Computer Algebra for Functional Equations in Combinatorics and Physics

Liste des contributions

1. Creative Telescoping by Shaoshi Chen, Manuel Kauers & Christoph Koutschan. 9:30-12:00. IHP, Amphitheater Hermite

27/11/2023 09:30

6. A generating function method for the determination of differentially algebraic integer sequences modulo prime powers by Christian Krattenthaler. 15:00-17:00. IHP, Amphitheater Hermite

27/11/2023 15:00

2. Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite

28/11/2023 09:30

7. A generating function method for the determination of differentially algebraic integer sequences modulo prime powers by C. Krattenthaler. 15:00-17:00. IHP, Amphitheater Hermite

28/11/2023 15:00

3. Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite

29/11/2023 09:30

47. General audience presentation The Zeta Team by Wadim Zudilin, Radboud University. 15:00-16:00. IHP, Amphitheater Hermite

29/11/2023 15:00

Abstract. An innocent story of finding recursions for the sums of powers of binomial coefficients transformed into an intrigue involving creative telescoping and irrationality. The goalscorers performed as a dream team.

8. General audience presentation. On Fermat’s penultimate theorem by Xavier Caruso, IMB, Bordeaux. 16:00-17:00. IHP

29/11/2023 16:00

Abstract. While reading books, Fermat often added notes in the margin for himself. The most famous one is certainly what is called nowadays Fermat's Last Theorem which stipulates that the equation xn+yn=zn has no nontrivial solution as soon as n>2. More than three centuries of effort were needed to eventually write down a proof of this result. In this presentation, I will discuss another...

4. Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite

30/11/2023 09:30

9. Special session of the Differential Seminar. Values of E-functions are not Liouville numbers by Tanguy Rivoal, Institut Fourier. 15:00-16:00. IHP, Amphitheater Hermite

30/11/2023 15:00

Abstract. A Liouville number is an irrational real number with an infinite irrationality exponent. Almost all real numbers are not Liouville numbers but it can be difficult in practice to prove that a given real number is not a Liouville one. In this talk, I will explain how a combination of results due to André, Beukers and Shidlovskii enables to prove that the real values of E-functions at...

48. Special session of the Differential Seminar. Rounding error analysis of linear recurrences using generating series by Marc Mezzarobba, LIX, Palaiseau. 16:00-17:00. IHP, Amphitheater Hermite

30/11/2023 16:00

Abstract. In the computation of linearly recurrent sequences, round-off errors arising from each step tend to “cancel out” rather than just accumulate. This phenomenon is crucial to consider when aiming to establish realistic error bounds. That requires a close examination of how a given round-off error propagates through the remaining iterations of the recurrence. Doing so using traditional...

5. Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite

01/12/2023 09:30

10. Special session of the Differential Seminar. Computation of sums and integrals by reduction-based creative telescoping by Bruno Salvy, Inria, ENS Lyon. 15:00-16:00. IHP, Amphitheater Hermite

01/12/2023 15:00

Abstract. Creative telescoping is an algorithmic method initiated by Zeilberger to compute definite sums or integrals by synthesizing summands or integrands that telescope, called certificates. We describe a creative telescoping algorithm that computes telescopers for definite sums or integrals of D-finite functions as well as the associated certificates in a compact form. In the integral...

49. Special session of the Differential Seminar. Combinatorics and Transcendence: Applications of Inhomogeneous order 1 iterative functional equations by Marni Mishna, Simon Fraser University. 16:00-17:00. IHP, Amphitheater Hermite

01/12/2023 16:00

The problem of understanding the structure of transcendental objects has fascinated mathematicians for well over a century. Combinatorics provides an intuitive framework to study power series. A combinatorial family is associated to a power series in $\mathbb{Q}[[t]]$ via its enumerative generating function wherein the number of objects of size $n$ is the coefficient of $t^n$. Twentieth...

11. Welcome coffee

04/12/2023 09:30

16. Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials by Kilian Raschel

04/12/2023 10:00

Abstract. We establish exact formulae for the (positivity) persistence probabilities of an autoregressive sequence with symmetric uniform innovations in terms of certain families of polynomials, most notably a family introduced by Mallows and Riordan as enumerators of finite labeled trees when ordered by inversions. The connection of these polynomials with the volumes of certain polytopes is...

17. Stretched exponentials and beyond by Michael Wallner

04/12/2023 11:00

Abstract. The appearance of a stretched exponential term $\mu^{n^{\sigma}}$ with $\mu>0$ and $\sigma \in (0,1)$ in a counting sequence $(c_n)_{n\geq0}$ is not common, although more and more examples are appearing lately. Proving that a sequence has a stretched exponential is often quite difficult. This is in part because such a sequence cannot be "very nice": its generating function cannot be...

18. Summation Tools for Combinatorics and Elementary Particle Physics by Carsten Schneider

04/12/2023 15:00

Abstract. The summation theory of difference rings provides general and efficient tools to derive linear recurrences for definite sums based on parameterized telescoping and to solve recurrences within the class of indefinite nested sums defined over hypergeometric products, q-hypergeometric products and more generally, mixed hypergeometric products and their nested versions. In particular,...

19. Coffee break

04/12/2023 16:00

20. What is beyond D-finite? by Igor Pak

04/12/2023 16:30

Abstract: I will review some classes of GFs that are of interest in Enumerative Combinatorics but remain understudied. This talk will be accessible to the general audience.

12. Welcome coffee

05/12/2023 09:30

21. Galois group for large steps walks by Charlotte Hardouin

05/12/2023 10:00

Abstract. In this talk, we will present some Galois theoretic tools to study large steps walks confined in the quadrant. We generalize in particular the notion of group of the walk introduced by Bousquet-Mélou and Mishna for small steps walk to the large steps framework. This allows to develop algorithms and criteria to test the existence of invariants and decoupling functions. This is a...

22. Quadratizations of differential equations by Gleb Pogudin

05/12/2023 11:00

Abstract. Quadratization problem is, given a system of ODEs with polynomial right-hand side, transform the system to a system with quadratic right-hand side by introducing new variables. Such transformations have been used, for example, as a preprocessing step by model order reduction methods and for transforming chemical reaction networks. We will present a recent algorithm for computing such...

23. New Software for Analytic Combinatorics by Stephen Melczer

05/12/2023 15:00

Abstract. This talk surveys some recently developed software for analytic combinatorics, including an extension to the Sage ore_algebra package for the asymptotics of P-recursive sequences with explicit error terms (used for certifying sequence positivity), and the new sage_acsv package for rigorous multivariate asymptotics using the tools of Analytic Combinatorics in Several Variables (ACSV).

24. Coffee break

05/12/2023 16:00

26. Submodule approach to creative telescoping by Mark van Hoeij

05/12/2023 16:30

Abstract.This talk proposes ideas to speed up the process of creative telescoping, particularly when the telescoper is reducible. One can interpret telescoping as computing an annihilator $L$ in $D$ for an element $H$ in a D-module $M$. The main idea is to look for submodules of $M$. For a non-trivial submodule $N$, constructing the minimal operator $R$ of the image of $H$ in $M/N$ gives a...

13. Welcome coffee

06/12/2023 09:30

28. Some problems I’d like solved, from a user of computer algebra by Alan Sokal

06/12/2023 10:00

Abstract. A matrix $M$ of real numbers is called {\em totally positive}\/ if every minor of $M$ is nonnegative. Gantmakher and Krein showed in 1937 that a Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$ of real numbers is totally positive if and only if the underlying sequence $(a_n)_{n \ge 0}$ is a Stieltjes moment sequence, i.e.~the moments of a positive measure on $[0,\infty)$. Moreover, this...

29. Efficient algorithms for differential equations satisfied by Feynman integrals by Pierre Vanhove

06/12/2023 11:00

Abstract. Feynman integrals are a type of mathematical function that are important for precision measurements in physics. They are notoriously difficult to evaluate, and a lot of effort has been devoted to developing efficient analytic and numerical methods for doing so. In this talk, we will present a new algorithm for determining the minimal order Picard-Fuchs operator associated with a...

30. Cocktail. IHP, Perrin building, 2nd floor

06/12/2023 18:30

Emmy Noether salon located at the second floor of the Perrin building (IHP). We'll need to have a precise list of participants at the entrance to the building. If you were not registered before December or if you have any doubts, please register or contact the organizers.

14. Welcome coffee

07/12/2023 09:30

Special Day, joint with the Flajolet Seminar.

46. Computer algebra in my combinatorics life by Mireille Bousquet-Mélou

07/12/2023 10:00

Abstract. Many of my papers would just not exist without computer algebra. I will describe how CA has become an essential tool in my research in enumerative combinatorics. The point of view will be that of a (sometimes naive) user, not of an expert. Many examples and questions will be taken from a joint paper with Michael Wallner dealing with the enumeration of king walks avoiding a quadrant...

32. Self-avoiding walks in a square and the gerrymander sequence by Tony Guttmann

07/12/2023 11:00

Abstract. We give an improved algorithm for the enumeration of self-avoiding walks and polygons within an N×N square as well as SAWs crossing a square. We present some proofs of the expected asymptotic behaviour as the size N of the square grows, and then show how one can numerically estimate the parameters in the asymptotic expression. We then show how the improved algorithm can be adapted to...

31. Computer algebra for the study of two-dimensional exclusion processes by Arvind Ayyer

07/12/2023 15:00

Abstract. We define a new disordered asymmetric simple exclusion process (ASEP) with two species of particles, first-class and second-class, on a two-dimensional toroidal lattice. The dynamics is controlled by first-class particles, which only move horizontally, with forward and backward hopping rates $p_i$ and $q_i$ respectively if the particle is on row $I$. The motion of second-class...

27. Coffee break

07/12/2023 16:00

34. A new proof of Viazovska's modular form inequalities for sphere packing in dimension 8 by Dan Romik

07/12/2023 16:30

Maryna Viazovska in 2016 found a remarkable application of the theory of modular forms to a fundamental problem in geometry, obtaining a solution to the sphere packing problem in dimension 8 through an explicit construction of a so-called "magic function" that she defined in terms of classical functions, the Eisenstein series and Jacobi thetanull functions. The same method also led shortly...

15. Welcome coffee

08/12/2023 09:30

35. Partition identities, functional equations and computer algebra by Jehanne Dousse

08/12/2023 10:00

A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. A partition identity is a theorem stating that for all n, the number of partitions of n satisfying some conditions equals the number of partitions of n satisfying some other conditions. In this talk, we will show how functional equations and computer algebra can be used to prove such...

36. Beyond Painlevé: The need for computational tools to reveal hidden structure by Nicholas Witte

08/12/2023 11:00

Abstract. The simplest examples of integrability in mathematical physics - spectral distributions of fundamental random matrix ensembles or the diagonal spin-spin correlations of the square lattice Ising model - to give just two examples, reveal the importance and practical utility of the six Painlevé equations in our understanding of these models. Few aspects of this understanding escape the...

37. Opening remarks for Topical day: Elimination for Functional Equations. IHP, Amphitheater Darboux

11/12/2023 09:45

38. Differential Equation Invariance Axiomatization by André Platzer

11/12/2023 10:00

25. Coffee break

11/12/2023 10:45

39. Differential elimination ideals and spectral curves by Sonia Rueda

11/12/2023 11:15

Abstract. Commuting pairs of ordinary differential operators have been related to plane algebraic curves at least since the work of Burchnall and Chaundy a century ago. This talk is devoted to the revision of some classical results, using now a differential algebra framework and, oriented to the development of algorithms based in the computation of differential resultants. The new concept of...

40. Geometry-driven algorithms for combinatorial functional equations by Hadrien Notarantonio

11/12/2023 14:00

Abstract. Enumerative combinatorics contains a vast landscape of problems that could hardly be solved without the consideration of special functional equations called “Discrete Differential Equations”. Among these problems, the enumeration of walks, planar maps carrying hard particles, etc. These functional equations relate formal power series in n variables with specializations of them to...

41. Examples of use of functional equations obtained from the elimination theory in nonlinear models by Nathalie Verdière

11/12/2023 14:30

Abstract. This talk addresses the use of functional equations obtained from computer algebra in the structural properties study of nonlinear dynamical models. For several years, they were exploited in the context of identifiability and diagnosticability studies. A final application was recently developed and concerns an a priori study for the reconstruction of some variables of interest in...

42. Coffee break

11/12/2023 15:15

43. On some functional equations for maps by Alexandros Singh

11/12/2023 15:45

Abstract. The goal of this exploratory talk is to present various functional equations related to the enumeration of maps and the study of various combinatorial parameters thereof. In particular, we'll concern ourselves with questions such as: how are these equations related and can they be inter-derived by combinatorial or differentially-algebraic methods? This presentation draws from...

44. Solving differential elimination problems with Thomas decomposition by Daniel Robertz

11/12/2023 16:15

Abstract. This talk gives an introduction to the Thomas decomposition method for systems of nonlinear partial differential equations. A Thomas decomposition is a finite family of so-called simple differential systems, each of which is formally integrable, and such that the solution set of the given PDE system is the disjoint union of the solution sets of the simple systems. This versatile...

45. Closing remarks

11/12/2023 17:00

33. Proving positivity for P-recursive sequences by Veronika Pillwein

Abstract. In this talk we consider the problem of automatically proving inequalities involving sequences that are only given in terms of their defining recurrence relations. We will consider sequences satisfying linear recurrences with constant coefficients (C-recursive), linear recurrences with polynomial coefficients (P-recursive or holonomic), or certain systems of polynomial non-linear...