geometry, topology, and all of your wildest dreams will come true
DESCRIPTION
In this light talk, I give a high level view of some of my recent research in using ideas from mesh generation to lower the complexity of computing persistent homology in geomemtric settings. Because this talk is for a general audience, I will focus on three related applications (where related is interpreted loosely) that I think have the widest appeal. The applications are: 1. Winning Nobel Peace Prizes2.Winning Olympic Gold Medals3.Finding True LoveTRANSCRIPT
Geometry, Topology and all your wildest dreams will come true
Don Sheehy
I do theory.
Computational Geometry(geometric approximation algorithms)
Computational Topology(geometric inference)
Applications
Surface reconstruction
Manifold learning
Topological data analysis
Winning Nobel Peace Prizes
Winning Gold Medals in the Olympics
Finding True Love
Computer Scientists want to know the shape of data.
Clustering
Principal Component Analysis
Convex HullMesh Generation
Surface Reconstruction
Point sets have no shape...so we have to add it ourselves.
Distance functions add shape to data.
dP (x) = minp!P
|x ! p|
P! = d!1
P[0, !]
=!
p!P
ball(p, !)
In Persistent Homology, we look at the changes in the shape over time.
Use a simplicial complex rather than the union of balls.
(Think graphs plus triangles, tetrahedra, etc.)
Previous methods build complexes of size nO(d).
We can do this with complexes of size O(n).
nO(d) O(n)
Previously, we had to stop early.
Topology is not Topography
(But in our case, there are some similarities)
Sublevel sets
Nobel Peace Prize!
Mesh generation
Gold Medals!
Where do we get geometric data?
True Love!
Pittsburgh!
Thanks!