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CAM Colloquium: Charles Van Loan (CS, Cornell University) - Untangling Random Polygons

Friday, August 24, 2018 at 3:30pm

Frank H. T. Rhodes Hall, 655

Charles Van Loan is the 2018 recipient of The John von Neumann Lecture prize in recognition of his pioneering contributions to research in numerical linear algebra and to the exposition of the subject. SIAM’s highest honor and flagship lecture, The John von Neumann Lecture, is awarded annually to recognize outstanding and distinguished contributions to the field of applied mathematical sciences and the effective communication of these ideas to the community. Van Loan delivered The John von Neumann Lecture at the SIAM Annual Meeting in Portland, Oregon on July 10, 2018, and will present a version of it for this Colloquium.

My experience with the polygon averaging problem is one of my favorites because it involves most of what I like about teaching and research in matrix computations. As a story it has everything:  my favorite matrix decomposition (the singular value decomposition), my favorite matrix dimension (N=2), and my favorite structured matrix (the “upshift matrix”). On top of all that, the whole thing was motivated by an assignment that I gave in CS 1112, a Matlab-based introduction to computing course. This is a reminder about just how much there is to learn when teaching at that level.

Here’s the problem that I will talk about. Suppose we are given a random polygon P0 with vertices (x1, y1), . . . , (xn, yn) that have centroid (0,0). If we connect the midpoints of its edges, then we obtain a new polygon. Assume that the x-values and y-values are scaled after each update so that we always have kxk2=1 and kyk2=1. This update process can obviously be repeated to produce a sequence of polygons {Pk}. The students were required in the assignment to plot a reasonable number of the Pk. And then something truly amazing was observed. No matter how “criss-crossy” the initial polygon P0, the Pk eventually “untangle,” and their vertices head towards an ellipse with a 45-degree tilt.

Charles Van Loan has been a faculty member in the Department of Computer Science since 1975. He received his BS (1969), MS (1970), and PhD (1973) in mathematics from the University of Michigan. His research area is matrix computations, a practical field that has an important role  throughout the physical, biological, and social sciences.  Van Loan’s teaching (and textbook writing) range from freshman-level programming to upper level scientific computing. His research text Matrix Computations (with G.H. Golub) is one of the most widely cited books in all of mathematics and computer science. He is a Fellow of the Society for Industrial and Applied Mathematics.

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Computer Science, Center for Applied Mathematics





Charles Van Loan

Speaker Affiliation

Computer Science, Cornell University

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