Session III.6 - Symbolic Analysis
Talks
Monday, June 19, 14:00 ~ 14:30
Computing the fundamental invariants and equivariants of a finite group
Evelyne Hubert
Inria Méditerranée, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
For a finite group, we present three algorithms to compute a generating set of invariant simultaneously to generating sets of basic equivariants, i.e., equivariants for the irreducible representations of the group. The main novelty resides in the exploitation of the orthogonal complement of the ideal generated by invariants; Its symmetry adapted basis delivers the fundamental equivariants. Fundamental equivariants allow to assemble symmetry adapted bases of polynomial spaces of higher degrees, and these are essential ingredients in exploiting and preserving symmetry in computations. They appear within algebraic computation and beyond, in physics, chemistry and engineering. \url{https://doi.org/10.1090/mcom/3749}
Joint work with Erick Rodiguez Bazan.
Monday, June 19, 14:30 ~ 15:00
Integrable systems of the Boussinesq type and their formal diagonalisation
Rafael Hernandez Heredero
Polytechnic University of Madrid, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
We will deal with the symmetry approach to integrability for multi-component evolution systems $\mathbf{u}=\mathbf{\Phi}(\mathbf{u},\mathbf{u}_1, \dots , \mathbf{u}_n)$, or in components \[ u^i_t = \phi^i(\mathbf{u},\mathbf{u}_1, \dots , \mathbf{u}_n),\quad i=1,\ldots,m \] where $\mathbf{u}=(u^1, \dots , u^m)$ and $\mathbf{u}_i=\partial\mathbf{u}/\partial x^i$. Formal recursion $\mathbf{R}$ and symplectic $\mathbf{S}$ operators are matrix pseudo-differential series satisfying the equations \begin{gather*}{\bf R}_t=[\mathbf{\Phi}_*,\,\mathbf{R}], \\ \mathbf{S}_t + \mathbf{S} \,\mathbf{\Phi}_* +\mathbf{\Phi}_*^ + \, \mathbf{S} = 0. \end{gather*} where $\mathbf{\Phi}_*$ denotes the Fréchet derivative of $\mathbf{\Phi}$. Integrability amounts to the existence of such operators for a given system.
Non-degenerate systems are those having its separant matrix $\mathbf{\Sigma}$ (with entries $\sigma_{ij}=\partial \phi^i/\partial u_n^j$) invertible and without multiple eigenvalues at a generic point. It is well known that starting from the transformation that diagonalises the separant, a transformation that formally diagonalises the whole system can be built, helping to solve the equations for recursion and symplectic operators, i.e. to establish integrability.
We will explore in this talk the possibility of diagonalising degenerate systems, i.e. with non diagonalisable separants. As an illustration, we will show some classifications of integrable systems of the form \[\begin{aligned} u_t&=v,\\ v_t&=u_4+f(u,u_1,\ldots,u_{3},v,v_1). \end{aligned} \] The computational component of this work consist in developing and implementing the formal calculus of pseudo-differential expressions in the jet space.
Joint work with Vladimir V. Sokolov (Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia).
Monday, June 19, 15:00 ~ 15:30
Regular singular differential equations and free proalgebraic groups
Michael Wibmer
Graz University of Technology, Austria - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $S$ be a finite subset of the Riemann sphere. It is an immediate consequence of the Riemann-Hilbert correspondence that the differential Galois group of the family of all regular singular differential equations with singularities inside $S$ is the proalgebraic completion of the free group on $|S|-1$ generators . In this talk we will discuss generalizations of this statement to infinite $S$. In particular, we will determine the differential Galois group of the family of all regular singular differential equations on the Riemann sphere.
Monday, June 19, 15:30 ~ 16:00
On Normal Forms in Differential Galois Theory for the Classical Groups
Matthias Seiß
Universität Kassel, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
In classical Galois theory there is the well-known construction of the general polynomial equation over $\mathbb{Q}$ with Galois group the symmetric group $S_n$. Shortly recalling this construction we consider a rational function field $\mathbb{Q}(\boldsymbol{x})$ in $n$ indeterminates $\boldsymbol{x}=(x_1,\dots,x_n)$ over $\mathbb{Q}$ on which the symmetric group acts by permuting the variables $\boldsymbol{x}$. The fix field under this action is generated over $\mathbb{Q}$ by the elementary symmetric polynomials \[\boldsymbol{s}(\boldsymbol{x})=(s_1(\boldsymbol{x}),\dots,s_n(\boldsymbol{x}))\] and has transcendence degree $n$ over $\mathbb{Q}$. Moreover, $\mathbb{Q}(\boldsymbol{x})$ is a Galois extension of $\mathbb{Q}(\boldsymbol{s}(\boldsymbol{x}))$ with Galois group $S_n$ and its defining equation is the polynomial equation of degree $n$ whose coefficients are (up to sign) the polynomials $\boldsymbol{s}(\boldsymbol{x})$. Every algebraic extension of $\mathbb{Q}$ which is defined by a polynomial of degree $n$ is obtained as a specialization by substituting the roots for $\boldsymbol{x}$ in the general equation.
In differential Galois theory there is a similar construction of a general differential equation with differential Galois group the general linear group $\mathrm{GL}_n$. More precisely, for an algebraically closed field $C$ of characteristic zero one considers here a differential field $C\langle \boldsymbol{y} \rangle$ which is generated by $n$ differential indeterminates $\boldsymbol{y}=(y_1,\dots,y_n)$. Now the group $\mathrm{GL}_n(C)$ acts on $C\langle \boldsymbol{y} \rangle$ by linearly transforming the indeterminates $\boldsymbol{y}$. For a new differential indeterminate $Y$ the general differential equation is defined as the quotient of the two Wronskians \[ \frac{wr(Y,y_1,\dots,y_n)}{wr(y_1,\dots.y_n)} =: Y^{(n)} + c_{n-1}(\boldsymbol{y})Y^{(n-1)} + \dots + c_0(\boldsymbol{y}) Y=0. \] The coefficients $\boldsymbol{c}=(c_{n-1}(\boldsymbol{y}),\dots,c_0(\boldsymbol{y}))$ of the general equation are differentially algebraically independent over $C$ and as the elementary symmetric polynomials they generate the fixed field of $C\langle \boldsymbol{y} \rangle$ under $\mathrm{GL}_n(C)$ over the constants. Clearly, $C\langle \boldsymbol{y} \rangle$ is a Picard-Vessiot extension of $C\langle \boldsymbol{c} \rangle$ with differential Galois group $\mathrm{GL}_n(C)$ and it is generic extension. Indeed, let $E$ be a Picard-Vessiot extension of a differential field $F$ with constants $C$ defined by a linear scalar differential equation of order $n$. If $\boldsymbol{\eta}=(\eta_1,\dots,\eta_n)$ are the linearly independent solutions of this equation, then $E/F$ is obtained as a specialization of the general equation by substituting the solutions $\boldsymbol{\eta}$ for the indeterminates $\boldsymbol{y}$. Generalizations to groups other than $\mathrm{GL}_n(C)$ were obtained in [1] and [2]. In all these cases the general differential equation involves $n$ differential indeterminates over $C$ (apart from $Y$).
An analogue construction of a general linear differential equation for the classical groups was presented in [5]. This approach combines the geometric structure of a classical group $G(C)$ of Lie rank $l$ with Picard-Vessiot theory. As in the case of the general equation with group $\mathrm{GL}_n(C)$ the construction starts with a differential field $C\langle \boldsymbol{v} \rangle$ generated by $l$ differential indeterminates $\boldsymbol{v}=(v_1,\dots,v_l)$ over $C$, but this time the general extension field is a Liouvillian extension $\mathcal{E}$ of $C\langle \boldsymbol{v} \rangle$ with differential Galois group a fixed Borel group $B^-(C)$ of $G(C)$. Choosing a Chevalley basis of the Lie algebra $\mathfrak{g}(C)$ of $G(C)$ the Liouvillian extension is defined by a matrix $A_{\mathrm{Liou}}(\boldsymbol{v})$ which is the sum of the Cartan subalgebra parametrized by the indeterminates $\boldsymbol{v}$ and the basis elements of the root spaces corresponding to the negative simple roots. In order to define a group action of $G(C)$ on $\mathcal{E}$ one needs to construct a fundamental matrix $\mathcal{Y}$ and let $G(C)$ act on it by right multiplication which will then induce an action of $G(C)$ on $\mathcal{E}$. To this end, let $b$ be a fundamental matrix of $A_{\mathrm{Liou}}(\boldsymbol{v})$ in $B^-(\mathcal{E})$, $\overline{w}$ a representative of the longest Weyl group element and let $u(\boldsymbol{v}, \boldsymbol{f})$ be the product of matrices of all negative root groups where the matrices corresponding to the negative simple roots are parametrized by $\boldsymbol{v}$ and the matrices corresponding to all remaining negative roots depend on differential polynomials $\boldsymbol{f}$ in $C\{ \boldsymbol{v}\}$. These differential polynomials are chosen in such a way that the logarithmic derivative of $\mathcal{Y} = u \overline{w} b$ is the matrix $A_G(\boldsymbol{s}(\boldsymbol{v}))$ constructed in [4] and [5], where $\boldsymbol{s}(\boldsymbol{v})$ are $l$ differential polynomials in $C\{ \boldsymbol{v}\}$. Analogue to the cases of the symmetric group and $\mathrm{GL}_n(C)$ presented above, the $\boldsymbol{s}(\boldsymbol{v})$ are differentially algebraically independent over $C$. Multiplying $\mathcal{Y}$ from the right with elements of $G(C)$ and then recomputing the Bruhat decomposition of the product defines an action on $\boldsymbol{v}$, $\boldsymbol{f}$ and on the generators of the Liouvillian extension, i.e. the entries of $b$, and so induces an action of $G(C)$ on $\mathcal{E}$. The fixed field under this induced action is $C\langle \boldsymbol{s}(\boldsymbol{v}) \rangle$ and one can show that the field $\mathcal{E}$ is a Picard-Vessiot extension of $C\langle \boldsymbol{s}(\boldsymbol{v}) \rangle$ with differential Galois group $G(C)$ for the differential equation defined by $ A_G(\boldsymbol{s}(\boldsymbol{v}))$. The construction is only generic for Picard-Vessiot extensions of $F$ with defining matrix gauge equivalent to a matrix in \emph{normal form}, i.e.\ a specialization of $A_{G}(\boldsymbol{s}(\boldsymbol{v}))$. Deciding such a gauge equivalence is non-trivial as a consequence of the fact that $\mathcal{E}$ and $C\langle \boldsymbol{s}(\boldsymbol{v}) \rangle $ have differential transcendence degree $l$ over $C$.
This talk is dedicated to the question of the genericity properties of the extension $\mathcal{E}$ over $C\langle \boldsymbol{s}(\boldsymbol{v}) \rangle$. We consider the problem of gauge equivalence of a generic element of the Lie algebra to a matrix in normal form. More precisely, let $d$ be the dimension of the classical group $G$ and let $\boldsymbol{a}=(a_1,\dots,a_d)$ be differential indeterminates over a differential field $F$ with constants $C$. Further let $A(\boldsymbol{a})$ be a generic element in the Lie algebra $ \mathfrak{g}(F\langle \boldsymbol{a} \rangle)$ obtained from parametrizing the Chevalley basis from above with the indeterminates $\boldsymbol{a}$. It is known (cf. [3]) that the differential Galois group of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $F \langle \boldsymbol{a} \rangle$ is $G(C)$. We present the construction of a differential field extension $\mathcal{L}$ of $F \langle \boldsymbol{a} \rangle$ such that the field of constants of $\mathcal{L}$ is $C$, the differential Galois group of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $\mathcal{L}$ is still the full group $G(C)$ and $A(\boldsymbol{a})$ is gauge equivalent over $\mathcal{L}$ to a specialization of $A_G(\boldsymbol{s}(\boldsymbol{v}))$, i.e. to a matrix in normal. In the special case of $G=\mathrm{SL}_3$ we show how one obtains an analogous result for specializations of the coefficients of $A(\boldsymbol{a})$.
$\textbf{References}$
[1] L. Goldman, Specialization and Picard-Vessiot theory. Transactions of the American Mathematical Society, 85:327–356, 1957.
[2] L. Juan and A. Magid, Generic rings for Picard-Vessiot extensions and generic differential equations. Journal of Pure and Applied Algebra, 209(3):793–800, 2007.
[3] L. Juan, Pure Picard-Vessiot extensions with generic properties. Proceedings of the American Mathematical Society, 132(9):2549–2556, 2004.
[4] M. Seiss, Root Parametrized Differential Equations for the Classical Groups. https://arxiv.org/abs/1609.05535.
[5] M. Seiss, On the Generic Inverse Problem for the Classical Groups. https://arxiv.org/abs/2008.12081.
Joint work with Daniel Robertz (RWTH Aachen University, Germany).
Monday, June 19, 16:30 ~ 17:00
Syzygies and constant rank operators
Lisa Nicklasson
Università di Genova, Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.
The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the commutative algebra notion of primary decomposition is another important tool for studying such system of PDEs. In this talk we will explore the connection between these two concepts. Along the way, a decomposition of our differential operator into a controllable and an uncontrollable part will play an important role.
Joint work with Marc Härkönen (Georgia Institute of Technology, USA) and Bogdan Raiță (Scoula Normale Superiore di Pisa, Italy).
Monday, June 19, 17:00 ~ 17:30
Algebraic consequences of the fundamental theorem of calculus in differential rings
Georg Regensburger
University of Kassel, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we discuss the fundamental theorem of calculus and its consequences from an algebraic point of view [1]. In particular, for functions with singularities, this leads to a generalized notion of evaluation. We present properties of such integro-differential rings and discuss several examples. We outline the construction of the corresponding integro-differential operators and provide normal forms using rewrite rules. These rewrite rules are then used to derive several identities and properties in a purely algebraic manner, generalizing well-known results from analysis. In identities such as shuffle relations for nested integrals and the Taylor formula, additional terms are obtained to account for singularities. Another focus lies on treating the basics of linear ordinary differential equations (ODEs) within the framework of integro-differential operators. These operators can have matrix coefficients, enabling the treatment of systems of arbitrary size in a unified manner.
[1] Clemens G. Raab and Georg Regensburger. The fundamental theorem of calculus in differential rings. arXiv:2301.13134 [math.RA] (2023)
Joint work with Clemens Raab (Johannes Kepler University Linz, Austria).
Monday, June 19, 17:30 ~ 18:00
Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory
Jaques-Arthur Weil
Université de Limoges, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Darboux developed an ingenious algebraic mechanism to construct infinite chains of ``integrable" second order differential equations as well as their solutions. After a surprisingly long time, Darboux's results were rediscovered and applied in many frameworks, for instance in quantum mechanics (where they provide useful tools for Supersymmetric Quantum Mechanics), in soliton theory, Lax pairs and many other fields involving hierarchies of equations. In this work, we propose a method which allows us to generalize the Darboux transformations algorithmically for tensor product constructions on linear differential equations or systems. We obtain explicit Darboux transformations for third order orthogonal systems ($\mathfrak{so}(3, C_K)$ systems) as well as a framework to extend Darboux transformations to any symmetric power of $\mathrm{SL}(2,\mathbb{C})$-systems.
Joint work with Primitivo Acosta-Humanez (Universidad Autónoma de Santo Domingo, Dominican Republic), Moulay Barkatou (Université de Limoges, France), Raquel Sanchez-Cauce (Universidad Nacional de Educación a Distancia, Spain).
Tuesday, June 20, 14:00 ~ 14:30
The generating function of DYZ-like numbers is algebraic
Sergey Yurkevich
Inria Saclay and University of Vienna, Austria - This email address is being protected from spambots. You need JavaScript enabled to view it.
In a recent work Don Zagier mentions a mysterious integer sequence $(a_n)_{n \geq 0}$ which arises from a solution of a topological ODE discovered by Marco Bertola, Boris Dubrovin and Di Yang. In my talk I show how to conjecture, prove and even quantify that $(a_n)_{n \geq 0}$ actually admits an algebraic generating function which is therefore a very particular period. Moreover, I define and explore eight other alike sequences with very similar origins and also algebraic generating functions. The methods are based on experimental mathematics, numerics, and algorithmic ideas in differential Galois theory.
Joint work with Alin Bostan (Inria Saclay) and Jacques-Arthur Weil (University of Limoges).
Tuesday, June 20, 14:30 ~ 15:00
Fast computation of coefficients of algebraic power series over finite fields
Alin Bostan
Inria, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
We address the following algorithmic question: given an algebraic power series with coefficients in a finite field, either univariate or multivariate, how fast can one compute a selected coefficient in its expansion? After revisiting several classical algorithms, we describe a new method, based on Christol’s 1979 theorem that connects algebraicity of power series and automaticity of their coefficients sequences. We provide new proofs of Christol’s theorem, both in the univariate and in the multivariate cases. These proofs are elementary, effective, and allow for rather sharp estimates. In particular, they are the basis for faster algorithms for computing coefficients of algebraic power series over finite fields. The talk is based on two joint works: one with Xavier Caruso, Gilles Christol and Philippe Dumas, the other with Boris Adamczewski and Xavier Caruso.
Tuesday, June 20, 15:00 ~ 15:30
Mahler discrete residues and summability for rational functions
Yi Zhang
Xi'an Jiaotong-Liverpool University, China - This email address is being protected from spambots. You need JavaScript enabled to view it.
We construct Mahler discrete residues for rational functions and show that they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p \gt 1$. This extends to the Mahler case the analogous notions, properties, and applications of discrete residues (in the shift case) and $q$-discrete residues (in the $q$-difference case) developed by Chen and Singer. Along the way we define several additional notions that promise to be useful for addressing related questions involving Mahler difference fields of rational functions, including in particular telescoping problems and problems in the (differential) Galois theory of Mahler difference equations.
Joint work with Carlos E. Arreche (The University of Texas at Dallas, USA).
Tuesday, June 20, 15:30 ~ 16:00
Galois groups for linear integrable systems of differential and difference equations over elliptic curves
Carlos Arreche
UT Dallas, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $C$ be an algebraically closed field of characteristic zero, let $E$ be an elliptic curve defined over $C$, and let $K$ be the field of rational functions on $E$. Let $\delta$ be an invariant derivation on $K$ (unique up to a constant multiple), and $\sigma$ denote the automorphism on $K$ induced by addition by a fixed non-torsion $C$-point of $E$ under the elliptic group law. We consider a linear system \[\delta(Y)=AY; \qquad \sigma(Y)=BY;\] where $A\in\mathfrak{gl}_n(K)$ and $B\in\mathrm{GL}_n(K)$ satisfy the integrability condition \[\delta(B)=\sigma(A)B-BA.\] There are several (five!), a priori different and seemingly incomparable, Galois groups that one can attach to such a system. We explain why some of them must be abelian, and why (conjecturally) all of them must be solvable.
Joint work with Matthew Babbitt (UT Dallas).
Tuesday, June 20, 16:30 ~ 17:00
Hahn series and Mahler equations
Julien Roques
Université Lyon 1, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Unlike most linear functional equations such as linear differential equations or q-difference equations, Puiseux series are not sufficient to study linear Mahler equations. Instead, a fundamental role is played by the Hahn series (a generalization of the Puiseux series allowing arbitrary exponents of the indeterminate as long as the set that supports them forms a well-ordered set). In this talk, we will explain the role played by Hahn series in the theory of Mahler equations and discuss the concrete calculation of Hahn series solutions of linear Mahler equations. This talk is based on a joint work with Colin Faverjon.
Tuesday, June 20, 17:00 ~ 17:30
Differential transcendance and Galois theory
Thomas Dreyfus
CNRS, université de Strasbourg, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we consider meromorphic solutions of difference equations and prove that very few among them satisfy an algebraic differential equation. The basic tool is the difference Galois theory of functional equations.
Tuesday, June 20, 17:30 ~ 18:30
Symbolic Invariant Calculus
Francis Valiquette
Monmouth University, United State of America - This email address is being protected from spambots. You need JavaScript enabled to view it.
The theory of equivariant moving frames is a modern reformulation of Cartan's method of moving frames with the distinctive advantage that many computations can be performed symbolically, without requiring coordinate expressions for the invariants or the moving frame. This is possible thanks to the recurrence formulas, which encapsulate the extend by which the invariantization process that emerges from the construction of a moving frame does not commute with differentiation in differential geometry or shift maps in the discrete setting. The symbolic invariant calculus that emerged from these recurrence relations has led to a wide range of new results in geometry, the study of geometric invariant curve flows, the calculus of variations, the integration of differential equations, and much more.
In the first half of my presentation I will introduce the method of equivariant moving frames and the important recurrence formulas. In the second half of the talk I will survey recent applications of the resulting symbolic invariant calculus.
Wednesday, June 21, 14:00 ~ 15:00
Spectral Picard-Vessiot theory and its applications
Sonia L. Rueda
Universidad Politécnica de Madrid, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
We investigate new techniques in differential and noncommutative algebra to develop the differential Galois theory for spectral problems, and apply these abstract theories to concrete differential spectral problems of integrable systems from Physics.
Given an ordinary differential operator $L$, with coefficients in a differential field, we study the spectral problem $LY=\lambda Y$, for a constant spectral parameter $\lambda$. We are interested in algebro-geometric linear differential operators, characterized by having nontrivial centralizers, and a spectral parameter governed by the famous spectral curve. In this situation we have coupled-spectral problems \[LY=\lambda Y, \,\,\, AY=\mu Y,\] for a non trivial ordinary differential operator $A$ commuting with $L$. For instance, in the case of algebro-geometric Schrödinger operators $\partial^2+u$, the potentials $u$ are solutions of the KdV-hierarchy.
The recently defined spectral Picard-Vessiot field of a second order operator L provided a new approach to the factorization problem of ordinary differential operators in terms of parameters. The generalization to the case of prime-order operators appears naturally. Already the case of third-order operators, which are associated to the Boussineq integrable hierarchy, is an interesting and challenging problem, which has not been approached by differential Galoisian methods before.
In this talk I will present recent results and open questions related with this problems, in the framework of the project Algorithmic Differential Algebra and Integrability (ADAI).
Joint work with Maria-Angeles Zurro (Universidad Autónoma de Madrid), Juan J. Morales-Ruiz (Universidad Politécnica de Madrid) and Emma Previato (Boston University).
Wednesday, June 21, 15:00 ~ 15:30
A New Equivalence Method
Joël Merker
Paris-Saclay University, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
To upgrade the naive approach through power series expanded at only one point, the talk will show how to import concepts and principles of \'Elie Cartan's G-structured method of equivalence. Quite unexpectedly, basic linear representation theory happens to become of central use within highly nonlinear jet bundles.
Two specific illustrations will be presented: Homogeneous models of 5D PDE systems under fiber-preserving transformations; Affinely homogeneous surfaces $S^2$ in $R^4$.
Joint work with Julien Heyd (Paris-Saclay University).
Wednesday, June 21, 15:30 ~ 16:00
Developing an Algebraic Theory of Integral Equations
Richard Gustavson
Manhattan College, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we will study the algebraic structure underlying Volterra integral operators and their corresponding equations. While the operator satisfies the Rota-Baxter identity when the kernel of the operator only depends on the variable of integration, we show that when the kernel is more generally separable, a twisted Rota-Baxter identity is satisfied. We will then discuss the development of an algebraic theory of general integral equations that allows for both arbitrary kernels and limits of integration using bracketed words and decorated rooted trees. As an application, we will show how any separable Volterra integral equation is equivalent to one that is operator linear, that is, contains only iterated integrals.
Joint work with Li Guo (Rutgers University - Newark) and Yunnan Li (Guangzhou University).