I and my research group Computer
algebra
for
Quantum
Field Theory, a subgroup of the RISC
combinatorics group, develop
Computer algebra algorithms for nested sums and
products
with the task to apply them, e.g., to combinatorial problems,
number theory,
numerics,
and particle
physics
More information
can be found, e.g., on the publication
list of my group, on MathSciNet,
or on Google
Scholar. In short, I focus currently on two main
aspects:
1. Multisummation in difference rings
Within difference fields and rings I am developing
summation theories that enable one to simplify definite
nested multisums to representations in terms of
indefinite nested sums and products. Special emphasis is
put on representations which are optimal, e.g., concerning
their nested depth. The resulting algorithms of the
underlying summation theory are implemented in the
summation package Sigma.
Among other features, the following tasks can be
handled:
 Compute
recurrences (based on the paradigm of Z's
creative telescoping) for definite sums with summands
given in terms of indefinite nested sums and products.
 Solve
recurrences in terms of all solutions that
are expressible in terms of indefinite nested sums and
products (d'Alembertian solutions).
 Simplify
indefinite nested sums and products to
representations with optimal nesting depth.
All three components combined deliver strong tools in
order to compute closed forms of summation problems. For
more details see, e.g.,
C. Schneider. Symbolic Summation Assists
Combinatorics. Sem. Lothar. Combin. 56, pp. 136. 2007. ISSN 12864889. Article B56b. [url] [ps]
[pdf]
[bib]
C. Schneider. Simplifying Multiple
Sums in Difference Fields. In: Computer Algebra in Quantum Field
Theory: Integration, Summation and Special Functions, J. Bluemlein, C. Schneider (ed.), Texts and Monographs in Symbolic
Computation. 2013. Springer. To appear. [url]
[bib]
2. Exploring properties of indefinite nested
sums and products
In order to simplify multisums (with infinite summation
ranges) to indefinite nested sums and products, algebraic
and analytic properties of indefinite nested sums are
crucial. In particular, I am interested, e.g., in
the following problems:
 Prove algebraic
independence of indefinite nested sums and products
exploiting difference ring theory.
 Find algebraic
relations of infinite nested sums (multiple
zeta values and generalizations of them) using the so
far understood algebraic relations induced, e.g., by
shuffle algebras, stuffle algebras, duplication
relations, etc.
 Compute asymptotic
expansions of indefinite nested sums by
exploiting, e.g., integral representations (Mellin
transform).
Currently, I am applying these tools to the following two
major research fields.
Long term
cooperation: Quantum Field Theory
From 2007 on my group Computer
algebra
for
Quantum
Field Theory applies these summation tools in
particle physics. This long term project is carried out in
close and intensive cooperation with the theory group (Johannes
Blümlein) at Deutsches ElektronenSynchrotron (DESY Zeuthen, a research
centre of the German Helmholtz association). In this
interdisciplinary project we try to deal with challenging
problems in the field of particle physics and perturbative
quantum field theory with the help of our summation
technology. In particular, the interaction between the
RISCDESY cooperation inspires new summation and special
function technologies that are also of interest for other
sciences and mathematical disciplines, like combinatorics,
number theory, or numerics.
SFB Algorithmic
and Enumerative Combinatorics
I am Principal Investigator (and cospeaker) of the
Special Research Program (SFB, in short for the German
name SpezialForschungsBereich) Algorithmic and
Enumerative Combinatorics which will start in March
2013. It is a special effort of three institutes, the Faculty of Mathematics (University of Vienna),
the Institute for Discrete
Mathematics and Geometry (Technical University Vienna),
and the Research Institute for Symbolic Computation
(Johannes Kepler University Linz) funded by the Austrian
Science Funds (FWF).
In the project part "Computer algebra of nested sums and
products" the research topics mentioned above will be
pushed forward in the context of combinatorial problems.
