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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 hyperbolic 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 to 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, simple 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 graphs, 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 between 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 time 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 d'homologie.
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 the result of Bangert-Franks-Hingston on the existence of an infinite number of geodesics for any Riemannian metric on S 2 , discussing 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, implicit 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: 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. Boothby, An introduction to differentiable manifolds and Riemannian geometry , Academic Press.
Tu, Differential forms in algebraic topology , Springer. A compact oriented surface is homeomorphic either to a sphere, a torus obtained from gluing opposite sides of a square, or to a surface obtained from gluing the sides of a hyperbolic polygon. The goal of the course is to indicate a similar classification in dimension 3. We will start following the notes of Hatcher, where two classical decompositions prime and JSJ of a 3-manifold into smaller pieces are described.
We will then follow the book of Thurston edited by Levy. Thurston conjectured that each piece carries one of the 8 model geometries.
We will discuss each of the geometries, provide examples, and study their fundamental groups. The most ubiquitous geometry is the hyperbolic geometry, on which we will focus at the end of the course. Some of the examples will be geometrically finite hyperbolic groups obtained as surfaces from gluing finite polyhedra , and quasifuchsian groups.
We will finish with a sketch of the proof of Mostow's rigidity theorem saying that for a compact hyperbolic 3-manifold, its fundamental group determines its isometry type. Voici une courte description du contenu:. Les notes seront en principe suffisantes. On pourra par exemple consulter les livres suivants: I, II, Interscience Pub. Harris, Representation Theory, Springer-Verlag.
Dimension de Krull des anneaux. Localisation dans les anneaux. Paysage du local au global. La question est cependant largement ouverte en dimensions plus grandes. Cambridge University Press, Cambridge, Notes taken by Meike Akveld. This will be an introductory course in algebraic topology.
As such, I am planning to cover a selection of the following topics: Two classical approaches to credibility theory are discussed: Topics covered include American, Bayesian and exact credibility. Generalized linear models and the issue of robustness will also be discussed. H Panjer, and G.
Willmot, Wiley, 4 th Edition, or the 3rd Edition, Gisler, Springer Universitext, It also covers more advanced material, as needed to use modern credibility with real insurance data. 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 is designed for the study of numerical methods in finance, with an emphasis on numerical methods for the pricing of contingent claims.
The reading material covers earlier developments as well as current research issues such as quasi random numbers, Markov chain approximations, finite elements and Monte Carlo methods for the pricing of American securities. Some of the material will be presented in class by the instructors.
However, the course will generally involve presentations and discussions of the assigned readings by the students. The course will be taught in both languages French and English. Utilisation des mesures de risque. This course gives a brief introduction of fluctuation theory for spectrally negative Levy processes. Some applications in risk theory will be discussed. Risk theory forms the core part of Property-Casualty Insurance mathematics. The course gives an introduction to classical models and applies them to some common problems of interest in risk theory.
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, homogeneous and non-homogeneous discrete-time Markov chain models, Poisson processes, coinsurance, effects of inflation on losses, risk measures VaR, TVaR. The problem of fitting probability distributions to loss data is studied.
In practice, heavy tailed distributions are used 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 and of sampling designs.
The course also covers more advanced material, as needed to use modern loss models with real insurance data. It also covers more advanced material, as needed to use modern statistics, such as GLMs with real insurance data. 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.
Dans le cadre du cours, nous aborderons les sujets suivants: Une base de programme leur sera fournie afin de faciliter l'accomplissement de ce projet. The course is a general introduction to non-cooperative and cooperative static and dynamic game theory.
The main concepts are defined rigorously and illustrated by a series of examples, exercises and applications to different problems in management which are of interest to the participants. 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. The objective of this course is to introduce students to optimization models in a dynamic contest; more specifically, the students will be acquainted with the major tools used in dynamic optimization, that is, dynamic programming, Markov decision problems and optimal control.
In addition, the course will cover many applications where the use of such tools is called for. At the end of the course, the students should be able to:. The aim of the course is to present the most important aspects of mathematical optimization applied to network flow problems. On the one hand, we study the specialized algorithms, and on the other hand, we take a look at the numerous applications in this field. This course covers strategic, tactical and operational planning in distribution management systems.
Long term decisions relate mainly to the location of major installations, namely transportation infrastructures. Tactical planning includes medium term operations such as route design in inter-city planning and warehouse location. Operational planning covers the design of daily pickup and delivery routes and the location of light facilities such as mail boxes.
In several transportation areas operations may have to be planned in real time, like in pickups and deliveries of letters and packages in fast courier operations, in dial-a-ride services for handicapped people, and in ambulance relocation.
This course introduces the main methods and applications encountered in distribution management. It is partly based on some real cases published in recent scientific articles. Numerical solution of initial and boundary value problems in science and engineering: Topics include Runge Kutta and linear multistep methods, adaptivity, finite elements, finite differences, finite volumes, spectral methods.
Random variables and their expectations. Convergence of random variables in Lp. Independence and conditional expectation. Conditional probability and conditional expectation, generating functions. Branching processes and random walk. Birth and death processes, renewal theory, queueing theory.
Introduction to Brownian motion, martingales. Further topics as time permits. Les objectifs principaux du cours sont: Il apprendra aussi le langage macro de SAS et s'en servira. If you suffer from occasional or chronic lower back pain, patented Jox Athletic Shorts can provide the relief you have been seeking. Lower back pain can result from injuries, poor posture, excess weight, osteoarthritis, and a range of other conditions.
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