Course Learning Outcomes:

1. Understand the fundamental ideas of linear algebra, including linear systems, vector spaces, matrices, linear transformations, eigenvalues and eigenvectors, diagonalization, orthogonality, and diverse applications.
2. Give rigorous mathematical proofs of basic theorems.
3. Solve concrete problems in linear system, giving algorithmic solutions.

Course Learning Outcomes:

1. Apply the above skills to complex problems (e.g., error correcting codes, dynamical systems, games on graphs and probability).
2. Compute eigenvalues and eigenvectors and understand their utility.
3. Manipulate matrix equations and compute their determinants and inverses.
4. Solve systems of linear equations and visualize the related geometry.
5. Visualize and express algebraically the geometry of lines and planes.
6. Work with linear and affine transformations and relate them to matrices.

Course Learning Outcomes:

1. Use the ideas in the course fluently. Indicators of fluency include: using the ideas in a new situation; using the ideas in a different order or manner than they have been presented; constructing minor extensions and variations of the ideas.
2. Write clear, clean, and well-reasoned mathematical arguments.
3. Understand the standards for such arguments.
4. Work through and solve more difficult problems, particularly those which may seem confusing at first and require time to digest and understand.
5. Demonstrate mastery of the underlying concepts of the course: limits, continuity, differentiation, integration, convergence.
6. Compute limits, derivatives, integrals, and infinite sums.

Course Learning Outcomes:

1. Apply differential and integral calculus procedures and techniques in technical problems.
2. Extend the one-variable analysis to multi-variable functions.
3. Work with a variety of standard models and applications, using the tools of calculus to gain new understanding.
4. Communicate results in writing using appropriate mathematical format and notation.

Course Learning Outcomes:

1. Understand and perform the basic techniques of differential and integral calculus.
2. Apply these techniques to solve problems in the areas of biology, physics, chemistry, economics, and social sciences.
3. Solve basic problems in differential equations, multivariable differential calculus, and sequences and series.

Course Learning Outcomes:

1. Apply differential and integral calculus concepts and procedures in technical problems.
2. Extend the one-variable analysis to multi-variable functions.
3. Work with a variety of standard models and applications, using the tools of calculus to gain new understanding.
4. Communicate results in writing using appropriate mathematical format and notation.

Course Learning Outcomes:

1. Use the ideas and methods of the differential and integral calculus to study living systems.
2. Work with the basic functions of differential calculus, in particular the exponential and logarithm functions, and solve optimization problems.
3. Work with the fundamental processes of integral calculus and differential equations.
4. Work with multivariable functions, contour diagrams and phase-plane analysis.
5. Work with probabilistic processes, and random variables in both discrete and continuous spaces, involving independent events and conditional probability.

Course Learning Outcomes:

1. Understand curriculum in terms of content, pedagogy and structure.
2. Work with the Ontario math curriculum expectations in Grades 7-10.
3. Experience teaching as guided play.
4. Work with the existing literature in terms of curriculum design.
5. Experience the power and beauty of mathematics.

Course Learning Outcomes:

1. Perform accurate and efficient computations with integers and polynomials involving quotients, remainder, divisibility, greatest common divisors, primality, irreducibility, and factorization.
2. Define and illustrate basic concepts in ring theory using examples and counterexamples.
3. Describe and demonstrate an understanding of equivalence classes, ideals, quotient rings, ring homomorphisms, and some standard isomorphisms.
4. Recognize and explain a hierarchy of rings that includes commutative rings, unique factorization domains, principal ideal domains, Euclidean domains, and fields.
5. Write rigorous solutions to problems and clear proofs of theorems.

Course Learning Outcomes:

1. Computing eigenvalues and eigenvectors of endomorphisms.
2. Computing Jordan Normal Forms of endomorphisms.
3. Computing Taylor and Laurent expansions of complex analytic functions.
4. Proving properties of homomorphisms between vector spaces, as well as properties of eigenvalues and eigenvectors of endomorphisms.
5. Working with the properties of inner-products and Hilbert spaces.

Course Learning Outcomes:

1. Calculate and work with gradients, divergence and curl, and spherical and cylindrical coordinates.
2. Integrate vector functions and evaluate path integrals, surface integrals and volume integrals.
3. Perform calculations using Green's and Stokes' Theorems, and the Divergence Theorem.
4. Work with multi-variable functions and vectors in dimensions 2 and 3.

Course Learning Outcomes:

1. Model a mass-spring-damper system or RLC circuit using differential equations.
2. Model interconnected fluid reservoirs using differential equations.
3. Solve basic initial value problems.
4. Solve linear constant coefficient differential equations.
5. Use the Laplace transform to solve differential equations.

Course Learning Outcomes:

1. Compute harmonic conjugates of harmonic functions.
2. Compute Taylor and Laurent expansions of complex analytic functions.
3. Evaluate contour integrals using residue theory.
4. Prove basic properties of complex analytic functions.
5. Solve algebraic equations involving complex numbers.
6. Understand basic properties of complex mappings.

Course Learning Outcomes:

1. Solve and manipulate ordinary differential equations.
2. Learn techniques to solve scalar first-order equations.
3. Learn techniques to solve linear scalar higher-order equations.
4. Learn techniques to solve linear vector first-order equations.
5. Use qualitative analysis to understand the behaviour planar systems of (not necessarily linear) ordinary differential equations.

Course Learning Outcomes:

1. Computing areas of regions bounded by simple closed curves.
2. Computing path integrals.
3. Computing potential functions for conservative vector fields.
4. Determining whether or not a given force field is conservative.
5. Determining whether or not a given vector field can be the curl of another vector field.
6. Evaluating the work done by a force field along a path.
7. Using Green's theorem for computing contour integrals.

Course Learning Outcomes:

1. Determining convergence or divergence of a sequence of real numbers.
2. Determining uniform/pointwise convergence or divergence of a sequence of functions.
3. Proving properties of limits of sequences of functions.
4. Using definitions to prove relationships between types of subsets of Euclidean space.
5. Using the definition of continuity to prove properties of continuous functions.

Course Learning Outcomes:

1. Create mathematical models of simple biological interactions.
2. Use numerical and graphical methods to find approximate solutions to discrete-time and continuous-time models.
3. Analyze the long-term behaviour of nonlinear planar differential equation models.
4. Make biological inferences from the analysis of nonlinear differential equation and difference equation models.

Course Learning Outcomes:

1. Work with axioms of an abstract group, different examples of finite and infinite groups in geometric, combinatorial and algebraic settings.
2. Work with group action on sets and its orbits, the orbit-stabilizer theorem.
3. Work with homomorphism, automorphism, and isomorphism of groups, as well as all three isomorphism theorems for groups.
4. Work with Lagrange's theorem, Euler's theorem, Fermat's Little theorem, Cauchy's theorem, and their applications.
5. Work with subgroups, generators, cosets, conjugacy classes, quotient groups, as well as examples of these notions in different settings.
6. Work with Sylow theorems and their application.

Course Learning Outcomes:

1. Gained an in introduction to advanced concepts that are taken up again in higher level courses.
2. Gained proficiency in congruence arithmetic.
3. Worked with different applications to cryptography, especially in the context of RSA encryption.

Course Learning Outcomes:

1. Computing path integrals of analytic functions using residue theory.
2. Computing Taylor and Laurent expansions for analytic functions of a complex variable.
3. Proving results on analytic functions.
4. Proving results on roots of polynomials using complex function theory.
5. Solving algebraic equations involving complex numbers.
6. Work with precise formulation of basic definitions and results on analytic functions.
7. Work with precise use of mathematical definitions in proving results on analytic functions.

Course Learning Outcomes:

1. Understand and distinguish various topological notions for subsets of the real line and of the n-dimensional Euclidean space, such as interior and boundary points, open/closed sets, cluster points, isolated points, nowhere dense sets, compact sets, connected sets, G-delta and F-sigma sets.
2. Understand and apply the concept of limit, continuity for functions of several real variables, and their ramifications, such as the intermediate value theorem and the extreme value theorem. Understand and apply the concept of uniform continuity and its ramification, such as the Heine-Cantor theorem. Understand and distinguish sets of continuity and their properties.
3. Understand and apply the concept of differentiability for (possibly vector-valued) functions of several real variables and its ramifications. For instance, apply the chain rule, the inverse function theorem, and the implicit function theorem in concrete examples. Understand differentiability and gradients in terms of partial derivatives.
4. Understand and apply the concept of relative extrema for functions of several variables and their relation to the Jacobian and the Hessian. Understand and apply the concept of constrained extrema and the Lagrange Multiplier Theorem in concrete examples.
5. Understand and apply the Picard-Lindelöf theorem to study the existence and uniqueness of solutions of systems of ordinary differential equations.

Course Learning Outcomes:

1. Understand the basic theory behind existence and uniqueness of solutions for ODEs.
2. Explore the solution space of general systems of ODEs, both linear and selected non-linear.
3. Work with the notions of equilibrium and stability of dynamical systems.
4. Work with unstable, stable and centre manifolds.
5. Work with one-parameter bifurcations.

Course Learning Outcomes:

1. Computing the Fourier transform of a signal.
2. Solving a difference equation using the z-transform.
3. Proving results on the Fourier transform.
4. Proving results on distributions.
5. Investigating the possibility of signal representations through polynomials, Haar wavelets and harmonic signals.
6. Mathematical formulation of lowpass filtering and noise removal.
7. Mathematical analysis of signal sampling.
8. Using mathematics to develop algorithms for noise removal.

Course Learning Outcomes:

1. Computing optimal policies via dynamic programming in applications such as inventory control.
2. Understanding stability of queuing models.
3. Using mathematics to establish optimal decision and planning via dynamic programming.
4. Rigorously proving results in Markov chains and optimal planning/control using tools from mathematical analysis.

Course Learning Outcomes:

1. Computing expected payoffs of games.
2. Using backward induction to find solutions to games in extensive form.
3. Using iterated elimination of dominated strategies to find a solution to games in normal form.
4. Using the theorem of mixed strategy Nash equilibria to find solutions to games in normal form.
5. Finding the Nash equilibria of a game.

Course Learning Outcomes:

1. Use the ideas in the course fluently. Indicators of fluency include: being able to use the ideas in a new situation; being able to use the ideas in a different order or manner than they have been presented; being able to construct minor extensions or use minor variations of the ideas.
2. Write clear, clean, and well-reasoned mathematical arguments.
3. Understand the standards for such arguments.
4. Master the ideas of topological spaces, subspaces, product and quotient spaces, continuity, compactness, separation and countability conditions, metric spaces and metrizability, the fundamental group and some applications.

Course Learning Outcomes:

1. Define and recognize the definitions of annuity-immediate, annuity due, perpetuity, m-thly payable, continuous annuity, level payment annuity, arithmetic increasing/decreasing annuity, geometric increasing/decreasing annuity, term of annuity. Given sufficient information of immediate or due, present value, future value, current value, interest rate, payment amount, and term of annuity, student is able to calculate any remaining item.
2. Understand different type of interest rates: effective rate, nominal rate, discount rate, simple rate and simple discount, real and inflation rates, yield rate, and be able to set up the equation of values and perform calculations relating to present value, current value, and accumulated value.
3. Understand key concepts of bonds: term of bond, bond price, book value, redemption value, face value, yield rate, coupon, coupon rate, term of bond, callable bond, amortization of bond. Given sufficient information of bond, be able to calculate the remaining item(s).
4. Understand key concepts of cash flows: cash-in, cash-out, net cash flow, yield rates, net present value, and internal rate of return, measure of duration and convexity, cashflow matching and immunization. Be able to perform related calculations.
5. Understand key concepts of loans: amortization, term of loan, outstanding balance, principal repayment, interest amount/payment, payment period, refinancing. Given sufficient information of loans, be able to calculate any remaining item(s).

Course Learning Outcomes:

1. Understand the mathematical theory behind life contingencies.
2. Work with survival models, life insurance, life annuities, benefit premiums, benefit reserves, multiple life functions, multiple decrement models and insurance models including expenses.
3. Gain a significant start on preparations for the Society of Actuaries MLC examination.

Course Learning Outcomes:

1. Experience the development of other topics such as Ramsey theory, spectral methods, or random graphs.
2. Use an inquiry-based approach to explore bipartite graphs, trees and connectivity, Euler and Hamiltonian paths, graph matchings and colourings, and planar graphs.
3. Work with the fundamental concepts of graph theory (cycles, regular graphs, matrix representations, isomorphisms, etc.).

Course Learning Outcomes:

1. Understand the role of generating functions in combinatorial analysis.
2. With the study of examples and proofs, interact with a variety of concepts and techniques from enumerative combinatorics.
3. Work with counting techniques, permutations, partitions, cardinality, and Fibonacci and Catalan numbers.

Course Learning Outcomes:

1. Do matrix manipulations for linear codes and to compute decoding errors.
2. Rigorously prove results on error correction and detection.
3. Understand the structure of finite fields and to do computations in these fields.
4. Understand various methods for encoding and decoding messages for the purpose of error-correction and to perform the necessary computations.

Course Learning Outcomes:

1. Create and present rigorous solutions to problems and coherent proofs of theorems.
2. Define and illustrate the correspondence between ideals and varieties by translating between algebraic and geometric statements.
3. Describe and demonstrate a basic understanding of projective geometry.
4. Execute accurate and efficient calculations with ideals in a multivariate polynomial ring involving Gröbner bases, membership, intersections, and quotients.
5. Explain and use elimination theory to solve systems of polynomial equations.

Course Learning Outcomes:

1. Design a cryptographic system with given real-world parameters.
2. Perform analysis of cryptographic protocols.
3. Perform computations and constructions in finite fields.
4. Perform probability analysis of primality tests.
5. Perform reasoning on elliptic curves.
6. Rigorously prove results in number theory.
7. Test various designs to determine which performs the best.
8. Understand the algebraic structure of a group.

Course Learning Outcomes:

1. Explore a variety of different types of dynamical systems: flows and semiflows, differential and difference equations, vector fields, linear versions of the above.
2. Investigate various classes of low-dimensional dynamical systems, and consider in detail what behaviours can be exhibited by these.
3. Study recurrent behaviour, indecomposability, invariance and stability.
4. Understand what a dynamical system is.

Course Learning Outcomes:

1. Have experience with extensions to the mathematics of Quantum Mechanics, the Schrödinger equations, and the harmonic oscillator.
2. Have foundational experience with metric spaces, Banach spaces, inner product spaces, Hilbert spaces, operators and their spectrum.
3. Work with extending the concepts of Linear Algebra and Calculus to the infinite dimensional setting.
4. Work with problems from Mathematical Physics and in particular with the mathematical foundations of Quantum Mechanics.

Course Learning Outcomes:

1. Compute tractions, principal stresses, and principal axes of stress of a body.
2. Derive and analyze the differential equations governing the motion of a body.
3. Develop engineering solutions for real-world problems concerned with stress of materials, deformation of elastic bodies and motion of fluids.
4. Understand definitions and techniques of tensor algebra and tensor calculus.
5. Use the methods of tensor algebra and calculus for the description of the deformation of a body.
6. Use the methods of tensor algebra and calculus to describe the motion of a body.

Course Learning Outcomes:

1. Computing necessary conditions for optimality.
2. Solving constrained optimization problems.
3. Understanding the mathematical properties of convex sets and convex functions.
4. Rigorously using separation theorems for solving optimization problems.
5. Using numerical methods in the study of optimization problems.
6. Solving resource allocation problems using duality theory.

Course Learning Outcomes:

1. Solve and analyze the partial differential equations modeling transport phenomena.
2. Solve and analyze the partial differential equations modeling diffusion phenomena.
3. Solve and analyze the initial-boundary value problems involving Laplace equation and Poisson equation.
4. Solve and analyze the partial differential equations modeling waves and vibrations.
5. Use the method of characteristics to solve first-order quasilinear equations.
6. Apply analytical tools to solve nonlinear partial differential equations.

Course Learning Outcomes:

1. Model physical systems using methods of geometric mechanics.
2. Translate physical concepts to differential geometric concepts.
3. Understand the definitions and constructions from differential geometry.
4. Use the methods of differential geometry.

Course Learning Outcomes:

1. Understand stochastic stability notions for controlled and control-free Markov chains, including recurrence, positive Harris recurrence, and transience and stationarity.
2. Establish structural and existence results on optimal control policies through dynamic programming and properties of conditional expectation.
3. Understand the theory of martingales and their use in optimal stochastic control.
4. Rigorously use tools from stochastic analysis and stochastic control theory.
5. Compute optimal policies and costs for stochastic control problems via various analytical, numerical, and computational methods (including machine learning and reinforcement learning methods).
6. Design controllers leading to various forms of stability for a controlled stochastic system.

Course Learning Outcomes:

1. Computing Shannon's information measures (entropy, Kullback-Leibler distance and mutual information).
2. Computing the capacity of communication channels.
3. Reasoning about the properties of Shannon's information measures (entropy, Kullback-Leibler distance and mutual information).
4. Using mathematical tools to infer properties of coding and communication systems.
5. Working with probabilistic modeling of communication systems for source and channel coding purposes.
6. Using tools from probability theory to analyze communication systems.
7. Working with metric assessment of data compression code designs.

Course Learning Outcomes:

1. Compute distortion in source quantization.
2. Prove rigorously the optimality of lossless and lossy source codes.
3. Work with mathematical formulation of source coding.
4. Use probabilistic tools to understand the effect of data compression on random sources.

Course Learning Outcomes:

1. Have a rigorous understanding of the key elements of the theory of martingales (sub- and super-).
2. Rigorously use key results from martingale theory in order to establish properties of stochastic processes, and, in particular, of Brownian motion.
3. Rigorously construct the Itô stochastic integral of a predictable process with respect to a right-continuous square-integrable martingale.
4. Rigorously apply the Itô stochastic calculus to solve problems in stochastic analysis.
5. Have a rigorous understanding of stochastic differential equations.
6. Rigorously apply tools from stochastic analysis to solve problems in mathematical finance.