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Linear Algebra and Geometry.

Terms offered: Summer 2025

Systems of linear equations, matrices, inverses, determinants; geometric vectors in three dimensions, dot product, cross product, lines and planes; introduction to vector spaces, linear dependence and independence, bases. Linear transformations. Eigenvalues and diagonalization.

Calculus 1.

Terms offered: Summer 2025

Review of functions and graphs. Limits, continuity, derivative. Differentiation of elementary functions. Antidifferentiation. Applications.

Calculus 2.

Terms offered: Summer 2025

The definite integral. Techniques of integration. Applications. Introduction to sequences and series.

Foundations of Programming.

Terms offered: Summer 2025

Introduction to computer programming in a high level language: variables, expressions, primitive types, methods, conditionals, loops. Introduction to algorithms, data structures (arrays, strings), modular software design, libraries, file input/output, debugging, exception handling. Selected topics.

Introduction to Software Systems.

Terms offered: this course is not currently offered.

Comprehensive overview of programming in C, use of system calls and libraries, debugging and testing of code; use of developmental tools like make, version control systems.

Introduction to Computer Science.

Terms offered: this course is not currently offered.

Mathematical tools (binary numbers, induction,recurrence relations, asymptotic complexity,establishing correctness of programs). Datastructures (arrays, stacks, queues, linked lists,trees, binary trees, binary search trees, heaps,hash tables). Recursive and non-recursivealgorithms (searching and sorting, tree andgraph traversal). Abstract data types. Objectoriented programming in Java (classes andobjects, interfaces, inheritance). Selected topics.

Algorithms and Data Structures.

Terms offered: this course is not currently offered.

Data Structures: priority queues, balanced binary search trees, hash tables, graphs. Algorithms: topological sort, connected components, shortest paths, minimum spanning trees, bipartite matching, network flows. Algorithm design: greedy, divide and conquer, dynamic programming, randomization. Mathematicaltools: proofs of asymptotic complexity and program correctness, Master theorem.

Introduction to Computer Systems.

Terms offered: this course is not currently offered.

Number representations, combinational and sequential digital circuits, MIPS instructions and architecture datapath and control, caches, virtual memory, interrupts and exceptions, pipelining.

Programming Languages and Paradigms.

Terms offered: this course is not currently offered.

Programming language design issues and programming paradigms. Binding and scoping, parameter passing, lambda abstraction, data abstraction, type checking. Functional and logic programming.

Theory of Computation.

Terms offered: this course is not currently offered.

Finite automata, regular languages, context-free languages, push-down automata, models of computation, computability theory, undecidability, reduction techniques.

Numerical Computing.

Terms offered: this course is not currently offered.

Computer representation of numbers, IEEE Standard for Floating Point Representation, computer arithmetic and rounding errors. Numerical stability. Matrix computations and software systems. Polynomial interpolation. Least-squares approximation. Iterative methods for solving a nonlinear equation. Discretization methods for integration and differential equations.

Algorithm Design.

Terms offered: this course is not currently offered.

Advanced algorithm design and analysis. Linear programming, complexity and NP-completeness, advanced algorithmic techniques.

Calculus 3.

Terms offered: Summer 2025

Taylor series, Taylor's theorem in one and several variables. Review of vector geometry. Partial differentiation, directional derivative. Extreme of functions of 2 or 3 variables. Parametric curves and arc length. Polar and spherical coordinates. Multiple integrals.

Linear Algebra.

Terms offered: this course is not currently offered.

Review of matrix algebra, determinants and systems of linear equations. Vector spaces, linear operators and their matrix representations, orthogonality. Eigenvalues and eigenvectors, diagonalization of Hermitian matrices. Applications.

Algebra 1.

Terms offered: this course is not currently offered.

Sets, functions and relations. Methods of proof. Complex numbers. Divisibility theory for integers and modular arithmetic. Divisibility theory for polynomials. Rings, ideals and quotient rings. Fields and construction of fields from polynomial rings. Groups, subgroups and cosets; homomorphisms and quotient groups.

Algebra 2.

Terms offered: this course is not currently offered.

Linear equations over a field. Introduction to vector spaces. Linear mappings. Matrix representation of linear mappings. Determinants. Eigenvectors and eigenvalues. Diagonalizable operators. Cayley-Hamilton theorem. Bilinear and quadratic forms. Inner product spaces, orthogonal diagonalization of symmetric matrices. Canonical forms.

Analysis 1.

Terms offered: this course is not currently offered.

A rigorous presentation of sequences and of real numbers and basic properties of continuous and differentiable functions on the real line.

Advanced Calculus.

Terms offered: this course is not currently offered.

Derivative as a matrix. Chain rule. Implicit functions. Constrained maxima and minima. Jacobians. Multiple integration. Line and surface integrals. Theorems of Green, Stokes and Gauss. Fourier series with applications.

Numerical Analysis.

Terms offered: this course is not currently offered.

Error analysis. Numerical solutions of equations by iteration. Interpolation. Numerical differentiation and integration. Introduction to numerical solutions of differential equations.

Probability.

Terms offered: Summer 2025

Sample space, events, conditional probability, independence of events, Bayes' Theorem. Basic combinatorial probability, random variables, discrete and continuous univariate and multivariate distributions. Independence of random variables. Inequalities, weak law of large numbers, central limit theorem.

Statistics.

Terms offered: this course is not currently offered.

Sampling distributions, point and interval estimation, hypothesis testing, analysis of variance, contingency tables, nonparametric inference, regression, Bayesian inference.

Applied Regression.

Terms offered: this course is not currently offered.

Multiple regression estimators and their properties. Hypothesis tests and confidence intervals. Analysis of variance. Prediction and prediction intervals. Model diagnostics. Model selection. Introduction to weighted least squares. Basic contingency table analysis. Introduction to logistic and Poisson regression. Applications to experimental and observational data.

Principles of Statistics 2.

Terms offered: this course is not currently offered.

The concept of degrees of freedom and the analysis of variability. Planning of experiments. Experimental designs. Polynomial and multiple regressions. Statistical computer packages (no previous computing experience is needed). General statistical procedures requiring few assumptions about the probability model.

Introduction to Statistical Computing.

Terms offered: this course is not currently offered.

Basic data management. Data visualization. Exploratory data analysis and descriptive statistics. Writing functions. Simulation and parallel computing. Communication data and documenting code for reproducible research.

Fundamentals of Statistical Learning.

Terms offered: this course is not currently offered.

Theory and application of various techniques for the exploration and analysis of multivariate data: principal component analysis, correspondence analysis, and other visualization and dimensionality reduction techniques; supervised and unsupervised learning; linear discriminant analysis, and clustering techniques. Data applications using appropriate software.

Matrix Numerical Analysis.

Terms offered: this course is not currently offered.

An overview of numerical methods for linear algebra applications and their analysis. Problem classes include linear systems, least squares problems and eigenvalue problems.

Discrete Mathematics.

Terms offered: this course is not currently offered.

Discrete Mathematics and applications. Graph Theory: matchings, planarity, and colouring. Discrete probability. Combinatorics: enumeration, combinatorial techniques and proofs.

Honours Discrete Mathematics .

Terms offered: this course is not currently offered.

Discrete mathematics. Graph Theory: matching theory, connectivity, planarity, and colouring; graph minors and extremal graph theory. Combinatorics: combinatorial methods, enumerative and algebraic combinatorics, discrete probability.

Problem Seminar.

Terms offered: this course is not currently offered.

Seminar in Mathematical Problem Solving. The problems considered will be of the type that occur in the Putnam competition and in other similar mathematical competitions.

Majors Project.

Terms offered: this course is not currently offered.

A supervised project.

Statistical Quality Control.

Terms offered: this course is not currently offered.

Introduction to quality management; variability and productivity. Quality measurement: capability analysis, gauge capability studies. Process control: control charts for variables and attributes. Process improvement: factorial designs, fractional replications, response surface methodology, Taguchi methods. Acceptance sampling: operating characteristic curves; single, multiple and sequential acceptance sampling plans for variables and attributes.

Introduction to Stochastic Processes.

Terms offered: this course is not currently offered.

Conditional probability and conditional expectation, generating functions. Branching processes and random walk. Markov chains, transition matrices, classification of states, ergodic theorem, examples. Birth and death processes, queueing theory.

Generalized Linear Models.

Terms offered: this course is not currently offered.

Exponential families, link functions. Inference and parameter estimation for generalized linear models; model selection using analysis of deviance. Residuals. Contingency table analysis, logistic regression, multinomial regression, Poisson regression, log-linear models. Multinomial models. Overdispersion and Quasilikelihood. Applications to experimental and observational data.

Nonparametric Statistics.

Terms offered: this course is not currently offered.

Distribution free procedures for 2-sample problem: Wilcoxon rank sum, Siegel-Tukey, Smirnov tests. Shift model: power and estimation. Single sample procedures: Sign, Wilcoxon signed rank tests. Nonparametric ANOVA: Kruskal-Wallis, Friedman tests. Association: Spearman's rank correlation, Kendall's tau. Goodness of fit: Pearson's chi-square, likelihood ratio, Kolmogorov-Smirnov tests. Statistical software packages used.

Sampling Theory and Applications.

Terms offered: this course is not currently offered.

Simple random sampling, domains, ratio and regression estimators, superpopulation models, stratified sampling, optimal stratification, cluster sampling, sampling with unequal probabilities, multistage sampling, complex surveys, nonresponse.

Statistical Data Science Practicum.

Terms offered: this course is not currently offered.

The holistic skills required for doing statistical data science in practice. Data science life cycle from a statistics-centric perspective and from the perspective of a statistician working in the larger data science environment. Group-based projects with industry, government, or university partners. Statistical collaboration and consulting conducted in coordination with the Data Science Solutions Hub (DaS^2H) of the Computational and Data Systems Initiative (CDSI).

Statistical Data Science Practicum.

Terms offered: this course is not currently offered.

See MATH 527D1 for course description.

Introduction to Time Series Analysis.

Terms offered: this course is not currently offered.

Stationary processes; estimation and forecasting of ARMA models; non-stationary and seasonal models; state-space models; financial time series models; multivariate time series models; introduction to spectral analysis; long memory models.

Design of Experiments.

Terms offered: this course is not currently offered.

Introduction to concepts in statistically designed experiments. Randomization and replication. Completely randomized designs. Simple linear model and analysis of variance. Introduction to blocking. Orthogonal block designs. Models and analysis for block designs. Factorial designs and their analysis. Row-column designs. Latin squares. Model and analysis for fixed row and column effects. Split-plot designs, model and analysis. Relations and operations on factors. Orthogonal factors. Orthogonal decomposition. Orthogonal plot structures. Hasse diagrams. Applications to real data and ethical issues.

Bayesian Theory and Methods.

Terms offered: this course is not currently offered.

Subjective probability, Bayesian statistical inference and decision making, de Finetti’s representation. Bayesian parametric methods, optimal decisions, conjugate models, methods of prior specification and elicitation, approximation methods. Hierarchical models. Computational approaches to inference, Markov chain Monte Carlo methods, Metropolis—Hastings. Nonparametric Bayesian inference.

Numerical Analysis 1.

Terms offered: this course is not currently offered.

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.

Topics in Probability and Statistics.

Terms offered: this course is not currently offered.

This course covers a topic in probability and/or statistics.

Artificial Intelligence.

Terms offered: this course is not currently offered.

Introduction to search methods. Knowledge representation using logic and probability. Planning and decision making under uncertainty. Introduction to machine learning.

Computational Biology Methods.

Terms offered: this course is not currently offered.

Application of computer science techniques to problems arising in biology and medicine, techniques for modeling evolution, aligning molecular sequences, predicting structure of a molecule and other problems from computational biology.

Matrix Computations.

Terms offered: this course is not currently offered.

Designing and programming reliable numerical algorithms. Stability of algorithms and condition of problems. Reliable and efficient algorithms for solution of equations, linear least squares problems, the singular value decomposition, the eigenproblem and related problems. Perturbation analysis of problems. Algorithms for structured matrices.

Cryptography and Data Security.

Terms offered: this course is not currently offered.

This course presents an in-depth study of modern cryptography and data security. The basic information theoretic and computational properties of classical and modern cryptographic systems are presented, followed by a cryptanalytic examination of several important systems. We will study the applications of cryptography to the security of systems.

Applied Machine Learning.

Terms offered: this course is not currently offered.

Selected topics in machine learning and data mining, including clustering, neural networks, support vector machines, decision trees. Methods include feature selection and dimensionality reduction, error estimation and empirical validation, algorithm design and parallelization, and handling of large data sets. Emphasis on good methods and practices for deployment of real systems.

Advanced Computational Biology Methods and Research.

Terms offered: this course is not currently offered.

Fundamental concepts and techniques in computational structural biology, system biology. Techniques include dynamic programming algorithms for RNA structure analysis, molecular dynamics and machine learning techniques for protein structure prediction, and graphical models for gene regulatory and protein-protein interaction networks analysis. Practical sessions with state-of-the-art software.

Discrete Optimization 1.

Terms offered: this course is not currently offered.

Use of computer in solving problems in discrete optimization. Linear programming and extensions. Network simplex method. Applications of linear programming. Vertex enumeration. Geometry of linear programming. Implementation issues and robustness. Students will do a project on an application of their choice.

Discrete Optimization 2.

Terms offered: this course is not currently offered.

Formulation, solution and applications of integer programs. Branch and bound, cutting plane, and column generation algorithms. Combinatorial optimization. Polyhedral methods. A large emphasis will be placed on modelling. Students will select and present a case study of an application of integer programming in an area of their choice.