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!
