### Project 10: "Subdivision in nonlinear geometries"

*This project is running since 2010.*

Principal investigator: Johannes Wallner

Graz University of Technology, Austria.

*Mentor for*: Ferizovic, Moßhammer; Boiko, Frei.

Associated scientist: Michael Kerber (since 2015)

Graz University of Technology, Austria.

*Mentor for*: del Guidice, Huening, Vogel.

### DK Students

**Second phase of the doctoral program:****Svenja Hüning**(Germany, since April 2016)**First phase of the doctoral program:**-
**Oliver Ebner**(Austria; July 2010–July 2012)

Personal homepage;*Email*: o.ebner@tugraz.at

*Mentors*: Rainer Burkard, Wolfgang Woess.

*PhD Defense*: July 20, 2012.

*Referees*: Ph. Grohs (Zürich), K. Jetter (Stuttgart), J. Wallner.

*Examiners*: Ph. Grohs (Zürich), J. Wallner.

*Mentors*: Michael Kerber, Birgit Vogtenhuber

*Thesis Title*: "Convergence analysis of nonlinear refinement rules".

### Associated Students

**Second phase of the doctoral program:****Leonardo Alese**(Italy; since November 2015)

*Email*: alese@tugraz.at

*Mentors*: Peter Grabner, Daniele D'Angeli.

*Thesis Title*: "Curved-crease surfaces".**Caroline Moosmüller**(Austria; since November 2013)

*Email*: moosmueller@tugraz.at

*Mentors*: Mihyun Kang, Roswitha Rissner.

*Thesis Title*: "Smoothness analysis of linear and nonlinear Hermite subdivision schemes".

*PhD Defense*: February 17, 2017.

*Referees*: J. Wallner, T. Sauer (Passau), C. Conti (Florence)

*Examiners*: J. Wallner, T. Sauer (Passau).**First phase of the doctoral program:**-
**Wolfgang Carl**(Austria; since December 2012)

Personal homepage;*Email*: carl@tugraz.at

*Mentors*: Mihyun Kang, Franz Lehner.

*Thesis Title*: "Semidiscrete surfaces".

*PhD Defense*: June 3, 2016.

*Referees*: J. Wallner, T. Hoffmann (München), W. Rossmann (Kobe).

*Examiners*: J. Wallner, T. Hoffmann (München). -
**Florian Lehner**(Austria; July 2011–July 2014)

Personal homepage;*Email*: f.lehner@tugraz.at

*Mentors*: Wilfried Imrich, Bettina Klinz, Wolfgang Woess.

*PhD Defense*: July 4, 2014.

*Referees*: W. Imrich (Leoben), S. Klavzar (Ljubljana), J. Wallner.

*Examiners*: W. Imrich (Leoben), S. Klavzar (Ljubljana).

### Project description

Subdivision means the refinement of discrete data arranged either over
regular grids or over more general meshes. Nonlinear data, which have
values in manifolds or more general metric spaces are not accessible by
the well-established linear theory. A well-researched special case of such
data given by the cone of positive definite symmetric matrices which arise
in diffusion-tensor imaging and whose treatment is facilitated by a
symmetric space structure and a natural metric with the Cartan-Hadamard
property. Our work revolves around proving convergence of refinement
processes and investiating continuity properties of limits. We strive to
establish a body of results which covers large classes of subdivision
processes, recognizes sensible properties imposed on admissible input
data, and is valid for a large classes of geometries.

Showcases for possible PhD themes can be found here.