This workshop is dedicated to interactions between nonlinear algebra and the study of differential equations, with a particular focus on computations and applications. Topics include linear PDE with constant coefficients, algebraic dynamical systems, D-modules, control theory, and the Ritt-Kolchin theory of differential polynomials.
The event will be streamed, too. Streaming information will be distributed a few days before the workshop.
We discuss a framework for defining and detecting singularities of arbitrary fully nonlinear systems of ordinary or partial differential equations with polynomial nonlinearities. It combines concepts from differential topology with methods from differential algebra and algebraic geometry. With its help, we provide for the first time a general definition of singularities of partial differential equations and show that it is at least meaningful in the sense that generic points are regular. Our definition is then extended to the notion of a regularity decomposition of a differential equation at a given order and the existence of such decompositions is proven by presenting an algorithm for their effective determination (with an implementation in Maple). Finally, we rigorously define the notion of a regular differential equation (a fundamental concept in the geometric theory of differential equations). We show that our algorithm automatically extracts one provably regular differential equation from each prime component of a given equations and thus provides an effective answer to an old problem in the geometric theory of differential equations.
Given a system of linear or nonlinear partial differential equations, various tasks like determining all power series solutions, finding all compatibility conditions, or deciding whether another given equation is a consequence of the system, require formal manipulation of the system. The Thomas decomposition method lends itself to answering such questions. It splits a differential system into finitely many so-called simple differential systems whose solution sets form a partition of the original solution set. This talk gives an introduction to this method and presents recent advances in a similar technique for difference equations and applications to the study of finite difference schemes.
We describe recent developments in commutative algebra that lead to a characterization of ideal membership in terms of differential conditions. These results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis-Palamodov. The main concept we introduce is that of differential primary decompositions.
Provably correct software is one of the key challenges in our software-driven society. While formal verification establishes the correctness of a given program, the result of program synthesis is a program which is correct by construction. In this talk I overview some of our results for both of these scenarios when analysing programs with loops. The class of loops we consider can be modelled by a system of algebraic recurrence equations, importantly with constant coefficients. I first describe an algorithmic approach for synthesising all polynomial equality invariants of such non-deterministic numeric single-path loops. By reverse engineering invariant synthesis, I then present an automated method for synthesising program loops satisfying a given set of polynomial loop invariants. Our results have applications towards proving partial correctness of programs, compiler optimisation and generating number sequences from algebraic relations.
Given a first order autonomous algebraic ordinary differential equation, i.e. an equation of the form $F(y,y')=0$ with polynomial $F$ and complex coefficients, we present a method to compute all formal Puiseux series solutions. By considering $y$ and $y'$ as independent variables, results from Algebraic Geometry can be applied to the implicitly defined plane curve. This leads to a complete characterization of initial values with respect to the number of distinct solutions extending them. Furthermore, the computed formal solutions are convergent in suitable neighborhoods and for any given point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point. Recently we generalized some of these results to systems of ODEs which implicitly define a space curve and to differential equations of the form $F(y,y^{(r)})=0$. Moreover, we present a method to compute local solutions with real or even rational coefficients only. This is a joint work with Jose Cano, Rafael Sendra and Daniel Robertz.
In the initial courses of calculus, when dealing with the computation of primitives, it is shown how certain integrals, whose integrands involve radicals of polynomials, can be transformed by a change of variable into integrals whose integrand are rational functions; and, hence, handleable algebraically. If the indefinite integral is seen as a differential equation, the natural question is whether we could develop a similar strategy to transform differential equations, whose coefficients involve radicals of polynomials, into differential equations whose coefficients are now rational functions. Illustrating examples of this are \[4(y')^3\sqrt{x+1}^{\,3} -(x+1)yy' + y^2 = 0,\] or \[ \left(-\sqrt{x_2}\,\dfrac{\partial y(\overline{x})} {\partial x_3} +2\,\dfrac{\partial y(\overline{x})} {\partial x_1} \right)\sqrt{x_1+\sqrt{x_2}} + 2\sqrt{x_2}\,\dfrac{\partial y(\overline{x})} {\partial x_2} - y(\overline{x})^2 - \dfrac{\partial y(\overline{x})} {\partial x_1}=0,\] or even $$\sqrt{x}\,y''-\sqrt{y^{3}} = 0.$$In this talk we will recall some basic facts on radical varieties, and we will present an algorithm to transform, if possible, a given differential equation, with radical function coefficients, into one with rational function coefficients by means of a rational change of variables so that solutions correspond.
This talk will explain a few questions that arise in the control of systems described by PDE, and how the algebraic structure (injectivity, flatness) of the space in which the solutions of the PDE are located help answer these questions in the case of linear constant coefficient PDE. It will describe some partial results in the setting of periodic solutions/Sobolev spaces, and finally pose some 'open problems'.
Linear systems of PDEs can be viewed comprehensively through an algebraic geometric lens. However, most conditions on linear PDEs used in Analysis have a strong Linear Algebra flavor and are difficult to describle using tools of Nonlinear Algebra. We use differential primary decompositions to characterize some of these conditions. In doing so, we prove new properties of the evaluations of polynomial matrices. We aim to use these insights in Analysis applications: weak continuity/lower semicontinuity in Compensated Compactness/Calculus of Variations. Joint work with Marc Härkönen and Lisa Nicklasson.
We consider the interplay between Gaussian processes, a classical stochastic framework with recent fame in machine learning, with differential algebra. We construct Gaussian process priors that concentrate their probability mass on solutions of certain linear differential equations, which yields a strong inductive bias in the learning algorithm. Technically, Gröbner basis algorithms yield a parametrizations of the solution sets of such differential equations, with is used to push forward a suitable Gaussian process. If time permits, we discuss control, boundary values, and parameter identification as applications.
I will present some results on how to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate. I will then explain how this result can be extended to linear hybrid automata which contain a mixture of linear discrete and continuous (ODE) dynamics. Finally, I will mention some recent results on the complexity of computing those invariants and obtaining bounds on the degree of the algebraic closure of finitely generated groups of matrices.
In this talk I will present several open problems about differential equations of algebraic and combinatorial nature, mostly inspired by applications. The main emphasis will be on the problems that do not require deep knowledge of differential algebra and can be illustrated by interesting specific examples.
We will discuss our current progress and open problems in the parameter identifiability/estimation for differential equations with parameters and the role of algebra in tackling these problems.
Participants
Rida Ait El Manssour
Max Planck Institute for Mathematics in the Sciences
Fuensanta Aroca
Universidad Nacional Autónoma de México
Lukas Barth
Max Planck Institute for Mathematics in the Sciences
Lara Bossinger
UNAM (National Autonomous University of Mexico)
Yairon Cid-Ruiz
Ghent University
Elena Derunova
Max Planck Institute of Microstructure Physics
Ruiwen Dong
University of Oxford
Marzieh Eidi
Max Planck Institute
Sebastian Falkensteiner
RISC Linz
Paul Görlach
Otto von Guericke University Magdeburg
Marc Härkönen
Georgia Institute of Technology
Heather Harrington
University of Oxford
Claudio Iuliano
University of Leipzig & MPI-MiS
Laura Kovacs
TU Wien
Kaushal Kumar
Heidelberg University
Pierre Lairez
Inria
Markus Lange-Hegermann
inIT / TH OWL
Julia Lindberg
Max Planck Institute
Saiei-Jaeyeong Matsubara-Heo
Kumamoto university
Stefano Mereta
Swansea University & Université Grenoble Alpes
Sayan Mukherjee
University of Leipzig and MPI MiS
Xianglong Ni
University of California, Berkeley
Joël Ouaknine
Max Planck Institute for Software Systems
Alexey Ovchinnikov
CUNY Queens College and Graduate Center
Gleb Pogudin
Institut Polytechnique de Paris
Amaury Pouly
Universite de Paris, CNRS, IRIF
Bogdan Raita
Scuola Normale Superiore Pisa
Georg Regensburger
University of Kassel
Ángel David Rios Ortiz
Max Planck Institute for Mathematics in the Sciences
Daniel Robertz
RWTH Aachen
Anna-Laura Sattelberger
MPI MiS
Werner Seiler
University of Kassel
Matthias Seiss
Universität Kassel
Juan Rafael Sendra
Universidad de Alcalá
Shiva Shankar
Indian Institute of Technology Bombay
Mima Stanojkovski
RWTH Aachen University / MPI MiS Leipzig
Bernd Sturmfels
Max Planck Institute for Mathematics in the Sciences
Libby Taylor
Stanford University
Bertrand Teguia Tabuguia
MPI for Mathematics in the Sciences
Charles Wang
Harvard University
James Worrell
University of Oxford
Gentian Zavalani
Center for Advanced Systems Understanding (CASUS)/ Technische Universität Dresden
Scientific Organizers
Rida Ait El Manssour
Max Planck Institute for Mathematics in the Sciences
Marc Härkönen
Georgia Institute of Technology
Bernd Sturmfels
Max Planck Institute for Mathematics in the Sciences
Administrative Contact
Saskia Gutzschebauch
Max Planck Institute for Mathematics in the Sciences
Contact by email
