### Project 17: "Computational geometry and topology"

*This project is running since 2019.*

Principal investigator: Michael Kerber

Graz University of Technology, Austria.

*Mentor for*: Wulf, Missethan; Hüning, Del Giudice.

Associated scientist: Cesar Ceballos

Graz University of Technology, Austria.

### DK Student

**Third phase of the doctoral program:****Matthias Söls**(Austria; since April 2022)

*Email*: soels@tugraz.at

*Mentors*: Oswin Aichholzer, Barbara Giunti.

**Bianca Boeira Dornelas**(Brazil; since October 2019)

*Email*: bdornelas@tugraz.at

*Mentors*: Mihyun Kang, Cesar Ceballos.

### Associated Students

**Third phase of the doctoral program:****Jan Jendrysiak**(Germany; since April 2022)

*Email*: jendrysiak@tugraz.at

*Mentors*: Daniel Smertnig, Cesar Ceballos.**Florian Russold**(Austria; since November 2021)

*Email*: russold@student.tugraz.at

*Mentors*: Alfred Geroldinger, Olga Diamanti.**René Corbet**(Germany; October 2019–July 2020)

*Email*: corbet@tugraz.at

*Mentors*: Alfred Geroldinger, Mickael Buchet.

*Thesis Title*: Improvements to the Pipeline of Multiparameter Persistence.

*PhD Defense*: July 21, 2020

*Referees*: W. Chachólski (KTH Stockholm), H. Edelsbrunner (IST Austria), M. Kerber.

*Examiners*: H. Edelsbrunner (IST Austria), M. Kerber.

### Project description

My research focuses on the area of applied algebraic topology.
The theory of **persistent homology** permits the usage of
classical homology theory in the context of noisy data. This has lead to
an entirely new discipline named **topological data analysis**
with applications to applied fields such as neuroscience,
material science, medical science, and many others.
See this survey for a more thorough introduction.

Thanks to the wide-spread application domains of the developed theory,
there is an ever-increasing demand for computational methods
to apply persistent homology to larger (and high-dimensional) data sets.
My primary interest is the development of efficient methods
for the computation of topological descriptors
applicable to data analysis.
Here, ``efficient'' means both theoretically efficient in terms of
worst-case asymptotic complexity
(connecting to theoretical computer science),
and practically efficient in terms of runtime and memory consumption
(reaching into algorithm engineering and software development).

I am especially interested in the above questions
in the context of geometric input data.
Besides being an important special case,
the geometric perspective often
provides intuition and leads to approaches that can later be generalized
to more general inputs. Another aspect of my research is
to apply efficient methods from computational geometry in
topological contexts. More often than not, the application of such methods
requires the development of additional theory in geometry first.

Showcases for possible PhD themes can be found here.