geometry, topology, and all of your wildest dreams will come true

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!