To register for an ISM course, you must first have you course selection approved by your supervisor and departmental Graduate Program Director. You may then register for the course using the electronic form available on the BCI website (the BCI is the organization that handles inter-university registration). The form will then be sent to the home and host universities' Registrars for approval.
Additional procedures for non-McGill students registering for a course at McGill University:
Once the registration through the BCI website is complete, the student will receive a confirmation. The student must then register for the course at McGill University through the MINERVA registration system.
Important deadlines: Concordia, Laval, McGill, Université de Montréal, UQAM, UQTR, Université de Sherbrooke
Course Schedules:
Online Open Access Courses
Henri Darmon, Université McGill
Modular Forms and Orthogonal Groups
Course site
Antonio Lei, Université Laval
Modular Forms and Elliptic Curves
Course site
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Melina Mailhot, Université Concordia
Risk Measures
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Javad Mashreghi, Université Laval
Reproducing Kernel Hilbert Space of Analytic Functions
Course site
Iosif Polterovich, Université de Montréal
Geometric Spectral Theory
Course site
This course will be an introduction to the theory of modular forms over the complex number. We shall cover the following topics: the modular group and the upper half-plane, Eisenstein series, Hecke operators, L-functions, modular curves, geometric interpretation of modular forms. If time allows it, further topics (Galois representations or Eichler--Shimura relations) will be considered. Knowledge of complex analysis, Riemann surfaces, and sheaves is useful but not necessary. Further reading on p-adic or Drinfeld modular forms will be available for motivated students.
This is a first course in the study of algebraic number fields. In the first part of the course, we will concentrate on proving the two main basic results in the subject: the ideal class group is finite and the unit group is finitely generated. Other topics will include: the distribution of ideals, the Dedekind zeta function and the class number formula.
The goal of this course is to study Hiday theory and Iwasawa theory of modular forms. We will first review basic arithmetic properties of modular forms and elliptic curves. We will study L-functions, Hecke operators and Galois representations attached to these objects. We will study the theory of deformation theory of modular forms, Hecke algebras and Hida theory, as well as the Iwasawa main conjecture of modular forms. We will explore different elements and techniques used in the proofs of different versions of the Iwasawa main conjectures developed in recent years.
Théorie de Galois : théorie de base sur les extensions de corps, groupe de Galois d'un polynôme (groupe de permutations), correspondance de Galois, corps finis.
Algèbre commutative : idéaux premiers, primaires; anneaux noethériens, de Dedekind; radicaux; anneaux simples, semi-simples, premiers, semi-premiers. Modules libres, projectifs, injectifs et indécomposables. Suites exactes. Foncteurs Hom et produit tensoriel.
Représentations et module d'un groupe G, représentations équivalentes, sous-module. Représentations indécomposable, réductible, irréductible. Théorème de Maschke. Morphisme, lemme de Schur.
Algèbre de groupe, fonctions sur cette algèbre, fonction de classe. Caractères, relations d'orthogonalité, tables de caractères. Représentation régulière. Analyse de Fourier sur les groupes finis, identité de Parseval, théorème de Wedderburn.
Nombres algébriques, théorème de la dimension, théorème de Burnside. Action de groupe, lemme de Burnside, paires de Gelfand.
Représentations induites, théorème de réciprocité de Frobenius, critère d'irréductibilité de Mackey.
Marche aléatoire sur les groupes finis. Modèles de Gilbert–Shannon–Reeds, théorèmes de Diaconis.
This course will give an introduction to modular forms with emphasis on its connections with quadratic forms. Topics may include representations of integers by quadratic forms. theta functions, the Weil representation and the theta correspondence, as well as Hilbert and Siegel modular forms as forms on orthogonal groups.
Introduction to the theory of rings and modules: rings, ideals, quotients, ring of fractions, euclidean domains, principal ideals domain, unique factorisation domains, polynomial rings; modules and vector spaces.
The main reference is chapter 2 of Hartshorne's book: Algebraic Geometry. We will study the category of sheaves of abelian groups and rings on a topological space. Then, we will attach to every commutative ring such a sheaf of rings and the pair consisting of the commutative ring and its sheaf of rings will be called an affine scheme. A scheme is a ringed space which is locally an affine scheme. Then we will study the main properties of schemes: open and closed immersions, fiber product of schemes, separated and proper schemes. An important part of the course will be to solve the many exercises at the end of each section in Hartshorne's book.
Anneaux et modules: k-algèbre et anneaux de division, modules à gauche, à droite et bimodules, R-modules simples, semi-simples, cycliques et indécomposables, Théorème de structure pour les anneaux semi-simples et théorème du double centralisateur.
Théorie des représentations: Représentations linéaires, théorème de structure pour les algèbres de groupes C[G], idempotents, relations d'orthogonalités, caractères et fonctions centrales, théorème de la dimension et théorème pq de Burnside.
Background: Background in algebraic geometry and schemes is assumed (say at the level of a first course in each), as well as algebraic number theory. A highly motivated student may be able to acquire this background on their own in preparation for the course. Please contact the instructor if you are not sure you have sufficient background.
Syllabus: Topics in the theory of Shimura varieties. The exact syllabus depends on the audience. Some topics I am considering are:
(1) A tour of Shimura varieties of low dimension - modular curves, Hilbert-Blumenthal surfaces, Picard modular surfaces, quaternionic modular surfaces, Siegel modular threefolds.
(2) Toric varieties and toroidal compactifications of Shimura varieties of low dimension.
(3) Group schemes, Dieudonné modules and stratification of moduli spaces of abelian varieties.
(4) Dimension formula for modular forms.
(5) Deformation theory of abelian varieties and local models.
Automorphic representations generalize the theory of classical modular forms and their Hecke operators, and have become both a central object of study and an indispensable tool in number theory. This course will be an introduction to automorphic representations. We will emphasize examples, and some structural theorems will be taken as black boxes. Some familiarity with algebraic number theory, modular forms, and Lie algebras will be assumed.
We will study the integer and rational points on linear, quadratic and cubic curves, culminating in the proof of Mordell's Theorem, which describes a surprising structure amongst the rational points on elliptic curves.
Anneaux commutatifs et leurs modules. Localisation : idéaux premiers, racine d'un idéal, anneaux et modules de fractions, anneaux locaux. Dépendance entière: clôture intégrale, théorème de montée. Anneaux et modules noethériens, anneaux de polynômes sur un anneau noethérien. Ensembles algébriques affines, théorème des zéros de Hilbert, ensembles algébriques irréductibles et idéaux premiers, propriétés des courbes planes, dimension des variétés. Applications.
Projective, injective, and flat modules, exact sequences, chain complexes, homology of a chain complex; derived functors, Tor and Ext functors, universal coefficient theorems, Künneth formula. Other topics and examples will be drawn from commutative algebra, non-commutative algebra, and algebraic topology.
The course will cover the basics of Fourier analysis: convergence of Fourier series on the circle; the Hardy-Littlewood maximal function; harmonic functions, Poisson integrals and the conjugate function; Fourier transforms on the line and in Euclidean space; the Schwartz space and tempered distributions; the Poisson Summation Formula.
Additional topics from among the following will be covered if time permits, as well as in student projects or presentations: interpolation; Hardy spaces and BMO; singular integrals; Littlewood-Paley theory; distributions, Sobolev spaces and applications to PDE; spherical harmonics; Fourier analysis on groups, the discrete Fourier transform and applications; wavelets.
RKHS is a fast-growing topic with numerous connections to other areas of mathematics, as well as profound applications in other basic sciences and engineering. This theory plays a decisive role in complex analysis, statistics, probability, functional analysis and integral operator theory, matrix theory, group representations, machine learning, signal processing and telecommunication, etc. In this course, the emphasize is on the abstract theory of RKHS with a glance to analytic function spaces such as Hardy, Dirichlet, Bergman, model and de Branges-Rovnyak spaces as special cases.
References:
-V. Paulsen, M. Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, Cambridge Studies in Advanced Mathematics 152, Cambridge University Press, 2016.
-J. Agler, J. McCarthy, Pick Interpolation and Hilbert function spaces, Graduate Studies in Mathematics 44, American Mathematical Society, 2002.
-J. Mashreghi, Representation Theorems in Hardy Spaces, London Mathematical Society Student Texts (74), Cambridge University Press, 2013.
Review of theory of measure and integration; product measures, Fubini's theorem; Lp spaces; basic principles of Banach spaces; Riesz representation theorem for C(X); Hilbert spaces; part of the material of MATH 565 may be covered as well.
Banach and Hilbert spaces, theorems of Hahn-Banach and Banach-Steinhaus, open mapping theorem, closed graph theorem, Fredholm theory, spectral theorem for compact self-adjoint operators, spectral theorem for bounded self-adjoint operators.
Ensembles mesurables, mesure de Lebesgue, théorèmes de Lusin et de Egorov, intégrale de Lebesgue, théorème de Fubini, espaces Lp, éléments de la théorie ergodique, mesure et dimension de Hausdorff, ensembles fractals.
The course will cover the fundamentals of geometric spectral theory as well as some recent developments in the area. Main topics include: spectral theorem for the Laplacian on Euclidean domains and Riemannian manifolds, geometric eigenvalue inequalities, nodal geometry of eigenfunctions, spectral invariants, "Can one hear the shape of a drum?", Steklov eigenvalue problem, spectral asymptotics.
Basic knowledge of PDEs and differential geometry will be assumed.
Topics from differentiable manifolds, tangent space, cotangent space, immersions, orientation, vector fields, differentiable forms, integration on manifolds, Riemannian metrics, curvature tensors, Bonnet-Myers Theorem and Synge Theorem, fundamental group, manifolds of negative curvature, the sphere theorem, eigenvalues of Riemannian manifolds, and de Rham theory.
Measure and integration, measure spaces, convergence theorems, Radon-Nikodem theorem, measure and outer measure, extension theorem, product measures, Hausdorf measure, LP-spaces, Riesz theorem, bounded linear functionals on C(X), conditional expectations and martingales.
The course is devoted to the theory of unbounded operators in Hilbert spaces. The main themes are extensions of symmetric operators and criteria of self-adjointness, proofs of the spectral theorem for unbounded operators, applications to PDE. As an additional topic I am planning to include some versions of the adiabatic theorem for time-dependent Hamiltonians and elements of Berry phase theory.
Linear and quasilinear 1-st order equations. Transport equation. Shock waves and rarefactions. D'Alembert solution to the one-dimensional wave equation. Infinite, semiinfinite and finite string. Separation of variables, Fourier method for the 1-d wave equation. Solution of the wave equation in 2-d and 3-d. Duhamel formula. Energy method, finite speed of propagation. Laplace and Poisson equations in 2-d and 3-d. Green's formula. Hydrodynamical interpretation. Properties of harmonic functions. Maximum principle, mean value theorem, Liouville and Harnack's theorems. Dirichlet's and Neumann's problems for the Laplace equation. Variational method. Heat equation. Solution in the whole space. Energy method for the proof of existence and uniqueness of solution.
Séries de Fourier, théorèmes de convergence, théorème de Fejér. Transformée de Fourier, théorème de convolution, théorème d'inversion, théorème de Plancherel, formule de Poisson. Transformée de Fourier rapide. Espaces de Hilbert : bases orthonormées, polynômes orthogonaux, ondelettes de Haar. Théorie des ondelettes : analyses multirésolutions, ondelettes père et mère, transformée d'ondelette rapide, intégrabilité et moments. Exemples.
Continuation of topics from MATH 564. Signed measures, Hahn and Jordan decompositions. Radon-Nikodym theorems, complex measures, differentiation in Rn, Fourier series and integrals, additional topics.
Espaces d’Hilbert, de Banach, théorèmes de Hahn-Banach, de Banach-Steinhaus et du graphe fermé, topologies faibles, espaces réflexifs, décomposition spectrale des opérateurs auto-adjoints compacts.
Espaces de Hilbert, espaces de Banach, algèbres de Banach. Étude particulière de l'algèbre des opérateurs sur un espace de Hilbert. Espace de Banach des fonctions à variation bornée et intégrale de Stieltjes. Fonctionnelles linéaires. Théorème de représentation de Riesz. Théorèmes de Hahn-Banach, de la borne uniforme et du graphe fermé. Topologies faibles. Convexité : théorèmes de séparation, inégalité de Jensen, théorème de Krein-Milman.
Examples of applications of statistics and probability in epidemiologic research. Sources of epidemiologic data (surveys, experimental and non-experimental studies). Elementary data analysis for single and comparative epidemiologic parameters.
Foundations of causal inference in biostatistics. Statistical methods based on potential outcomes; propensity scores, marginal structural models, instrumental variables, structural nested models. Introduction to semiparametric theory.
Statistical methods for multinomial outcomes, overdispersion, and continuous and categorical correlated data; approaches to inference (estimating equations, likelihood-based methods, semi-parametric methods); analysis of longitudinal data; theoretical content and applications.
Common data-analytic problems. Practical approaches to complex data. Graphical and tabular presentation of results. Writing reports for scientific journals, research collaborators, consulting clients.
Multivariable regression models for proportions, rates, and their differences/ratios; Conditional logistic regression; Proportional hazards and other parametric/semi-parametric models; unmatched, nested, and self-matched case-control studies; links to Cox's method; Rate ratio estimation when "time-dependent" membership in contrasted categories.
Advanced applied biostatistics course dealing with flexible modeling of non-linear effects of continuous covariates in multivariable analyses, and survival data, including e.g. time-varying covariates and time-dependent or cumulative effects. Focus on the concepts, limitations and advantages of specific methods, and interpretation of their results. In addition to 3 hours of weekly lectures, shared with epidemiology students, an additional hour/week focuses on statistical inference and complex simulation methods. Students get hands-on experience in designing and implementing simulations for survival analyses, through individual term projects.
Représentations et module d'un groupe G, représentations équivalentes, sous-module. Représentations indécomposable, réductible, irréductible. Théorème de Maschke. Morphisme, lemme de Schur.
Algèbre de groupe, fonctions sur cette algèbre, fonction de classe. Caractères, relations d'orthogonalité, tables de caractères. Représentation régulière. Analyse de Fourier sur les groupes finis, identité de Parseval, théorème de Wedderburn.
Nombres algébriques, théorème de la dimension, théorème de Burnside. Action de groupe, lemme de Burnside, paires de Gelfand.
Représentations induites, théorème de réciprocité de Frobenius, critère d'irréductibilité de Mackey.
Marche aléatoire sur les groupes finis. Modèles de Gilbert–Shannon–Reeds, théorèmes de Diaconis.
Ce cours sert comme introduction au niveau des cycles supérieurs à la combinatoire algébrique et énumérative, avec une emphase sur les méthodes efficaces. Les sujets de base comprendront les séries formelles ordinaires et exponentielles, les objets classiques de la combinatoire (partages, chemins dans un réseau, graphes, permutations), méthodes asymptotiques, suites vérifiant des récurrences linéaires, et l'occurrence et le comportement des séries rationnelles, algébriques, et D-finies. Dépendant des intérêts des étudiants, on pourrait aussi regarder les séries formelles à plusieurs variables, l'application des logiciels pour la combinatoire, démonstrations algorithmique de transcendence des constants, questions de calculabilité et complexité dans l'énumération, et les liens avec les langues formelles
This course serves as a graduate introduction to enumerative and algebraic combinatorics, with a focus on effective methods. Our core topics include ordinary and exponential generating functions, classical combinatorics objects (partitions, lattice paths, graphs, permutations), asymptotic methods, sequences satisfying linear recurrence relations, and the occurrence and behaviour of rational, algebraic, and D-finite functions. Depending on student interest, more advanced topics could include multivariate generating functions, computer algebra tools for combinatorics, algorithmic transcendence of constants vs generating functions, computability and complexity questions in enumeration, and connections to formal languages.
Compléments sur la théorie des groupes:
- actions de groupes ; orbites ; stabilisateurs
- calcul dans les groupes de permutations:
- orbite et stabilisateur d'un élément;
- système de représentants du stabilisateur d'un élément;
- ordre d'un groupe de permutations;
- test d'appartenance à groupe de permutations;
- forme normale des éléments;
- algorithme de Todd-Coxeter.
Théorie des catégories:
- définition et exemples des catégories;
- foncteurs;
- transformations naturelles;
- propriétés universelles;
- (co)produits, produit fibré et somme amalgamée;
- équivalence des catégories.
Théorie des modules:
- modules artiniens et noethériens;
- modules simples, semisimples, indécomposables;
- théorèmes de Jordan-Holder, Krull-Schmidt et Artin-Wedderburn;
- notion de dimension d'un module;
- produit tensoriel de modules.
Enumerative combinatorics: inclusion-exclusion, generating functions, partitions, lattices and Moebius inversion. Extremal combinatorics: Ramsey theory, Turan's theorem, Dilworth's theorem and extremal set theory. Graph theory: planarity and colouring. Applications of combinatorics.
Partially ordered sets, linear extensions, lattices, and associated simplicial complexes. Sperner properties; theorems of Dilworth and Greene. Combinatorial aspects of algebraic topology. Hyperplane arrangements.
Le séminaire portera provisoirement sur le "pléthysme", ce qui peut être défini comme une opération sur les fonctions symétriques ; comme une opération sur les représentations des groupes linéaires GL(n) ; ou comme une opération sur les représentations des groupes symétriques. En plus de comprendre ces trois constructions, l'objectif du séminaire est de faire une étude approfondie des certains résultats récents.
L'objectif du cours est de présenter les structures discrètes standards et les principales méthodes d'énumération. Les sujets suivants seront présentés :
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.
Basic point-set topology, including connectedness, compactness, product spaces, separation axioms, metric spaces. The fundamental group and covering spaces. Simplicial complexes. Singular and simplicial homology. Part of the material of MATH 577 may be covered as well.
Préliminaires: topologie générale, théorème des fonctions implicites, et théorème fondamental des équations différentielles ordinaires (EDO) dans les espaces euclidiens. Variétés différentiables, formes différentielles, fibrés. Partitions de l’unité. Groupes à un paramètre de difféomorphismes et généralisation du théorème des EDO, dérivée et crochet de Lie. Intégration et théorème de Stokes. Cohomologie de De Rham. Éléments de géométrie riemannienne.
Note: In this course, I will speak in English and write in French!
Structures topologiques. Convergence de suites généralisées et axiomes de séparation. Fonctions continues. Espaces topologiques produits et topologie quotient. Plongement et métrisabilité. Espaces topologiques compacts et théorème de Tychonoff. Compactification de Stone-Cech. Structures uniformes et complétion. Espaces uniformes métrisables et théorème de Baire.
Groupe fondamental. Théorie des revêtements. Groupes d'homotopie de dimensions supérieures. Homologie singulière relative, homologie simpliciale, théorème d'approximation simpliciale. Relation entre le groupe fondamental et le premier groupe d'homologie. Théorème d'excision. Suite exacte de Mayer-Vietoris. Homologie des sphères, degré des applications entre sphères, applications. Théorème de Jordan-Brouwer. Complexes C.W. et discussion des théorèmes de base de la théorie de l'homotopie: théorème de Whithead, théorème de Hurewicz. Homologie cellulaire, caractéristique d'Euler. Le théorème de point fixe de Lefschetz.
This course will develop the basic elements of the theory of orderable groups and their applications to low-dimensional topology. In particular we will discuss the L-space conjecture which posits the equivalence between the left-orderability of the fundamental group of a 3-manifold and certain of its analytic and topological properties.
Basic properties of differentiable manifolds, tangent and cotangent bundles, differential forms, de Rham cohomology, integration of forms, Riemannian metrics, geodesics, Riemann curvature.
Actions on trees. Cayley graphs. The word problem. Hyperbolic groups. Quasi-isometry invariants.
Background: Background in algebraic geometry and schemes is assumed (say at the level of a first course in each), as well as algebraic number theory. A highly motivated student may be able to acquire this background on their own in preparation for the course. Please contact the instructor if you are not sure you have sufficient background.
Syllabus: Topics in the theory of Shimura varieties. The exact syllabus depends on the audience. Some topics I am considering are:
(1) A tour of Shimura varieties of low dimension - modular curves, Hilbert-Blumenthal surfaces, Picard modular surfaces, quaternionic modular surfaces, Siegel modular threefolds.
(2) Toric varieties and toroidal compactifications of Shimura varieties of low dimension.
(3) Group schemes, Dieudonné modules and stratification of moduli spaces of abelian varieties.
(4) Dimension formula for modular forms.
(5) Deformation theory of abelian varieties and local models.
Variétés, transversalité et degré. Théorème de Sard. Éléments de la théorie de Morse. Complexe de Morse. Théorème de Hopf-Poincaré. Cobordisme. Signature. Théorème de h-cobordisme. Classes caractéristiques. Espaces de Thom, groupes de cobordisme.
Définitions, exemples et propriétés de base des groupes et algèbres de Lie. Classification et structure des algèbres de Lie semi-simples. Décomposition de Cartan: algèbres de Lie réelles. Formule des caractères de Weyl. Représentations orthogonales et symplectiques.
Variétés complexes, variétés projectives, variétés de Kähler. Fibrés holomorphes hermitiens. Théorie de Hodge élémentaire. Positivité et théorème de plongement de Kodaira. Notions de courbure. Équation de Monge-Ampère complexe. Équation d'Hermite-Einstein. Correspondance de Kobayashi-Hitchin pour les fibrés. Noyau de Bergman. Applications à la géométrie algébrique complexe.
Le cours commencera avec des bases de la géométrie complexe visées pour l’étude du rôle des courbes complexes en dimension plus grande, telle que leurs implications pour la géométrie birationelle, topologie et arithmétique d’une variété algébrique en générale. On mettra accent sur appliquer les techniques de la géométrie (algébrique) complexe plutôt que la théorie derrière eux, ce qu’on va souvent prendre comme des boites noires en dimension plus grande pour ceux qui ne l'ont pas vu. Cependant, on traitera plus en profondeur des techniques de base des courbes complexes (i.e. surfaces de Riemann) comme leurs espaces de module, déformation et dégénérescence dans une famille, leur comportement ‘’arithmétique’’ et holomorphe, l’effet de la courbure dans la géométrie, surtout sur l’intersection avec une hyperplane, des courbes holomorphes (i.e. la théorie de distribution de valeurs) et dans l’arithmétique, etc. On touchera si possible à des conjectures de Vojta et son dictionnaire entre la théorie de distribution de valeur et la distribution des points rationnels dans une variété algébrique complexe.
Pré-requis minimaux: Analyse complexe, géométrie différentielle.
Une connaissance de la théorie de Surface de Riemann ou bien des courbes algébriques projective (en dimension deux) serait bon mais pas obligatoire.
This course focuses on computational aspects, implementation, continuous-time models, and advanced topics in Mathematical and Computational Finance. We shall cover the following topics (time permitting):
The course presents an introduction to statistical estimation techniques for insurance data. It is the natural continuation of Risk Theory, which discusses the probabilistic aspects of insurance portfolios. Two approaches to credibility theory are discussed: limited fluctuations and greatest accuracy. Topics covered include American, Bayesian and exact credibility. Bühlmann, Bühlmann-Straub, hierarchical and regression credibility models are derived. Generalized linear models and the issue of robustness will also be discussed. The course prepares for the Credibility part of the Society of Actuaries Exam STAM and the Casualty Actuarial Society Exam MAS II. It also covers more advanced material, as needed to use modern credibility with real insurance data. A grade of B or better is needed to apply to the Canadian Institute of Actuaries for exemption of Exam STAM (see Accredited Programs (concordia.ca).
This course is intended to deepen statistical and probabilistic concepts, related
to risk measures. The topics are:
(1) Definition of Distributional Univariate Risk Measures and Properties
(2) Risk Aggregation and Decomposition
(3) Optimization Based Univariate Risk Measures
(4) Systemic Risk
(5) Sensitivity Analysis
(6) Distorted Measures
(7) Backtesting, Regression, Depth Trimmed Regions
(8) Multivariate Framework and Extension of Univariate Risk Measures
Structures à terme, processus stochastiques, modèles et produits dérivés de taux d'intérêt, immunisation et appariement, produits dérivés de crédit, titres adossés à des créances hypothécaires, volatilité.
Tribus et variables aléatoires. Théorie de l'intégration: théorème de Lebesgue, espace Lp, théorème de Fubini. Construction de mesures, mesure de Radon. Indépendance. Conditionnement.
Mesures et comparaison des risques, Théorie de la ruine en temps discret et continu, Mouvement brownien et temps de premier passage, Modèles de risque de crédit, Concepts et mesures de dépendance, Copules, Applications des modèles de dépendance en actuariat et en finance.
The problem of fitting probability distributions to loss data is studied. In practice, heavy tailed distributions are used (i.e. skewed to the right) which require some special inferential methods. The problems of point and interval estimation, test of hypotheses and goodness of fit are studied in detail under a variety of inferential procedures (empirical, maximum likelihood and minimum distance) and of sampling designs (individual/grouped data, truncation and censoring). Loss data sets serve as illustration of the method. A reasonable understanding of undergraduate mathematical statistics is the only prerequisite for the course. The statistical package S-Plus or the (shareware) statistical software R or the spreadsheet EXCEL application will be used for data analysis. The course prepares for the Loss Models part of the Society of Actuaries (SOA) Exam STAM and the Casualty Actuarial Society (CAS) Exam MAS-I.
This course is a rigorous introduction to mathematical and computational finance. The focus is on the general theory through a thorough study of binomial models in finance. The topics covered include:
The emphasis is on the probabilistic aspects (stochastic processes) although some estimation (inference) questions will also be discussed. The topics include (but are not limited to) aggregate risk models, homogenous and nonhomogenous discrete-time Markov chain models, Poisson processes, coinsurance, effects of inflation on losses, copulas, risk measures (VaR, TVaR), claim reserving.
Ce cours offre une introduction aux méthodes d’inférence pour les modèles à chaîne de Markov cachée (hidden Markov models), aussi connus sous les appellations modèles à changement de régimes (regime-switching models) ou modèles espace-état (state-space models). Ces processus sont des modèles de séries temporelles faisant intervenir un signal markovien observé de façon imparfaite et bruité sous forme de données.
Le cours aborde les propriétés statistiques du modèle, l’estimation des paramètres et les techniques de filtrage, de lissage et de prédiction qui y sont reliées. Les méthodes suivantes seront notamment étudiées : filtre d’Hamilton, algorithme avant-arrière, filtre de Kalman, filtre particulaire, méthodes de Monte Carlo séquentielles et algorithme espérance-maximisation.
Des applications en mathématiques financières seront présentées et l’étudiant sera appelé à implanter plusieurs algorithmes à l’aide du logiciel informatique R.
Modèles discrets. Stratégies de transaction. Arbitrage. Marchés complets. Évaluation des options. Problème d'arrêt optimal et options américaines. Mouvement brownien. Intégrale stochastique, propriétés. Formule d'Itô. Localisation. Introduction aux équations différentielles sotchastiques. Changement de probabilité et théorème de Girsanov. Représentation des martingales et stratégie de couverture. Modèle de Black et Scholes.
Modèle de Black-Scholes; Changements de numéraire; Options exotiques; Options américaines; Modèles à temps discret (GARCH, changements de régime, volatilité stochastique): inférence et simulation; Modèles à temps continu (sauts, volatilité stochastique): inférence et simulation; Modèles de taux d'intérêt: tarification, couverture, inférence et simulation.
The course begins with a review of some topics in multivariate calculus. Then we go on to discuss convex sets and functions, convex optimization, and (nonlinear) duality theory. An introduction to Nonsmooth Analysis is given. All course references and materials are available online.
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.
Introduction and analysis of basic models in mathematical biology using modern numerical simulation techniques and tools. Discrete and continuous dynamical systems, stochastic processes, statistical models, and numerical simulation. Journal club focused on recent articles in mathematical biology.
The course will accommodate anglophone students who are interested in taking it, as well as francophone students.
While graphs are intuitively and naturally represented by vertices and edges, such representations are limited in terms of their analysis, both theoretically and practically (e.g., when implementing graph algorithms). A more powerful approach is yielded by representing them via appropriate matrices (e.g., adjacency, diffusion kernels, or graph Laplacians) that capture intrinsic relations between vertices over the "geometry" represented by the graph structure. Spectral graph theory leverages such matrices, and in particular their spectral and eigendecompositions, to study the properties of graphs and their underlying intrinsic structure. This study leads to surprising and elegant results, not only from a mathematical standpoint, but also in practice with tractable implementations used, e.g., in clustering, visualization, dimensionality reduction, and manifold learning, and geometric deep learning. Finally, since nearly any modern data nowadays can be modelled as a graph, either naturally (e.g., social networks) or via appropriate affinity measures, and therefore the notions and tools studied in this course provide a powerful framework for capturing and understanding data geometry in general.
Accessibilité de langue : The course will accommodate anglophone students who are interested in taking it, as well as francophone students.
Topics will include (time permitting) concentration of sum or random variables, random vectors in high-dimensions, random matrices, symmetrization, random processes, chaining. We will also discuss applications in key data science areas, such as random graphs, community detection in networks, dimensionality reduction, statistical learning theory, and sparse recovery.
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.
Line search methods including steepest descent, Newton's (and Quasi-Newton) methods. Trust region methods, conjugate gradient method, solving nonlinear equations, theory of constrained optimization including a rigorous derivation of Karush-Kuhn-Tucker conditions, convex optimization including duality and sensitivity. Interior point methods for linear programming, and conic programming.
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.
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.
This topic course concerns the interplay between the Large Deviation Principle (LDP) of probability theory and the mathematical foundations of statistical mechanics (SM). Although this fundamental link goes back to the pioneering work of Boltzmann and has played a central role in the development of both subjects, it is rarely discussed at the introductory level. The goal of the course is to describe the basic theory of LDP and SM with an emphasis on the foundational link between them.
Topics to be covered:
LDP. Cramér’s theorem in the i.i.d. setting. General structure of LDP. Gärtner-Ellis theorem. Boltzmann-Sanov theorems. Method of Ruelle-Lanford’s functions. Varadhan’s Lemma. Applications.
SM of Lattice Gasses. Interactions and pressure. Entropy. Boltzmann and Gibbs equilibrium states. Equivalence of ensembles. Theory of Gibbs states. Hausdorff dimension and Boltzmann entropy. Information theory perspective. Beyond Gibbsianity.
Additional topics will include: LDP and SM in the general dynamical systems setting. Thermodynamic formalism of dynamical systems. Rotators, dynamics, and the 0- Law of Thermodynamics.
Prerequisites. Honours Analysis 3-Math 454, Honours Probability-Math 356, and willingness to pick up on the pre-requisite topics (which are of independent interest) as we proceed. The references, and in some cases pre-recorded videos with pre-requisites, will be provided. In exceptional cases (and this in particular applies to the Joint Honours Math. Phys students), the course can be taken with Math 454 and Math 356 or Phys 362 as co-prerequisites. If you are interested to do so, please contact the instructor.
Représentations et module d'un groupe G, représentations équivalentes, sous-module. Représentations indécomposable, réductible, irréductible. Théorème de Maschke. Morphisme, lemme de Schur.
Algèbre de groupe, fonctions sur cette algèbre, fonction de classe. Caractères, relations d'orthogonalité, tables de caractères. Représentation régulière. Analyse de Fourier sur les groupes finis, identité de Parseval, théorème de Wedderburn.
Nombres algébriques, théorème de la dimension, théorème de Burnside. Action de groupe, lemme de Burnside, paires de Gelfand.
Représentations induites, théorème de réciprocité de Frobenius, critère d'irréductibilité de Mackey.
Marche aléatoire sur les groupes finis. Modèles de Gilbert–Shannon–Reeds, théorèmes de Diaconis.
This course covers most of the materials in the first seven chapters of Probability and Random Processes by Grimmett and Stirzaker. In particular, it covers topics such as generating and characteristic functions and their applications in random walk and branching process, different modes of convergence and an introduction of martingales.
Probability spaces. Random variables and their expectations. Convergence of random variables in Lp. Independence and conditional expectation. Introduction to Martingales. Limit theorems including Kolmogorov's Strong Law of Large Numbers.
Espace de probabilité, variables aléatoires, indépendance, espérance mathématique, modes de convergence, lois des grands nombres, théorème central limite, espérance conditionnelle et martingales. Introduction au mouvement brownien.
Topics will include (time permitting) concentration of sum or random variables, random vectors in high-dimensions, random matrices, symmetrization, random processes, chaining. We will also discuss applications in key data science areas, such as random graphs, community detection in networks, dimensionality reduction, statistical learning theory, and sparse recovery.
This course introduces the basic ideas and methods of stochastic calculus. Topics covered include:
1. Martingales.
2. Brownian motion and Markov processes.
3. Stochastic integrals, Ito's formula and Girsanov theorem.
4. Stochastic differential equations.
If time allows additional topics might be covered.
Characteristic functions: elementary properties, inversion formula, uniqueness, convolution and continuity theorems. Weak convergence. Central limit theorem. Additional topic(s) chosen (at discretion of instructor) from: Martingale Theory; Brownian motion, stochastic calculus.
Random matrix theory is a relatively new branch of probability, which has arisen to answer questions in high-dimensional statistics, disordered quantum mechanical systems, the smoothed analysis of algorithms and machine learning. This course aims to give a broad foundation in random matrix theory and illustrate some of the applications.
Core course content:
We also briefly cover one or two of the topics below.
Prerequisite: measure-theoretic probability (in particular, familiarity with weak/almost sure/in-probability convergence), linear algebra
Corequisite: stochastic calculus
Suggested course: complex analysis.
Mouvement brownien, intégrale stochastique, formule d’Itô, équations différentielles stochastiques, théorèmes de représentation, théorème de Girsanov. Formule de Black et Scholes.
Révision de la théorie des probabilités. Espérances conditionnelles. Martingales à temps discret et théorème de convergence de Doob. Convergence étroite, tension et théorème de la limite centrale.
This course is an introduction to statistical inference for parametric models. The following topics will be covered:
1. Distribution of functions of several random variables (distribution function and change of variable techniques), sampling distribution of mean and variance of a sample from Normal distribution.
2. Distribution of order statistics and sample quantiles.
3. Estimation: unbiasedness, Cramér-Rao lower bound and efficiency, method of moments and maximum likelihood estimation, consistency, limiting distributions, delta-method.
4. Sufficiency, minimal sufficiency, completeness, UMVUE, Rao-Blackwell and Lehman-Scheffe theorems.
5. Hypothesis-testing: likelihood-ratio tests.
6. Elements of Bayesian estimation and hypothesis-testing.
Text: Introduction to Mathematical Statistics (6th, 7th or 8th Edition), by R.V. Hogg and A.T. Craig, Prentice Hall Inc., 1994. Recommended reading: (for problems, examples etc) Statistical Inference (2nd Edition), by G. Casella and R. L. Berger, Duxbury, 2002. Evaluation: Assignments (4), Midterm exam, Final exam.
This course introduces multivariate statistical analysis, both theory and methods. The goal of this course is to help students develop the statistical skills to approach and analyse multivariate data correctly in an applied, as opposed to theoretical, context. Topics covered include:
This course is an introduction to reinforcement learning techniques. It requires extensive programming with the R language. Topics covered include: Multi-armed bandit problem, Markov Decision Problems, Dynamic Programming, Monte-Carlo solution methods, Temporal difference methods, Multi-period Approximation methods, Policy gradient.
Multivariate normal and chi-squared distributions; quadratic forms. Multiple linear regression estimators and their properties. General linear hypothesis tests. Prediction and confidence intervals. Asymptotic properties of least squares estimators. Weighted least squares. Variable selection and regularization. Selected advanced topics in regression. Applications to experimental and observational data.
Conditional probability and Bayes’ Theorem, discrete and continuous univariate and multivariate distributions, conditional distributions, moments, independence of random variables. Modes of convergence, weak law of large numbers, central limit theorem. Point and interval estimation. Likelihood inference. Bayesian estimation and inference. Hypothesis testing.
Processus stochastiques (généralités). Description et caractéristiques des séries chronologiques. Transformées de Fourier. Analyse statistique des séries chronologiques. Analyse spectrale des processus linéaires. Lissage des estimateurs spectraux.
Notions fondamentales de probabilités appliquées à divers domaines de l’intelligence artificielle. Réseaux bayésiens, champs markoviens, diverses méthodes d’inférence (variationnelle, par maximum a posteriori, recuit simulé, etc.), échantillonnage et méthodes de Monte Carlo par chaînes de Markov, séries chronologiques, partitionnement spectral et modèles à variables latentes. Applications en imagerie, en analyse de textes et sur les réseaux de neurones.
Principes de l’analyse bayésienne; loi à priori et à postériori, inférence statistique et théorie de la décision. Méthodes computationnelles; méthodes de Monte Carlo par chaînes de Markov. Applications.
Étude du « bootstrap ». Estimation du biais et de l'écart-type. Intervalles de confiance et tests. Applications diverses, incluant la régression et les données dépendantes. Étude du « jackknife », de la validation croisée et du sous-échantillonnage.
Notions de probabilités. Comportement asymptotique des moments et quantiles échantillonnaux. Normalité asymptotique de transformation; stabilisation de la variance. Loi asymptotique du test du khi-deux. Théorie asymptotique en inférence paramétrique.
Tableaux de contingence. Mesures d'association. Risque relatif et rapport de cote. Tests exacts et asymptotiques. Régression logistique, de Poisson. Modèles log-linéaires. Tableaux de contingence à plusieurs dimensions. Méthodes non paramétriques.
Étude des distributions échantillonnales classiques: T2 de Hotelling; loi de Wishart; distribution des valeurs et des vecteurs propres; distribution des coefficients de corrélation. Analyse de variance multivariée. Test d'indépendance de plusieurs sous-vecteurs. Test de l'égalité de matrices de covariance. Sujets spéciaux.
Nombre aléatoire. Simulation de lois classiques. Méthodes d'inversion et de rejet. Algorithmes spécifiques. Simulation des chaines de Markov à temps discret et continu. Solution numérique des équations différentielles ordinaires et stochastiques. Méthode numérique d'Euler et de Runge-Kutta. Formule de Feynman-Kac. Discrétisation. Approximation faible et forte, explicite et implicite. Réduction de la variance. Analyse des données simulées. Sujets spéciaux.
Rappel sur les principales notions de statistique mathématique et sur la statistique asymptotique. Introduction à la théorie des copules. Description des modèles de dépendance bidimensionnels et multidimensionnels les plus populaires et exploration exhaustive des propriétés de ces copules. Inférence statistique dans les modèles de copules : estimation de paramètres, copule empirique, tests d'adéquation et tests d'hypothèses composites. La méthode delta fonctionnelle et ses nombreuses applications, notamment en inférence de copules. Survol des avancées récentes, incluant les tests de rupture, l'étude de la dépendance conditionnelle, la modélisation de la dépendance spatiale et l'utilisation de la fonction caractéristique. Les objectifs spécifiques de ce cours sont : de maîtriser la théorie des copules, de connaître les principales méthodes d'inférence concernant les copules, d'être au fait des principaux développements récents, de bien connaître la littérature sur les copules, d'être capable de mettre en oeuvre les méthodes statistiques avec le logiciel Matlab (estimation de la puissance de tests, analyse de jeux de données).
This course is an introduction to simulation and Monte Carlo estimation. The following topics will be covered:
1. Simulation of random variables/vectors from their (joint) probability mass function/density function: methods of inverse-transform, accept-reject, composition and factorization (for random vectors).
2. Simulation of homogeneous and non-homogeneous Poisson processes in 1-dimension: methods of inverse-transform and thinning.
3. Some discrete-event simulation models, e.g., 1-server and 2-server queues, insurance-risk model, machine-repair model.
4. Some variance-reduction techniques: methods of anti-thetic variables, control variables, conditional expectation, stratified sampling.
The software R will be extensively used to write simulation codes and will be demonstrated over a few classes.
Text: Simulation, 5th Edition, by Sheldon M. Ross. Recommended reading: A first course in statistical programming with R, 2nd Edition, by W. John Braun and Duncan J. Murdoch (Cambridge University Press). Evaluation: Assignments (4), Midterm exam, Final exam.
This course is an introduction to statistical learning techniques. Topics covered include cross-validation, regression methods, classification methods, tree-based methods, introduction to neural networks, unsupervised learning.
Statistical analysis of time series in the time domain. Moving average and exponential smoothing methods to forecast seasonal and non-seasonal time series, construction of prediction intervals for future observations, Box-Jenkins ARIMA models and their applications to forecasting seasonal and non-seasonal time series. A substantial portion of the course will involve computer analysis of time series using computer packages (mainly MINITAB). No prior computer knowledge is required.
This course is an introduction to basic experimental designs and analysis of linear statistical models related to them. The following topics will be covered:
1. Review of estimation and hypothesis-testing in Normal error-based linear models.
2. Analysis of completely randomized design (CRD), randomized complete block design (RCBD), balanced incomplete block design (BIBD), Latin Square design (LSD), Graeco-Latin Square design (GLSD).
3. Factorial experiments: 2-factor and 3-factor designs, confounding, fractional replication.
4. Response-surface models.
Text: Design and Analysis of Experiments, 10th Edition, by Douglas C. Montgomery (John Wiley). Evaluation: Assignments (4), Midterm exam, Final exam.
Réduction de la dimensionnalité : par exemple, analyse en composantes principales et analyse canonique des corrélations. Classification non supervisée : classification hiérarchique, non hiérarchique et sur la base de modèles, évaluation de la qualité et choix du nombre de groupes. Classification supervisée : classifieurs linéaires et non linéaires, évaluation de la qualité des classifieurs.
Exponential families, link functions. Inference and parameter estimation for generalized linear models; model selection using analysis of deviance. Residuals. Contingency table analysis, logistic regression, multinomial regression, Poisson regression, log-linear models. Multinomial models. Overdispersion and Quasilikelihood. Applications to experimental and observational data.
Simple random sampling, domains, ratio and regression estimators, superpopulation models, stratified sampling, optimal stratification, cluster sampling, sampling with unequal probabilities, multistage sampling, complex surveys, nonresponse.
Stationary processes; estimation and forecasting of ARMA models; non-stationary and seasonal models; state-space models; financial time series models; multivariate time series models; introduction to spectral analysis; long memory models.
Analyse en composantes principales. Analyse des corrélations canoniques et régression multidimensionnelle. Analyse des correspondances. Discrimination. Classification. Analyse factorielle d'opérateurs.
Rappels sur la régression linéaire multiple (inférence, tests, résidus, transformations et colinéarité), moindres carrés généralisés, choix du modèle, méthodes robustes, régression non linéaire, modèles linéaires généralisés.
Comparaison de plusieurs populations. Représentations graphiques. Analyse en composantes principales, factorielle, des correspondances, canonique, discriminante. Classification. Mesures de redondance.
Espérance conditionnelle. Prédiction. Modèles statistiques, familles exponentielles, exhaustivité. Méthodes d'estimation: maximum de vraisemblance, moindres carrés etc. Optimalité: estimateurs sans biais à variance minimum, inégalité de l'information. Propriétés asymptotiques des estimateurs. Intervalles de confiance et précision. Éléments de base de la théorie des tests. Probabilité critique, puissance en relation avec la taille d'échantillon. Relation entre tests et intervalles de confiance. Tests pour des données discrètes.
Théorie des modèles linéaires généraux. Théorie des modèles linéaires généralisés. Régression logistique. Modèles log-linéaires.