Graduate Colloquium Fall 2010
For further information please contact Zhe Xu at zxxu@ucsc.edu
October 21, 2010
Understanding Real Derivatives of Complex-Valued Maps
Victor Dods
I'll start by talking about embedding the complex numbers into the 2x2 real matrices to establish an identification of C with a subspace of M_2(R). Then under certain identifications, the expected rules of differentiation will be proven to still hold. I'll end with example calculations, including calculating the tangent map of the famous Hopf fibration. This talk will interest you if you like linear algebra and/or geometry involving complex variables.
November 18, 2010
Growth Vectors and the Monster Tower
Corey Shanbrom
Due to a misinterpretation of the work of E. Cartan, for over 60 years all Goursat distributions were believed to be equivalent to the canonical distribution associated to the jet bundle over the reals. The first singularities were discovered in 1978, and their complete classification was published in January 2010. I will discuss the history of the subject, and the discovery of the universal space for Goursat distributions: the Monster tower. I will also discuss the basic invariant of a Goursat distribution, its growth vector, and its relationship with a geometric coding of points in the Monster tower.
December 2, 2010
Curve Singularities and the Monster Tower
Wyatt Howard
In a recent paper titled Points and Curves in the Monster Tower(2009), Richard Montgomery and Misha Zhitomirskii were able to classify all of the points in a tower of manifolds, built from the plane R^{2}. They called this structure the "Monster Tower". The motivation for doing this stems from Montgomery's interest in trying to classify all possible Goursat distributions(meaning a subbundle of the tangent bundle). He demonstrated that there was a correspondence between Goursat distributions and points in the Monster Tower. In this talk I will discuss the construction and properties of the Monster tower when built from C^{n} for n > 2. I will also present some of the basic classification results for the Monster Tower for the case of C^{3}.
