Geometry for Teachers

This course will assume a year of linear algebra. It will combine explorations of plane and spherical geometry, use of dynamic geometry programs (in particular The Geometer’s Sketchpad) and projects generated by students. The marks will be base on in-class work, assignments and a project (which you are encouraged to do in a group).

Text: David Henderson, Experiencing Geometry on Plane and Sphere, Prentice-Hall, 1996.

Instructor: W. Whiteley, S616 Ross, ext. 33971, e-mail: whiteley@mathstat.yorku.ca

Mixed Models

Mixed (random and fixed) effects models have been developed under different guises in many different fields: hierarchical models, multi-level linear models, random-effects models, random-coefficient regression models, covariance component models, etc. These models have very wide applications to: longitudinal data analysis, unbalanced nested designs, hierarchical data structures, meta analysis, etc. This course will consider both the theory and the practical aspects of the use of these models. As with most new methods, mixed models present many potential pitfalls. The course will stress model interpretation, diagnostics and dealing with numerical problems. A major component of the course will be the analysis of a suitably complex real data set.

Prerequisites

An intermediate course in linear models such as MATH3330 or MATH3033, an introduction to likelihood.

Text: J.C. Pinheiro, D.M. Bates. 2000. Mixed Effects Models in S and S-Plus. Springer, New York

Instructor: Professor G. Monette, 263 SSB, Ext. 77164

Mathematical Modelling

Mathematical modelling is the basis of almost all applied mathematics. A ‘real-world' problem is dissected and phrased in a mathematical setting, allowing it to be simplified and ultimately solved. In this course, models in a variety of applications are derived, simplified, and analyzed.

Using examples from industrial, environmental, biological, and financial applications, we discuss the uniformity of the approach used by applied mathematicians in these different contexts and various analytical and numerical techniques. Only a basic mathematical background in calculus

and analysis is required. The course is designed to develop and improve problem solving skills for graduate students. Students will be encouraged to attend one of the industrial problem solving workshops/study groups.

Prerequisites

Calculus & Analysis such as Math3210 and Differential Equations (e.g. Math2270) or equivalent. Some basic programming skills and knowledge of Partial Differential Equations will be helpful.

Text: A.C. Fowler, Mathematical Models in the Applied Sciences, Cambridge University Press, 1996.

Instructor: H. Huang, S622, Ext. 66090

Introduction to Lie Algebras and their Representations

This is an introductory course on Lie algebras and representation theory. Lie Algebra is a very elegant area which involves many parts of mathematics and has a lot of applications in both mathematics and physics. Elementary structural properties of semi-simple Lie algebras over the complex field, finite irreducible root systems, Weyl groups, and sl-theory will be studied. Although the primary subject in this course is finite dimensional Lie algebras, some basics about certain important infinite dimensional Lie algebras such as the Heisenberg algebra, Virasoro algebra and Kac-Moody Lie algebra will be touched upon as well.

Prerequisites: A good knowledge of Linear Algebra (Math 2022, or equivalent) and some familiarity with Abstract Algebra (Math 3020, or equivalent).

Instructor: Y. Gao, S624 Ross, ext. 33952

Robust Statistics

Robust statistics are those which provide protection against violation of assumptions underlying the statistical procedure. In this course, basic robustness concepts including sensitivity, influence function and breakdown points of estimates and tests will be discussed. Classical procedures will be evaluated in terms of robustness and alternative techniques will be developed. M-estimator, L-estimator, R-estimator, LMS estimator, LTS estimator and other robust estimators will be introduced and their efficiency compared with the classical estimators will be investigated. Besides, the resistent diagnostics will be described. Starting from the location problem, we will move on to regression problems. The statistical software package S-PLUS will be used.

Prerequisites

Math 6621 or equivalent and Math 6620B or equivalent

Instructor: Y. Wu, N609 Ross, Ext. 88604

Differential Geometry

The course will begin with an introduction to differentiable manifolds, including also matrix groups as an important class of examples. The tangent bundle to a smooth manifold will be defined and there will be some discussion of vector bundles and smooth bundles. Included in this section is the discussion of vector fields on a manifold and the differential map on the corresponding tangent bundles that is induced by a smooth map betwen manifolds.

A second section of the course introduces differential forms on a smooth manifold including also a discussion of differential forms with values in a vector space. Integration of differential forms, the Poincare Lemma and Stokes' Theorem will be studied.

A third section of the course introduces the theory of connections on a manifold, with some discussion of affine connections on a vector bundle. Riemannian manifolds, the curvature tensor and Riemannian connections will be discussed. Time permitting, some properties of the curvature tensor will be developed, and generalized versions of Green's Theorem on a Riemannian manifold will be discussed.

Prerequisites:

(i) A good knowledge of advanced calculus of several variables, linear algebra and some modern algebra. (ii) A basic understanding of point set topology, or permission of the instructor.

Reading List:

There is no one textbook that covers adequately all the topics above in an introductory manner, with applications. I will use the following book as a REQUIRED TEXT:

M. do Carmo, "Riemannian Geometry", Birkhauser (1992).

OTHER BOOKS that are useful sources for some of the above topics are:

1. M. do Carmo, " Differential Forms and Applications", Springer Verlag (Universitext) (1991).

2. M.W. Hirsch, "Differential Topology", Springer Verlag, Graduate texts in mathematics.

3. F.W. Warner, "Foundations of Differentiable Manifolds and Lie Groups", Springer-Verlag, Graduate texts in Mathematics.

4. G.E. Bredon, "Topology and Geometry", Springer-Verlag, Graduate Texts in Mathematics.

5. S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry, Volume I", J. Wiley and Sons.

Instructor: D. Spring, Glendon College, York Hall 351, 487-6731 ext. 66815