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Michi’s blog » archive for December, 2009

 Coordinatization with hom complexes

  • December 7th, 2009
  • 10:55 pm

These are notes from a talk given at the Stanford applied topology seminar by Gunnar Carlsson from 9 Oct 2009. The main function of this blog post is to get me an easily accessible point of access for the ideas in that talk.

Coordinatization

First off, a few words on what we mean by coordinatization: as in algebraic geometry, we say that a coordinate function is some X\to\mathbb R or possibly some X\to\mathbb C, with all the niceness properties we’d expect to see in the context we’re working.

A particularly good example is Principal Component Analysis which yields a split linear automorphism on the ambient space that maximizes spread of the data points in the initial coordinates.

Topological coordinatization

The core question we’re working with right now is this:
Given a space (point cloud) X, and a (persistent) view of H_*(X), can we use some map H_*(X)\to H_*(Y) to generate a map X\to Y inducing that map?

 [MATH198] Lecture 10 (last lecture) posted

  • December 2nd, 2009
  • 11:17 pm

Now up: Lecture 10 with the definition of a topos and a derivation of internal, inutitionistic logic within a topos.