Mathematics Liberal Program - Core Science Component (B.Sc.) at McGill University

Entry requirements

Students entering the Core Science Component in Mathematics are normally expected to have completed the courses below or their equivalents. Otherwise, they will be required to make up any deficiencies in these courses over and above the 45 credits required for the program. 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. Review of functions and graphs. Limits, continuity, derivative. Differentiation of elementary functions. Antidifferentiation. Applications. The definite integral. Techniques of integration. Applications. Introduction to sequences and series. Guidelines for Selection of Courses The following informal guidelines should be discussed with the student's adviser. Where appropriate, Honours courses may be substituted for equivalent Major courses. Students planning to pursue graduate studies are encouraged to make such substitutions. Students interested in computer science are advised to choose courses from the following and to complete the Computer Science Minor: Error analysis. Numerical solutions of equations by iteration. Interpolation. Numerical differentiation and integration. Introduction to numerical solutions of differential equations. Propositional logic: truth-tables, formal proof systems, completeness and compactness theorems, Boolean algebras; first-order logic: formal proofs, Gödel's completeness theorem; axiomatic theories; set theory; Cantor's theorem, axiom of choice and Zorn's lemma, Peano arithmetic; Gödel's incompleteness theorem. An overview of numerical methods for linear algebra applications and their analysis. Problem classes include linear systems, least squares problems and eigenvalue problems. Transformation groups of the plane. Inversions and Moebius transformations. The hyperbolic plane. Tilings in dimension 2 and 3. Group presentations and Cayley graphs. Free groups and Schreier's theorem. Coxeter groups. Dehn's Word and Conjugacy Problems. Undecidability of the Word Problem for semigroups. Regular languages and automatic groups. Automaticity of Coxeter groups. Discrete Mathematics and applications. Graph Theory: matchings, planarity, and colouring. Discrete probability. Combinatorics: enumeration, combinatorial techniques and proofs. An introduction to linear optimization and its applications: Duality theory, fundamental theorem, sensitivity analysis, convexity, simplex algorithm, interior-point methods, quadratic optimization, applications in game theory. Students interested in probability and statistics are advised to take 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. Sampling distributions, including Normal, Chi-squared, Student’s t, and F distributions. Sampling properties of the sample mean and variance. Point estimation: bias, variance, mean squared error, consistency, sufficiency, UMVUE. Method of moments and maximum likelihood estimation. One- and two-sample inference. Confidence intervals and sample size determination. Hypothesis testing, Type 1 and Type 2 errors, power, and p-values. The Neyman Pearson framework and likelihood ratio tests. Linear models used as illustrative examples, linking them to topics in estimation and hypothesis testing. One-way analysis of variance (ANOVA) and contingency tables. 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. 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. 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. 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. Students interested in applied mathematics should take Error analysis. Numerical solutions of equations by iteration. Interpolation. Numerical differentiation and integration. Introduction to numerical solutions of differential equations. MATH 319Partial Differential First order equations, geometric theory; second order equations, classification; Laplace, wave and heat equations, Sturm-Liouville theory, Fourier series, boundary and initial value problems. Sampling distributions, including Normal, Chi-squared, Student’s t, and F distributions. Sampling properties of the sample mean and variance. Point estimation: bias, variance, mean squared error, consistency, sufficiency, UMVUE. Method of moments and maximum likelihood estimation. One- and two-sample inference. Confidence intervals and sample size determination. Hypothesis testing, Type 1 and Type 2 errors, power, and p-values. The Neyman Pearson framework and likelihood ratio tests. Linear models used as illustrative examples, linking them to topics in estimation and hypothesis testing. One-way analysis of variance (ANOVA) and contingency tables. Linear systems of differential equations, linear stability theory. Nonlinear systems: existence and uniqueness, numerical methods, one and two dimensional flows, phase space, limit cycles, Poincare-Bendixson theorem, bifurcations, Hopf bifurcation, the Lorenz equations and chaos. An overview of numerical methods for linear algebra applications and their analysis. Problem classes include linear systems, least squares problems and eigenvalue problems. An introduction to linear optimization and its applications: Duality theory, fundamental theorem, sensitivity analysis, convexity, simplex algorithm, interior-point methods, quadratic optimization, applications in game theory. Students considering a career in secondary school teaching are advised to take Propositional logic: truth-tables, formal proof systems, completeness and compactness theorems, Boolean algebras; first-order logic: formal proofs, Gödel's completeness theorem; axiomatic theories; set theory; Cantor's theorem, axiom of choice and Zorn's lemma, Peano arithmetic; Gödel's incompleteness theorem. Egyptian, Babylonian, Greek, Indian and Arab contributions to mathematics are studied together with some modern developments they give rise to, for example, the problem of trisecting the angle. European mathematics from the Renaissance to the 18th century is discussed, culminating in the discovery of the infinitesimal and integral calculus by Newton and Leibnitz. Demonstration of how mathematics was done in past centuries, and involves the practice of mathematics, including detailed calculations, arguments based on geometric reasoning, and proofs. Divisibility. Congruences. Quadratic reciprocity. Diophantine equations. Arithmetical functions. Points and lines in a triangle. Quadrilaterals. Angles in a circle. Circumscribed and inscribed circles. Congruent and similar triangles. Area. Power of a point with respect to a circle. Ceva’s theorem. Isometries. Homothety. Inversion. Students interested in careers in business, industry or government are advised to select courses from the following list: Error analysis. Numerical solutions of equations by iteration. Interpolation. Numerical differentiation and integration. Introduction to numerical solutions of differential equations. MATH 319Partial Differential First order equations, geometric theory; second order equations, classification; Laplace, wave and heat equations, Sturm-Liouville theory, Fourier series, boundary and initial value problems. An overview of numerical methods for linear algebra applications and their analysis. Problem classes include linear systems, least squares problems and eigenvalue problems. Simple and compound interest, annuities certain, amortization schedules, bonds, depreciation. An introduction to linear optimization and its applications: Duality theory, fundamental theorem, sensitivity analysis, convexity, simplex algorithm, interior-point methods, quadratic optimization, applications in game theory. 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. Introduction to concepts of price and hedge derivative securities. The following concepts will be studied in both concrete and continuous time: filtrations, martingales, the change of measure technique, hedging, pricing, absence of arbitrage opportunities and the Fundamental Theorem of Asset Pricing. 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. 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. 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. Prerequisite courses: MATH 133L inear Algebra and Geometry; MATH 140C alculus 1; MATH 141C alculus 2; MATH 317N umerical Analysis; MATH 318M athematical Logic; MATH 327M atrix Numerical Analysis; MATH 335G roups, Tilings and Algorithms; MATH 417L inear Optimization; MATH 204P rinciples of Statistics 2; MATH 324S tatistics; MATH 423A pplied Regression; MATH 447I ntroduction to Stochastic Processes.