It is known that a finite system of algebraic difference equations has a solution in the ring of sequences if and only if the difference ideal it generates contains 1. Moreover, there exists an algorithm that determines whether or not 1 lies in this difference ideal. In this talk we discuss various possibilities to generalize this result. Most of our findings are negative, for example, we show that the radical difference ideal membership problem is undecidable. This is joint work with Thomas Scanlon and Gleb Pogudin.

Strong minimality is a central notion in model theory which has an interpretation in differential algebra as a functional transcendence statement. We will talk about some new proofs of strong minimality for differential equations which build on existing results with two main examples - Painleve equations and Fuchsian equations. We will give a new proof of the strong minimality of PVI with generic coefficients. In the case of Fuchsian equations, we will explain how triangle groups let you prove the strong minimality of Fuchsian differential equations over the rationals of any genus. The Painleve work is joint with Nagloo, and the work on Fuchsian equations is joint with Basque-Sanz, Casale, and Nagloo.

In his monograph "Systèmes Orthogonaux" (Lecons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, Paris, 1910), Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of PDEs, where for each unknown function a certain subset of partial derivatives is prescribed and the values of the unknown functions are prescribed along transversal coordinate affine subspaces. The more general of the theorems, Theorem III, was proved by Darboux only for the cases of 2 and 3 independent variables. In a recent paper by Michael Benfield, Kris Jenssen and myself, published in the Journal of Geometric Analysis, we formulate and prove a generalization of Theorem III. Instead of partial derivatives, we prescribe derivatives of the unknown functions along vector fields. The values of the unknown functions are prescribed along arbitrary transversal submanifolds. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the set of vector fields and on the initial manifolds. SCC is automatically met in the case considered by Darboux. Assuming the SCC and the relevant integrability conditions are satisfied, we establish local existence and uniqueness of a \(C^1\)-smooth solution via Picard iteration for any number of independent variables. If the SCC is not satisfied, we show by an example that the uniqueness may fail.

Diagonals of rational functions naturally occur in many applications, study several families of rational functions in three or four variables and investigate the nature of their diagonals (hypergeometric \({}_{2}F_{1}\) functions, Heun functions, modular forms, functions associated with Shimura curves, etc.). It is interesting to compare them with the solutions of the telescopers associated to these rational functions. The connection with creative telescoping leads us to eliminate some of the potential counterexamples to Christol's conjecture that were proposed in the literature. This is joint work with Youssef Abdelaziz, Salah Boukraa, and Jean-Marie Maillard.

Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. We extend Hermite reduction to arbitrary linear differential operators instead of the pure derivative, and develop efficient algorithms for this reduction. We then apply the generalized Hermite reduction to the computation of linear operators satisfied by single definite integrals of D-finite functions of several continuous or discrete parameters. The resulting algorithm is a generalization of reduction-based methods for creative telescoping. Based on joint work with A. Bostan, P. Lairez, and B. Salvy.

We work over the field \(\mathbb{C}((x)\) equipped with the derivation \(\frac{\mathrm{d}}{\mathrm{dx}}\) and the shift operator \(s\) with \(s(f(x)) =f(x+1)\). The difference Galois group of a differential equation over \(\mathbb{C}((x)\) measures the algebraic relations among the solutions and its transforms under \(s\) and higher powers of \(s\). It is a difference algebraic group, i.e., a matrix group defined by polynomial equations inthe matrix entries and their transforms under \(s\). In my talk, I will give an introduction to this theory and present recent results on the inverse Galois problem in this theory. This is joint workwith Michael Wibmer.

We study algebraic structures underlying Volterra integral operators, in particular the operator identities satisfied by such operators. While the operator satisfies the Rota-Baxter identity when the kernel of the operator only depends on the phantom (dummy) variable, we show that when the kernel is more generally separable, the operator satisfies a generalized Reynolds identity which, in its original form, can be tracked back to the famous study of Reynolds in fluid mechanics in the late 19th century. Furthermore a generalized differential operator arises which combined with the generalized Reynolds operator provides an algebraic context, called \(D\)-Reynolds algebra, to study Volterra operators and equations. For this purpose, completions and free objects of \(D\)-Reynolds algebras are constructed. As applications, linearity of a class of integral equations is established. This is a joint work with Richard Gustavson and Yunnan Li.

We view difference algebra as the study of algebraic objects in the topos of difference sets, i.e., as `ordinary algebra’ in a new universe. The methods of topos theory and categorical logic enable us to develop difference homological algebra, identify a solid foundation for difference algebraic geometry, and cohomology theory of difference schemes.

In this talk, we explore the prominent method of creative telescoping. This is a pow-erful method pioneered by Zeilberger in the 1990s and has now become the cornerstone forfinding closed forms for definite sums (and definite integrals) in computer algebra. After a brief overview of the method, the talk is divided into two parts. In the first part, we restrict our attention to the case of rational functions and introduce two most recent algorithms for creative telescoping in this context. In the second part, these two algorithms will be exploited under the more general framework of hypergeometric terms. Difficulties for this generalization will be discussed then. This talk is based on joint papers with Shaoshi Chen, Hao Du, Mark Giesbrecht,Qing-Hu Hou, Manuel Kauers, George Labahn, Ziming Li, Rong-Hua Wang and Eugene Zima.

In joint work with Alexey Ovchnnikov, Anand Pillay, and Gleb Pogudin we study the problem of whether and how parameters in input-output equations may be recovered from experiments. Specifically, we relate the problem to a problem in the model theory of differential fields of computing a canonical base of a type from a realization. From this analysis, we find examples of such systems in which the canonical parameters cannot be recovered from a single experiment. On the other hand, on general grounds, we conclude that they may be computed from a bounded number of independent solutions and then convert the general method to an explicit algorithm.

Mazur conjectured, after Faltings's proof of the Mordell conjecture, that the number of rational points on a curve depends only on the genus, the degree of the number field and the Mordell-Weil rank. This conjecture was established in a few cases. In this talk I will explain how to prove this conjecture and some of its generalization. I will focus on how functional transcendence and unlikely intersections on mixed Shimura varieties, where o-minimality is vastly used, are applied. This is joint work with Vesselin Dimitrov and Philipp Habegger.

We study an extension of a theorem of Cantat, which says that if \(\varphi \colon X \to X\) is adominant rational self-map then the number of totally invariant hypersurfaces \(C\) (that is, hyper-surfaces for which \(\varphi^{-1}(C) = C\)) is finite unless \(\varphi \circ f = \varphi\) for some non-constant rational map \(f \colon X \to \mathbb{P}^1\). We relate this to the question of the Dixmier-Moeglin equivalence for twisted ho-mogeneous coordinate rings and other algebras that "come from geometry". This is joint work with Rahim Moosa and Adam Topaz.

It is a theorem from Landis and Petrovskii from the fifties that for most planar algebraic vector fields, all the algebraic solutions of the vector field are stationary at some zero of the vector field. For such a vector field, the description of a non-stationary solution therefore involve certain transcendental functions but few things are known about the nature of these transcendental functions. For example, can they be expressed using only "classical" transcendental functions such as exponentials, logarithms, Weierstrass's elliptic functions?
I will describe a stronger non-integrability result - namely that most planar algebraic vector fields define a strongly minimal and disintegrated differential equation - based on a combination of linearization techniques for planar algebraic vector fields and methods from model-theory.

Lee A. Rubel proved in 1981 that there exists a universal fourth-order algebraic differential equation \[P(y,y',y'',y''')=0 \;\;\;\;\;\; (1)\] and provided an explicit example. It is universal in the sense that for any continuous function \(f\) from \(\mathbb{R}\) to \(\mathbb{R}\) and any continuous positive function \(\epsilon\) from \(\mathbb{R}\) to \((0,+\infty)\), there exists an infinitely smooth solution \(y\) to \((1)\) such that \(|F(t)-y(t)|\leq\epsilon(t)\) for all real \(t\). In other everywhere. However this result suffers from a big shortcoming, identified as an open question by Rubel in his paper: "It is open whether we can require that the solution that approximates f be the unique solution for its initial data." Indeed, the solution to \((1)\) is not unique when ones adds the condition that \(y(0)=y_0\) for example. In fact, one can add a finite number of such conditions and still not get uniqueness of the solution. This is not surprising because the construction of Rubel fundamentally relies on the non-uniqueness of thesolution to work.
Our main result is to answer Rubel's question positively. More precisely, we show that there a polynomial \(P\) of degree \(k\) such that for any continuous function \(f\) from \(\mathbb{R}\) to \(\mathbb{R}\) and any continuous positive function \(\epsilon\) from \(\mathbb{R}\) to \((0,+\infty)\), there exist real \(y_0, y'_0, ..., y^{(k - 1)}_0\) such that the unique analytic solution \(y\) to \(P(y,y',...,y^{(k)})=0\) and \(y(0)=y_0, y'(0)=y'_0,..., y^{(k?1)}(0)=y^{(k - 1)}\) is such that \(|f(t)-y(t)|\leq\epsilon(t)\) for all real \(t\). The major difference with Rubel's result is that we get unicity by requiring that the solution be analytic. This requires a completely different and more involved construction. A by-product of the proof is that \(y_0\) is constructive and in fact computable from \(f\) and \(\epsilon\).

Firstly, we will show that if \(L\) is a Fucshian differential operator with coefficients in \(\mathbb{Q}(z)\), whose monodromy group is rigid and the exponents are rational numbers at singular points, then \(L\) has a strong Frobenius structure for almost all prime numbers.
Secondly, we will study the link between the existence of a strong Frobenius structure for a prime number \(p\) and the algebraicity modulo \(p\) of the solutions of the corresponding operator.

Harmonic functions play an important role in probability theory and are strongly related to the enumeration of walks. Doob \(h\)-transform is a way to build conditioned random processes in cones from a random process and a positive harmonic function vanishing on the boundary of the cone. Finding positive harmonic functions for random processes is therefore a natural objective in the study of confined random walks. There are very few ways to compute discrete harmonic functions. In this talk we are interested in positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane - resolution of a functional equation via boundary value problem using a conformal mapping - to the three-quarter plane applying the strategy of splitting the domain into two symmetric convex cones. We obtain a simple explicit expression for the algebraic generating function of harmonic functions associated to random walks avoiding a quadrant.

Elliptic hypergeometric functions were introduced roughly 10 years ago as a generalization of classical hypergeometric functions and q-hypergeometric functions. These special functions enjoy remarkable symmetry properties, like their more classical counterparts, and find applications in representation theory and mathematical physics. After interpreting one of these symmetries as a linear difference equation over an elliptic curve, we apply differential Galois theory of difference equations to show that these functions are always differentially transcendental for "generic" values of the parameters. This is joint work with Thomas Dreyfus and Julien Roques.