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4000-level Courses

A project of a pure or applied nature in mathematics or statistics under the supervision of a faculty member. The project allows the student to apply mathematical or statistical knowledge to problems of current interest. A report is required at the conclusion of the project. In order to satisfy the prerequisites for this course, students taking MATH 4000 must:

  • have a cumulative GPA of B (6.00) or higher
  • have completed of all Core courses
  • have no more than 6.00 credits in total from MATH 4000/4300
  • have successfully completed 24 credits in the Department of Mathematics and Statistics.

The student works under the supervision of a faculty member who is selected by the student.

  • The student and the faculty member must agree on a written description of the course, its content, and its method of evaluation before enrolment in the course.
  • The supervising faculty member must submit this description for approval by the Curriculum Coordinating Committee.
  • Copies must be deposited with the Curriculum Coordinating Committee, and the student and faculty member should each retain a copy.
  • The student must also supply the supervising faculty member with a transcript to be submitted, by the faculty member, to the Curriculum Coordinating Committee to verify the GPA and completed courses requirement.
  • The amount of work expected of the student is approximately ten hours per week for the duration of the course. This is the equivalent of a standard full-year (MATH 4000 6.0) or half-year (MATH 4000 3.0) course.
  • The supervisor is expected to spend about one or two hours per week with the student averaged over the duration of the project.
  • In addition to the  final report, regular short written progress reports will be expected from the student at de nite times during the course.
  • The final grade will be based upon the  nal report as well as the interim progress reports.

Please go to "Need Advising?" for the contact information of our program coordinators.

Metric spaces, norms, metric topology, continuity, connectedness, completeness, Baire category, compactness of metric spaces, Stone-Weierstrass Theorem, Heine-Borel Theorem, Banach Contractive Mapping Theorem, Hilbert spaces.

Prerequisites: SC/MATH 3001 3.00, SC/MATH 2022 3.00.

Groups, Homomorphisms, Subgroups, Normal subgroups and internal equivalence relations, Free groups, Abelian groups, Product of groups, Cayley's theorem, The fundamental theorem of finitely generated Abelian groups, Sylow theorems, Application of group theory in topology. We aim to look at familiar concepts of group theory from the perspective of category theory and universal algebra.

References:

Prerequisite: SC/MATH 3020 6.00 or SC/MATH 3022 3.00 or permission of the course coordinator.

Course credit exclusions: SC/MATH 4020 6.00, SC/MATH 4241 3.00.

Note: This course is a prerequisite for SC/MATH 4022 3.00.

Crosslisted to: LE/ESSE 4020 3.00 and SC/PHYS 4060 3.00.

Treatment of discrete sampled data involving correlation, convolution, spectral density estimation, frequency, domain filtering, and Fast Fourier Transforms.

Prerequisites: LE/EECS 1011 3.00 or equivalent programming experience; SC/MATH 2015 3.00; SC/MATH 2271 3.00.

Course credit exclusions: LE/CSE 3451 4.00, SC/CSE 3451 4.00 LE/CSE 3451 3.00, SC/CSE 3451 3.00, SC/MATH 4130B 3.00, SC/MATH 4930C 3.00.

This course is an introduction to the theory of general topological spaces.  A topological space is a set together with a collection of subsets which describe which points in the space are close together.  Given a topological space, one can generalize the notion of continuous function as seen in previous analysis courses and thereby develop a richer function theory. From graph theory to functional analysis, topological methods illuminate a number of diverse areas of modern mathematics.             

In this course, we will cover the most fundamental topics in general topology, such as metric spaces, compactness, connectedness (which yields the Intermediate Value Theorem), separation and countability axioms, compactifications, Baire Category Theorem, Stone-Weierstrass Theorem, Tietze Extension Theorem, Tychonoff Theorem, nets and filters, and metrization theorems. 

Prerequisites: SC/MATH 3210 3.00 or SC/MATH 3001 3.00 or permission of the course coordinator.

This course will introduce the student to traditional and newer methods of mathematical modelling. The topics include population models, service systems stochastic modeling (single server and queueing networks), kinetic reactions simulations, methods in data-based model identification and sensitivity analysis, and elements of game theory. These subjects will be studied analytically and computationally.

Prerequisites: Registration in an Honours Program in Mathematics and Statistics and the completion of all specified core courses in that Program as well as SC/MATH 3241 3.00 and SC/MATH 3271 3.00.

This course provides opportunities for students to examine in-depth specific ideas in mathematics as well as themes and theories in mathematics education. The main focus will be on exploring different ways to unpack, repack and communicate concepts in mathematics, and to think critically and reflectively about how mathematics can be learnt, taught, and understood. Students will be en­couraged to work with multiple representations and ap­proaches and reflect on how peers also do the mathematics. We will look at sample concepts from a wide area of mathematics including both pure mathematics and applied mathematics, as well as concepts which are central to the Ontario curriculum. The course is designed as a ‘capstone’ course for students preparing to become teachers, but is relevant to anyone interested in reflecting on the learning of mathematics. We recommend you take the course in your final semester.

Prerequisites: A minimum of 21 credits in SC/MATH courses without second digit "5"; permission of the course coordinator.

Crosslisted to: SC/PHYS 4120 3.00.

This course treats basic continuum mechanics and the fundamental laws of fluid motion; inviscid incompressible flows and potential theory; free surface flows, water waves and perturbation methods; dimensional analysis, Navier-Stokes equations, viscous flow and boundary layer theory; gas dynamics, compressible flows and shock waves. Applications from aerodynamics, geophysics and atmospheric science are discussed.

Prerequisites: SC/PHYS 2010 3.00 or LE/ESSE 2470 3.00; SC/MATH 2271 3.00.

Integrated with GS/MATH 6633 3.00.

In this course, we will study many statistical techniques for the analysis of time series data. The core topics include time dependence and randomness, trend, seasonality and error, stationary processes, ARMA and ARIMA processes, multivariate time series models and state-space models.

Prerequisites: either SC/MATH 3033 3.00 or SC/MATH 3330 3.00; SC/MATH 3131 3.00; or permission of the course coordinator.

Course credit exclusions: LE/CSE 3451 4.00, LE/EATS 4020 3.00, SC/MATH 4830 3.00, SC/PHYS 4060 3.00, SC/PHYS 4250 3.00.

Integrated with GS/MATH 6641 3.00.

This course provides students with an introduction to the statistical methods for analyzing censored data which are common in medical research, industrial life-testing and related fields. Topics include accelerated life models, proportional hazards model, time dependent covariates. Computer/Internet use is essential for course work.

Prerequisites: SC/MATH 3131 3.00; either SC/MATH 3033 3.00 or SC/MATH 3330 3.00.

Integrated with GS/MATH 6651 3.00 and GS/PHYS 5070A 3.00.

Numerical methods for solving ordinary differential equations; optimization problems: steepest descents, conjugate gradient methods; approximation theory: least squares, orthogonal polynomials, Chebyshev and Fourier approximation, Padé approximation.

Prerequisite: SC/MATH 2270 3.00; LE/CSE 3122 3.00 or SC/MATH 3242 3.00.

This course introduces the basic concepts and numerical methods in computational finance. The topics include an introduction to mathematical finance, basics in numerical computations; option pricing and risk management by lattice methods and Monte-Carlo simulations.

We will use MATLAB to carry out the numerical computations and illustrations in the class.

The final grade will be based on assignments, a mid-term, a group term project and a final exam.

Prerequisites: One of SC/MATH 2015 3.00 or SC/MATH 2310 3.00; SC/MATH 1131 3.00; SC/MATH 2030 3.00; One of LE/CSE 1530 3.00, LE/CSE 1540 3.00 or SC/MATH 2041 3.00.

Textbook:

  • Options, Futures, and Other Derivatives by J. C. Hull.
  • Optional: However, you may find its student solution manual quite useful for the course and for future references: Student Solutions Manual for Options, Futures, and Other Derivatives by J. C. Hull 2017.

We learn how to count in this course. Methods used to enumerate finite sets include bijections, the principle of inclusion-exclusion, generating functions, recurrence relations and Polya’s theory. We will apply these methods to study the occupancy problem, derangements, partitions of integers, Catalan numbers, and simple graphs.

Prerequisites: SC/MATH 2022 3.00 or SC/MATH 2222 3.00; six credits from 3000-level mathematics courses without second digit 5; or permission of the course coordinator.

Crosslisted to: LE/EECS 4161 3.00.

In cryptography, our objective is to keep information secret from everyone except for those who are authorized to see it. We will start with classical codes such as Caesar shift and Vigenere. We will introduce symmetric key encryption and public key encryption, with examples like DES and RSA, respectively. We will learn the background in probability theory, information theory and number theory needed for the analysis of these cryptographic systems. Other topics include digital signature, message authentication and hash function.

Prerequisites: At least 12 credits from 2000-level (or higher) MATH courses (without second digit "5", or second digit "7"); or LE/EECS 3101 3.00 or permission of the Instructor.

The course introduces the fundamentals of convex analysis and its roles in modern optimization. We study the KKT theorem and LICQ constraint qualification in depth. Iterative optimization algorithms, including Lagrangian method, are introduced.

AMPL is a user friendly programming language, which makes optimization modelling much easier, and is used widely in industries and research institutes. We will learn and use AMPL extensively in the course. Selected optimization models, including the financial portfolio optimization, renewable energy power optimization, transportation scheduling, etc., will be used as examples and assignments throughout the course. Previous programming experience is not required.

Prerequisites: SC/MATH 2015 3.00 or SC/MATH 2310 3.00; SC/MATH 1021 3.00 or SC/MATH 1025 3.00 or SC/MATH 2221 3.00.

Course credit exclusion: SC/MATH 4170 6.00.

Textbook:

  • AMPL: A modelling Language for Mathematical Programming by R. Fourier, D. M. Gay and B. W. Kernighan. The free PDF version is available at: http://ampl.com/resources/the‑ampl‑book/;
  • Operations Research: Applications and Algorithms by W. L. Winston.

This course introduces the theory and applications of the following operations research decision models: Decision Tree Analysis, Game Theory, Inventory Models, and Dynamic Programming.

Prerequisites: SC/MATH 2015 3.00 or SC/MATH 2310 3.00; SC/MATH 2030 3.00; CSE 1560 3.00 or equivalent.

Course credit exclusion: SC/MATH 4170 6.0

The course offers an introduction to dynamical systems at an advanced undergraduate level. Upon completing the course material, students will be prepared to use analytical and qualitative techniques of dynamical systems and bi­furcations widely applicable in analysis and in ap­plications.

Prerequisites: SC/MATH 2270 3.00; SC/MATH 1021 3.00 or SC/MATH 2221 3.00 or SC/MATH 1025 3.00.

This course, together with MATH 4281 3.00, is part of the risk theory sequence of the BA Honours Math for Commerce Actuarial Stream Program. Students who complete this course, along with the sequence MATH 4281 3.00, MATH 3131 3.00, MATH 3132 3.00 and MATH 4430 3.00 (or MATH 4431 3.00) should be adequately prepared to pass the Society of Actuaries Exam C.

This course focuses on mathematical modelling and analysis of the way that funds flow out of an insurance system due to the payment of insurance benefits. The main topics of the course include (a) loss models: severity, frequency, and aggregate models of loss, and (b) risk measures: Value-at-Risk and Tail-Value at Risk, deductibles, coinsurance, limits.

Prerequisite: SC/MATH 2131 3.00.

This course, together with MATH 4280 3.00, is part of the risk theory sequence of the BA Specialized Honours Math for Commerce Actuarial Stream Program. Students who complete this course, along with the sequence MATH 4280 3.00, MATH 3131 3.00, MATH 3132 3.00 and MATH 4430 3.00 (or MATH 4431 3.00) should be adequately prepared to pass the Society of Actuaries Exam C.

The course focuses on the mathematical analysis of models for the probability that an insurer's claims will be so severe as to cause ruin (ie insolvency), as well as on measuring the credibility of the data on which prices and forecasts for insurance are based. The main topics of the course include (a) Ruin Theory: probability of ruin at finite and infinite horizons, adjustment coefficient, Lundberg’s inequality, Cramer’s asymptotic ruin; (b) Credibility Theory: limited fluctuation credibility theory, greatest accuracy credibility theory, Buhlmann and Buhlmann-Straub models.

Prerequisite: SC/MATH 2131 3.00.

A student may arrange to do independent study with a member of the Mathematics and Statistics Department. Such an arrangement must have the approval of the Curriculum Coordinating Committee. One term: 3 credits. Two terms: 6 credits.

  1. Students may wish to pursue intensive work with a particular faculty member on a topic of study not offered in a particular academic session.
  2. Students taking MATH 4300 must:
    • have a cumulative GPA of B (6.00) or higher
    • have completed of all Core courses
    • have no more than 6.00 credits in total from MATH 4000/4300
    • have successfully completed 24 credits in the Department of Mathematics andStatistics.
  3. The student and the faculty member must agree on a written description of the course, its content, and its method of evaluation at the time of enrolment in the course.
  4. The supervising faculty member must submit this description for approval by the Curriculum Coordinating Committee.
  5. Copies must be deposited with the Curriculum Coordinating Committee, and the student and faculty member should each retain a copy.
  6. The student must also supply the supervising faculty member with a transcript to be submitted, by the faculty member, to the Curriculum Coordinating Committee to verify the GPA and completed courses requirement.

Please go to "Need Advising?" for the contact information of our program coordinators.

One of the major goals of this course is to study the important extensions of the ideas of linear regression, as seen in MATH 3330, to cases in which the response variable is categorical or integer valued. We also consider models for multi­variate categorical responses.

Prerequisite: SC/MATH 3131 3.00; SC/MATH 3330 3.00.

Course credit exclusion: SC/MATH 3034 3.00.

Note: This course is a prerequisite for SC/MATH 4939 3.00.

The course covers selected mathematical topics from ancient times up to the 20th century. The relationship of mathematical work to historical context and institutional structures will be explored, as will the impact of mathematical discovery on society and culture at various points in history. In addition to lectures on particular mathematical results and methods, students will be guided in research methods, and in oral and written presentation skills.

The grade will be determined by a combination of assignments, tests, written projects and oral presentations.

Note: 36 credits required from mathematics courses without second digit 5, including at least 12 credits at or above the 3000 level. (Twelve of the 36 credits may be taken as co-requisites.).

Integrated with GS/MATH 6602 3.00.

This course begins by reviewing conditional expectations and other key topics from probability theory. We then discuss counting processes and discrete time Markov chains. We consider the classification of states, first step analysis, invariant measures, first passage times, as well as applications in science and business. Moving to continuous time, we consider diffusion processes and Brownian motion. We will treat both analytical results and stochastic simulation, the latter using the R programming language.

Prerequisite: SC/MATH 2030 3.00.

Integrated with GS/Math 6604 3.00.

Probability theory has been used to describe and analyze many kinds of real-world phenomena. This course will begin with a review and introduction to techniques of conditional expectation and random variables and analytical methods in probability-generating functions and Laplace transforms. The course will focus on techniques of ab-initio derivation of probabilistic models, passages between low-copy number models and large-copy number stochastic approximations, and computational and algorithmic interfaces with statistical inference and machine learning approaches. Models from a variety of disciplines in physics and the life sciences and in business and engineering will be considered and analyzed in project work, drawing on studies in the original scientific literature.

Prerequisite: SC/MATH 2030 3.00.

Integrated with GS/MATH 6632 3.00.

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 standard multivariate inference theory. Sufficient theory will be developed to facilitate an understanding of the main ideas. This will necessitate a good background in matrix algebra, and some knowledge of vector spaces as well. Computers will be used extensively, and familiarity with elementary use of SAS will be assumed. Topics covered will include multivariate normal population, inference about means and linear models, principal component analysis, canonical correlation analysis, and some discussion of discriminate analysis, and factor analysis and cluster analysis, if term permits.

Prerequisites: SC/MATH 3131 3.00; SC/MATH 3330 3.00; SC/MATH 2022 3.00 or SC/MATH 2222 3.00.

Crosslisted to: LE/ENG 4650 3.00.

Control theory stems from the fundamental idea of modifying a dynamical system to achieve a desired goal. From this, many questions arise such as modeling a tractable control system, designing an appropriate control objective, finding an optimal control, and testing the performance and robustness of the controlled system, which lends to a rich course in control theory. This course is an introduction to control theory from a mathematical and engineering perspective. In fact, this course is cross-listed with Lassonde/ESSE so the class demographic will consist of a mixture of math students and engineering students. Topics include representation and solutions of control systems, defining controllability, stability analysis of a controlled system, controller design and optimal control. Applications are addressed throughout the course including time spent in a laboratory setting. Knowledge of complex analysis, ordinary differential equations and linear algebra is needed. Knowledge of MATLAB is an asset.

Prerequisites: LE/ENG 4550 or the following combination of courses: SC/MATH 3410 3.00; SC/MATH 2270 3.00 or SC/MATH 2271 3.00; SC/MATH 2022 3.00.

Experimental design is the process of planning an experiment so that appropriate data will be collected which may be analysed by statistical methods, resulting in valid and meaningful conclusions. This includes the choice of treatments, the required sample size, the random al­loca­tion of experimental units to treatments, the method of estimation, and a consideration of how the data will be analyzed once collected.

We will study various experimental situations in this course, considering how the principles of design can be applied to each to create a design that is appropriate to the objectives of the experiment. We will examine ap­propriate procedures for the analysis of the resulting data, including the underlying assumptions and limita­tions of the procedures. Students will use the statistical software SAS for data analysis.

The final grade will be based on two midterm exams, a final exam, and a presentation.

Prerequisite: SC/MATH 3330 3.00, or permission of the course coordinator.

Note: This course is a prerequisite for SC/MATH 4939 3.00.

This course concentrates on the statistical aspects of analyzing complex sample surveys obtained by using the basic sampling designs of simple random sampling, stratification, and cluster sampling with equal and unequal probabilities of selection. The use of sampling weights and design effects will be discussed as well as what to do if there is nonresponse. Several methods for estimating variances of various statistics will be described as well as how to perform chi-squared tests and regression analyses using data from complex surveys.

Prerequisites: SC/MATH 3430 3.00 or permission of the course director.

Textbook: Sampling: Design and Analysis (2nd Ed.) by S.L. Lohr.

This course provides a comprehensive coverage of the modern practice of statistical quality control from basic principles to state‑of‑the‑art concepts and applications.

Prerequisite: SC/MATH 3330 3.00.

Co-requisite: SC/MATH 4730 3.00.

The term “Monte Carlo” refers to a broad class of numerical algorithms which rely on repeated random sampling. Since its beginnings in the late 1940s at the Los Alamos National Laboratory, Monte Carlo has continued to gain in importance in scientific use. The continued growth of computing power coupled with a drastic decrease in price in the last decade, means that Monte Carlo methods are now more practical than ever.

In this course, we will discuss what Monte Carlo methods are, and we will look at their varied applications. The three main topics we will cover are (a) random number generation, (b) “basic” Monte Carlo integration, and (c) bootstrap methodology. Markov chain Monte Carlo will be covered if time permits. Applications will be taken from various sciences including statistics, operations research, and actuarial science.

A significant portion of this course will be spent in the computer lab, using the statistical software R to perform Monte Carlo simulations. Previous experience with computing will be an asset, but is not required.

Prerequisite: SC/MATH 3330 3.00 and LE/EECS 1560 3.00 and SC/MATH 2030.

Course credit exclusion: LE/SC/EECS 3408 3.00, SC/MATH 4930B 3.00.

Textbook: Introducing Monte Carlo Methods with R by C.P. Robert and G. Gasella (available in electronic form from York’s library system).

This is a capstone course in statistics, the course where everything comes together. You will use all the concepts and skills you have learned in previous courses to solve real problems using real data — where the solution can’t be found in the previous chapter of the text but uses everything you have learned — and more.

We will learn:

  • a lot of R and some SAS and how to combine R and SAS to wrangle real data that always come in more complex forms than simple data sets,
  • how the real meaning of statistical results depends on so many factors beyond the analysis and the data itself and we will learn principles we use to find accurate and meaningful interpretations of our results,
  • how all the assumptions that underlie statistical methods are never satisfied with real data but understanding those assumptions and developing good judgment about the degree to which they may be violated and the implications for interpretations is vital,
  • in addition to basic R and SAS, we will learn about additional essential tools, like regular expressions, SQL, and packages like ‘dplyr’.

Team projects and assignments are an important component of this course. Attendance and punctuality are required.

Prerequisites: MATH 3131 3.00, MATH 3330 3.00 and MATH 4330 3.00 (these prerequisites are strictly enforced).