## 2017-18 Courses

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To register for apprendre option binaire quebec ISM course, you must first have you course selection approved by your supervisor and departmental Graduate Program Director. Additional procedures to register for a course at McGill University: Number Fields and Ideals.

Dedekind domains, unique factorization of ideals, apprendre option binaire quebec class groups. Geometry of numbers, finiteness of the class number and the unit theorem. Special Fields quadratic, cyclotomic, etcapplications to Fermat's last theorem. Analytic Methods, Zeta and L-functions, analytic continuation, density theorems. N umber Fields by Daniel A. The emphasis will be on harmonic analysis as used in analytic number theory. The unorthodoxy will consist in that we will follow the natural development and inter-relation of various techniques rather than focus on the primes, which are typically given apprendre option binaire quebec in introductory courses.

The course will follow chapter II of Hartshorne's book and will develop the basic tools of the theory of schemes: The course is devoted to the study of formal groups and their applications. More precisely, we will mostly focus apprendre option binaire quebec one-dimensional commutative formal groups. Understanding of the basic theory of formal groups how they arise, universal group laws, deformation theory. Applications to elliptic curves, class field theory, topology perhaps…p-adic dynamics.

Period maps on the moduli space of formal groups. SMF, 94 Equivariant vector bundles on the Lubin-Tate moduli space, Contemp.

The grade will be given on the basis of submitted exercise sets, which some students will be able to substitute for by giving presentations. There will also be assigned reading to cover additional background material. Anneaux commutatifs et leurs modules.

The goal apprendre option binaire quebec this course will be to study algebraic methods for the estimation of exponential sums over finite fields, and the applications of the latter in number theory.

A particular emphasis will be put on ideas around the Weil conjectures for curves and varieties, and Deligne's generalization to weights of l-adic sheaves. 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 may be covered as well.

The course will cover the following topics, as far as time permits: The topics that will be covered include Functional and Spectral Analysis, Representations and States, von Neumann algebras, elements of Tomita-Takesaki modular theory, quantum spin systems.

Systems of conservation laws and Riemann invariants. Cauchy- Kowalevskaya theorem, powers series solutions. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators.

Second order parabolic and **apprendre option binaire quebec** equations. Further advanced topics may be included. 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. Statistical methods for multinomial outcomes, overdispersion, and continuous and categorical correlated data; approaches apprendre option binaire quebec inference estimating equations, likelihood-based methods, semi-parametric methods ; analysis of longitudinal data; theoretical content and applications.

Practical approaches to complex data. Graphical and tabular presentation of results. Writing reports for scientific journals, research collaborators, consulting clients. Introduction to practical Bayesian methods. Topics will include Bayesian philosophy, apprendre option binaire quebec Bayesian models including linear and logistic regression, hierarchical models, and numerical techniques, including an introduction to the Gibbs sampler. Advanced applied biostatistics course dealing with flexible modeling of non-linear effects of continuous covariates in multivariable analyses, and survival data, including e.

Focus on the concepts, limitations and advantages of specific methods, and interpretation of their results. Students get hands-on experience in designing and implementing simulations for survival analyses, through individual term projects. Bayesian design and analysis with applications specifically geared towards epidemiological research.

Topics may include multi-leveled hierarchical models, diagnostic tests, Bayesian sample size methods, issues in clinical trials, measurement error and missing data problems. 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. 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. Foncteurs et transformations naturelles: Sommes et produits directs, modules libres.

Modules de type fini sur un anneau principal et applications aux formes canoniques des matrices. Extension du semi-anneau de coefficients. Graph Theory meets Linear Algebra on the street. Let X be a graph. Graph Theory squints, and asks: OK, but can you also tell me something about X? Can you help me in difficult problems such as colouring or measuring X?

I can do that, too. Finite Field Theory and Group Theory, overhearing the exchange, join in: The aim of this course is to take this silly dialogue seriously. Spectral graph theory is concerned with eigenvalues of matrices associated to apprendre option binaire quebec, more specifically, with the interplay between spectral properties and graph-theoretic properties.

It often feeds on graphs built from groups or finite fields, and this is the direction we will emphasize. In a somewhat larger sense, this course aims to be an introduction to algebraic graph theory. In an even larger sense, this course aims to braid together several strands of interesting mathematics. A basic familiarity with linear algebra, finite fields, and groups, but not necessarily with graph theory. The course is, however, fairly self-contained and very much accessible to senior undergraduate students.

Les grandes lignes sont les suivantes:. Dynamical systems, phase space, limit sets. Review of linear systems. Stable manifold and Hartman-Grobman theorems.

Local bifurcations, Hopf bifurcations, global bifurcations. Sarkovskii's theorem, periodic doubling route to chaos. Basic point-set topology, including connectedness, compactness, product spaces, separation axioms, metric spaces. The fundamental group and covering spaces. Singular and simplicial homology. Part of the material of MATH may be covered as well. The aim of the course is to cover the basic theory of compact Riemann surfaces, or, alternately, one dimensional smooth projective curves over the complex numbers.

The two terminologies reflect a very useful tension apprendre option binaire quebec analysis, on one hand, and geometry on the other, and the course aims to both understand these tensions and exploit them a third, more algebraic aspect, of thinking of affine Riemann surfaces as field extensions of transcendence degree one of the complex numbers, will only be covered more tangentially.

The course could at the same time serve as an introduction to some basic ideas of algebraic geometry, such as sheaves and their cohomology. If apprendre option binaire quebec permits, I would hope to give some elements of the theory of theta-functions. Lie groups, Lie algebras and their representations play an important role in many areas of pure and applied mathematics, ranging from differential geometry and geometric analysis to classical and quantum mechanics.

Our goal is to give a motivated introduction to the representation theory of compact Lie groups and their Lie algebras, the essentials of which go back to the classical works of Elie Cartan and Hermann Weyl. Relation entre le groupe fondamental et le premier groupe apprendre option binaire quebec.

Suite exacte de Mayer-Vietoris. I plan to discuss questions about the existence of closed geodesics on Riemannian manifolds. The first goal is to present **apprendre option binaire quebec** result of Bangert-Franks-Hingston on the existence of an infinite number of geodesics for any Riemannian metric on S 2discussing the relevant 2-dimensional dynamics. Time permitting, further topics may include homological methods and applications of non-linear Cauchy-Riemann equations to questions of this kind.

Differentiable manifolds, tangent and cotangent spaces, smooth maps, submanifolds, tangent and cotangent bundles, apprendre option binaire quebec function theorem, partition of unity. Examples include real projective spaces, real Grassmannians and some classical matrix Lie groups.

Differential forms and de Rham cohomology: Review of exterior algebra, the exterior differential and the definition 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. An introduction to Riemannian geometry: