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 Prizes 2.Winning Olympic Gold Medals 3.Finding True Love

1. Geometry, Topology and all your wildest
dreams will come true
Don Sheehy

2. I do theory.
Computational Geometry
(geometric approximation algorithms)
Computational Topology
(geometric inference)

3. Applications
Surface reconstruction
Manifold learning
Topological data analysis
Winning Nobel Peace Prizes
Winning Gold Medals in the Olympics
Finding True Love

4. Computer Scientists want to know
the shape of data.
Clustering
Principal Component Analysis
Convex Hull
Mesh Generation
Surface Reconstruction

5. Point sets have no shape...
so we have to add it ourselves.

6. Distance functions add shape to data.
dP (x) = min |x − p|
p∈P
α
P = dP [0, α]
−1
= ball(p, α)
p∈P

7. In Persistent Homology, we look at the
changes in the shape over time.
(Think graphs plus triangles, tetrahedra, etc.)
Use a simplicial complex rather
than the union of balls.

8. Previous methods build complexes of size nO(d).
We can do this with complexes of size O(n).
nO(d) O(n)

9. Previously, we had to stop early.

10. Topology is not Topography
Sublevel sets
(But in our case, there are some similarities)

11. Nobel Peace Prize!

12. Mesh generation
Gold Medals!

13. Where do we get geometric data?
True Love!

14. Pittsburgh!
Thanks!