Symbolic Computation - Research Topics

Designing mechanical devices, called linkages, that perform a prescribed motion has been a topic that interested engineers and mathematicians for hundreds of years. We apply techniques from algebraic geometry and symbolic computation to various problems in this area, for example the classification of closed 6R linkages, the study of pentapods and hexapods [2], or the construction of planar linkages [3]. In this context, the concept of motion polynomial and the theory of bonds have been developed, which are powerful tools to answer questions arising in kinematics.

[1] Zijia Li, Josef Schicho (2013): Classification of Angle Symmetric 6R Linkages. Mechanism and Machine Theory, Bd. 70, S. 372-379.
[2] Josef Schicho, Matteo Gallet, Georg Nawratil (2015): Bond theory for pentapods and hexapods. Journal of Geometry, Bd. 106 (2), S. 211-228.
[3] http://www.koutschan.de/data/link/

The field of Computer Aided Geometric Design (CAGD) deals with the mathematical construction and representation of shapes for applications in industrial developments and production processes. One of the most effective tools in CAGD to approximate surfaces is the use of splines, which are piecewise polynomial functions on a partition of a real domain with a determined order of smoothness at the places where the polynomial pieces connect. Splines are also universally recognized as one of the most powerful tools for approximating the solution of partial differential equations, and therefore essential in novel fields such as Iso-geometric analysis.

The study of splines involves an interplay between the underlying combinatorics and geometry of the partition, and the algebraic properties of the resulting functions. A fundamental question is whether a surface given by a polynomial equation can be parametrized by rational functions. An algorithm for deciding this, and for computing a parametrization in the affirmative case, has been designed and implemented in a cooperation with the University of Sydney in Magma.

[1] Villamizar, Nelly; Mantzaflaris, Angelos; Juettler, Bert (2016, online: 2015): Characterization of bivariate hierarchical quartic box splines on a three-directional grid. Computer Aided Geometric Design, Bd. 41, S. 1-15.
[2] http://magma.maths.usyd.edu.au/magma/handbook/algebraic_surfaces

Many elementary and special functions arising in mathematics and physics are solutions to linear differential equations, and similarly, many combinatorial sequences satisfy linear recurrence equations. If such functions satisfy further technical conditions, they are called holonomic. A proof theory for holonomic functions was developed in [1], which adresses identities involving summation quantifiers and integrals. An algorithmic ingredient to proving such identities is creative telescoping. We are concerned with inventing fast creative telescoping algorithms, implementing them in computer algebra systems, and applying them to various problems. The application areas of these techiques include combinatorics, special functions, statistical physics, knot theory, numerical analysis, engineering, and many more; see [2] for a detailed survey.

[1] Herbert S. Wilf, Doron Zeilberger (1992): An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities. Inventiones Mathematicae, 108(1), S. 575-633.
[2] Christoph Koutschan (2013): Creative telescoping for holonomic functions. In: Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, Carsten Schneider, Johannes Blümlein (editors), Texts & Monographs in Symbolic Computation, S. 171-194, Springer, Wien.