Kevin Vixie's Analysis Seminar Talk for February 16, 2011
Title: A Deformation Theorem for Simplicial Complexes
Abstract: In this talk I will give an intuitive, yet complete picture of the deformation theorem from geometric measure theory and its recent extension in work with Bala Krishnamoorthy and Sharif Ibrahim to the case in which the polyhedral complex is not the usual cubic complex. This result came about as a side effect of a paper that we are writing on computation of the flat norm on simplicial complexes. Very briefly, there is a critical integral that we recompute for the case of a general simplex obtaining as a corollary the precise way in which the regularity of the simplicial complex effects the constants in the theorem. The fact that this is all very geometric will allow me to illustrate everything with pictures that contain all the ideas in the proof.
Background: See Frank Morgan's "Geometric Measure Theory: A Beginner's Guide" for a good introduction to the deformation theorem and Krantz and Parks "Geometric Integration Theory" for details.