### Project 04: "Point distributions and fractal analysis"

*This project is running since 2010.*

Principal investigator: Peter Grabner

Graz University of Technology, Austria.

*Mentor for*: Spanos, Zalla; Alese, Zhang; Krenn, Minervino, Raseta.

### DK Students

**Third phase of the doctoral program:****Michael Henry**(USA; since February 2021)

*Email*: michael.henry@tugraz.at

*Mentors*: Christoph Aistleitner, Cesar Ceballos.

**Second phase of the doctoral program:****Damir Ferizovic**(Bosnia and Herzegovina; January 2017-June 2021)

*Email*: damir.ferizovic@tugraz.at

*Mentors*: Jörg Thuswaldner, Johannes Wallner, Mario Weitzer.

*Thesis Title*: "Uniform Point Distributions and Energy Estimates on Compact Riemannian Manifolds".

*Examiners*: Carlos Beltran Alvarez (Cantabria), Dmytro Olexandrovych Bilyk (Minnesota).

**First phase of the doctoral program:**-
**Florian Greinecker**(Austria; October 2010-April 2016)

*Email*: greinecker@math.tugraz.at

*Mentors*: Clemens Heuberger, Jörg Thuswaldner.

### Project description

P. Grabner is interested in the construction and analysis of well
distributed point sets and point sequences on manifolds, especially on
the sphere. Such point distributions have applications in numerical
analysis in the larger context of Quasi-Monte Carlo methods. He is a
member of a special research programme with this title. The techniques
to obtain such point distributions involve digital expansions, number
theoretic methods, modular forms, ergodic theory, and harmonic
analysis. Probabilistic models for such point distributions use
determinantal point processes. Furthermore, he is interested in
fractal and ergodic properties of digital expansions, which occur in
this context.

Showcases for possible PhD themes can be found here.