Symbolic Summation in Difference Rings and Challenging Applications

Speaker:
Date: 30/06/2021
Time: 11:30 - 12:20

Location: online via Zoom

ABSTRACT: My main research interest is computer algebra: I develop algorithmic theories in algebra, implement efficient and stable software packages based on these sophisticated theories, and apply them to non-trivial problems in mathematics and interdisciplinary research areas. In the first part of the talk I will focus on my theoretical and algorithmic contributions. Starting with Karr's summation algorithm (1981) I have developed in the last 20 years a general summation framework that can deal with indefinite and definite multi-sum expressions. The underlying backbone is a new algorithmic difference ring theory that is strongly connected to the Galois theory of difference equations. This general machinery enables one to rephrase expressions in terms of indefinite nested sums and products completely automatically to canonical form representations. As a by-product, the computed expressions are built by sums and products that are algebraically independent among each other. Together with general and highly efficient difference ring algorithms for recurrence finding and solving, one can simplify in non-trivial cases gigantic expressions of multi-sums and related multi-integrals to certain classes of special functions defined by iterated sums and integrals. Summarizing, the derived algorithms for symbolic summation and special functions provide a flexible toolbox that can tackle challenging problems coming, e.g., from combinatorics, special functions, number theory, statistics, and particle physics (a long-term collaboration with DESY, Deutsches Elektronen-Synchrotron). In the second part of my talk I will illustrate how computer algebra in general, and in particular with the developed tools in my group, can be utilized to solve such complicated problems that are hard to solve or that are completely unreachable with other methods.