always the case? Only very basic notions from Riemannian geometry will be needed. Undergraduates are welcome. Artem Dudko October 27, 2009 I'll start my talk with some simple examples of divergent series, like 1-1+1-1+..., for which you can, I hope, guess what the "sum" should be. Then I'll talk about more complicated examples, like Euler series x-1!x2+2!x3-3!x4+..., which in some sense converge much better than many convergent series. One of the main aims is to show how divergent series can be used to solve or estimate solutions of difference and differential equations. Ehsan Kamalinejad October 13, 2009 In mathematical image processing one of the interesting problems is to find robust methods of noise reduction. Mumford and Shah proposed a method using piecewise smooth variational model which is effective but computationally expensive. We will study the Mumford-Shah energy functional and some of PDE approaches and approximations to the problem, and also some of the examples and results. Yuri Burda September 26, 2009 We will examine a heuristic technique of inventing and "proving" polynomial identities, which was extensively used by classics, but looks shocking to a modern mathematician. Using this technique, named "Umbral Calculus" by J. Sylvester, classics were able to "prove" such theorems as Euler-Maclaurin formula, Vandermonde identity, Lagrange inversion theorem and many more. After working out several examples, we will see how this technique can be made rigorous with little more than Linear Algebra. Maybe just a little more: structures like algebras, coalgebras and groups will also play a role in showing the power of Umbral Calculus. Finally we will compare the classical non-rigorous, but beautiful, method with its modern formulation and see how they compliment each other. The only technical knowledge that will needed to enjoy the talk is knowledge of linear algebra. |