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 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).

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 algebraic, and it can only be D-finite if it has an irregular singularity. Previously, the saddle-point 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 in-depth study of limit shapes and open many new avenues for further research.

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, one can use these techniques to simplify definite multi-sums 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 non-trivial applications coming from enumerative combinatorics and elementary particle physics.

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.

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 collaboration with Pierre Bonnet (Labri).

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 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.

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).

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 right-factor 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.

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 holds if and only if the ordinary generating function $\sum_{n=0}^\infty a_n t^n$ can be expanded as a Stieltjes-type 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 Hankel-totally 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 Hankel-totally 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.

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 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 Griffiths-Dwork 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 Griffiths-Dwork reduction is applied for determining the Picard-Fuchs 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 Picard-Fuchs 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 Pichon-Pharabod, Charles Doran, and Andrew Harder.

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 (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.

Abstract. We give an improved algorithm for the enumeration of self-avoiding 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.

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 particles depends on the relative position of these with respect to the first-class 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 first-class 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).

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 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.

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 identities. In particular we will discuss a semi-automatic method using recurrences and $q$-difference equations, and what would be needed to make it fully automatic.

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 presence of these equations, whether it is the application of Riemann-Hilbert 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 off-diagonal 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, Fuji-Suzuki, Sasano and matrix-Painlevé systems - has been sketched out but there are gaps in our understanding and our tool-box is missing critical tools that one needs. In some senses the complexity has grown and it is often impractical to do hand-calculations and when computer-assisted 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, model-adapted algorithms for doing what are essentially algebraic geometry calculations. A number of examples, including those mentioned previously, will illustrate this state of affairs.