Algorithms for asymptotic differential algebra (M/F)

Job Description

Thesis Subject

In a similar way as differential algebra was introduced to study solutions of differential equations in a purely formal way, the theory of H-fields aims at studying the asymptotic behavior of such solutions from a formal perspective.

Important examples of H-fields are the field of transseries 𝕋 and Hardy fields. A transseries is a generalized power series that can recursively involve exponentials and logarithms. For instance,
∫ e^(e^x) = e^(e^x - x) + e^(e^x - 2x) + 2 e^(e^x - 3x) + ...
W(x) = (x e^x)^(inv) = log x - log log x + (log log x)/(log x) + ...
are transseries as x → ∞. A Hardy field is a differential subfield of the ring of germs of differentiable real functions at infinity. The theory of H-fields offers a unified framework for both examples despite their distinct nature (purely formal vs. analytic).

The goal of the PhD is to develop an effective counterpart of the theory of H-fields, while also investigating applications to the main two examples of transseries and Hardy fields.
The language of H-fields consists of the field operations (0, 1, +, -, ×), differentiation (∂), and the ordering (≤). The axioms for an H-field K include the usual ordered field and differential field axioms, plus two special axioms expressing compatibility between derivation and ordering: Let
C = {f ∈ K | f' = 0} (constants),
o = {f ∈ K : |f| ⪇ c for all c ∈ C, c ⪈ 0} (infinitesimals),
O = {f ∈ K : |f| ≤ c for some c ∈ C} (bounded elements).
We require: f ⪈ C ⇒ f' ⪈ 0 (for all f ∈ K), and O = C + o.
The ordering on K induces asymptotic relations:
f ≼ g iff f = O(g) (f is bounded by g up to a constant),
f ≺ g iff f = o(g) (f is negligible compared to g).
For this reason, H-fields are a suitable framework for 'asymptotic differential algebra'.

Algorithms exist for asymptotic computations in specific H-fields (e.g., 'exp-log' functions or Hardy fields). The PhD aims to generalize this theory abstractly while applying it to transseries and Hardy fields. We first define an effective H-field as one where elements are computer-representable and algorithms exist for +, -, ×, ∂, ≤, and ≼. Early tasks include showing that the real closure of an effective H-field remains effective, and that we can effectively close under integration and exponentiation.

Next, we study general asymptotic differential equations. Every H-field K embeds into an H-closed extension K̂, where any solvable asymptotic differential equation over K̂ has a solution in K̂. The ultimate goal is a general algorithm to solve such equations over K within its H-closure. For effective transseries or Hardy fields, we also aim to derive formal solutions and analyze their analytic properties.

We expect that constructive analysis of growth orders in H-fields will enable rigorous, algorithmic study of multi-scale dynamics. In dynamical modeling, time-scale separation simplifies systems by splitting behaviors into 'slow' and 'fast' components (e.g., quasi-steady-state approximation, QSSA). Understanding when such approximations hold is challenging. Since 'slow vs. fast' subdivisions align naturally with asymptotic differential algebra, we will explore how H-fields can address this challenge.

Your Work Environment

The MAX team is searching for PhD candidates on the themes of the ERC ODELIX project: solving differential equations fast, precisely, and reliably. The present proposal concerns the resolution of differential equations using asymptotical methods. The PhD will be co-supervised by Gleb Pogudin.

Compensation and benefits

Compensation

2300 € gross monthly

Annual leave and RTT

44 jours

Remote Working practice and compensation

Pratique et indemnisation du TT

Transport

Prise en charge à 75% du coût et forfait mobilité durable jusqu’à 300€

About the offer

About the CNRS

The CNRS is a major player in fundamental research on a global scale. The CNRS is the only French organization active in all scientific fields. Its unique position as a multi-specialist allows it to bring together different disciplines to address the most important challenges of the contemporary world, in connection with the actors of change.

CNRS

The research professions