Math 5100.06 MATHEMATICAL LITERATURE SEMINAR FOR TEACHERS
(F 1997, TR 6-9, S128 Ross)
The course deals with a variety of mathematical issues, and is intended to convince the students that mathematics is meaningful, that some of its problems are profound, and that the evolution of some of its ideas is an exciting chapter of intellectual history. Students are encouraged to present material in class, and one of the key objectives of the course is to develop in students the ability to read independently and critically in the relevant mathematical literature.
| Course director: | T.B.A. |
Math 5450.06 GEOMETRY FOR TEACHERS
(W 1998, MR 6-9, S525 Ross)
Geometry is an important part of 'classical mathematics'. Recently, interest in geometry has revived for a variety of additional reasons:
The course will include a variety of 'geometries' which arise in these contexts: Euclidean geometry, projective geometry, spherical geometry, some elementary topology (graph theory, surfaces) etc. The choice of topics will be influenced by current discussions of curriculum reform for high school geometry across North America.
We will begin with spherical geometry, and the comparisons between plane geometry and spherical geometry. The final selections will evolve from the interests expressed by those registered in the course.
Reforms in mathematics education promote the use of explorations with manipulatives, with computer programs, and with pencil and paper. While explorations can provide intuition and conjectures, formal mathematical proofs provide an additional insight - and generate new question for exploration. The van Hiele model of Geometry education will be followed as this course uses all these approaches to geometry. In addition, we make use of information on the internet (the Math Forum news groups, web resources) to find out how other high school teachers approach geometry education.
This course will use a variety of methods of assessment and evaluation, including group work, oral presentations, journals, and written projects. There will not be major exams.
We will use portions of the following text:
David Henderson, Exploring Geometry on Plane and Sphere, Prentice Hall, paperback.
| Course director: | W. Whiteley, S616 Ross |
| 736-5250 Ext. 33971 | |
| Whiteley@mathstat.yorku.ca |
Math 6003A.03W CONWAY'S THEORY OF GAMES
(W 1998, T12:30-2:30, R 12:30-1:30, 115 Winters)
An overview of the theory of two-person games as developed by Conway.
Special consideration will be given to recent applications of this theory to Error Correcting Codes, Cryptography and Computational Complexity.
Game theoretical constructions of the Real and Surreal numbers as well as constructions of algebraic structures.
Students will be expected to do a project on one or several games.
The course is open in the sense that special emphasis will be given to those topics that the students find most interesting.
Prerequisite: Introductory abstract algebra (such as Math 3020).
Text: John Conway, On Numbers and Games, Academic Press.
| Course director: | A. Trojan, 532 Atkinson |
| 736-5232 |
Math 6004.00 MATHEMATICS SEMINAR
This course provides students with a chance to work independently and to present the results of their work to other students. Each student gives two one-hour seminars on topics arranged with one or two faculty members. The topics may be related to other courses the student is taking, but should not actually be covered in those courses. They may be in the same field or two different fields. Students are expected to submit a written report prior to presenting each seminar. The seminars are graded separately and the course is graded on a pass/fail basis. Students in the course are expected to attend all seminars.
| Course director: | Faculty |
Math 6030.03F Mathematical Logic
(Math 4290.03F) (F 1996, TR 2:30-4:00, 129 CCB)
Logic is the study of reasoning. Mathematical Logic studies the kind of reasoning practiced by mathematicians. The modern use of the title suggests the inclusion of proof theory, model theory, recursion theory, and set theory.
Under the first subtitle the objects of study are (mathematical) proofs. In particular, questions such as "what statements are `reachable' by proofs in a given axiomatic theory", and "is there an algorithm which separates the theorems from the non theorems of a particular theory" are dealt with.
Model theory studies the connection between the syntax and semantics of mathematical theories. G”del's completeness theorem as well as the G”del-Mal'cev compactness theorem belong here. One of the most beautiful applications of model theory is the legalization of infinitesimals by Robinson.
Recursion theory, narrowly speaking, addresses the question of what is, and what is not, "mechanically", or "effectively", computable (once it gives a formal definition of what the terms in quotation marks ought to mean). Broadly speaking, it studies recursive (or inductive) definitions on all sorts of mathematical structures.
Set theory is the subtitle of logic that is most familiar to the working mathematician. An oversimplifed view of the subject would be "the study of the properties of collections of mathematical objects" (for more details consult MATH 6040.03).
In this course we shall mostly deal with proof theory and recursion theory, and we will also do a fair bit of elementary model theory as outlined below. Our primary goal will be a thorough understanding of G”del's incompleteness theorems as well as the related undecidability result of Church. We shall address these results at several levels of abstraction.
Recursion theory will be studied to the extent that it will serve the above goal. Aspects of model theory (compactness, completeness and L”wenheim-Skolem results, ultrafilters) will be covered, with a quick introduction to nonstandard analysis … la Robinson.
Prerequisites: Nothing specific, except "mathematical maturity" such as the one a 3rd--4th year undergraduate or 1st year graduate mathematics student would normally have.
Previous courses in recursion theory and/or (undergraduate) "logic" (this is, usually, mostly proof theory) would make life easier.
Work load/grading: Several homework assignments, equally weighted. No exams/tests in class. As this course is integrated with MATH 4290.03 in 1997--98, there will be a differential in workload between graduate and undergraduate students taking the course, as it is required by Senate. Graduate students will be guided to more esoteric aspects of the literature for study (in recursion, proof and model theory) and also will do approximately 2\% more homework problems than the undergraduate students (on G”del's incompleteness theorems, Robinson's nonstandard analysis, and some work on reducibility).
References:
| Course director: | G. Tourlakis, 518 Atkinson College |
| 736-2100 ext. 66674 | |
| E-mail: tgeorge@yorku.ca (NeXT or MIME mail OK) |
Math 6120.06 MODERN ALGEBRA
(FW 1997-98, MW 1:00-2:30, S128 Ross)
This course presents the most important elements of modern algebra, as described by the following headings: group theory (Sylow theorems, finitely generated abelian groups); ring- and module theory (unique factorization domains, free modules, finitely generated modules over a PID); field theory (finite fields, Galois theory).
Final grades are based on homework assignments, class test and possibly an oral presentation.
Text: P.B. Bhattacharya, S.K. Jain, S.R. Nagpaul, Basic Abstract Algebra, Cambridge University Press.
| Course director: | N. Bergeron, N626 Ross |
| 736-5250 |
Math 6280.03F MEASURE THEORY
(F 1996, MWF 10:30-11:30, S128 Ross)
A measure is a type of set function, which assigns numbers to subsets of a given set. The motivation comes from two main sources. One is geometric, whereby measures generalize such notions as length, area, and volume. The other is from probability theory, where the measure of a set is the probability of the associated event. Measure theory is intimately connected with integration theory, since integrals are constructed by means of measures, and conversely, integrals give rise to measures.
In this course, we will first cover the classical theory of Lebesgue measure and the Lebesgue integral on the real line. We will then look at the general theory on abstract spaces.
Topics include: algebras, measure spaces, measurable functions, outer measure and measurability, the Caratheodory extension theorem, integration, convergence theorems, signed measures, Hahn-Jordan decomposition, Radon-Nikodym theorem, product measures, Fubini Theorem, the Daniel integral.
Prerequisite: An undergraduate course in real analysis.
Texts:
| Course director: | E.J. Janse van Rensburg, 215 Petrie |
| 736-5248 Ext. 33837 |
Math 6340.03W ORDINARY DIFFERENTIAL EQUATIONS
(Math 4110.03W) (W 1998, TR 10:00-11:30, 224
Stong)
This is an advanced introduction to a number of topics in ordinary differential equations. The topics will be chosen from the following: existence and uniqueness theorems, qualitative theory, oscillation and comparison theory, stability theory, bifurcation, dynamical systems, boundary value problems, asymptotic methods. The lectures will survey the above topics and students will be expected to make an in-depth study of some of them by doing assignments and projects.
Students should have a thorough knowledge of undergraduate analysis and linear algebra to the level of MATH 2220 and MATH 3210. It would be desirable but not essential that they have taken an undergraduate course in differential equations. Some exposure to real analysis, complex analysis and topology would be desirable also.
The probable textbook is J.K. Hale, Ordinary Differential Equations, Krieger, Malabar, Florida, 1980.
References:
| Course director: | Y. Yang |
| 736-5250 |
Math 6350.03F PARTIAL DIFFERENTIAL EQUATIONS
(F 1997, MWF 9:30-10:30, S128 Ross)
This is primarily a basic course in the modern theory of partial differential equations using pseudo-differential operators. We shall begin the course with a self-contained treatment of Fourier analysis and the theory of a standard class of pseudo-differential operators, and then proceed to use the techniques so developed to study the existence and regularity of weak solutions of elliptic partial and pseudo-differential equations.
The main prerequisite for the course is familiarity with the theory of Lebesgue measure and integration.
The final grade will be based on several homework assignments.
Text: M.W. Wong, An Introduction to Pseudo-Differential Operators, World Scientific, 1991.
| Course director: | M.W. Wong, S518 Ross |
| 736-5250 |
Math 6461.03W FUNCTIONAL ANALYSIS I
(W 1998, MWF 10:30-11:30, S128 Ross)
Functional Analysis is a subject with connections to both pure and applied mathematics, as well as many branches of physics. It deals primarily with spaces of functions rather than individual functions. These spaces usually have both a natural metric ("distance between two functions") as well as a natural linear space structure. Of special importance are Banach spaces (complete normed linear spaces) and Hilbert spaces (complete inner product spaces). This course is intended as an introduction to the study of Banach and Hilbert spaces and the bounded linear functionals and operators on these spaces. Topics include: the Hahn-Banach theorem, the uniform boundedness principle, the open mapping theorem, the closed graph theorem, and the Krein Milman theorem.
The main prerequisite for the course is a good undergraduate course in real analysis, such as MATH 4010.06.
Text: T.B.A.
| Course director: | J. W. Pelletier, N534 Ross |
| 736-5250 |
Math 6540.03F GENERAL TOPOLOGY I
(F 1997, MWF 11:30-12:30, S128 Ross)
This course will examine some of the basic concepts of topology, concentrating on metrization theorems, consequences of compactness and connectedness. An attempt will be made to explain certain aspects of the historical development of the subject as it evolved under the impetus of various discoveries in real analysis at the turn of the century. Some attention will be paid to the role of topology in the wider context of mathematics and, time permitting, some applications and connections with other areas like algebra and functional analysis will be discussed.
The book Topology by Munkres will serve as a text for the course. An upper year undergraduate course in real analysis is a prerequisite for this course.
| Course director: | H.-P. Kunzi, S511 Ross |
| 736-5250 |
Math 6550.03W ALGEBRAIC TOPOLOGY I
(W 1998, TR 2:30-4:00, S701 Ross)
Algebraic topology solves problems in topology by studying algebraic invariants associated to topological spaces. This point of view has been useful for studying certain problems in analysis and algebra as well as in topology.
We will begin by studying the fundamental group using covering spaces. Then we will define singular homology and determine its basic properties. Cell complexes will be introduced to make computations. Applications will be made to the study of surfaces, the generalized Jordan curve theorem and invariance of domain. If time permits we will study singular cohomology and prove various duality theorems.
Prerequisite: Mathematics 6540.03 or a knowledge of basic general topology; undergraduate algebra including the fundamental theorem of abelian groups.
Text: Greenberg and Harper, Algebraic Topology, A First Course, Addison-Wesley Publishing Co., 1981.
| Course director: | S.O. Kochman, N510 Ross |
| 736-5250 |
Math 6602.03W STOCHASTIC PROCESSES I
(Math 4430.03MW) (W 1998, MWF 11:30-12:30,
S128 Ross)
This course is an introduction to stochastic, or random, processes. Stochastic processes are models which represent phenomena that change in a random way over time. Simple examples are (a) the amount of money a gambler has after each play of a game and (b) the number of people waiting for service at a bank at various times. This course studies some of the most basic stochastic processes, including Markov chains and Poisson processes. A Markov chain is a stochastic process in which predictions for the future depend only on the present state of affairs, but not on knowledge of the past behaviour of the process. Markov chains are relatively easy to analyze, and they have been used as models in many areas of science, management, and social science. A Poisson process is a model for the occurrence of random events (such as oil spills in the Atlantic Ocean). This course will treat both the theory and applications of these stochastic processes.
The text will be An Introduction to Stochastic Modelling (revised edition) by Howard M. Taylor and Samuel Karlin (Academic Press).
Students should have a reasonable knowledge of basic undergraduate probability before taking this course.
The final grade is likely to be based on assignments (20%), two tests (20% each), and a final exam (40%).
| Course director: | N. Madras, N623 Ross |
| 736-5250 |
Math 6620.06 MATHEMATICAL STATISTICS
(FW 1997-98, MWF 12:30-1:30, 112 Founders)
The course presents a detailed theoretical development on foundations of mathematical statistics including likelihood, Bayes, minimax and conditional inference. The topics consist of statistical models, exponential families, sufficiency, completeness, properties of estimators, optimal tests and confidence regions, and elements of decision theory.
Text: P.J. Bickel and K.A. Doksum (1977). Mathematical Statistics: Basic Ideas and Selected Topics, Holden-Day.
References:
| Course director: | P. Song, N636 Ross |
| 736-5250 |
Math 6621.03F LINEAR MODELS AND REGRESSION
(F 1997, MWF 3:30-4:30, S128 Ross)
This course will assume a familiarity with multiple regression, as covered, for instance, in MATH 3033 or MATH 3330, and with basic mathematical statistical theory, at the level of Hogg and Craig or higher and basic matrix algebra. A theoretical basis in linear models is provided first, including standard topics as distribution theory, least square point estimation and hypothesis testing in the full or non-full rank cases, multiple comparison procedures and analysis of variance.
Theory will be applied to a variety of data sets with intensive use of statistical computing packages. Regression diagnostics including the identification of outliers and model selection procedures will be developed.
The grade in the course will be determined by a combination of exams, homework assignments and projects.
All students will need access to the departmental computer lab in N604 Ross. The lab has a variety of terminals and manuals for faculty/student use.
Texts:
| Course director: | Y. Wu, N609 Ross |
| 736-5250 |
Math 6622.03W STATISTICAL TECHNIQUES
(W 1998, MWF 3:30-4:30, S128 Ross)
The following topics in analyzing catagorical data will be covered:
Texts:
References:
| Course director: | C. Czado, N621B Ross |
| 736-5250 |
Math 6625.03W MULTIVARIATE STATISTICS
(Math 4630.03W) (W1998, TR 10:00-11:30, 112 MC)
This course will assume a familiarity with the contents covered in Math 3131, Math 3033, Math 3034, Math 2022.
We will study methods of analysis for data which consist of observations on a number of variables. The primary aim will be interpretation of the data, starting with the multivariate normal distribution and proceeding to the standing multivariate inference theory. Sufficient theory will be developed to facilitate an understanding of the main ideas. Computers will be used extensively, and familiarity with elementary use of SAS will be assumed. Grades will be based on a combination of class test and final examination, plus routine homework. Topics covered will include multivariate normal population, inference about means and covariance, multivariate linear models, principal component analysis, canonical correlation analysis, and possibly some discussion of discriminant and classification, factor analysis and cluster analysis, if time permits.
| Course director: | Y. Wu, N609 Ross |
| 736-5250 |
Math 6626.03F EXPERIMENTAL DESIGN
(Math 4730.03AF) (F 1997, T 12:30-1:30, R 12:30-2:30; 203 Founders)
An examination of the statistical issues involved in ensuring that an experiment yields relevant information. Topics include randomized block, factorial, fractional factorial, nested, Latin square and related designs. Further topics as time permits. The emphasis is on applications.
Good experimental design is the founding block of generation of good data. It can be viewed as selecting the best feasible experiment to achieve some specific objective. This includes the choice of treatments, and the random allocation of experimental units to them. Method of estimation is an important component of the determination of a design. One must consider how the data will be analysed after the experiment is carried out and data are collected. With the analytical procedure in mind, a proper choice of experiment is then determined to achieve that goal.
Various designs will be discussed in this course through definition of objectives, analytical procedures, and feasibility of experimental constraints.
The text will be announced later.
The final grade may be based on assignments, test(s), and a final examination. Details will be announced.
Prequisites: A second 6 credits in statistics, including either AS/SC/MATH3033.03, or both AS/SC/AK/ MATH 3230.03 and AS/SC/AK/MATH3330.03, or permission of the course coordinator.
| Course director: | P. Ng, N601B |
| 736-5250 |
Math 6638.03W PRACTICUM IN STATISTICAL CONSULTING
Many graduates from programmes in statistics enter jobs in which initial success depends on their effectiveness as statistical consultants - and they find themselves needing skills very different from those they developed while doing traditional course work.
They have learned to expect that problems are well posed and contain all the information necessary for a correct statistical solution. In the real world of consulting clarifying and identifying the actual problem is often half the problem (and half the solution), the other half being the study design and statistical analysis of the data and its interpretation.
The purpose of the practicum course is to prepare students for the transition to the "real world" of consulting:
Course content: The student will be given a number (3 or 4) of projects in which s/he will act as a statistical consultant for clients with problems that can be handled most of the time with standard statistical techniques (undergraduate regression, analysis of variance, basic contingency tables, generalized linear models).
Sometimes, the student will also explore the literature on non standard issues arising in the course of their work on the projects. For each consultation the student will prepare a brief but effective report using terms and concepts that would be understood by a client. Graphs and charts are strongly encouraged. The student may also be called upon to participate in and/or lead discussions in the statistical consulting service's regular meetings on various topics including: the client-consultant interaction, model finding, report preparation, the role of the statistician in scientific inference, principles and ethics, applying a deductive descipline to inductive problems.
Pre and Corequisites: Before taking this practicum students should have had a good undergraduate background in statistics, including a full course in mathematical statistics, a full course in linear models (regression and analysis of variance) and the equivalent of two half-courses in more advanced topics such as time series, experimental design, sample survey methods, non-parametric methods or multivariate analysis. Students must have taken or must take concurrently the courses MATH 6621.03F: Linear Models and Regression and MATH 6622.03: Statistical Techniques.
Ph.D. students in statistics should enrol in this course to fulfil part of their degree requirements (see page 7 under "Breadth Requirement and Comprehensive Examinations").
Meetings: The class will meet every second Wednesday, starting September 17, 1997 (9:30-11:30 -101A MC) for two terms.
| Course director: | P. Ng, N601B Ross |
| 736-5250 |
Math 6639A.03F NON PARAMETRIC STATISTICS
(Math 4230.03AF) (F 1997, W 5:00-8:00, S128 Ross)
Order statistics; general rank statistics; one-sample, two-sample, and k-sample problems; Kolmogorov-Smirnov statistics; tests of independence and relative efficiencies.
Survey of basic nonparametric test procedures together with the related theory for permutation, rank, and related techniques.
Text: E.L. Lehmann, Nonparametric Methods based on ranks, McGraw Hill.
Prerequisite: AS/SC/MATH3030.03 (taken before 1993/94) or AS/SC/AK/MATH3131.03; AS/SC/ MATH3031.03 or AS/SC/AK/MATH3132.03 is rec-ommended but not required.
The final grade will be determined 20% by assignment, 30% by test, 50% by final exam.
| Course director: | D.A.S. Fraser, S128 Ross |
| 736-5250 |
Math 6639B.03W ADVANCED TOPICS IN STATISTICS: BAYESIAN STATISTICS
(4130.03W) (W 1998, TR 2:30-4:00, 208 CC)
We first present the Bayesian approach to single and multiparameter statistical problems and link it to major concepts of non-Bayesian statistics. We then study some hierarchical models and regression models using a Bayesian approach with theory and examples. The main topics covered in the course are:
Each topic will be illustrated by at least one example.
| Course director: | T.B.A. |
| Course director: | A. Stauffer, 223 Petrie |
| 736-5248 Ext. 77742 |
Math 6652.03W NUMERICAL ANALYSIS II
(4142.03MW) (W 1998, MWF 2:30-3:30, S128 Ross)
Review of linear and nonlinear partial differential equations; boundary conditions; finite-difference approximations; solution of parabolic equations; boundary-value problems for elliptic equations; iterative methods for solution; direct methods; finite-element approximations of elliptic equations; solution of hyperbolic equations; error analysis.
The most important reference for this course is: W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipies, 2nd edition, Cambridge, 1992.
| Course Director: | J.G. Laframboise, 228 Petrie |
| 736-5248 Ext. 55621 or 736-5621 |
Math 6900.03F OPERATIONS RESEARCH I
(Math 4170.06AY) (F 1997, TR 8:30-10:00, 106
Founders)
This course deals with deterministic and probabilistic models based on optimization. The following topics will be discussed: 1) game theory (how to find the best strategies in a confrontation between two players with opposite interests); 2) decision theory (how to act in order to minimize the loss subject to the available data); 3) simulation (how to sample from a probability distribution and accurately approximate multiple integrals using random numbers); 4) relia-bility theory (how to evaluate the lifetime of a system consisting of many interacting subsystems). Each chapter contains a specific optimization problem and methods and algorithms for solving it. The course is rich in examples.
There is no textbook and the lecture notes are essential. Useful books are:
| Course director: | S. Guiasu, N530 Ross |
| 736-5250 |
Math 6901.03W OPERATIONS RESEARCH II
(Math 4170.06AY) (W 1998, TR 8:30-10:00, 106
Founders)
This course deals with deterministic and probabilistic models based on optimization. The following topics will be discussed:
There is no textbook and the lecture notes are essential. Useful books are:
| Course director: | S. Guiasu, N530 Ross |
| 736-5250 |