2002-2003 FALL/WINTER COURSE OUTLINES
NOTE: Faculty of Graduate Studies regulations regarding integrated courses (6000/5000 level course with a 4000 level course):
Masters students who are enrolled in a thesis option must complete at least one full course (or equivalent) which is not integrated with an undergraduate course. Masterís students who are enrolled in a course work or research-review option must complete at least one and a half (or equivalent) courses, which are not integrated with an undergraduate course. Doctoral candidates shall not receive credit towards the Ph.D. degree for more than one full integrated course.
This course aims
to develop the student's problem
As well as specific mathematical techniques, we will discuss general approaches, typically a list such as: "Guess and Check", "Look for a Pattern", "Make A Systematic List", "Make A Drawing Or Model", "Simplify the Problem". There will be an emphasis throughout the course on teaching problem solving and items in the school curriculum will be looked at from a problem?solving point of view. We will use the MAPLE computer algebra to generate special solutions to help in discovering solutions in more general situations.
The following books will be helpful in giving an idea of the contents of the course:
P. D'Angelo and Douglas B. West, Mathematical Thinking: Problem-Solving
and Proofs, Prentice?Hall, 1997
Evaluation: There will be regular assignments (approximately every two weeks) and a major project. Performance on these will constitute the major items in determining course grades.
will be posted at the course web site:
Prior to the 19th century algebra meant the study of solutions of polynomial equations. In the 20th century algebra became the study of abstract, axiomatic systems such as groups, rings, and fields. The transition from the "classical algebra" of polynomial equations to the "modern algebra" of axiom systems occurred in the 19th century.
Modern algebra came into existence because mathematicians were unable to solve classical problems by classical (pre-19th-century) means. The major theme of this course will be to show how "abstract" algebra can shed light on some of these "concrete" (classical) problems, thus providing a confirmation of Whitehead's paradoxical dictum that "the utmost abstractions are the true weapons with which to control our thought of concrete fact".
Algebraic systems that will be studies are groups, rings and fields.
: A. Pietrowski, N617 Ross
This course is a continuation of General Topology (Math 6540). We focus now on compact Hausdorff spaces. We study Stone duality as a method of applying Boolean algebra to topology and vice versa. We study the Stone?Cech compactification of the integers and the kinds of ultrafilters which can be found there. We also investigate continuous images of powers of 2, the cardinalities needed to determine convergence and the compacta which lie in Banach spaces.
of Set?Theoretic Topology, K. Kunen, J. Vaughan, eds., North?Holland,
: S. Watson, N610 Ross
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.
It is the responsibility of students to check the course web page weekly for course related information.
Computability or recursion theory 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.
This course discusses fundamental issues of computability and uncomputability and the close affinity between computations and formal proofs, and concludes with a brief study of computational complexity (that deals with questions of "efficiency": that is, why some mechanically solvable problems seem to require enormous amounts of computational resources for their solution, while others do not).
Topics include computable and semi-computable functions and relations; universal function and S?m?n theorems; recursion theorem; unsolvable problems and Rice's Theorem; "strong" reducibilities; productive and creative sets; the connection between uncomputability and unprovability (Godel's first incompleteness theorem, and Church's undecidability of Hilbert's Entscheidungs problem); "oracle computations" and Turing reducibility.
We also sample topics from computational complexity, including Blum's abstract complexity axioms and some of their consequences; polynomial time reducibilities; NP-hard and NP-complete problems; and Cook's theorem. We conclude by revisiting creative sets and their connection with "inevitably long" formal proofs of "trivial" theorems.
Prerequisites: As published in the graduate calendar. Practically, the essential prerequisite for the course is a certain degree of proficiency in following and formulating combinatorial arguments. Such proficiency should be normally acquired by a student who has successfully immersed herself or himself in the "proof culture" of courses such as COSC 2001.03, COSC 3101.03, or MATH 2090.03. Students who have not completed any of the above courses, but (strongly) believe nevertheless that they have the equivalent background, should seek special permission to enroll in consultation with the me.
NB. The text is available via the Reserve Steacie Library since it is out of print.
· J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison?Wesley.
For more on Primitive Recursive and (total) Recursive functions see:
· R. Péter, Recursive Functions, Academic Press (Number-theoretic approach, rigorous, fairly difficult).
For more on unsolvability of concrete problems of combinatorial or number theoretical nature, see:
· M. Davis, Computability and Unsolvability, McGraw-Hill (Turing approach, rigorous, fairly difficult).
proof of the unsolvability of Hilbert's 10th problem, using methods
initiated by Davis
Y.I. Manin, A Course in Mathematical Logic, Springer-Verlag (quite difficult).
For more on recursion theory, in general, see:
· H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw?Hill (Semi-formal using Church's thesis throughout; difficult).
Finally, for more on machines (at an elementary level), see:
M. Minsky, Computation;
Finite and Infinite Machines, Prentice-Hall.
This course presents the most important elements of modern algebra, as described by the following headings: group theory (Sylow theorems, finitely generated abelian groups), category theory (adjoint functors, limits), ring- and module theory (unique factorization domains, free modules, finitely generated modules over a PID, tensor products, exactness); field theory (finite fields, Galois theory).
Final grades will be based on class tests, and a final exam.
Text: Thomas W. Hungerford, Algebra, Springer-Verlag.
The course provides an introduction to theory of Coxeter groups. It begins with concrete examples of reflection groups in two and three dimensions including the infinite groups. Students will have the opportunity to develop and improve the geometric reasoning through a project dealing with these groups. Further topics include orthogonal transformation, crystallographic groups, Coxeter graphs, and classification of Coxeter groups, polynomial invariants of finite reflection groups.
Text: L.C. Grove and C.T. Benson, Finite Reflection Groups, Graduate Texts in Mathematics, Springer?Verlag, 2nd edition (1985)
: A.I. Weiss, N518 Ross
A measure is a function that assigns numbers to subsets of a given set. For example, the Lebesgue measure of a subset of the plane is the area of the subset for those sets that have a classically defined area, and it also extends the definition of area to a much wider class of subsets. Measures are also important in probability theory, where events correspond to subsets and measures assign probabilities to events. More generally, measure theory is central to analysis, since measures are used to construct integrals and conversely integrals give rise to measures.
This course begins with the classical theory of Lebesgue measure and Lebesgue integration on the real line. We will then examine the general theory of measures on abstract spaces.
Topics include: Sigma-algebras, measure spaces, measurable functions, outer measure and measurability, extension theorems, integration, convergence theorems, signed measures, Hahn-Jordan decomposition, Radon-Nikodym theorem, product measures, Fubini theorem.
Text: H.L. Royden, Real Analysis (Third Edition), Macmillan, 1988.
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,
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.
Perko, Differential Equations and Dynamical Systems, Springer-Verlag,
H. Amann, Ordinary Differential Equations: an introduction to non?linear
analysis, Berlin ; New York : W. de Gruyter, 1990.
: H. Zhu, N621 Ross
Math 6350 3.0FPARTIAL DIFFERENTIAL EQUATIONS
(F 2002, MWF 12:30-1:30, 312 FS)
This is a basic
course in partial differential equations. We shall begin the course
with a self?contained treatment of Fourier analysis, which will be
used in the constructions of the heat kernels, the Newtonian potentials
and the wave kernels for the heat equation,
The main prerequisite is a basic course in real analysis.
The final grade will be based on several homework assignments (60%) and a final examination (40%).
: M.W. Wong, S518 Ross
Functional Analysis is a subject of great importance, with connections to both pure and applied mathematics, as well as many branches of physics. It deals primarily with linear functionals and operators on infinite dimensional linear spaces. The subject was motivated by certain problems in classical analysis, which considered various spaces of functions, together with integral and differential operators on these spaces.
This course provides an introduction to the subject. We consider general topological vector spaces, but with an emphasis on the more concrete examples of Banach spaces and Hilbert spaces. Topics include: the Hahn-Banach Theorem, duality theory, uniform boundedness principle, open mapping and closed graph theorems, spectral theory of compact operators.
The main prerequisite for the course is a good undergraduate course in real analysis (such as the York course Math 4010.06). Students should also be familiar with basic concepts of linear algebra. A knowledge of measure theory will be helpful, but it is not essential.
The evaluation in the course will depend on written assignments (65%) and a final examination (35%).
There is no official textbook. Following is alist of references which cover most of the material. They have been placed on reserve in the Steacie Library.
Differential Geometry is a subject with a long wonderful history . It is a subject which allows students to see mathematics for what it is - not the standard courses of a compartmentalized university curriculum - but rather a unified whole which mixes together geometry, calculus, linear algebra, differential equations, even complex variables, and various notions from science. Differential geometry has immediate relevance in areas ranging from applied areas such as computer aided design, computer aided manufacturing, computer aided engineering to a studies of the foundational studies in differential manifolds and algebraic topology.
This course is very much a first course in Differential Geometry. With a minimum of prerequisite, advanced calculus and linear algebra, the course will focus on developing geometric understanding. Differential Geometry begins with a the study of curves and surfaces in normal Euclidean space and before one branches out into the study of differential manifolds and spaces that are perhaps not Euclidean, it is essential that one start at the beginning - that the foundations be strong. In order to properly visualize various aspects of theory as they are developed, we will make extensive use the computer algebra system MAPLE . Full tutorial style instruction will be given in the use of MAPLE so students with no previous computing knowledge will not be disadvantaged. In the latter part of the course we will examine generalizations to higher dimensions. A complete study of the abstract foundations should form the content of a subsequent course.
It is hoped that the course will be conducted in a seminar style with students presenting some portion of the course material.
Text: John Oprea, Differential Geometry and its Applications, Prentice Hall, 1997.
: M. Walker, 538 Atkinson
3.0F GENERAL TOPOLOGY I
General topology is an important foundation for most areas of mathematics. This course will examine the fundamental operations on topological and metric spaces. We will study the basic concepts of topology such as compactness, connectedness, separation and countability. We will study the role that real?valued continuous functions play in topology and how they lead to key theorems about metrization and compactification.
: P. Szeptycki, 536 Atkinson
This course is an introduction to stochastic, or random, processes. Stochastic processes are mathematical models which represent phenomena that change in a random way over time. The focus of this course is on Markov processes, 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. The simplest examples, Markov chains, are purely discrete models. 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). Brownian motion is a continuous model that has been used to study diffusion in physics as well as stock prices in finance. This course will treat both the theory and applications of these stochastic processes.
Text: Howard M. Taylor and Samuel Karlin, An Introduction to Stochastic Modelling (third edition), 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 (40%), a "take home" test (30%), and a final exam (30%).
: N. Madras, N623 Ross
Topics include: Review undergraduate level Mathematical Statistics such as MATH 3131 and MATH 3132 offered at York University, exponential family and group family, minimal sufficiency, completeness and Basu's Theorem, decision theory, UMVU and equivariant estimators, Baysian and minimax estimators, admissibility and Stein paradox, limiting behaviour of MLE, theory of hypothesis testing, and confidence regions. As time and interest permit, further related topics may also be covered.
Text: P.J Bickel & K.A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics Vol. I, 2nd edition, 2001.
The topics of the course include: Introduction to measure theoretic probability, conditional expectation, sufficiency, convergence theorems, methods of large sample theory, order statistics and U?statistics, estimating equation, advanced theory of hypothesis testing, confidence regions. As time and interest permit, further related topics may also be covered.
Prerequisite: Math 6620 3.0 or equivalent, or with permission of instructor.
Text: J. Shao, Mathematical Statistics, Wiley.
: H. Massam, N618 Ross
3.0F GENERALIZED LINEAR MODELS
The course covers many aspects of the generalized linear models. The topics include: Classical linear models and regression; analysis of contingency tables: log linear models, linear logit models; generalized linear models: model specification, link functions, measures of discrepancy, fitting algorithms, examples of applications to continuous, binary and polytomous data.
Prerequisites: Math 3033 3.0 and Math 3034 3.0 or equivalent, or with permission of the instructor.
Course director :
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.
Students 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 asked to attend and report on three (likely) consulting sessions between clients and faculty consultants and engage in a project in which s/he will act as a statistical consultant for a client with problems that can be handled most of the time with standard statistical techniques (multiple regression, analysis of variance, basic contingency tables, generalized linear models). Students are expected to learn more advanced techniques as required pertaining to the study objectives. The student will prepare a formal report for his/her client.
Sometimes, the student will also explore the literature on non standard issues arising in the course of their work on the project. The student will prepare an 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 MATH6621 3.0F: Linear Models and Regression and MATH6622 3.0: Statistical Techniques.
Ph.D. students in statistics should enrol in this course to fulfil part of their degree requirements (see page 8 under "Breadth Requirement and Compre-hensive Examinations"). The Regular M.A. Programme also includes this course in one of the sets of core course requirements.
Meetings: Every 2nd. Wednesday 1:30-3:30 pm., starting September 18.
Course director :
The aim of this
course is to increase the computing proficiency of the students and
to help them to develop important statistical skills for data analysis.
A cursory overview of programming languages and statistical software
including C, Fortran, S-Plus and SAS will be given at the very beginning
of this course. The focus is on advanced statistical techniques such
as Bootstrap, EM algorithm, Monte Carlo Markov Chain, Fourier Analyses
and Wavelets. Examples and projects will be chosen from a wide variety
of disciplines such as image analysis, data mining and bioinformatics.
Prerequisites: Undergraduate Mathematical Statistics course (such as MATH 3132 3.0) or permission of instructor.
: S. Wang, N625 Ross
3.0W APPLIED STATISTICS II
This course is a continuation of Applied Statistics I. The objective of this course is to familiar students with the application of statistical techniques and methods to data analysis using real life data and statistical software. Examples will be chosen from a wide range of disciplines such as behaviourial science, economics, health and medical science, and social science.
Possible topics include: Multivariate techniques, longitudinal data analysis, sample survey, applied social statistics, mixed models, hierarchical models, multiple imputation, classification and regression trees, biostatistics, survival analysis, time series, advanced experimental design (e.g. clinical trials), advanced nonparametric methods, structural equation models, and genetic statistics.
Prerequisites: Math 6622, Math 6620, or with permission of instructor.
Text: Harrell, Frank E., Jr. (2001), Regression Modeling Strategies, Springer?Verlag, N.Y.
: G. Monette, N626 Ross
3.0F MULTIVARIATE STATISTICS
methods of analysis for data which consist of observations on a number
of variables will be studied.
Math 3131 3.0, Math 3031 3.0, Math 3034 3.0, Math 2022 3.0, or their
equivalent courses which must be approved by the course instructor.
Text: Richard A. Johnson & Dean W. Wichern, Applied Multivariate Statistical Analysis, 5th edition.
Course director: H.
Massam, N618 Ross
This course is intended to provide a systematic presentation of many statistical concepts and techniques for time series analysis. The core topics include time dependence and randomness, trend, seasonality and error, stationary processes, ARMA and ARIMA processes, multivariate time series and state?space models.
: S. Chamberlin, N628 Ross
This course introduces students to statistical methods which are commonly used in medical research and epidemiology. Topics include: 1. Planning of statistical investigations: surveys and clinical trials; 2. Descriptive epidemiology, vital statistics, and risk estimation; 3. Bayes theorem, sensitivity and specificity of medical tests; 4. Bioessay; 5. Analysis of frequencies: association between exposure to risk factors and disease status; 6. Description and comparison of Failure time distributions and estimation of survival functions: parametric and nonparametric approaches.
: P. Ng, 260 Atkinson
3.0F ADVANCED NUMERICAL METHODS
Optimization problems: simplex, conjugate directions, steepest descents, and conjugate gradient methods; matrix eigenvalues: power method, Householder's method and the QR algorithm, singular-value decomposition; approximation methods: least squares, orthogonal polynomials, Chebyshev approximations, Pade approximations, fast Fourier transforms.
The final grade will be based on assignments, tests, a project and a final examination.
Prerequisite: A previous course in Numerical Methods.
3.0W NUMERICAL SOLUTIONS TO PARTIAL DIFFERENTIAL EQUATIONS
Review of partial differential equations; well-posed boundary-value problems; finite-difference approximations of derivatives. Parabolic equations: reduction to dimensionless form; solution by explicit method, Crank-Nicholson method, and hopscotch method. Elliptic equations: review of Jacobi and Gauss-Seidel methods; successive overrelaxation method; non-Cartesian grids; finite-element grids; Bubnov-Galerkin method; direct solution of elliptic boundary-value problems. Hyperbolic equations: linear wave equation; method of characteristics. Alternating-Direction Implicit method; two-dimensional hopscotch method; Incomplete Cholesky-Conjugate Gradient method; finite-volume method. Convergence and stability of solution methods.
Although less elegant than the analytic methods studied in MATH 6269 3.0, Advanced Topics in Analysis, the numerical methods studied here are applicable to a much wider variety of spatial domains, and are therefore of wide use in industrial calculations of heat flows, diffusion, fluid motions, stresses in solids, and electromagnetic fields and waves.
There is no textbook, and the lecture notes are essential.
Useful references are:
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). 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: (a) F.S. Hillier and G.J. Liberman, Introduction to Operations Research, 4th ed. (Holden-Day); (b) H.A. Taha, Operations Research, An Introduction, 4th ed. (MacMillan).
Course director :
This course deals with deterministic and probabilistic models based on optimization. The following topics will be discussed:
1) reliability theory (how to evaluate the lifetime of a system consisting of many interacting subsystems); 2) queuing theory (how to assess what may happen in a system where the customers arrive randomly, wait in line, and then get served); 3) uncertainty (how to measure the amount of uncertainty contained by a probabilistic experiment, with applications in classification, pattern-recognition, and forecasting.) 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: (a) F.S. Hillier and G.J. Liberman, Introduction to Operations Research, 4th ed.
(Holden-Day); (b) H.A. Taha, Operations Research, An Introduction, 4th ed. (MacMillan).
Stochastic Calculus is the mathematical foundation of the theory of no-arbitrage asset pricing, in the continuous-time setting. This course carries the developemnt of the subject to the point that students can understand the mathematics of risk-neutral asset valuation and the Black-Scholes-Merton option pricing formula. No prior exposure to measure theory or stochastic processes is assumed, so probabilistic and analytic background material will be introduced as needed. The prerequisites are calculus and some elementary probability or statistics.
The emphasis of the couse is not on a rigorous derivation of the mathematical theory, but rather on the basic ideas, the pitfalls, and the computational aspects of the subject.
Topics will include: Probability models, sigma-fields and filtrations, Markov chains, stochastic processes, Brownian motion, martingales, stochastic integrals, Ito's formula, stochastic differential equations, hitting probabilities and their relation to partial differential equations, the Black-Scholes-Merton formula, diffusions, Girsanov transformations, risk-neutral valuation, no-arbitrage option pricing, interest rate models.
Text: Michael J. Steele, Stochastic Calculus and Financial Applications, Springer-Verlag, 2001.
Background materials in partial differential equations: classiciations of elliptic, parabolic and hyperbolic equations; examples of exact solutions including the Black-Schole equations and perpetual puts; information flow and propogation for problems in finance. Finite difference methods for parabolic equations; explicit methods; implicit methods, including Backward Euler method and Crank-Nicolson method; issues of stability and convergence; applications to finance, including the effects of boundary conditions, dividends and transaction costs; degeneracy and Asian option, higher order discretization techniques.
Text: P. Wilmott, J. Dewynne, S. Howison, The Mathematics of Financial Derivatives, Cambridge University Press, 1995.
Course director :
6931 3.0W MATHEMATICAL MODELING
The course is designed to develop and improve problem-solving skills for students. Models are derived, simplified, and analyzed using examples from industrial, environmental, biological, and financial applications. The topics include dimensional analysis, asymptotic and perturbation analysis and other basic methods; classical and advanced models in the following areas: heat transfer, fluid mechanics, enzyme kinetics, chemical reactors, groundwater and flow in porous medium, alloy solidification, chemosensory respiratory control, population dynamics, neural networks, data analysis, and option pricing.
Fowler, Mathematical Models in the Applied Sciences, Cambridge University
: J. Wu, N613 Ross
The modeling and numerical practicum which takes place every two weeks in the first two terms will be based on problems from industry or other applications. Each time, a problem will be presented to students in class either by an industrial researcher or a faculty member. The students are required to use the methods they have been learning from Math 6931 (Mathematical Modeling) to derive a reasonable mathematical model, to analyze and solve the model numerically. Students will be encouraged to work in groups. Evaluation will be based on individual reports.
There are no fixed textbook and references. However, students are expected to learn necessary techniques as required pertaining to the study objectives.
: H. Huang, S622 Ross