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:
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:
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.
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.
Honours level introduction to linear optimization and its applications: duality theory, fundamental theorem, sensitivity analysis, convexity, simplex algorithm, interior point methods, quadratic optimization, applications in game theory.
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.
Représentation et analyse des graphes par la décomposition spectrale des matrices dérivées de leurs topologies. Analyse harmonique sur les graphes. Applications au traitement de signal sur les graphes et à l’apprentissage profond géométrique.
The course will introduce some of the most widely used numerical methods for solving problems in linear algebra and differential equations. In linear algebra, topics of interest include solving linear systems, computing eigenvalues, and matrix factorizations. On the differential equations side, the course will cover methods for solving ordinary and partial differential equations, including approaches based on finite differences and finite elements. Time permitting, we will also explore topics such as modern approximation theory and physics-informed machine learning. The course will include a programming component, preferably in Python.
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.
Foundations of game theory. Computation aspects of equilibria. Theory of auctions and modern auction design. General equilibrium theory and welfare economics. Algorithmic mechanism design. Dynamic games.
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.
Examen de modèles fondamentaux utilisés en biologie mathématique et de leur analyse utilisant des outils modernes de calcul scientifique. Systèmes dynamiques discrets et continus, procédés stochastiques, modèles statistiques et simulation numérique.
Virgule flottante. ÉDOs. Modélisation et simulations. Méthodes directes et itératives pour la résolution de systèmes linéaires et non-linéaires. Optimisation sans contraintes. Valeurs propres. Décomposition en valeurs singulières. ÉDPs elliptiques et paraboliques. Équation de Black-Scholes.
L'objectif du concours est de présenter les notions principales de résolution des équations aux dérivées partielles (EDP). Dans ce cours, nous présentons les sujets suivants :