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)

- Peter Bartello (McGill)
- Anne Bourlioux (UdeM)
- Jason Bramburger (Concordia)
- Simone Brugiapaglia (Concordia)
- Rustum Choksi (McGill)
- Morgan Craig (UdeM)
- Jean Deteix (Laval)
- Eusebius Doedel (Concordia)
- Nicolas Doyon (Laval)
- André Fortin (Laval)
- Robert Guénette (Laval)
- Tim Hoheisel (McGill)
- Tony Humphries (McGill)
- Félix Kwok (Laval)
- Guillaume Lajoie (UdeM)
- Jessica Lin (McGill)
- 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.

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 branchement : modèles de Wright-Fisher, de Moran. Modèles à une infinité d’allèles, de sites. Facteurs d’évolution: sélection, mutation, migration, recombinaison, apparentement. Reconstruction et inférence de réseaux génétiques.

Formal and analytic approaches for modeling intrinsic geometries in data. Algorithms for constructing and utilizing such geometries in machine learning. Applications in classification, clustering, and dimensionality reduction. The course will accommodate anglophone students who are interested in taking it, as well as francophone students.

This course will illustrate some of the most popular numerical methods used to solve linear algebra problems and differential equations. On the linear algebra side, problems of interest include solving linear systems, computing eigenvalues, and matrix factorizations. On the differential equations’ side, the course will present methods for solving ordinary and partial differential equations such as Euler’s method, Runge-Kutta, finite differences, and finite elements. The course includes a computational component in MATLAB/Octave language.

Concentration inequalities, PAC model, VC dimension, Rademacher complexity, convex optimization, gradient descent, boosting, kernels, support vector machines, regression and learning bounds. Further topics selected from: Gaussian processes, online learning, regret bounds, basic neural network theory.

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.

Numerical solution of initial and boundary value problems in science and engineering: ordinary differential equations; partial differential equations of elliptic, parabolic and hyperbolic type. Topics include Runge Kutta and linear multistep methods, adaptivity, finite elements, finite differences, finite volumes, spectral methods.

Systems of conservation laws and Riemann invariants. Cauchy-Kowalevskaya theorem, powers series solutions. Distributions and transforms. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators. Second order parabolic and hyperbolic equations. Further advanced topics may be included.

Flots discrets et continus. Équations différentielles non linéaires, techniques classiques d’analyse de dynamique, existence et stabilité de solutions, variétés invariantes, bifurcations, formes normales, systèmes chaotiques. Applications moderne.

Virgule flottante. ÉDOs. Modélisation et simulations. Méthodes directes et itératives pour la résolution de systèmes linéaires et non-linéaires. Valeurs propres et valeurs singulières. Optimisation sans contraintes. ÉDPs elliptiques et paraboliques. Équation de Black-Scholes.