Michi’s blog » archive for December, 2009

Coordinatization with hom complexes

  • December 7th, 2009

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?

This is a preview of Coordinatization with hom complexes. Read the full post (732 words, 64 images)

[MATH198] Lecture 10 (last lecture) posted

  • December 2nd, 2009

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

(21 words)

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Michi is a recent PhD working in homological algebra and applied algebraic topology. This blog is his outlet for texts with some manner of thought put into them. Over at his LiveJournal intimate details and streams of consciousness might be found.
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