# Workshop on Symbolic Combinatorics and Computational Differential Algebra

September 14 - 19, 2015, The Fields Institute

## Overview

This workshop is devoted to algorithmic developments in Combinatorics and Differential Algebra with a particular focus on the interaction of these two areas.

* Symbolic Combinatorics:* Symbolic algorithms and software have recently been developed that allow researchers to discover and prove combinatorial identities as well as understand analytic and algebraic properties of generating functions. These functions seldom have closed form solutions and even when they do, direct evaluation may be intractable. Recent work in Symbolic Combinatorics has focused on representing such functions by annihilating differential or difference operators and then using techniques from differential and difference algebra, as well as analysis, to analyze these equations and their solutions. Besides methods relating to creative telescoping and verication of function identities, other areas of emphasis will include effective methods in difference algebra, the Galois theories of difference equations, and the interaction of differential and difference algebra in combinatorics.

* Computational Differential Algebra* is an approach to nonlinear or linear differential equations and differential structures, focusing not only on finding explicit closed form solution, but also on simplifying such equations to yield preconditioning for subsequent numerical solution, as well clearer understanding of the qualitative behavior of solutions. Highlighted aspects will include differential-elimination completion algorithms; geometric differential algebra, moving frames and differential invariants; differential Galois theory and integrability; complexity and open problems.

## Schedule

09:00 |
Rational Invariants of Finite Abelian Groups and their use in solving polynomial systems
George Labahn, University of Waterloo |

10:10 |
Partial Integral Operators and the Hopf Algebra of Linear Substitutions
Markus Rosenkranz, University of Kent |

11:30 |
Targeted overdetermined functional decomposition and applications to differential equations
Austin Roche, Maplesoft |

14:00 |
Creative Telescoping via Hermite Reduction
Manuel Kauers, Johannes Kepler University |

15:30 |
Algebras of Differential Invariants
Peter J. Olver, University of Minnesota |

09:00 |
Quasi-optimal computation of the p-curvature
Alin Bostan, Inria |

10:10 |
On the computation of the difference-differential Galois group for a second-order linear difference equation
Carlos Arreche, North Carolina State University |

11:30 |
Combinatorics on polynomial equations: do they describe nice varieties?
Joachim von zur Gathen |

14:00 |
A primer on lattice path combinatorics
Rika Yatchak, Johannes Kepler University |

15:10 |
The combinatorics behind lattice path asymptotics
Marni Mishna, Simon Fraser University |

09:00 |
Lattice based multispace and applications to moving frames
Elizabeth Mansfield, University of Kent |

10:10 |
Algorithms for computing Galois groups of parameterized differential equations
Alexey Ovchinnikov, CUNY Queens College |

11:30 |
Computer Algebra Algorithms for Modular Functions
Peter Paule, Johannes Kepler University |

09:00 |
A moment matrix approach to symmetric cubatures
Evelyne Hubert, INRIA Méditerranée |

10:10 |
To Think in a Symbolic-Computational Way...
Doron Zeilberger, Rutgers University |

11:30 |
Darboux transformations for differential operators on the superline
Ekaterina Shemyakova, State University of New York at New Paltz |

14:00 |
Uncoupling of Linear Difference/Differential Systems of Higher Order
Johannes Middeke, Johannes Kepler University |

15:10 |
Explicit generating series for small-step walks in the quarter plane
Frédéric Chyzak, INRIA Saclay Île-de-France |

09:00 |
From algebraic to differentialy-algebraic functions in combinatorics: impact of positive (integer) coefficients
Cyril Banderier, Univ Paris 13 |

10:10 |
On Apparent Singularities of Systems of Linear Differential Equations with Rational Function Coefficients
Moulay A. Barkatou, XLIM, CNRS, University of Limoges |

11:30 |
Symbolic-Numeric Differential Algebra and Applications
Greg Reid, Western University |

14:00 |
The generalized inverses of Ore matrices and applications
Yang Zhang, University of Manitoba |

15:10 |
Effective Analytic Combinatorics in Several Variables, With Applications to Lattice Path Enumeration
Stephen Melczer, ENS Lyon and University of Waterloo |

09:00 |
Two problems at the interface of combinatorics and symbolic computation
Christian Krattenthaler, Universitaet Wien |

10:10 |
Formal Reduction of Singularly-Perturbed Linear Differential Systems
Sumayya Suzy Maddah |

11:30 |
Computational and algebraic aspects of integro-differential operators
Georg Regensburger, Austrian Academy of Sciences |