SCU Mathematics Colloquium Series Schedule

Winter 2002

Talks will be at 4:00pm Tuesdays in O'Connor 107. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.

**Jan. 22**Sergio Zarantonello, Santa Clara UniversityTitle: A regularized solution of a Cauchy problem for the Laplace equation in the unit disk

Abstract: The steady state temperature distribution in a homogeneous circular plate satisfies the Laplace equation and can thus be determined when the boundary temperatures are known. In this talk we investigate a variant of this problem, that of determining the steady state temperature on the plate when the temperature and its normal derivative are known on only a part of the boundary. The temperature on the plate is then uniquely determined, but small perturbations in the boundary data can lead to large errors in the solution. We relate this problem to that of reconstructing analytic functions in the Hardy space H

^{2}from their boundary values on subsets of the circle of measure greater than zero, and discuss a regularization procedure of D. J. Patil that restores stability to the solution.

**Jan. 29**Francisco Samaniego, UC DavisTitle: From the X Files (and Y Files) of the Statistical Laboratory at the University of California, Davis

Abstract: Several recent projects "from the files of the Statistical Laboratory at UC Davis" will be discussed as examples. Applied problems in the areas of conservation, traffic engineering and the assessment of school effectiveness will be featured. This talk includes some mathematical ideas and derivations, but it is aimed primarily at upper division math majors and is geared toward revealing what most statisticians do for a living, i.e., apply mathematical tools to real-world problems.

**Feb. 5**Anne Shepler, UC Santa CruzTitle: Reflection groups and Hyperplanes

Abstract: Cut a huge wheel of cheese with 10 slices. How many pieces can you get? We can slice the cheese with planes in 3-space and ask "how many regions result? what is the maximum possible?" What if the planes were mirrors instead of knives? During this talk, we will discuss how an arrangement of planes in space can give rise to a group generated by mirror reflections. We will define a "reflection group" and talk about symmetry groups of regular polytopes (e.g., dodecahedron, complex polygons). We will indicate some recent results in the theory of reflection groups and invariant theory.

**Feb. 12**Charles Hamaker, Saint Mary's CollegeTitle: Constrained Optimization for a Quadratic Form with Linear Term

Abstract: In lattice gauge theory, optimal solutions are obtained by iteration of local gauge transformations. Such a transformation is determined by the solution of:

For a positive-definite symmetric n x n matrix A and a real vector b, find the unit vector x which maximizes f(x) = x

^{t}A x + b^{t}x.The conventional solution involves diagonalizing A, requiring O(n

^{4}) operations. An iterative method is proposed which converges effectively except in a singular case and requires O(n^{2}) operations per iteration.

**Feb. 26**Thomas Banchoff, Brown UniversityTitle: The Fourth Dimension: Interactive Geometry on the Internet

Interactive computer graphics techniques make it possible to investigate surfaces from four-dimensional space in dramatically new ways. This talk will feature virtual art galleries and library exhibits, electronic posters and journals, and Internet-based paperless classes in multivariable calculus and differential geometry.

**Mar. 5**Fred Wu, Northeastern UniversityTitle: Dimer statistics and spanning trees

Abstract: This talk reports some recent progress on dimer and spanning tree enumerations. We use the Kasteleyn formulation to enumerate close-packed dimers on a simple-quartic net embedded on non-orientable surfaces, and obtain solutions in the form of double products. For spanning trees the enumeration is carried out by evaluating the eigenvalues of the Laplacian matrix associated with the lattice. In two dimensions a bijection due to Temperley relates spanning tree and dimer configurations on two related lattices. We use this bijection to enumerate dimers on a net with a vacancy on the boundary. It is found that the occurrence of a vacancy induces a sqrt(N) correction to the enumeration, where N is the linear size of the lattice.

If you have a disability and require a reasonable accommodation, please call Rick Scott rscott@math.scu.edu, 1-408-554-4460 or 1-800-735-2929 (TTY - California Relay).

The list of talks from previous quarters are available via this archive link.

*Last Updated: 13 October 2001*
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