Summary: Shapes, radius functions and persistent homology

This is a summary of a talk by Professor Herbert Edelsbrunner, IST Austria. The PDF slides can be found here: persistent homology slides.

Biogeometry (2:51 in video)

We can think of proteins as a geometric object by replacing every atom by a sphere (possibly different radii). Protein is viewed as union of balls in \mathbb{R}^3.

Decompose into Voronoi domains V(x), and take the nerve (Delaunay complex).

Inclusion-Exclusion Theorem: \displaystyle Vol(\bigcup B)=\sum_{Q\in D_r(x)}(-1)^{\dim Q}Vol(\bigcap Q).
Volume of protein (\bigcup B) is alternating sum over all simplices Q in Delaunay complex.

Nerve Theorem: Union of sets have same homotopy type as nerve (stronger than having isomorphic homology groups).

Wrap (14:04 in video)

Collapses: 01 collapse means 0 dimensional and 1 dimensional simplices disappear (something like deformation retract).

Interval: Simplices that are removed in a collapse (always a skeleton of a cube in appropriate dimension)

Generalised Discrete Morse Function (Forman 1998): Generalised discrete vector field = partition into intervals (for acyclic case only)

Critical simplex: The only simplex in an interval (when a critical simplex is added, the homotopy type changes)

Lower set of critical simplex: all the nodes that lead up to the critical simplex.

Wrap complex is the union of lower sets.

Persistence (38:00 in video)

Betti numbers in \mathbb{R}^3: \beta_0: \# components, \beta_1:\# loops, \beta_2: \# voids.

Incremental Algorithm to compute Betti numbers (40:50 in video). [Deffimado, E., 1995]. Every time a simplex is added, either a Betti number goes up (birth) or goes down (death).

\alpha is born when it is not in image of previous homology group.

Stability of persistence: small change in position of points leads to similar persistence diagram.

Bottleneck distance between two diagrams is length of longest edge in minimizing matching. Theorem: \displaystyle W_\infty(Dgm(f),Dgm(g))\leq\|f-g\|_\infty. [Cohen-Steiner, E., Hares 2007]. One of the most important theorems in persistent homology.

Expectation (51:30 in video)

Poisson point process: Like uniform distribution but over entire space. Number of points in region is proportional to size of region. Proportionality constant is density \rho>0.

Paper: Expectations in \mathbb{R}^n. [E., Nikitenko, Reitones, 2016]

Reduces to question (Three points in circle): Given three points in a circle, what is the probability that the triangle (with the 3 points as vertices) contains the center of the circle? Ans: 1/4 [Wendel 1963].

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