Dates: November 27 –
December 1 (Special week),
December 4–8 (Workshop),
December 11 (Topical day)
Place: Amphitheater Hermite at the IHP
Registration
Schedule
Live stream
How to come to IHP
Some restaurants close to IHP
Workshop: Computer Algebra for Functional Equations in Combinatorics and Physics
December 4 to 8, 2023 (Amphitheater Hermite)
Organizers: A. Bostan, J. Bouttier, T. Cluzeau, L. Di Vizio, C. Krattenthaler, P. Lairez, J.M. Maillard
In many areas of pure and applied mathematics, as well as in computer science and in theoretical physics, functional equations form either the object of study or important tools for applications. We are currently experiencing increasingly strong interactions between theory and applications, many common actions having taken place over the past ten years. By functional equations, we mean mainly ordinary differential equations, with differences, with $q$differences, Mahlerian, linear or algebraic, possibly multivariate. For instance, nonlinear algebraic differential equations emerge naturally in integrable models in physics (Painlevé equations, Schlesinger systems, KdV equations, etc., associated with Lax pairs, YangBaxter equations,…). All these types of functional equations have been and are still very actively studied from many points of view, using algebraic, arithmetic and geometric tools. A recent trend is that computer algebra algorithms are more and more used to solve functional equations arising in enumerative combinatorics and in statistical physics. Notable examples come from questions related to lattice walks. In combinatorics, basic objects like trees, maps, permutations, and Young tableaux can be represented by models of walks confined to cones. In physics, many objects, including polymers and queueing models, are accurately modeled by walks on lattices, particularly those evolving in cones with several boundaries. This workshop brings together representatives from the three different communities (computer algebra, combinatorics and theoretical physics) to discuss longstanding conjectures, to learn each other’s techniques and to plan the directions for the future.
Invited speakers
 Arvind Ayyer (Bangalore, India): Computer algebra for the study of twodimensional exclusion processes
 Mireille BousquetMélou (Bordeaux, France): Computer algebra in my combinatorics life
 Jehanne Dousse (Geneva, Switzerland): Partition identities, functional equations and computer algebra
 Tony Guttmann (Melbourne, Australia): Selfavoiding walks in a square and the gerrymander sequence
 Charlotte Hardouin (Toulouse, France): Galois group for large steps walks
 Mark van Hoeij (Tallahassee, Florida, USA): Submodule approach to creative telescoping
 Stephen Melczer (Waterloo, Canada): New Software for Analytic Combinatorics
 Igor Pak (Los Angeles, USA): What is beyond Dfinite
 (Veronika Pillwein (Hagenberg, Austria): Proving positivity for Precursive sequences. Cancelled.)
 Gleb Pogudin (Palaiseau, France): Quadratizations of differential equations
 Kilian Raschel (Angers, France): Persistence for a class of orderone autoregressive processes and MallowsRiordan polynomials
 Dan Romik (California, USA): A new proof of Viazovska’s modular form inequalities for sphere packing in dimension 8
 Carsten Schneider (Hagenberg, Austria): Summation Tools for Combinatorics and Elementary Particle Physics
 Alan Sokal (London, UK): Some problems I’d like solved, from a user of computer algebra
 Pierre Vanhove (Saclay, France): Efficient algorithms for differential equations satisfied by Feynman integrals
 Michael Wallner (Vienna, Austria): Stretched exponentials and beyond
 Nicholas Witte (Wellington, New Zealand): Beyond Painlevé: The need for computational tools to reveal hidden structure
Program of the workshop  All talks in Amphitheater Hermite
Monday December 4th, 2023  

9:3010:00  Welcome coffee  
10:0011:00  Kilian Raschel  Persistence for a class of orderone autoregressive processes and MallowsRiordan polynomialsAbstract. 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 also discussed. Two further results provide factorizations of general autoregressive models, one for negative drifts with continuous innovations, and one for positive drifts with continuous and symmetric innovations. The second factorization extends a classical universal formula of Sparre Andersen for symmetric random walks. Our results also lead to explicit asymptotic estimates for the persistence probabilities. This is a joint work with Gerold Alsmeyer, Alin Bostan and Thomas Simon (Adv. Appl. Math., 2023). 
11:0012:00  Michael Wallner  Stretched exponentials and beyondAbstract. 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 algebraic, and it can only be Dfinite if it has an irregular singularity. Previously, the saddlepoint method was the only generic method for proving such a phenomenon, but it requires detailed information about the generating function. Recently, together with Andrew Elvey Price and Wenjie Fang, we have developed a new method at the level of recurrences to prove stretched exponentials. I will introduce the basics of this method and show how it can be extended to other problems. Then I will summarize recent progress (new bijections, limit laws, etc.) in the study of compacted trees, a subclass of directed acyclic graphs. Finally, I will give an outlook on how these results now allow an indepth study of limit shapes and open many new avenues for further research. 
12:0015:00  Lunch break  
15:0016:00  Carsten Schneider  Summation Tools for Combinatorics and Elementary Particle PhysicsAbstract. 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, qhypergeometric products and more generally, mixed hypergeometric products and their nested versions. In particular, one can use these techniques to simplify definite multisums to representations in terms of the class indefinite nested sums defined over indefinite nested products. A special feature is that the representation of the arising sums and products are optimal in the sense that the objects interpreted as sequences (except root of unity products) do not satisfy any algebraic relations among each other. This leads not only to compact expressions in the final output, but also speeds up significantly the underlying summation algorithms. In particular, one gains a general (Galois) machinery to prove algorithmically algebraic independence of big classes of sums and products. We will illustrate this algorithmic difference ring toolbox by nontrivial applications coming from enumerative combinatorics and elementary particle physics. 
16:0016:30  Coffee break  
16:3017:30  Igor Pak  What is beyond DfiniteAbstract. 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. 
Tuesday December 5th, 2023  
9:3010:00  Welcome coffee  
10:0011:00  Charlotte Hardouin  Galois group for large steps walksAbstract. 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 BousquetMélou 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 collaboration with Pierre Bonnet (Labri). 
11:0012:00  Gleb Pogudin  Quadratizations of differential equationsAbstract. Quadratization problem is, given a system of ODEs with polynomial righthand side, transform the system to a system with quadratic righthand 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 transformations and its extensions including systems with control and of varying dimension. The talk is based on joint works with Andrey Bychkov, Opal Issan, and Boris Kramer. 
12:0015:00  Lunch break  
15:0016:00  Stephen Melczer  New Software for Analytic CombinatoricsAbstract. This talk surveys some recently developed software for analytic combinatorics, including an extension to the Sage ore_algebra package for the asymptotics of Precursive 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). 
16:0016:30  Coffee break  
16:3017:30  Mark van Hoeij  Submodule approach to creative telescopingAbstract. 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 Dmodule $M$. The main idea is to look for submodules of $M$. For a nontrivial submodule $N$, constructing the minimal operator $R$ of the image of $H$ in $M/N$ gives a rightfactor of $L$ in $D$. Then $L = L' R$ where $L'$ is the telescoper of $R(H)$. To expedite computing $L'$, compute the action of $D$ on a natural basis of $N$, then obtain the telescoper $L'$ for $R(H)$ with a cyclic vector computation. The next main idea is that when $N$ has automorphisms, use them to construct submodules. An automorphism with distinct eigenvalues can be used to decompose $N$ as a direct sum of submodules $N_1, \ldots, N_k$. Then $L' = \text{LCLM}(L_1, \ldots, L_k)$ where $L_i$ is the telescoper of the projection of $R(H)$ on $N_i$. An LCLM can greatly increase the degrees of the coefficients, so $L'$ and hence $L$ can be much larger than the factors $L_1, \ldots, L_k$ and $R$. Examples show that computing each factor $L_i$ and $R$ separately can save a lot of CPU time compared to computing the full telescoper $L$ all at once with standard creative telescoping. 
Wednesday December 6th, 2023  
9:3010:00  Welcome coffee  
10:0011:00  Alan Sokal  Some problems I’d like solved, from a user of computer algebraAbstract. 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 holds if and only if the ordinary generating function $\sum_{n=0}^\infty a_n t^n$ can be expanded as a Stieltjestype continued fraction with nonnegative coefficients. So totally positive Hankel matrices are closely connected with the Stieltjes moment problem and with continued fractions. Here I will introduce a generalization: a matrix $M$ of polynomials (in some set of indeterminates) will be called {\em coefficientwise totally positive}\/ if every minor of $M$ is a polynomial with nonnegative coefficients. And a sequence $(a_n)_{n \ge 0}$ of polynomials will be called {\em coefficientwise Hankeltotally positive}\/ if the Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$ associated to $(a_n)$ is coefficientwise totally positive. It turns out that many sequences of polynomials arising naturally in enumerative combinatorics are (empirically) coefficientwise Hankeltotally positive. In some cases this can be proven using continued fractions, by either combinatorial or algebraic methods; in other cases this can be done using a more general algebraic method called {\em production matrices}\/. However, in a very large number of other cases it remains an open problem. Along the way I will mention some problems in computer algebra, the solution of which would be helpful to this research. 
11:0012:00  Pierre Vanhove  Efficient algorithms for differential equations satisfied by Feynman integralsAbstract. 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 PicardFuchs operator associated with a given Feynman integral. This operator is a differential operator that governs the analytic behavior of the Feynman integral. We will first discuss an implementation of the GriffithsDwork algorithm for the case of Feynman integrals in integer spacetime dimension. In that case, the integrand of the Feynman integral is a rational differential form to which the GriffithsDwork reduction is applied for determining the PicardFuchs operator. We will then extend this algorithm to the case of generic spacetime dimension. In this case, one needs to consider twisted cohomology. We will show how the knowledge of the PicardFuchs operator can be deduced by Hodge theoretic considerations on the variation of the mixed Hodge structure associated to the Feynman integral. This talk is based on work with Pierre Lairez, Eric PichonPharabod, Charles Doran, and Andrew Harder. 
Free afternoon  
18:30  Reception. Emmy Noether reception room 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.  
Thursday December 7th, 2023  
Special Day, joint with the Flajolet Seminar  
9:3010:00  Welcome coffee  
10:0011:00  Mireille BousquetMélou  Computer algebra in my combinatorics lifeAbstract. 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 (arXiv 2021, to appear). My hope is that some of the questions that I will raise will have an immediate answer ("yes, this is done") and/or that some people in the audience will find a question interesting enough to take it back home. 
11:0012:00  Tony Guttmann  Selfavoiding walks in a square and the gerrymander sequenceAbstract. We give an improved algorithm for the enumeration of selfavoiding walks and polygons within an $N \times N$ 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 count gerrymander sequences (OEIS A348456), and prove that the asymptotics of the gerrymander sequence is similar to that of SAWs crossing a square. This work has been done in collaboration with Iwan Jensen, and in part with Aleks Owczarek. 
12:00  Group photo (1) Group photo (2)  
12:0015:00  Lunch break  
15:0016:00  Arvind Ayyer  Computer algebra for the study of twodimensional exclusion processesAbstract. We define a new disordered asymmetric simple exclusion process (ASEP) with two species of particles, firstclass and secondclass, on a twodimensional toroidal lattice. The dynamics is controlled by firstclass 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 secondclass particles depends on the relative position of these with respect to the firstclass ones, and can be both horizontal and vertical. In the first part of my talk, I will illustrate how we used computer algebra software, specifically Mathematica and SageMath, to understand the stationary distribution of this process. We computed the partition function, as well as densities and currents of all particles in the steady state. We observed a novel mechanism we call the Scott Russell phenomenon: the current of second class particles in the vertical direction is the same as that of firstclass particles in the horizontal direction. In the second part of my talk, I will show how we simulated the process and realized that the Scott Russell phenomenon also holds out of equilibrium. This is partly joint work with P. Nadeau (European Journal of Combinatorics, 2022). 
16:0016:30  Coffee break  
16:3017:30  Dan Romik  A new proof of Viazovska's modular form inequalities for sphere packing in dimension 8Maryna 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 socalled "magic function" that she defined in terms of classical functions, the Eisenstein series and Jacobi thetanull functions. The same method also led shortly afterwards to the solution of the sphere packing in dimension 24 by her and several collaborators. One component of Viazovska's proof consisted of proving a pair of inequalities satisfied by the modular forms she constructed. Viazovska gave a proof of these inequalities that relied in an essential way on computer calculations. In this talk I will present a new proof of Viazovska's inequalities that uses only elementary arguments that can be easily checked by a human. 
Friday December 8th, 2023  
9:3010:00  Welcome coffee  
10:0011:00  Jehanne Dousse  Partition identities, functional equations and computer algebraA partition of a positive integer $n$ is a nonincreasing 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 identities. In particular we will discuss a semiautomatic method using recurrences and $q$difference equations, and what would be needed to make it fully automatic. 
11:0012:00  Nicholas Witte  Beyond Painlevé: The need for computational tools to reveal hidden structureAbstract. The simplest examples of integrability in mathematical physics  spectral distributions of fundamental random matrix ensembles or the diagonal spinspin 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 presence of these equations, whether it is the application of RiemannHilbert methods to the asymptotic limits as matrix ranks or spin separations tend to infinity or the relationships inherited from the affine Weyl group symmetries of the Painlevé equations. This understanding operates in both directions: fundamental understanding of the Painlevé equations provides the most powerful and rigorous tools to characterise a variety of statistics applied to the models, and these applications are driving and motivating a lot of studies into the pure mathematics at the core of these equations. However today we coming up against numerous examples where one can pose questions just outside the simplest models: e.g. the singular value distribution of a product of two Ginibre random matrices, or the diagonal correlations of the triangular Ising model or even the offdiagonal correlations of the square lattice model, and one is beyond the six Painlevé set or even the discrete Sakai scheme. Integrability is still present and some of this territory  the Garnier, FujiSuzuki, Sasano and matrixPainlevé systems  has been sketched out but there are gaps in our understanding and our toolbox is missing critical tools that one needs. In some senses the complexity has grown and it is often impractical to do handcalculations and when computerassisted systems are employed our best results so far cannot be analysed by hand. There are hints that simplicity is achievable yet the standard computer assisted algorithms do not find them. So the challenge is to develop some flexible, modeladapted algorithms for doing what are essentially algebraic geometry calculations. A number of examples, including those mentioned previously, will illustrate this state of affairs. 
End of the workshop 
Special week
November 27 to December 1, 2023 (Amphitheater Hermite)

Creative Telescoping (long course, Monday to Friday, 9:3012:00), Shaoshi Chen, Manuel Kauers and Christoph Koutschan
Notes SC (Mon); Notes SC (Tue); Notes SC (Wed); Notes SC (Thu); Notes SC (Fri);
Demo MK (Mon); Slides MK (Tue); Slides MK (Wed); Slides MK (Thu); Slides MK (Fri);
Slides CK (Mon); Slides CK (Tue); Demo CK (Tue); Demo CK (Tue); Slides CK (Wed); Notes CK (Wed); Demo CK (Wed); Demo CK (Wed); Slides CK (Thu); Notes CK (Thu); Slides CK (Fri); Demo CK (Fri); Demo CK (Fri). 
A generating function method for the determination of differentially algebraic integer sequences modulo prime powers (short course, Monday and Tuesday, 15:0017:00), Christian Krattenthaler
Slides (Mon); Slides (Tue).
 General audience presentations (Wednesday 29th, 15:0017:00)
15:0016:00, The Zeta Team, Wadim Zudilin, Radboud University, Nijmegen, Netherlands
NotesAbstract. 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.
16:0017:00, On Fermat's penultimate theorem, Xavier Caruso, IMB, Bordeaux, France
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 $x^n + y^n = z^n$ 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 margintheorem by Fermat. It is less known but still quite impressive as it took almost two centuries to the community to obtain a full proof, opening as a byproduct new topics of research.
 Special session of the Differential Seminar (Thursday and Friday, 15:0017:00)
Nov. 30th, 15:0016:00, Values of Efunctions are not Liouville numbers, Tanguy Rivoal, Institut Fourier, Grenoble, France
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 Efunctions at algebraic points are not Liouville numbers. This is a joint work with Stéphane Fischler (Université Paris Saclay).
Nov. 30th, 16:0017:00, Rounding error analysis of linear recurrences using generating series, Marc Mezzarobba, LIX, Palaiseau, France
SlidesAbstract. In the computation of linearly recurrent sequences, roundoff 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 roundoff error propagates through the remaining iterations of the recurrence. Doing so using traditional “sequence” notations results in intricate calculations involving nested sums. However, in other contexts, such as enumerative combinatorics, the use of generating series to encode sequences simplifies these calculations while simultaneously granting access to powerful new analytic tools. In this talk, I will show how the idea of using generating series can be adapted to the error analysis of numerical algorithms based on linear recurrence relations.
Dec. 1st, 15:0016:00, Computation of sums and integrals by reductionbased creative telescoping, Bruno Salvy, Inria, ENS Lyon, France
SlidesAbstract. 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 Dfinite functions as well as the associated certificates in a compact form. In the integral case, the algorithm relies on a generalization of the Hermite reduction in symbolic integration. In the sum case, the algorithm relies on a discrete analogue of the generalized Hermite reduction, or equivalently, a generalization of the AbramovPetkovsek reduction. We present a Maple implementation with good timings on a variety of examples.
Dec. 1st, 16:0017:00, Combinatorics and Transcendence: Applications of Inhomogeneous order 1 iterative functional equations, Marni Mishna, Simon Fraser University, Burnaby, Canada
SlidesThe 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 century combinatorics and theoretical computer science provided characterizations of classes with rational and algebraic generating functions. Finding natural extensions of these correspondences has been a motivating goal of enumerative combinatorics for several decades. This talk will focus on differentially transcendental functions. In particular, I will present recent work completed with Lucia Di Vizio and Gwladys Fernandes which characterizes solutions $f(t)$ of order 1 iterative equations of the form $f(R(t))=a(t)f(t)+b(t)$ where $R,a$, and $b$ are rational functions. These appear in the study of complete trees, walks on selfsimilar graphs (eg. the Sierpinski graph), and pattern avoiding permutations. The proof strategy is inspired by the Galois theory of functional equations, and relies on the property of the dynamics of $R(t)$, LiouvilleRosenlicht's theorem and Ax' theorem. This program and has led to progress in identifying the differential transcendence of combinatorial generating functions arising in the literature, and indeed generally.
Topical day: Elimination for Functional Equations
December 11, 2023 (Amphitheater Darboux)
Organizer: G. Pogudin
Speakers: Hadrien Notarantonio, André Platzer, Daniel Robertz, Sonia Rueda, Alexandros Singh, Nathalie Verdière
Schedule:

9:4510:00 Opening remarks
 10:0010:45 André Platzer
Differential Equation Invariance Axiomatization Slides
Abstract.

10:4511:15 Coffee break
 11:1512:00 Sonia Rueda
Differential elimination ideals and spectral curves Slides
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 BurchnallChaundy ideal of a commuting pair will be presented, as the ideal of all constant coefficient bivariate polynomials satisfied by the pair. This prime ideal will be proved equal to the radical of a differential elimination ideal and the defining ideal of a plane algebraic curve, the spectral curve of a commuting pair. We are motivated by the development of a PicardVessiot theory for spectral problems, in the case of algebrogeometric ordinary differential operator, which are intrinsically linked to the study of integrable hierarchies.

12:0014:00 Lunch
 14:0014:30 Hadrien Notarantonio
Solving equations from combinatorics via computer algebra Slides
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 some of the variables (the specializations being generating functions related to the enumeration of interest). When the involved variables are “nested”, a celebrated result by Popescu (1986) implies algebraicity of the solutions. In 2006, BousquetMélou and Jehanne provided an elementary proof of algebraicity of the solutions in the case $n = 2$. Their proof yields an algorithm, and it has been the stateoftheart in enumerative combinatorics for solving these equations since then. In this talk, I will present a recent approach, based on the intensive use of effective algebraic geometry, in order to solve more efficiently such equations in the case $n = 2$. Also, I will introduce and discuss recent advances in the case of systems of such equations. The talk is based on joint works with Alin Bostan, Mohab Safey El Din and Sergey Yurkevich.
 14:3015:15 Nathalie Verdière
Examples of use of functional equations obtained from the elimination theory in nonlinear models Slides
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 nonlinear complex networks with an application on a C. elegans chemotaxis network.

15:1515:45 Coffee break
 15:4516:15 Alexandros Singh
On some functional equations for maps Slides
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 interderived by combinatorial or differentiallyalgebraic methods? This presentation draws from material from ongoing joint work with Olivier Bodini and Konstantinos Tsagkaris.
 16:1517:00 Daniel Robertz
Solving differential elimination problems with Thomas decomposition Slides
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 socalled 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 technique allows solving differential elimination problems, for example, for the analysis of certain structural properties of nonlinear control systems. The Thomas decomposition method has been implemented in Maple.
 17:0017:15 Closing remarks