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 CRÉPUQ website (the CRÉPUQ 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 to register for a course at McGill University:
Once the registration through the CRÉPUQ Website is complete, the student will receive a confirmation from the CRÉPUQ. The student must then register for the course at McGill University through the MINERVA registration system.
Important deadlines: Concordia, HEC Montréal, Laval, McGill, Université de Montréal, UQAM, UQTR, Université de Sherbrooke
Course Schedules:
The course will develop the theory of the etale fundamental group of a connected scheme in parallel to the Galois theory of a field and the theory of the fundamental group of a topological space. Some knowledge of Algebraic Geometry would be helpful but not necessary as the course will be self contained.
This course will cover the basic theory of elliptic curves and algebraic curves. The prerequisites for this course is the standard background in abstract algebra (groups, rings, field extensions, Galois theory etc). A first course in algebraic number theory is recommended, but not mandatory. The course is open to all graduate students, either at the master or the Ph.D. level.
Recommended Textbook: J. Silverman, The Arithmetic of Elliptic Curves.
The course will describe the statement and proofs of the main results of class field theory, both local and global, following the treatment given in the textbook of Cassels-Frolich, which shall be followed fairly closely.
This is an introductory course to sieve methods and their applications. After reviewing some background material in probabilistic number theory, we will introduce and study the main objects of sieve theory. In particular, we will develop the combinatorial sieve and Selberg's sieve and use them to prove various estimates about prime numbers. We will then use sieve theory to study L-functions and establish Linnik's theorem. The proof of this result provides a natural entry point to the theory of bilinear sum estimates and the development of the Large Sieve. As an application of this circle of ideas, we will prove the Bombieri-Vinogradov theorem. The course will conclude with a discussion of the recent spectacular developments about bounded and large gaps between primes.
Prerequisites. Strong background in linear algebra. Familiarity with differential geometry (e.g. tangent bundles, Frobenius’s theorem on integrable subbundles of the tangent bundle), real analysis (e.g. Stone-Weierstrass theorem, Hilbert spaces, compact operators), and representation theory of finite groups would be very useful too.
Course textbooks. All optional.
Representations of Compact Lie Groups by Brocker and Tom Dieck.
Lie Groups by Bump.
Syllabus. Topics will include: examples of classical compact Lie groups and Lie algebras, Haar measure, the Peter-Weyl Theorem, the exponential map, maximal tori and their conjugacy, the Weyl group, Weyl integration formula, roots and root systems, Dynkin diagrams, highest weight theory, Weyl character formula.
Time permitting, we may also discuss applications of the above to random matrix theory.
Evaluation. To be determined after our organizational meeting.
This course will present the theory of p-adic and overconvergent modular forms as they appear in the work of J.-P. Serre, N. Katz and R. Coleman. Basic knowledge of classical modular forms (for example as they appear in "A course in Arithmetic" by J.-P. Serre) and of algebraic geometry is required.
This course will cover advanced topics in the theory of elliptic curves. It is intended as a continuation of the course “Elliptic Curves” to be taught by Prof. David in the Fall at Concordia University. Although it is not required to take David’s course, I will assume that students know the material covered in David’s course. The exact selection of topics will be determined once a more precise syllabus for David’s course becomes available, but they will mostly be chosen from Silverman’s books on elliptic curves.
The goal of this course is to give a complete proof of the prime number theorem, a proof that will help the student appreciate many of the important theorems in the subject. We will review in detail the motivation for the prime number theorem (and other conjectures and theorems about prime numbers), and focus on the background needed in both number theory and analysis (so that students feel comfortable with the techniques used). Then we will prove the prime number theorem and begin to appreciate the importance of the Riemann Hypothesis. Having gone slowly over this we will be ready to use these ideas in many different directions. Our only scheduled goal will be to prove Dirichlet's theorem, that there are infinitely many primes in each reduced residue class a mod q, though we will at least sketch how to estimate how many primes like in each such class. Moreover, if things go well, we will apply these ideas to primes in short intervals, in short arithmetic progressions, study least quadratic non-residues, ....
Tues and Thurs 10:30-12:30 in room 5183
Les sujets traités comprennent: Nombres p-adiques, corps p-adiques, topologie p-adique, théorie d'évaluation, théorie de ramification, groupe pro-fini, groupe pro-p, p-groupe puissant, groupe uniformément puissant, structure de Lie sur des groupes pro-p, algèbre d'Iwasawa, théorème de structure sur les modules d'Iwasawa.
Algèbres et morphismes, modules sur une algèbre, morphismes et suites exactes, modules de morphismes. Catégorie de modules, produits et sommes directs, équivalence de catégories. Foncteurs Hom, exactitude de foncteurs, modules projectifs et injectifs. Produits tensoriels de modules, théorèmes de Watts, algèbre tensorielle et extérieure. Algèbres des chemins d'un carquois. Modules artiniens et noethériens, suite de composition, théorème de Jordan-Hölder. Radicaux de modules, socle et coiffe.
Sources principales
• I Assem, Algèbres et modules, Masson et Les presses de l'Université d'Ottawa (1997).
• I Assem, D Simson, A Skowronski, Elements of the representation theory of associative algebras, Cambridge University Press (2006).
Starting with classical inequalities for convex sets and functions, the course aims to present famous geometric inequalities like the Brunn-Minkowski inequality and its related functional form, Prekopa-Leindler, the Blaschke-Santalo inequality, the Urysohn inequality, as well as more modern ones such as the reverse isoperimetric inequality, or the Brascamp-Lieb inequality and its reverse form. In the process, we will touch upon log-convex functions, duality for sets and functions and, generally, extremum problems.
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.
The course is devoted to the basics of the theory of operators in Hilbert space with emphasis on applications to Partial Differential Equations.
Prerequisites: MATH 564, MATH 565, and MATH 566.
Official Book: Haim Brezis, Functional Analysis, Sobolev spaces and partial differential equations. Since it is available for download, this book has not been ordered from the bookstore.
Please note that there are many sets of notes for functional analysis on the web. See the course webpage for a list and links.
Official Syllabus: Banach spaces. Hilbert spaces and linear operators on these. Spectral theory. Banach algebras. A brief introduction to locally convex spaces.
Proposed Syllabus: 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, Banach algebras and the Gelfand theory, Locally convex spaces. Additional topics to be chosen from: Lorentz spaces and interpolation, distributions and Sobolev spaces, The von Neumann-Schatten classes, symbolic calculus of Hilbert space operators, representation theory and harmonic analysis, semigroups of operators, Krein-Milman theorem, tensor products of Hilbert spaces and Banach spaces, fixed point theorems.
Assessment: 40% assignments, 60% final exam.
Exam: The final examination will be a take-home exam. There is no "additional work" option and the grade of incomplete will not be given. A supplemental exam will be available.
Course webpage: http://www.math.mcgill.ca/drury/teaching/math635f18/math635f18.php
Exam Viewing: The instructor reserves the right to set a specific time or times for the purpose of exam viewing. If such times are set, they will be announced on the course webpage.
Note: In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change.
Note: In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.
Academic Integrity: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).
Contenu du cours: ensembles mesurables, mesure de Lebesgue; principes de Littlewood, théorèmes de Lusin et de Egorov; intégrale de Lebesgue, théorème de Fubini, espaces L1 et L2; mesures absolument continues, théorème de Radon-Nikodym; éléments de la théorie ergodique; mesure et dimension de Hausdorff, ensembles fractales.
This graduate course is an introduction to the treatment of nonlinear differential equations, and more generally to the theory of dynamical systems. The objective is to introduce the student to the theory of dynamical systems and its applications. Firstly, classical dynamics analysis techniques will be presented: continuous and discrete flows, existence and stability of solutions, invariant manifolds, bifurcations and normal forms. Secondly, an introduction to ergodic theory and an overview of modern applications will be presented: chaotic dynamics, strange attractors, dynamic entropy, high-dimensional systems (e.g. networks), driven dynamics and information processing. Particular attention will be paid to computations performed by dynamical systems.
At the end of the course, the student will be able to apply dynamical systems analysis techniques to concrete problems, as well as navigate the modern dynamical systems literature. Several examples and applications making use of numerical simulations will be used. To take this course, the student must master, at an undergraduate level, notions of calculus, linear differential equations, linear algebra and probability.
Course schedule :
Tuesday : 10h30 - 12h00, 5183 Pav. Andre-Aisenstadt
Thursday : 10h30 - 12h00, 5183 Pav. Andre-Aisenstadt
Équation des ondes, problème de Sturm-Liouville, distributions et transformation de Fourier, équation de Laplace, espaces de Sobolev, valeurs et fonctions propres du laplacien, éléments de la théorie spectrale, équation de la chaleur.
The Non-Standard Analysis (NSA) is a new mathematical discipline created by Abraham Robinson in 1960-s. The mathematical universe of NSA is immensely wider than that of the classical mathematics. It contains infinite and infinitesimal numbers, and much more. The NSA provides the tools of mathematical constructions which are infinitely more powerful than the classical ones. However, the use of NSA requires a strong logical discipline; not all the classical constructions are permitted there. We cannot form viable sets as freely as we are used to.
The course is devoted to the systematic study of the basics of NSA from the beginning, i.e. the set theory and elements of mathematical logic. On this basis the fundamental notions of NSA are developed, like the Transfer Principle, Internal Sets, Overflow and Underflow, etc. By the end of the course I'm going to discuss some applications of NSA to the Geometry (Asymptotic spaces) and Dynamics, including some problems of Fluid Dynamics.
The course does not require special prerequisits beyond the elementary set theory and analysis.
This course will discuss recent developments in ergodic theory and infinite groups, with an emphasis on the orbit structure of group actions (a subject often referred to as orbit equivalence).
Topics will include:
• invariant means on L^infinity
• orbit equivalence
• von Neumann algebras
• the cost theorem
• entropy
• Kazhdan’s property T and rigidity theorems (of Zimmer, Furman, etc.) • percolation theory relevant to orbit equivalence
References: No textbook is required for this course. Some information can be found in the following references:
1. V.F.R. Jones. von Neumann algebras.
2. A. Kechris, B. Miller. Topics in orbit equivalence theory. 3. R.J. Zimmer. Ergodic theory and semisimple groups.
Prerequisites: Some topics from Algebra 3 (the basics of group theory) and Analysis 3 (namely, topological spaces and the basics of measure theory).
Grading: 40% Assignments + 60% Final report.
Ce cours est une introduction au flot de Ricci. Nous étudierons les propriétés de base du flot (formules de variations, existence et unicité en temps court) et des applications du flot en géometrie et topologie (uniformisation des surfaces, les variétés de dimension 3 à courbure de Ricci positive).
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.
This course will provide a basic introduction to methods for analysis of correlated, or dependent, data. These data arise when observations are not gathered independently; examples are longitudinal data, household data, cluster samples, etc. Basic descriptive methods and introduction to regression methods for both continuous and discrete outcomes.
Brève introduction aux concepts génétiques; épidémiologie génétique, concepts et introduction; études d’agrégation familiale; analyse de liaison; études d’association de population; analyse de liaison et études d’association pour traits quantitatifs.
7 au 23 novembre 2018
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.
Étudier certains algorithmes qui sont omniprésents en combinatoire ; et surtout pour comprendre leur rôle dans des interactions avec la géométrie et l’algèbre. On va développer les notions combinatoires et algorithmiques nécessaires, en particulier il n'y a pas de préalables formels (contrairement à ce qui est indiqué dans la description officielle du cours). Sujets : Représentation informatisée des structures combinatoires (permutations, partitions, compositions, etc.) ; génération exhaustive et aléatoire de ces structures; algorithme de Robinson-Schensted ; arbres binaires de recherche ; structures de données ; algorithmes sur les graphes.
Topics may be chosen from combinatorial set theory, Goedel's constructible sets, forcing, large cardinals.
Quiver representations are a beautiful subject, accessible with a minimum of background, while also providing a very important class of examples of finite-dimensional algebras whose representation theory can be understood very explicitly, with strong connections to root system combinatorics. Quiver representations also have important connections to geometric representation theory and cluster algebras. This course is in part intended to provide background for the thematic month on quiver varieties and representation theory which will be held at the Centre de Recherches Mathématiques in August 2019.
Le contenu du cours sera en partie précisé suivant les intérêts des étudiants. Les grandes lignes sont les suivantes :
Le cours portera sur les fonctions quasi-symétriques, en suivant le récent livre "An introduction to quasi-symmetric Schur functions" de Stephanie van Willigenburg et Karl Luoto.
Many physical processes are modelled by differential equations which involve delays. This course will provide an introduction to delay differential equations (DDEs) concentrating on the key tools needed to understand the behaviour of these equations, and also some of numerical techniques used to approximate solutions. Throughout we will emphasise the similarities and differences between DDEs and ordinary differential equations (ODEs).
Topics covered will include: DDEs as infinite dimensional dynamical systems, breaking points and smoothing of DDE solutions, continuous Runge-Kutta methods for ODEs and DDEs, linear stability of steady states, bifurcation theory. A selection of more advanced topics will also be covered. The choice of topics will depend on time and the preferences of the participants, but may include state-dependent delays, distributed delays, numerical continuation and bifurcation techniques.
This graduate course is an introduction to the treatment of nonlinear differential equations, and more generally to the theory of dynamical systems. The objective is to introduce the student to the theory of dynamical systems and its applications. Firstly, classical dynamics analysis techniques will be presented: continuous and discrete flows, existence and stability of solutions, invariant manifolds, bifurcations and normal forms. Secondly, an introduction to ergodic theory and an overview of modern applications will be presented: chaotic dynamics, strange attractors, dynamic entropy, high-dimensional systems (e.g. networks), driven dynamics and information processing. Particular attention will be paid to computations performed by dynamical systems.
At the end of the course, the student will be able to apply dynamical systems analysis techniques to concrete problems, as well as navigate the modern dynamical systems literature. Several examples and applications making use of numerical simulations will be used. To take this course, the student must master, at an undergraduate level, notions of calculus, linear differential equations, linear algebra and probability.
Course schedule :
Tuesday : 10h30 - 12h00, 5183 Pav. Andre-Aisenstadt
Thursday : 10h30 - 12h00, 5183 Pav. Andre-Aisenstadt
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.
In this course I plan to cover some basic material of Kähler geometry, roughly in line with the first chapter of the book of Griffiths and Harris, or Claire Voisin’s book on Hodge theory and Complex algebraic geometry (book 1).
Material should include:
-Rudiments of function theory of several complex variables,
-Complex manifolds, de Rham and Dolbeault cohomology,
- Sheaf theory and cohomology theory,
-Kähler metrics, connections and curvature,
-Harmonic theory: the Hodge theorem and the Hodge decomposition,
-The Lefschetz decomposition.
Closing, if time allows, with some material on periods and Hodge structures, or maybe on Hyperkähler manifolds.
Rappels de topologie et d’algèbre tensorielle. Variétés différentiables, espaces tangents, différentielle des fonctions, partitions de l’unité, tenseurs et formes différentielles, champs de vecteurs, théorème fondamental des EDO et dérivée de Lie. Intégration et théorème de Stokes, théorème de Fröbenius sur les distributions, cohomologie et théorème de DeRham. Métriques riemanniennes, connexions, dérivée covariante, géodésiques et courbure.
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.
1. Differentiable manifolds:
Differentiable manifolds, tangent and cotangent spaces, smooth maps, submanifolds, tangent and cotangent bundles, implicit function theorem, partition of unity. Examples include real projective spaces, real Grassmannians and some classical matrix Lie groups.
2. Differential forms and de Rham cohomology:
Review of exterior algebra, the exterior differential and the definition of de Rham cohomology. The Poincaré Lemma and the homotopy invariance of de Rham cohomology. The Mayer-Vietoris sequence, computation of de Rham cohomology for spheres and real projective spaces. Finite-dimensionality results for manifolds with good covers, the Kunneth formula and the cohomology of tori. Integration of differential forms and Poincare duality on compact orientable manifolds.
3. An introduction to Riemannian geometry:
Existence of Riemannian metrics, isometric immersions, parallel transport and the Levi-Civita connection, the fundamental theorem of Riemannian geometry, Riemannian curvature. Geodesics, normal coordinates, geodesic completeness and the Hopf-Rinow Theorem.
Textbooks:
W. Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press.
R. Bott and L. Tu, Differential forms in algebraic topology, Springer.
Textbook: Allen Hatcher, Algebraic Topology.
Syllabus: CW-complexes, cellular approximation theorem. Homotopy groups, long exact sequence for a fiber bundle. Whitehead theorem. Freudenthal suspension theorem. Singular and cellular homology and cohomology. Hurewicz theorem. Mayer-Vietoris sequence. Universal coefficients theorem. Cup product, Kunneth formula, Poincare duality.
Prerequisites: MATH 576 or equivalent or permission of instructor.
If you have taken or are taking the physics GR course, the two courses should complement each other nicely. In particular, there will not be much overlap. While a considerable part of the physics course is (probably) spent on introducing differential geometry, we will assume that the students are comfortable with basic differential geometry. Exact solutions with high degree of symmetry will be studied as prototypical examples of spacetimes, but our focus will be on the properties of realistic spacetimes with no or very little symmetry.
The following topics will be treated.
• Some exact solutions, including black hole and cosmological solutions.
• Lorentzian geometry, geodesic congruences, variational characterization of geodesics.
• Singularity theorems of Penrose and Hawking. These theorems are the highlight of the course, and basically show that spacetimes cannot avoid developing singularities.
• Cauchy problem, if time permits. This result says that the state of the universe "today" completely determines what happens in the future in a certain sense.
The grading will be based on a few homework, and a course project, where the student studies a special topic and gives a presentation.
Ce cours est proposé comme une introduction à la géométrie riemannienne. Nous couvrirons les sujets classiques suivants : Variétés riemanniennes, connexions, géodésiques. Exemples de variétés riemanniennes. Courbure sectionnelle, courbure de Ricci, courbure scalaire. Lemme de Gauss, application exponentielle, théorème de Hopf-Rinow. Transport parallèle, holonomie, théorème d'irréductibilité et de De Rham. Variations première et seconde, champs de Jacobi, cut locus. Théorème de Bonnet-Myers, théorème de Synge, théorème de Cartan-Hadamard. Théorème de comparaison de Rauch, Alexandrov et Toponogov. Submersion riemannienne, espaces homogènes riemanniens, espaces symétriques, l'exemple de l'espace projectif complexe. Théorème de Hodge-De Rham. Théorème de Bochner. Volume, théorèmes de Bishop et de Heintze-Karcher. Sous-variétés, seconde forme fondamentale, équation de Gauss. Inégalités isopérimétriques. Géométrie spectrale. Théorème de finitude de Cheeger.
L'objectif du cours est de présenter les concepts principaux de la théorie des courbes et des surfaces plongées dans des espaces multidimensionnels. Dans ce cours, nous présentons les sujets suivants :
Théorie générale au sens de Frenet sur les courbes plongées dans des espaces multidimensionnels. Procédure d'orthogonalisation de Gram-Schmidt, Repaire mobile, Théorème fondamentale de la théorie des courbes dans Rn.
Théorie générale des surfaces plongées dans des espaces multidimensionnels basée sur la théorie du repaire mobile. Formules de Gauss-Weingarten et de Gauss-Codazzi, Caractérisation au moyen des formes fondamentales des surfaces.
Propriétés intrinsèques des surfaces. Courbures et lignes géodésiques, Surfaces à courbure constante, Théorème de Bauss-Bonnet.
Propriétés extrinsèques des surfaces. Courbure normale, Courbure moyenne, Points umbiliques, Direction conjuguée et lignes asymptotiques, Courbures principales et l'indicateur de Dupin.
Propriétés globales et caractérisation des surfaces. Forme différentielle extérieure, Lemme de Cartan, Théorie du repaire mobile, Représentation d'Enneper-Weierstrass des surfaces.
Location: Monday and Wednesday 14h35-15h55 BURN 1214
This course will serve as an introduction to the Hamiltonian formulation of classical mechanics, and the underlying differential geometry of symplectic and Poisson manifolds. We will cover:
- examples of mechanical systems, e.g. oscillators, pendulums and tops - Hamilton's equations of motion
- definitions and basic properties of symplectic and Poisson manifolds
- Liouville's theorem on phase space volumes
- Lagrangian submanifolds and Weinstein's neighbourhood theorem - momentum, symmetries and symplectic reduction
- integrable systems and action-angle variables
Further advanced topics may be selected based on the tastes and background of the audience. Possibilities include local normal forms and stability of equilibiria; classification of toric integrable systems via Delzant polytopes; perturbations of integrable systems and the Kolmogorov-Arnold-Moser (KAM) theorem; Arnold's conjecture on periodic orbits and rudiments of Floer theory; or links with quantum mechanics via geometric/deformation quantization.
Ce cours est une introduction au flot de Ricci. Nous étudierons les propriétés de base du flot (formules de variations, existence et unicité en temps court) et des applications du flot en géometrie et topologie (uniformisation des surfaces, les variétés de dimension 3 à courbure de Ricci positive).
After a brief overview of the basic theory of one complex variable, we introduce Riemann surfaces as one dimensional complex manifold. We discuss visually the topological and geometric classification for compact Riemann surfaces, including the Poincaré-Hopf index theorem and its connection to the Gauss-Bonnet theorem. We state the uniformisation theorem after a look at the complex automorphism group of the three simply connected Riemann surfaces, which are the unit disk, the complex number plane and its extension at infinity: the Riemann sphere. Uniformisation for the case of elliptic curves, i.e., compact quotient of the complex number plane, will be proved in detail by the classical method of Weierstrass P functions. Analogous to the P functions, we prove the existence of meromorphic functions on an arbitrary Riemman surface with the help of uniformisation. Armed with this, we identify compact Riemann surfaces with nonsingular algebraic curves either as finite ramified cover of the Riemann Sphere as viewed by Riemann or as algebraic curves in the projective space and show a classical method of de-singularizing algebraic curves. A little Hodge theory will allow us to obtain the Riemann existence theorem without resorting to the uniformisation theorem and so, via the above identification, the Riemann-Roch theorem and Serre duality by algebraic means: divisors, line bundles, sheaves and their cohomologies. There follows readily the uniformisation theorem for genus zero and one as well as other important consequences. If time allows at the end, we may enter into the theory of Stein spaces and show that non-compact Riemann surfaces are Stein or into the Abel-Jacobi theory and its vector bundle generalization by Atiyah-Bott and others (Donaldson-Uhlenbeck-Yau in higher dimensions). We may even try to look at the Higgs bundle generalization which offers a modern proof à la Carlo Simpson of the uniformisation theorem for compact Riemann surfaces.
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):
Le but de ce cours est de couvrir les différentes méthodes de calcul numérique utilisées en ingénierie financière. Bien qu'une part théorique soit utile et nécessaire, l'emphase est sur la recherche de solutions pratiques aux problèmes, à travers des routines programmées soi-même ou à travers l'utilisation judicieuse de logiciels. On cherchera toujours une compréhension suffisante de la théorie pour pouvoir appliquer intelligemment les routines existantes, en les adaptant aux besoins d'applications particulières.
On traitera principalement des domaines de l'optimisation et de la résolution numérique des équations aux dérivées partielles, mais on discutera aussi de la résolution de systèmes d'équations, d'approximation de fonctions et d'intégration numérique.
Extreme events on financial markets are very difficult to predict and few models are capable of accounting for these characteristics. The theory of extreme values is an important statistical discipline allowing for a more proper modeling of rare events. In this course, we present the theory of extreme values necessary to solve problems in finance, economics and financial engineering. The analysis tools required to study such data are also studied. The proper analysis of extreme values, including methods of estimation, quantification of uncertainty, diagnostics, and maximal utilisation of available data are considered. We also make extensive use of R, a freely available language and environment for statistical computing and graphics.
Le cours est basé sur l'étude des principaux outils de la théorie de la probabilité qui sont utilisés en finance et en ingénierie financière. Bien que les applications soient liées à ces domaines et que de nombreux exemples seront étudiés en classe et lors des travaux, c'est un cours de mathématiques, ce qui implique la démonstration des résultats. Le principal objectif de ce cours est de rendre l'étudiant à l'aise avec les concepts mathématiques qu'il doit couramment employer en ingénierie financière : processus de diffusion, mesure neutre au risque, la structure de l'information, les martingales, etc.
Le cours est divisé en deux principaux blocs : le premier concernant les modèles à temps discret et le second traitant des modèles à temps continu. Chacune de ces parties est à nouveau subdivisée : une section plus théorique où l'on introduit les concepts mathématiques et une deuxième section dans laquelle ses outils mathématiques sont utilisés.
La complexité des modèles utilisés en ingénierie financière rend nécessaire l'utilisation de méthodes statistiques avancées. Dès qu'un modèle doit être mis en application, l'un des premiers problèmes rencontrés est l'estimation des paramètres du modèle. Se pose ensuite la question de la précision des estimations et de son influence sur les étapes subséquentes de l'implantation.
Le cours présente les outils statistiques permettant l'utilisation et l'implantation de modèles dans plusieurs aspects de l'ingénierie financière : évaluation d'options, risque de crédit, réplication de fonds de couverture, etc. Nous couvrirons les méthodes d'estimation (maixmum de vraisemblance, méthode de moments, estimation non-paramétrique, transformation de données), leur précision (intervalles de confiance, information de Fisher, rééchantillonnnage, méthode delta, quantiles), et ce dans le cadre de processus stochastiques couramment utilisés en ingénierie financière (mouvement brownien géométrique, processus avec sauts, modèles à volatilité aléatoire, modèles avec changement de régme, etc.) Nous verrons aussi l'estimation et l'ajustement de modèles de dépendance pour plusieurs facteurs de risque, ainsi que les méthodes de filtrage, permettant d'estimer les paramètres des modèles dont certaines des composantes ne sont pas observables, tel le bénéfice de détention, etc.
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.
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.
This course focuses on computational aspects, implementation, continuous-time models, and advanced topics in Mathematical and Computational Finance. We shall attempt to cover the following topics: stochastic calculus, Black-Scholes model, option pricing, Monte-Carlo methods, finite difference methods, volatility models, American options, exotic options, hedging, risk measurement and interest rate models.
This course covers the main tools of probability theory that are used in finance and financial engineering. Besides the theoretical concepts and proofs, many applications in finance are presented rigorously. The first half of the course is in discrete time, while the second half is about continuous time models. For each of these two parts, there is a theoretical component in which the basic concepts such as martingales, stochastic integrals and diffusion processes are introduced and a more applied segment where the mathematical tools are applied to financial problems.
La simulation de Monte Carlo est une technique numérique largement utilisée permettant de solutionner des problèmes généralement trop complexes pour qu'une solution analytique soit disponible. En ingénierie financière, elle est utilisée comme outil pour tarifer des produits dérivés, évaluer la distribution de la valeur d'un portefeuille comportant divers instruments, calculer des mesures de risque, etc.
Dans ce cours, nous aborderons les fondements mathématiques de cette méthode et nous l'appliquerons à des problèmes d'ingénierie financière. Comme certains problèmes sont complexes et nécessitent un effort de programmation important, certains cours seront substitués à des périodes en laboratoire où les étudiants pourront mettre en oeuvre la théorie vue en classe.
Le langage de programmation utilisé est Matlab.
Ce cours vise à permettre à l'étudiant de:
Ce cours vise à introduire les notions de segmentation des risques de la tarification en assurance, en utilisant divers outils statistiques. Les modèles de prévision pour le nombre et le coût des réclamations seront abordés afin d'inclure les caractéristiques du risque dans le calcul de la prime. La notion d'hétérogénéité en assurance et sa modélisation mathématique seront abordées, de même que les modèles hiérarchiques ou données longitudinales en assurance et en finance.
Ce cours présente un traitement académique des produits d’assurance-vie liés aux marchés (fonds distincts, variable annuities, equity-linked annuities) à travers la lecture et l’étude de récents articles scientifiques. Survol du marché des fonds distincts et des différentes garanties, évaluation et tarification des fonds distincts, principe de frais équitable, stratégies de couverture en marchés complets et incomplets, modélisation et évaluation du risque de mortalité, modélisation et évaluation du risque de rachat de contrat.
This is an introductory course on Lévy processes with an emphasis on fluctuation theory for spectrally negative Lévy processes (SNLPs), including compound Poisson with drift and Brownian motion with drift. Applications in ruin theory, operations research and/or financial mathematics will be discussed, in view of students interest.
Schedule: Tuesday and Thursday, 2-3:30pm (could be modified at the first lecture)
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.
The main focus of the course is going to be on linear first and second order equations, and Sobolev spaces. Rather than trying to build everything in full generality, we will study prototypical examples in detail to establish good intuition. Roughly speaking, most of the topics from the calendar description of Math 580 and some from that of Math 581 will be covered. More precisely, the planned topics are
• First order equations, method of characteristics
• Cauchy problem for heat and wave equations
• Duhamel's, Huygens, and maximum principles
• Green's identities, harmonic functions, Harnack inequality
• Fundamental solution, Green's function, Poisson's formula
• Dirichlet problem: Perron's method, barriers, boundary regularity
• Sobolev spaces, weak and strong derivatives, Dirichlet principle
• Poisson equations: Variational formulation, boundary conditions
• Elliptic regularity, Sobolev embedding
• Laplace eigenvalues and eigenfunctions (if time permits)
Many physical processes are modelled by differential equations which involve delays. This course will provide an introduction to delay differential equations (DDEs) concentrating on the key tools needed to understand the behaviour of these equations, and also some of numerical techniques used to approximate solutions. Throughout we will emphasise the similarities and differences between DDEs and ordinary differential equations (ODEs).
Topics covered will include: DDEs as infinite dimensional dynamical systems, breaking points and smoothing of DDE solutions, continuous Runge-Kutta methods for ODEs and DDEs, linear stability of steady states, bifurcation theory. A selection of more advanced topics will also be covered. The choice of topics will depend on time and the preferences of the participants, but may include state-dependent delays, distributed delays, numerical continuation and bifurcation techniques.
This course focuses on solving Markov Decision Problems (MDP) using methods of dynamic programming and reinforcement learning. Solution approaches to finite and infinite horizon MDP through the Bellman Equation, value function iteration and policy iteration are presented. Approximate dynamic programming ideas are then introduced to increase computational speed. Methods of reinforcement learning such as temporal-difference learning, online versus offline control and eligibility traces are then illustrated. Students will have to code extensively in R throughout the course.
Algorithmic methods for big data analysis. Complexity analysis, data structure, parallel and distributed computing.
The course is organized in 4 themes that will cover various aspects of algorithmic for big data, starting from sequential programming and ending with distributed computing. In the first part of the course, the student will learn to analyze an algorithm from the computational complexity and memory requirement. The second theme in the course deals with parallel computing with shared memory. The efficiency of the parallelization and memory safety will be discussed and analysed. In the third theme, the message passing interface (MPI) will be explored, which consists in simultaneous and collaborative parallel computing without shared memory. Finally, the basics of distributed computing, its strength and requirements will be introduced. The choice of the best approach toward the resolution of a problem will depend on the problem and the nature of the data.
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.
Kinematics. Dynamics of general fluids. Inviscid fluids, Navier-Stokes equations. Exact solutions of Navier-Stokes equations. Low and high Reynolds number flow.
The main focus of the course is going to be on nonlinear problems. Sobolev spaces, the Fourier transform, and functional analytic methods will be heavily used. The planned topics are
• Tempered distributions, convolution, Fourier transform
• Fourier analytic treatment of Sobolev spaces
• Problems in half-space, shades of hyperbolicity, parabolicity, and ellipticity
• Overview of elliptic theory, regularity
• Semilinear elliptic equations, monotonicity methods
• Variational problems, compactness methods
• Semilinear evolution equations, Duhamel's principle
• The Navier-Stokes equations and related turbulence models
• Semilinear elliptic problems with critical exponents (if time permits)
Introduction and examples of typical non-linear and hybrid control systems (respectively NLS and HCS). Specification of HCS via interlinked ODEs and automata. Controlled and autonomous discrete state switching. HCS trajectories: continuous and discrete state evolution. System linearization. Lyapunov stability theory. Regular and exotic trajectories of HCS. Hopf bifurcations. Basic topological dynamics and LaSalle stability theory for NLS and HCS. Switched systems. Controllability and stabilization of NLS and HCS. Controlled Lyapunov functions. The Hybrid Maximum Principle and Hybrid Dynamic Programming: optimal control theory and computational algorithms.
Course: ECSE 516 Department of Electrical and Computer Engineering,McGill University.
Note: this course is not restricted to engineering students.
Instructor P. E. Caines peterc@cim.mcgill.ca
Étude des algorithmes fondamentaux en calcul scientifique. Principes théoriques; programmation et application à des problèmes pratiques; utilisation scientifique de logiciels spécialisés.
Les thèmes principaux qui seront étudiés dans ce cours sont les quaternions, les algèbres de Clifford ainsi que la théorie des fonctions analytiques généralisées (fonctions pseudo-analytiques). Ces structures seront également utilisées pour considérer certaines applications, principalement en physique quantique. Pour toutes ces structures, nous allons porter une attention particulière aux généralisations des fonctions analytiques complexes. Dans le cas des quaternions et des algèbres de Clifford, les propriétés algébriques ainsi que géométriques seront considérées. La théorie des fonctions pseudo-analytiques généralise et préserve plusieurs caractéristiques de la théorie des fonctions analytiques complexes. Le système de Cauchy-Riemann est alors substitué par un système plus général, appelé équations de Vekua, qui apparaît dans plusieurs problèmes de la physique mathématique.
If you have taken or are taking the physics GR course, the two courses should complement each other nicely. In particular, there will not be much overlap. While a considerable part of the physics course is (probably) spent on introducing differential geometry, we will assume that the students are comfortable with basic differential geometry. Exact solutions with high degree of symmetry will be studied as prototypical examples of spacetimes, but our focus will be on the properties of realistic spacetimes with no or very little symmetry.
The following topics will be treated.
• Some exact solutions, including black hole and cosmological solutions.
• Lorentzian geometry, geodesic congruences, variational characterization of geodesics.
• Singularity theorems of Penrose and Hawking. These theorems are the highlight of the course, and basically show that spacetimes cannot avoid developing singularities.
• Cauchy problem, if time permits. This result says that the state of the universe "today" completely determines what happens in the future in a certain sense.
The grading will be based on a few homework, and a course project, where the student studies a special topic and gives a presentation.
Ce cours est une introduction à la théorie des surfaces de Riemann. Le préalable exigé est une connaissance de base de l'analyse complexe.
Contenu:
Surfaces de Riemann compactes. Structures complexes engendrées par une métrique. Applications holomorphes. Revêtements ramifiés de la sphère de Riemann, formule de Riemann-Hurwitz. Topologie et formes différentielles sur les surfaces de Riemann. Différentielles abéliennes, Jacobien. Fonctions méromorphes sur les surfaces de Riemann compactes. Théorème d'Abel. Théorème de Riemann-Roch. Fonctions théta, fonctions de Weierstrass. Aperçu des courbes algébriques.
Location: Monday and Wednesday 14h35-15h55 BURN 1214
This course will serve as an introduction to the Hamiltonian formulation of classical mechanics, and the underlying differential geometry of symplectic and Poisson manifolds. We will cover:
- examples of mechanical systems, e.g. oscillators, pendulums and tops - Hamilton's equations of motion
- definitions and basic properties of symplectic and Poisson manifolds
- Liouville's theorem on phase space volumes
- Lagrangian submanifolds and Weinstein's neighbourhood theorem - momentum, symmetries and symplectic reduction
- integrable systems and action-angle variables
Further advanced topics may be selected based on the tastes and background of the audience. Possibilities include local normal forms and stability of equilibiria; classification of toric integrable systems via Delzant polytopes; perturbations of integrable systems and the Kolmogorov-Arnold-Moser (KAM) theorem; Arnold's conjecture on periodic orbits and rudiments of Floer theory; or links with quantum mechanics via geometric/deformation quantization.
This course is a modern introduction to quantum information theory centred around the notions of quantum entropies. The presentation will be essentially self-contained. Pre-requisites are Honours Algebra II and Honours Analysis II, or the permission of the instructor. The topics to be covered are:
–Non-commutative probability
–Quantum entropies
–Quantum statistics (hypothesis testing and parameter estimation)
–Mathematical theory of quantum channels, finitely correlated states, and repeated quantum measurements
Additional topics may include quantum Shannon-McMillan-Breiman theorem and theory of quantum spin systems (time permitting).
The course will cover some of the most recent developments in quantum information theory and the lecture notes will be provided.
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.
Characteristic functions: Elementary properties, Inversion formula, Uniqueness, convolution and continuity theorems.
Weak convergence: Portmanteau theorem, Sequential compactness, tightness, Prohorov's theorem, Polish spaces, Central limit theorem, Skorokhod's representation theorem.
Stochastic processes: General theory. Kolmogorov Extension theorem, Kolmogorov continuity theorem. Regular probability spaces and conditional distributions, probability kernels. Construction of Brownian motion, Donsker's theorem
Exchangeability: De Finetti's theorem, The Aldous-Hoover theorem.
Other topics if time permits.
This is an introductory course on Lévy processes with an emphasis on fluctuation theory for spectrally negative Lévy processes (SNLPs), including compound Poisson with drift and Brownian motion with drift. Applications in ruin theory, operations research and/or financial mathematics will be discussed, in view of students interest.
Schedule: Tuesday and Thursday, 2-3:30pm (could be modified at the first lecture)
The theory of large deviations studies probabilities of events which, in a large sample, are exponentially rare in the number of samples. We will cover standard methods including large deviations for i.i.d. sequences, correlated variables, and occupation measures. This theory has found many applications in finance and risk management, simulation and sampling, as well as operations research and statistical mechanics. The main part of the course will cover: Introduction to large deviations, the large deviations principle, Sanov’s theorem and method of types, Cramer’s theorem, Gartner-Ellis theorem, concentration inequalities, large deviations for Markov chains. We will also discuss the application of the theory to constructing efficient sampling mechanisms, such as importance sampling. The prerequisites for the course are basic probability theory and stochastic processes.
Ce cours présente certaines des principales techniques d'analyse de grandes bases de données (data mining). Les technologies de l'intelligence d'affaires permettent aux entreprises, entre autres, d'analyser les données recueillies pour leurs opérations afin de mieux comprendre le comportement de leurs clients dans le but d'aider à anticiper la demande, accroître la rétention ou réduire la fraude. Différentes techniques de l'intelligence d'affaires, parmi les plus utilisées en pratique, seront donc présentées et illustrées à partir d'exemples concrets dans différents domaines de gestion.
This course introduces the main data mining and machine learning methods used in practice for the analysis of big data.
This course introduces the major data mining techniques used to analyze big data. Business intelligence technologies enable companies to analyze the large amount of data collected for their operations to, for example, better understand customer behavior in order to help anticipate demand or increase retention, reduce fraud, optimize preventive maintenance, etc.. Different data mining techniques, among the most widely used in practice, will therefore be presented and illustrated based on concrete examples in different management domains.
Les entreprises croulent littéralement sous le poids des données qu'elles ont à leur disposition. Ces données contiennent potentiellement une quantité importante d'informations pouvant être bénéfiques à l'entreprise si utilisées correctement. Sous le vocable « data mining », on retrouve différentes techniques statistiques utilisées pour explorer et analyser de grands ensembles de données. Ces techniques ont généralement pour but de développer des modèles prévisionnels, de réduire la taille des données, de faire de la segmentation ou bien de découvrir des associations pertinentes. L'analyse multidimensionnelle est à la base de plusieurs techniques de data mining et est utilisée dans plusieurs domaines de gestion dont le marketing.
Le but du cours analyse multidimensionnelle est de donner aux étudiants(e)s une formation de base en traitement de données multidimensionnelles. Plusieurs techniques statistiques seront présentées et on insistera surtout sur la compréhension intuitive, l'interprétation correcte et l'utilisation pratique de celles-ci. Par conséquent, l'emploi de concepts mathématiques sera réduit à son minimum et ces derniers ne serviront qu'à faciliter la compréhension des méthodes étudiées. Le logiciel SAS sera utilisé mais aucune connaissance préalable de celui-ci n'est requise. Par contre, une connaissance des concepts et méthodes statistiques (population, échantillon, estimation, test d'hypothèse) de base est requise.
L'étudiant apprendra à programmer en SAS et en R afin de nettoyer des jeux de données, de les représenter graphiquement et d'en faire une analyse statistique complexe. En plus de maîtriser le code de base de SAS, l'étudiant apprendra la syntaxe du module ODS qui permet de gérer le contenu des sorties. Il apprendra aussi le langage macro de SAS et s'en servira . afin de créer des fonctions permettant des analyses statistiques supplémentaires. En R, l'étudiant apprendra les bases du langage qui lui serviront à faire une analyse statistique des données .. II ·apprendra aussi à écrire des fonctions permettant l'analyse statistique de données et à construire une librairie de fonctions afin de partager les outils d'analyse qu'il aura codés. R et SAS sont basés sur des langages de programmation différents que l'étudiant devra apprendre à maîtriser.
L'objectif principal du cours est de fournir à l'étudiant les notions fondamentales de l’analyse et de l’inférence statistique ainsi que les méthodes statistiques avancées. En plus des concepts théoriques, ce cours mettra particulièrement l'accent sur les applications pratiques de ces méthodes dans des contextes de recherche.
L'étudiant découvrira les méthodes qui permettent d'analyser automatiquement un corpus de documents par des algorithmes classiques d'exploitation de données. Les textes étant avant tout destinés à la lecture par des humains, l'information qu'ils recèlent n'est pas structurée de manière appropriée à un traitement automatisé. Nous présenterons dans ce cours diverses techniques spécifiques grâce auxquelles un traitement automatisé des documents est possible.
Après avoir suivi ce cours, l'étudiant saura identifier les paramètres appropriés et utiliser de manière appropriée les principaux logiciels disponibles. Le cours est composé de 6 séances de 3 heures durant lesquelles les techniques sont présentées formellement d'abord, puis par l'entremise d'applications.
Le but du cours est de fournir aux étudiants les outils nécessaires à l'analyse de données longitudinales et de survie. Contrairement aux études transversales, la caractéristique principale de ces études est que les sujets sont suivis à travers le temps. Ceci permet d'étudier directement la façon dont évoluent les phénomènes à travers le temps. Par contre, ce type de données engendre aussi des difficultés supplémentaires comme de la dépendance entre les observations d'un même sujet ou la présence de censure. Le cours sera axé sur la compréhension des concepts ainsi que sur l'aspect pratique afin de rendre l'étudiant capable de procéder à l'analyse de données longitudinales et de survie. L'apprentissage se fera à l'aide d'exemples concrets provenant de plusieurs domaines de la gestion.
Extreme events on financial markets are very difficult to predict and few models are capable of accounting for these characteristics. The theory of extreme values is an important statistical discipline allowing for a more proper modeling of rare events. In this course, we present the theory of extreme values necessary to solve problems in finance, economics and financial engineering. The analysis tools required to study such data are also studied. The proper analysis of extreme values, including methods of estimation, quantification of uncertainty, diagnostics, and maximal utilisation of available data are considered. We also make extensive use of R, a freely available language and environment for statistical computing and graphics.
In this course, we will study machine learning models, a type of statistical analysis that focuses on prediction, for analyzing very large datasets ("big data"). In addition to standard models, we will study models for analyzing user behaviour and for decision making. Massive datasets are now common and require scalable analysis tools. Machine learning provides such tools and is widely used for modelling problems across many fields including artificial intelligence, bioinformatics, finance, marketing, education, transportation, and health.
In this context, we study how standard machine learning models for supervised (classification, regression) and unsupervised learning (for example, clustering and topic modelling) can be scaled to massive datasets using modern computation techniques (for example, computer clusters). In addition, we will discuss recent models for recommender systems as well as for decision making (including multi-arm bandits and reinforcement learning).
Through a course project students will have the opportunity to gain practical experience with the analysis of datasets from their field(s) of interest. A certain level of familiarity with computer programming will be expected.
This course has four main objectives: 1) to present the major experimental designs used for research in management and in the behavioral sciences; 2) to familiarize students with the statistical methods and software (e.g. PASW, formerly SPSS) used to analyze experimental data; 3) to interpret and present results from the statistical analyses and discuss the validity and limits of the methods; 4) to understand and to critic the methodology and statistical results of published articles in the research fields of the students.
Distribution free procedures for 2-sample problem: Wilcoxon rank sum, Siegel-Tukey, Smirnov tests. Shift model: power and estimation. Single sample procedures: Sign, Wilcoxon signed rank tests. Nonparametric ANOVA: Kruskal-Wallis, Friedman tests. Association: Spearman's rank correlation, Kendall's tau. Goodness of fit: Pearson's chi-square, likelihood ratio, Kolmogorov-Smirnov tests. Statistical software packages used.
Distribution theory, stochastic models and multivariate transformations. Families of distributions including location-scale families, exponential families, convolution families, exponential dispersion models and hierarchical models. Concentration inequalities. Characteristic functions. Convergence in probability, almost surely, in Lp and in distribution. Laws of large numbers and Central Limit Theorem. Stochastic simulation.
Overview: The purpose of this course is to (re)visit some of the main ideas of statistics which students might have seen, perhaps fleetingly, in previous courses. The emphasis will be on understanding rather than on breadth. Students will be assigned papers to read that convey these ideas and instruction will be through in-class student presentations and discussion. Most of the papers will be selected from those that are considered to be the historical “breakthrough papers.”
The topics to be covered, time permitting
1. Tests of hypotheses, with emphasis on the meaning of a (frequentist) p-value and the approach taken by Bayesians, as well as attempts to reconcile these two viewpoints.
Book Chapter by M.J.Bayarri and J.O.Berger: “Hypothesis Testing and Model Uncertainty.”
2. The origins Markov Chain Monte Carlo methods.
Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. (1953). “Equations of State Calculations by Fast Computing Machines.” Journal of Chemical Physics.
Hastings, W.K. (1970). "Monte Carlo Sampling Methods Using Markov Chains and Their Applications". Biometrika.
3. The origins of empirical Bayesian methods.
H.E. Robbins. “(1955). “An Empirical Bayes Approach to Statistics.” Third Berk. Symp. Statist. Prob.
4. The origins of the bootstrap.
Efron. B. (1979). “Bootstrap methods: another look at the jackknife.” Ann. Statist.,
5. The origins of generalized estimating equations (GEEs).
Liang, K.E. and Zeger, S.L. (1986). “Longitudinal Data Analysis using Generalized Linear Models.”
Prerequisites
The minimum prerequisites are MATH 556 and MATH 557 (Distribution theory and statistical inference) or equivalent, and a course in regression analysis. Naturally, the more exposure that one has had to different areas of statistics the easier the papers will be to read.
Comments
1. Often, at the first, second or third reading of a paper it appears to be opaque. Then, (hopefully) it begins to reveal its secrets.
2. Students will be seriously evaluated on their ability to clearly present the contents of the paper(s) for which they are mainly responsible: Evaluation will be based on in-class presentations and short written summaries. Novel explanations will be rewarded.
3. Some of these papers are long. The technical details are not always crucial to the thrust of the paper and may be omitted as long the reader can describe in broad terms, what they are.
The idea is for students to have fun and derive satisfaction from being able to brag that they have read “the original paper on.......” by “....”
General introduction to computational methods in statistics; optimization methods; EM algorithm; random number generation and simulations; bootstrap, jackknife, cross-validation, resampling and permutation; Monte Carlo methods: Markov chain Monte Carlo and sequential Monte Carlo; computation in the R language.
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.
Théorie des modèles linéaires généraux. Théorie des modèles linéaires généralisés. Régression logistique. Modèles log-linéaires.
Principes d'inférence; estimation ponctuelle et distribution des estimateurs, approximation normale, point de selle et « bootstrap »; tests d'hypothèses; robustesse, inférence bayésienne, pseudo- et quasi vraisemblance, estimation non paramétrique.
Rappels et compléments sur la théorie du modèle linéaire : moindres carrés, théorèmes de Gauss-Markov et de Cochran, inférence. Modèle à effets fixes et aléatoires. Plan incomplet. Plan à mesures répétées.
Rappels sur la régression linéaire multiple. Diagnostics. Transformations, moindres carrés pondérés, méthodes robustes, régression « ridge ». Régression non linéaire. Modèles spécifiques: logistique, probit, de Poisson.
Ce cours présente certaines des principales techniques d'analyse de grandes bases de données (data mining). Les technologies de l'intelligence d'affaires permettent aux entreprises, entre autres, d'analyser les données recueillies pour leurs opérations afin de mieux comprendre le comportement de leurs clients dans le but d'aider à anticiper la demande, accroître la rétention ou réduire la fraude. Différentes techniques de l'intelligence d'affaires, parmi les plus utilisées en pratique, seront donc présentées et illustrées à partir d'exemples concrets dans différents domaines de gestion.
Les entreprises croulent littéralement sous le poids des données qu'elles ont à leur disposition. Ces données contiennent potentiellement une quantité importante d'informations pouvant être bénéfiques à l'entreprise si utilisées correctement. Sous le vocable « data mining », on retrouve différentes techniques statistiques utilisées pour explorer et analyser de grands ensembles de données. Ces techniques ont généralement pour but de développer des modèles prévisionnels, de réduire la taille des données, de faire de la segmentation ou bien de découvrir des associations pertinentes. L'analyse multidimensionnelle est à la base de plusieurs techniques de data mining et est utilisée dans plusieurs domaines de gestion dont le marketing.
Le but du cours analyse multidimensionnelle est de donner aux étudiants(e)s une formation de base en traitement de données multidimensionnelles. Plusieurs techniques statistiques seront présentées et on insistera surtout sur la compréhension intuitive, l'interprétation correcte et l'utilisation pratique de celles-ci. Par conséquent, l'emploi de concepts mathématiques sera réduit à son minimum et ces derniers ne serviront qu'à faciliter la compréhension des méthodes étudiées. Le logiciel SAS sera utilisé mais aucune connaissance préalable de celui-ci n'est requise. Par contre, une connaissance des concepts et méthodes statistiques (population, échantillon, estimation, test d'hypothèse) de base est requise.
L'étudiant apprendra à programmer en SAS et en R afin de nettoyer des jeux de données, de les représenter graphiquement et d'en faire une analyse statistique complexe. En plus de maîtriser le code de base de SAS, l'étudiant apprendra la syntaxe du module ODS qui permet de gérer le contenu des sorties. Il apprendra aussi le langage macro de SAS et s'en servira . afin de créer des fonctions permettant des analyses statistiques supplémentaires. En R, l'étudiant apprendra les bases du langage qui lui serviront à faire une analyse statistique des données .. II ·apprendra aussi à écrire des fonctions permettant l'analyse statistique de données et à construire une librairie de fonctions afin de partager les outils d'analyse qu'il aura codés. R et SAS sont basés sur des langages de programmation différents que l'étudiant devra apprendre à maîtriser.
L'objectif principal du cours est de fournir à l'étudiant les notions fondamentales de l’analyse et de l’inférence statistique ainsi que les méthodes statistiques avancées. En plus des concepts théoriques, ce cours mettra particulièrement l'accent sur les applications pratiques de ces méthodes dans des contextes de recherche.
This course introduces classical time series concepts: trend and seasonal pattern identification, stationarity, autocorrelation and partial autocorrelation, ARMA processes, estimation and prediction, model diagnostics and possibly GARCH and regime-switching models.
The goal of this course is to provide basic notions of statistical analysis and inference as well as advanced statistical methods. In addition to the theoretical concepts, this course will focus on the practical applications of these methods.
Thèmes couverts
1) SAS software
2) ANOVA, t-tests
3) Linear models
4) Generalized linear models
5) Likelihood methods
6) Correlated data analysis
7) Non-parametric methods
Présentation des principales méthodes propres à la prévision nécessaire à la prise de décisions en présence l'incertitude. Grands principes des méthodes de prévision utilisées.
Les étudiants se familiariseront avec l'utilisation des principales techniques telles le lissage, la régression, les séries chronologiques et les réseaux de neurones. Les méthodes d'évaluation et de sélection de modèles, ainsi que les méthodes d'évaluation des erreurs de prévision, sont aussi au programme. Le logiciel R sera utilisé.
Presentation of the main forecasting methods necessary for decision making in the presence of uncertainty. General principles of forecasting methods used are outlined.
Students will become familiar with the use of key techniques such as smoothing, regression, time series and neural networks. Methods for model evaluation and selection, as well as methods for estimating forecast errors, are also on the program. The R software will be used.
Structural equation models and latent variables is a field of data analytics that has undergone substantial developments over the past two decades. These models allow to characterize and relate some factors that are not directly observable. The range of application of such models is very wide in social sciences, including marketing, management, IT and human resources. The course will be divided into several parts, including a review of the concepts of regression, correlation, causal relation, direct / indirect effects and correlation diagrams. We will then discuss some specific types of structural equation models such as exploratory/confirmatory factor analysis and we will study the general formulation of the model, characterized by a path diagram with latent variables. Finally, component-based structural equation models will also be discussed, such as partial least squares (PLS) and GSCA.
All the analyses seen in this course will be carried out using specialized software. For each type of model studied, we will focus on model identification and specification, parameter inference, model fit and interpretation of results through applied examples in administrative sciences.
Sondages avec probabilités inégales, stratifiés, en grappes, à plusieurs degrés. Estimation par le quotient et la régression, optimalité. Coûts; non-réponse; population de référence et population-mère; inférence bayésienne.
The course will complement the course 80-622 Analysis of extreme values with application to financial engineering that Debbie Dupuis will be teaching at HEC in the Fall 2018.
Topics to be covered include: Bayesian statistical inference and decision making; de Finetti’s representation; parametric methods; conjugate models; hierarchical models; computational approaches to inference; Monte Carlo methods; bootstrap methods; Markov chain Monte Carlo methods; Metropolis– Hastings; Hamiltonian Monte Carlo; sequential Monte Carlo; approximate Bayesian computation; nonparametric Bayesian inference; semiparametric Bayesian inference.