# Katharine Turner

I am an Aussie currently studying at the University of Chicago. I am doing my PhD in mathematics under the supervision of Shmuel Weinberger.

Like all good graduate students I have an office in the basement; Eckhart 14 to be precise. The Eckhart building is in the beautiful main quadragle of the University of Chicago.

My email is kate(at)math.uchicago.edu

## Applied Topology … ?

Topology is the study of space up to continuous deformation. The old joke is a topologist can't tell the different between a coffee cup and a bagel. How can it now be applied? One way is through persistent homology. Homology groups are algebraically described invariants of topology. They provide information such as the number of connected components, the dimension of the space of loops and higher dimensional analogs of loops. Instead of considering the homology of a single space we learn a lot more by looking at how the homology evolves over a filtration of a space. A filtration is a family of spaces $K_t$ such that $K_s\subset K_t$ whenever $s\leq t$. Often this parameter $t$ is the length scale making the process scale free. Although we are considering how the topology is changing we actually learn a lot about geometrical features because using a filtration has a quantifying effect. For example, if we used the filtration of all of $\mathbb R^3$ defined by the distance to the bagel we would be able to read the radius of the hole of the bagel from the persistent homology as it is at that radius that the space of loops collapses to being trivial. When considering persistent homology rather than just homology we can distinguish the coffee cup from the bagel. Persistent homology is often depicted by persistence diagrams which are sets of points in the plane - each point (b,d) corresponding to persistent homology class which is born at time b and dies at time d.

### Statistics and applied topology

A lot of my research involves performing standard statistics when given a set of persistence diagrams; sometimes characterizing statistical objects like the mean or the median and sometimes considering some technique such as null hypothesis testing. Using a functional variant on the persistence diagram called the persistent homology rank function we also develop a method of performing PCA.

### Persistent homology and Euler transforms for shape statistics

One approach to shape statistics is to try to find ways to reexpress information about shape that can be easier to compare and analyse. Most commonly a set of landmarks are picked and then local information is described at each of these landmarks. Combining this information gives a summary of the shape. I am interested in a completely different approach. Given a shape we can considers a whole family of filtrations of that shape by relevent functions (in particular we can use the height functions in different directions). These filtrations give persistence diagrams and Euler characteristic curves and we can analyse them.

### Reconstruction of compact spaces from point clouds

Sometimes data lie near some unknown but reasonably behaved set. In order to analyze the data it makes sense to try and understand what this underlying set is. In the case this unknown set is assumed to be a manifold this problem is manifold learning, when it is a surface it is called surface reconstruction. We can consider the problem of constructing a simplicial complex which is homotopic to the underlying unknown set.

## Papers

Persistent Homology Transform for Modeling Shapes and Surfaces,
with S Mukherjee, DM Boyer.
Information and Inference, in press. Available at: arXiv preprint arXiv:1310.1030.

Fréchet Means for Distributions of Persistence diagrams,
with Y Mileyko, S Mukherjee, J Harer.
Discrete & Computational Geometry 52 (2014), no. 1, 44–70.

Cone fields and topological sampling in manifolds with bounded curvature.
Foundations of Computational Mathematics 13 (2013), no. 6, 913–933.

Harmonic tori in De Sitter spaces $S^{2n}_1$,
with E Carberry.
Geometriae Dedicata 170 (2014), 143–155.

Hypothesis Testing for Topological Data Analysis,
with A Robinson.
arXiv preprint arXiv:1310.7467.

Means and medians of sets of persistence diagrams,
arXiv preprint arXiv:1307.8300.

Probabilistic Fréchet Means and Statistics on Vineyards,
with E Munch, P Bendich, S Mukherjee, J Mattingly, J Harer.
arXiv preprint arXiv:1307.6530.

Toda frames, harmonic maps and extended Dynkin diagrams,
with E Carberry.
arXiv preprint arXiv:1111.4028.

## Talks

Here are the slides to a selection of my talks:

PCA of persistent homology rank functions (EPFL, February 20, 2015)

Topological analogs of the Radon transform (John Hopkins, October 20, 2014)

## Code

My research sometimes involves coding so that the theory can be applied to actual data. Mostly this is in python. I am slowly trying to comment the code and make it presentable enough for public consumption from this website.

There are many different packages in many different languages for computing persistence diagrams from filtrations of simplicial complexes. This is not what I am trying to do. Complementing any code here with other TDA code is hardly a bad thing. I personally use Dmitry Morozov's code which is useable in python.

However, I will endeavour to make everything self-contained. As a result I will often restrict to the zeroth dimensional persistence, as this can be calculated quickly using a union find algorithm I can provide here when needed. I also have algorithms which use persistence diagrams as input.

### Persistent Homology Transform

This is code related to the paper:

Persistent Homology Transform for Modeling Shapes and Surfaces,
with S Mukherjee, DM Boyer.
Information and Inference, in press. Available at: arXiv preprint arXiv:1310.1030.

If you use any of this code please cite this paper.

The persistent homology transform provides a method of performing shape statistics. Given an object $M\subset \mathbb{R}^d$ and $v\in S^{d-1}$, let $X_k(M,v)$ be the $k$th dimensional persistence diagram corresponding to the height function in direction $v$ (i.e. $f(x)=x\cdot v$ for $x \in M$). Two objects $M$ and $M'$ are close if, for every direction $v$, the corresponding diagrams $X(M,v)$ and $X(M',v)$ are close.

The persistent homology transform of $M\subset \mathbb{R}^d$ is the function $$PHT(M): S^{d-1} \to \mathcal{D}^{d}$$ $$v\mapsto (X_0(M,v), X_1(M,v), \ldots, X_{d-1}(M,v)).$$ We then define an $L_1$ distance between $M$ and $M'$ by $d(M,M')=\sum_{k=1}^d \int_S^{d-1} d_1(X_k(M,v), X_k(M',v)) dv$

The code here only does $H_0$ for speed and self-contained reasons. It is further justified by results in the associated paper that the $H_0$ persistence diagrams are enough to reconstruct surfaces homeomorphic to $S^2$ or $S^1$ in $\mathbb{R}^2$ or $\mathbb{R}^3$.

The zeroth dimensional persistent homology transform of $M\subset \mathbb{R}^d$ is the function $$PH_0T(M): S^{d-1} \to \mathcal{D}$$ $$v\mapsto X_0(M,v).$$ With induced distance between $M$ and $M'$ by $d(M,M')=\int_S^{d-1} d_1(X_0(M,v), X_0(M',v)) dv.$

The code computes the distance matrix given a set of finite simplicial complexes. There are examples provided including ellipsoids and cut off hyperboloids in $\mathbb{R}^3$, and a selection of boundaries of silhouette shapes from a shape database.

#### Python and data files

python files which contain:

• PHTzero.py - the classes needed to compute the $PH_0T$ for shapes in $\mathbb{R}^3$ and the distances between them.
• PHTzeroinR2.py - extra classes to those in PHTzero.py needed to compute the $PH_0T$ for shapes in $\mathbb{R}^2$. This includes computing distances independent of alignment. It also has code for running on the silhouette data example in the paper.
• munkres.py - code (not written by me) for the Munkres assignment problem. This is used in computing the distance between two diagrams. (This software is released under a BSD license, adapted from Copyright (c) 2008 Brian M. Clapper)
• Examples_Ellipsoids_Hyperboloids.py - can be used to compute the ellipsoid and cut-off hyperboloids examples done in depth in the appendix.
boundaries of silhouette data. These boundary perimeters where obtained from Visionlab, http://visionlab.uta.edu/shape_data.htm

## Teaching

### Classes I am/have taught

Math 161, Section 32, Autumn 2014

Math 133, Section 26, Spring 2014

Math 132, Section 26, Winter 2014

Math 131, Section 26, Autumn 2013

Math 153, Section 27, Autumn 2012