Complex reflection groups comprise a generalization of Weyl groups of semisimple Lie algebras, and even more generally of finite Coxeter groups. They have been heavily studied since their introduction and complete classification in the 1950s by Shephard and Todd, due to their many applications to combinatorics, representation theory, knot theory, and mathematical physics, to name a few examples. For a given complex reflection group G, we explain a new recipe for producing integrable systems of linear differential equations whose differential Galois group is precisely G. We exhibit these systems explicitly for many (low-rank) irreducible complex reflection groups in the Shephard-Todd classification. 

This is joint work with Avery Bainbridge, Ben Obert, and Alavi Ullah. 

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.

Given two algebraic differential equations, how can one determine if there are relations between generic tuples of solutions? In this talk, I will use recent progress in understanding algebraic and differential-algebraic group actions with high generic degree of transitivity to give a sharp bound on the number of solutions one needs to consider. Along the way, I will explain how this falls out of the O'nan-Scott like classification of definably primitive groups of finite Morley rank of Macpherson and Pillay. This is joint work with James Freitag and Rahim Moosa.

In this talk, we show how differential elimination applies to effective computations for the closure properties of solutions to algebraic differential equations. By modelling the underlying operations as dynamical systems, we introduce radical dynamical systems and derive appropriate algorithms to compute their corresponding input-output equations. In contrast to the well-known Thomas decomposition and the Rosendfeld-Groebner algorithm which may derive such equations from the generators of a radical differential ideal, we describe an algorithm that builds the input-output equations directly from the ideal associated to the dynamical system. Our method relates to (Dong-Goodbrake-Harrington-Pogudin, 2023), and presents an interesting computational progress for differential elimination. 

Given an algebraic input-output equation, it is natural to ask whether it comes from a dynamical system in the state space form. Recovering such a system is known as the realizability problem. When the dynamical system is restricted to have right-hand sides that are rational functions, this problem can be treated with methods from differential algebra. In this talk we discuss the problem of rational realizability for single-input-single-output systems and focus on the case when the coefficients of the system are real. In particular, we discuss algorithmic approaches and show that the realizability  problem is closely connected a classical problem of rationally parametrizing hypersurfaces in the affine space. This is based on joint work with Gleb Pogudin, Sebastian Falkensteiner, and Rafael Sendra. 

Joint work with: Gabriela Jeronimo and Pablo Solernó. 

Under mass-action kinetics, biochemical reaction networks induce a polynomial autonomous system of differential equations. The problem of identifiability of the parameters of the system has been broadly studied under different approaches in the literature. We define the concept of identifiability from a reduced set of variables, inspired by (Craciun-Pantea, 2008), and analyze a family of biochemical networks where we are able to identify all the reaction constants from a few biologically relevant variables. In particular, we prove that all the parameters in a signaling cascade system can be identified from only one variable: the concentration of the fully phosphorylated substrate of the last layer. 

This talk will present a simple but meaningful example of the importance and application of identifiability analysis to compartmental models in a real-world context. Carbon stable isotope breath tests provide a dose of non-radioactive 13 C-labeled substrate, which is digested, absorbed, and metabolized, appearing on the breath as 13 CO2. These tests offer new opportunities to better understand gastrointestinal function in health and disease. However, it is often not clear how to isolate information about a gastrointestinal or metabolic process of interest from a breath test curve, and it is generally unknown how well summary statistics from empirical curve fitting correlate with underlying biological rates. We developed a framework that can be used to make mechanistic inference about the metabolic rates underlying a 13 C breath test curve, and we applied it to a pilot study of 13 C-sucrose breath test in 20 healthy adults. Starting from a standard conceptual model of sucrose metabolism, we determined the structural and practical identifiability of the model, using algebraic methods and profile likelihoods, respectively. We used these results to develop a reduced, identifiable model as a function of a gamma-distributed process; an exponential process; and a scaling term related to the fraction of the substrate that is exhaled as opposed to sequestered or excreted through urine. Our work develops a better understanding of how the underlying biological processes impact different aspects of 13C breath test curves, enhancing the clinical and research potential of these 13 C breath tests.

Abstract in pdf

Abstract in pdf

The concept of controllability describes the possibility of driving a dynamical system from an initial state to any point in its neighborhood. The methodology for analyzing the controllability of linear systems is well established; however, its generalization to the nonlinear case has proven elusive. Thus, a number of related but different properties have been defined to approach its study. A property that is similar to (but weaker than) controllability is accessibility or reachability. 

We are interested in determining the controllability and accessibility of nonlinear systems described by affine-in-inputs ordinary differential equations. Our motivation comes mainly from the study of biological processes, many of which can be described in this framework. This problem has been studied since the 1970s, but no generally applicable test exists yet; instead, several conditions have been proposed.  

In this talk we will present sufficient conditions to assess nonlinear controllability, as well as a necessary and sufficient condition for accessibility. These tests are not new; they can be found in the geometric control literature. Our main contribution is an algorithmic procedure to evaluate these conditions efficiently, along with its computational implementation. We will also show how the software tool can be applied to determine the accessibility and controllability of a number of biological systems.  

Let p ≥ 2 be an integer. A p-Mahler function is a power series f(z) with algebraic coefficients such that the orbit of f(z) under the map z → z^p spans a finite dimensional vector space over the rational functions.

The algebraic relations between M-values – that is the values of Mahler functions at algebraic points – has been the subject of many papers since the pioneering work of Mahler in the late 20’ and some rather satisfactory results have been obtained. For example we now have an algorithm to decide if a given M-value is algebraic or transcendental. Furthermore, the description of the relations between values of p-Mahler functions at a given non-zero algebraic point rely now entirely on our capacity in describing the relations between these functions. This is the subject of the Galois theory of Mahler equations.

In this talk we will focus on the algebraic relations between values of Mahler functions associated to distinct parameters p or evaluated at distinct algebraic points. We present a strong algebraic independence result which implies the following two facts:

• transcendental M-values associated to multiplicatively independent parameters p are algebraically independent ;

• transcendental M-values obtained as specializations of multiplicatively independent algebraic points are algebraically independent.

This result has important consequences regarding the expansion of real numbers in integer bases that we will discuss. It is a step toward the following general heuristic: the expansions of an irrational real number can not be too simple simultaneously in two multiplicatively independent bases.

This is a joint work with B. Adamczewski.