Hello,

I am (Priv. Doz. Dr.)  Carsten Schneider
 

You can reach me at the

Research Institute for Symbolic Computation

J. Kepler University Linz
Altenbergerstraße 69
A-4040 Linz, Austria

Tel.: ++43 732 2468 9966
Fax: ++43 732 2468 9930
E-Mail: Carsten.Schneider@risc.jki.at


 
 

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. Multi-summation in difference rings

Within difference fields and rings I am developing summation theories that enable one to simplify definite nested multi-sums 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. 1-36.2007.ISSN 1286-4889.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]
  • C. SchneiderModern Summation Methods for Loop Integrals in Quantum Field Theory: The Packages Sigma, EvaluateMultiSums and SumProduction. In: Proc. ACAT 2013, , J. Phys.: Conf. Ser.523/012037, pp. 1-17.2014.ISSN 1742-6596. arXiv:1310.0160 [cs.SC]. [url] [bib]

  • A recent tutorial (held at SYNASC'14, Timisoara, Romania):  [Slides] [Mathematica]

2. Exploring properties of indefinite nested sums and products

In order to simplify multi-sums (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 Elektronen-Synchrotron (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 RISC-DESY 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 co-speaker) 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.