Short Bio
I am currently employed as a postdoctoral researcher of
Ulrich Bauer» working on the SFB project
Discretization in Geometry and Dynamics». Before this I did my
PhD (defended December 2015) at the Norwegian University of Science and Technology under the supervision of
Nils Baas».
My CV can be found
here.
Seminars/Teaching
I am organizing the
Applied Topology Seminar and co-organizer of the course
Categories by Examples.
Research Interests
I work within the field of topological data analysis: a relatively recent branch of mathematics in which topological signatures are assigned to data. Enjoying being at both the pure and applied side of mathematics, my research includes pure elements such as category theory, representation theory of quivers and cosheaves, as well as more computational aspects, and applications towards the sciences. Roughly, my research can be divided into the following:
Multidimensional Persistent Homology
Ordinary persistent homology is concerned with (e.g.) the evolution of the topology (homology) of the sublevel filtration of a scalar function on a topological space. This evolution of topology is succinctly captured in what is known as a barcode (see e.g.
Ghrist» for an elementary introduction). The theoretical justification for this can be found in a classical theorem in representation theory by
Gabriel». However, in many instances - such as multi-parameter clustering - it is natural to consider a space equipped with multiple functions. As it turns out, when any pair of functions on an arbitrary CW complex is allowed, there exists no barcode construction which captures the evolution of the topology as in the one-dimensional case. Put loosely, 'understanding' such evolutions is as hard as understanding the module category of any finite dimensional algebra, or even the word problem for finitely presented groups; Representation theorists refer to these types of problems as
wild or
hopeless. Although the general case is hopeless, it does not mean that data obtained in nature inherits this complexity, and preliminary computations seem to suggest this. Positive results about multidimensional persistence, such as 'nice' behaviour in certain restricted settings, may carry a profound impact on (topological) data analysis.
I am currently involved in a project with Ulrich Bauer (TUM), Steffen Oppermann (NTNU) and Johan Steen (NTNU) in which we have identified certain spaces and functions for which a barcode-like invariant exists, and identified other efficiently computable invariants. We have also shown that certain scenarios for which one would expect to admit barcode-like structures, are just as hopeless as the general setting.
A video recording of this talk can be found here:
Webex».
Interleavings
The theory of interleavings lies at the very core of the theoretical foundations of persistent homology, and includes the important
algebraic stability theorem. This theorem has seen many applications, and perhaps most notably: 1) the modern way of showing that persistent homology is
stable, 2) allows us to give error bounds of approximative algorithms (see e.g.
my paper with Spreemann»). With rising interest in more generalized indexing categories (zigzag (see e.g.
my paper»), commutative ladders, circle valued maps, etc. ) comes the desire to extend this theory. In recent work with Michael Lesnick (Princeton U.) we proved an algebraic stability theorem for zigzag persistence. In fact, we prove an algebraic stability theorem for a nice class of multidimensional modules (see above), and then we used a category theoretical concept called Kan extensions to put our zigzags into this framework. As it turns out, this approach to interleavings can be applied to any poset and this is the focus of an ongoing project with Justin Curry (Duke) and Elizabeth Munch (U Albany). This project also has applications to another tool in topological data analysis called Mapper.
Another ongoing research project is to understand the computational complexity of the computation of these generalized interleavings (in particular in the multidimensional case).
Applications
Neuroscience has shown itself to be one of the most prominent avenues for the application of topological methods. In recent work with Gard Spreemann (EPFL/NTNU), Benjamin Dunn (Kavli Institute NTNU) and Nils Baas(NTNU) we have applied persistent homology to reveal topological signatures of the covariates driving the activity of certain kinds of neurons. I have also been involved in a project with Javiar Arsuaga (UC Davis) et al. in which we analyzed the folding patterns of chromosomes.