Katharine Turner

About me

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

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

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

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.

Persistent Homology Transform for Modeling Shapes and Surfaces,
with S Mukherjee, DM Boyer.
arXiv preprint arXiv:1310.1030.

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.

Teaching

Classes I have taught

Math 133, Section 26, Spring 2014

Math 132, Section 26, Winter 2014

Math 131, Section 26, Autumn 2013

Math 153, Section 27, Autumn 2012

Math 153, Section 20, Spring 2012

Math 152, Section 20, Winter 2012

Math 151, Section 20, Autumn 2011