Research interests of the members cover several closely connected areas which include dynamical systems and delay equations; physics of fluids and continua; material sciences; phase transitions and crystal growth; numerical methods in fluid dynamics and asymptotic analysis; shape and structural optimization; control of partial differential equations.

Two research centers are affiliated with the group:

- CRM Applied Mathematics Laboratory
- GIREF (Groupe interdisciplinaire de recherche en éléments finis)

- Anne Bourlioux (UdeM)
- Jason Bramburger (Concordia)
- Simone Brugiapaglia (Concordia)
- Rustum Choksi (McGill)
- Morgan Craig (UdeM)
- Jean Deteix (Laval)
- Eusebius Doedel (Concordia)
- Nicolas Doyon (Laval)
- Tim Hoheisel (McGill)
- Tony Humphries (McGill)
- Félix Kwok (Laval)
- Guillaume Lajoie (UdeM)
- Jessica Lin (McGill)
- David McLeod (Université de Montréal)
- Jean-Christophe Nave (McGill)
- Adam Oberman (McGill)
- Robert Owens (UdeM)
- Courtney Paquette (McGill)
- Ronald Stern (Concordia)
- Gantumur Tsogtgerel (McGill)
- José Urquiza (Laval)
- Adrian Vetta (McGill)

The objective of this program is a training in modern mathematics aimed at applications and in the use of computers as a tool in the analysis, optimization, and control of physical and technological systems. It welcomes strong graduate students with a variety of backgrounds (ranging from the physical sciences and engineering to mathematics) wishing to work in partial differential equations and their applications. The program is sufficiently broad to accomodate software development and physical modelling as well as topics requiring delicate techniques in functional analysis or partial differential equations.

It is intended to offer students the possibility of collaborative contact with several local government and industrial research groups such as the Canadian Space Agency and a variety of other organisations with which members of the group have been involved at various times.

The program covers several closely connected areas which include:

- Dynamical systems and delay equations.
- Physics of fluids and continua.
- Material sciences; phase transitions and crystal growth.
- Numerical methods in fluid dynamics and asymptotic analysis.
- Shape and structural optimization.
- Control of partial differential equations.

There are no formal programmatic requirements beyond the departmental requirements. However the following guidelines should be followed and courses must be selected in consultation with an adviser from the group.

- All students should take courses in partial differential equations: appropriate courses are MATH 580 and MATH 581 at McGill and MAT 6110 at U de M.
- It is essential that most (and desirable that all) students develop their computational skills by taking appropriate courses in numerical analysis. Beyond the introductory courses, generally at an undergraduate level, the essential courses cover computational mathematics (MATH 578 at McGill and MAT6470 at U de M) numerical differential equations (MATH 579 at McGill) finite difference methods (MAT 6165 at U de M) and finite element methods (MTH 6206/7 at Polytechnique and MAT6450 at U de M).
- Students should develop an understanding of neighbouring areas of physics such as fluids and continuum mechanics, thermodynamics, etc. Suitable courses include MATH 555 at McGill and MAT 6150 at U de M; other useful courses can be found in Physics or Engineering departments.
- Students involved in fluid mechanics or material sciences should take a course on asymptotic and perturbation methods: MATH 651 at McGill or MTH 6506 at Polytechnique.
- Students in shape optimization or control should take at least one course in optimization. The following courses are available: MATH 560 at McGill, MAT 6428, MAT 6439 (Optimisation et contrôle), MAT 6441 (Analyse et optimisation de forme) at U de M; MTH 6403 and MTH 6408 at Polytechnique.
- Students who wish to work on shape optimization or the control of distributed parameter systems will need to develop a strong background in real analysis and functional analysis.

We expect that future elaboration and formalization of this program will occur within the framework described above which allows also for the introduction of additional areas under the broad umbrella of the program title.

This course will cover the theory of differential equations from a rigorous graduate mathematics perspective. Topics related to ordinary differential equations to be covered include proving existence and uniqueness for nonlinear systems, examining linear systems, fundamental solutions, equilibria, periodic solutions, stability, invariant manifolds, and hyperbolic theory. We will be introduced to important theorems that underscore the discipline such as Floquet’s theorem, the Hartman-Grobman theorem, and the stable and centre manifold theorems. The final weeks of the course will be dedicated to boundary value problems and Sturm-Liouville theory.

This course introduces the mathematical foundations of data science. Topics covered tentatively include machine learning basics, rudiments of statistical learning theory, optimal recovery, compressive sensing, elements of optimization theory and deep learning. Although the course will focus on theoretical aspects, it will also include computational illustrations. We will primarily follow the book "Mathematical Pictures at a Data Science Exhibition" by S. Foucart (Cambridge University Press, 2022). The course will include a final individual research project.

Algorithmic and structural approaches in combinatorial optimization with a focus upon theory and applications. Topics include: polyhedral methods, network optimization, the ellipsoid method, graph algorithms, matroid theory and submodular functions.

Development, analysis and effective use of numerical methods to solve problems arising in applications. Topics include direct and iterative methods for the solution of linear equations (including preconditioning), eigenvalue problems, interpolation, approximation, quadrature, solution of nonlinear systems.

Classification and wellposedness of linear and nonlinear partial differential equations; energy methods; Dirichlet principle. Brief introduction to distributions; weak derivatives. Fundamental solutions and Green's functions for Poisson equation, regularity, harmonic functions, maximum principle. Representation formulae for solutions of heat and wave equations, Duhamel's principle. Method of Characteristics, scalar conservation laws, shocks.

Processus de modélisation mathématiques avancés : simulations, estimation de paramètres, interprétation. Utilisation des mathématiques dans un milieu multidisciplinaire (p. ex. oncologie, neurosciences, génétique). Étude de cas et projets appliqués.

The formulation and treatment of realistic mathematical models describing biological phenomena through such qualitative and quantitative mathematical techniques as local and global stability theory, bifurcation analysis, phase plane analysis, and numerical simulation. Concrete and detailed examples will be drawn from molecular, cellular and population biology and mammalian physiology.

Convex sets and functions, subdifferential calculus, conjugate functions, Fenchel duality, proximal calculus. Subgradient methods, proximal-based methods. Conditional gradient method, ADMM. Applications including data classification, network-flow problems, image processing, convex feasibility problems, DC optimization, sparse optimization, and compressed sensing.

Virgule flottante. ÉDOs. Méthodes directes et itératives pour la résolution de systèmes linéaires et non-linéaires. Valeurs propres. ÉDPs elliptiques et paraboliques. Équation de Black-Scholes. Optimisation sans contraintes (MAT 6473 uniquement), Décomposition en valeurs singulières (SVD, MAT 6473 uniquement).