Vector integral calculus-line integrals, surface integrals and vector fields, Green's theorem, the Divergence theorem, and Stokes' theorem. Applications include conservation laws, fluid flow and electromagnetic fields. An introduction to Fourier analysis. Fourier series and the Fourier transform. Parseval's formula. Frequency analysis of signals. Discrete and continuous spectra.
Prerequisites
Must have completed at least 1 of the following:
MATH237 - Calculus 3 for Honours Mathematics (0.50)
AMATH 242Introduction to Computational Mathematics0.50 units
A rigorous introduction to the field of computational mathematics. The focus is on the interplay between continuous models and their solution via discrete processes. Topics include pitfalls in computation, solution of linear systems, interpolation, discrete Fourier transforms, and numerical integration. Applications are used as motivation.
AMATH 250Introduction to Differential Equations0.50 units
Physical systems which lead to differential equations (examples include mechanical vibrations, population dynamics, and mixing processes). Dimensional analysis and dimensionless variables. Solving linear differential equations: first- and second-order scalar equations and first-order vector equations. Laplace transform methods of solving differential equations. Introduction to series solutions and special functions.
AMATH 251Introduction to Differential Equations (Advanced Level)0.50 units
AMATH 251 is an advanced-level version of AMATH 250. Basic techniques for ODEs: integrating factors, undetermined coefficients and variation of parameters. Introduction to existence and uniqueness theorems. Systems of n-dimensional coupled ODEs and high order ODEs. Matrix Exponential. Laplace Transform. Stability and nonlinear ODEs. Phase space analysis and chaos. Introduction to numerical methods. Series solutions and special functions.
AMATH 271Introduction to Theoretical Mechanics0.50 units
Newtonian dynamics, gravity and the two-body problem, introduction to Lagrangian mechanics, introduction to Hamiltonian mechanics, non-conservative forces, oscillations, introduction to special relativity.
Topology of Euclidean spaces, continuity, norms, completeness. Contraction mapping principle. Fourier series. Various applications, for example, to ordinary differential equations, optimization and numerical approximation.
Prerequisites
Must have completed at least 1 of the following:
MATH237 - Calculus 3 for Honours Mathematics (0.50)
Complex numbers, Cauchy-Riemann equations, analytic functions, conformal maps and applications to the solution of Laplace's equation, contour integrals, Cauchy integral formula, Taylor and Laurent expansions, residue calculus and applications.
Prerequisites
Must have completed at least 1 of the following:
MATH237 - Calculus 3 for Honours Mathematics (0.50)
AMATH 333Calculus on Manifolds for Applied Mathematics and Physics0.50 units
Manifolds and tensors, Lie derivatives and Lie groups, differential forms and applications to physics. This course covers the basic concepts of differential geometry from the perspective of its applications to physics. The course focuses on the concepts and the techniques that allow us to formulate physical problems in the powerful language of differential geometry including thermodynamics, classical mechanics, fluid dynamics, and relativity.
AMATH 342Computational Methods for Differential Equations0.50 units
An introduction to numerical methods for ordinary and partial differential equations. Ordinary differential equations: multistep and Runge-Kutta methods; stability and convergence; systems and stiffness; boundary value problems. Partial differential equations: finite difference methods for elliptic, hyperbolic and parabolic equations; stability and convergence. The course focuses on introducing widely used methods and highlights applications in the natural sciences, the health sciences, engineering, and finance.
Prerequisites
Complete all of the following
Must have completed at least 1 of the following:
AMATH242 - Introduction to Computational Mathematics (0.50)
AMATH 345Data-Driven Mathematical Models0.50 units
An introduction to data-driven mathematical methods for modelling and prediction of complex systems in science, medicine, and technology. Topics include singular value decomposition, sparsity and compressed sensing, calibration and parameter inference for differential equation models, as well as neural networks and data-driven methods for dynamical systems. Throughout the course, students will learn to use modern data science methods and apply recent advances in data-driven methods to a wide range of applications.
Prerequisites
Complete all of the following
Must have completed at least 1 of the following:
AMATH250 - Introduction to Differential Equations (0.50)
AMATH251 - Introduction to Differential Equations (Advanced Level) (0.50)
MATH228 - Differential Equations for Physics and Chemistry (0.50)
Must have completed at least 1 of the following:
CS114 - Principles of Computing for Science (0.50)
AMATH 350Differential Equations for Business and Economics0.50 units
First order ordinary differential equations. Applications to continuous compounding and the dynamics of supply and demand. Higher order linear ordinary differential equations. Systems of linear ordinary differential equations. Introduction to linear partial differential equations. The Fourier Transform and the diffusion equation. Discussion of the Black-Scholes partial differential equations, and solutions thereof.
AMATH 351Ordinary Differential Equations0.50 units
Linear differential equations with non-constant coefficients, Sturm comparison, oscillation and separation theorems, series solutions and special functions. Boundary value problems. Linear systems in Rn, an introduction to dynamical systems. Laplace transforms applied to linear systems, transfer functions, the convolution theorem. An introduction to dynamical systems and stability. Perturbation methods for differential equations. Applications are discussed throughout.
Prerequisites
Complete all of the following
Must have completed at least 1 of the following:
AMATH250 - Introduction to Differential Equations (0.50)
AMATH251 - Introduction to Differential Equations (Advanced Level) (0.50)
AMATH350 - Differential Equations for Business and Economics (0.50)
Must have completed at least 1 of the following:
MATH237 - Calculus 3 for Honours Mathematics (0.50)
AMATH 353Partial Differential Equations 10.50 units
Second order linear partial differential equations - the diffusion equation, wave equation, and Laplace's equation. Methods of solution - separation of variables and eigenfunction expansions, the Fourier transform. Physical interpretation of solutions in terms of diffusion, waves, and steady states. First order non-linear partial differential equations and the method of characteristics. Applications are emphasized throughout.
Stress and strain tensors; analysis of stress and strain. Lagrangian and Eulerian methods for describing flow. Equations of continuity, motion and energy, constitutive equations. Navier-Stokes equation. Basic equations of elasticity. Various applications.
An introduction to the mathematical description of the climate. Topics covered include solar radiation and how the Earth is heated, climate feedbacks, the role of the oceans and the Earth's rotation, climate data and climate variability, and simple models of human-climate interactions. Calculus and computation will be used throughout the course. [Offered in W of even years.]
Prerequisites
Complete all of the following
Must have completed at least 1 of the following:
AMATH250 - Introduction to Differential Equations (0.50)
AMATH251 - Introduction to Differential Equations (Advanced Level) (0.50)
AMATH350 - Differential Equations for Business and Economics (0.50)
MATH211 - Advanced Calculus 1 for Electrical and Computer Engineers (0.50)
MATH213 - Signals, Systems, and Differential Equations (0.50)
MATH218 - Differential Equations for Engineers (0.50)
MATH228 - Differential Equations for Physics and Chemistry (0.50)
Critical experiments and old quantum theory. Basic concepts of quantum mechanics: observables, wavefunctions, Hamiltonians, and the Schroedinger equation. Uncertainty, correspondence, and superposition principles. Simple applications to finite and extended one-dimensional systems, harmonic oscillator, rigid rotor, and hydrogen atom.
AMATH 382Computational Modelling of Cellular Systems0.50 units
An introduction to dynamic mathematical modelling of cellular processes. The emphasis is on using computational tools to investigate differential equation-based models. A variety of cellular phenomena are discussed, including ion pumps, membrane potentials, intercellular communication, genetic networks, regulation of metabolic pathways, and signal transduction.
AMATH 383Introduction to Mathematical Biology0.50 units
An introduction to the mathematical modelling of biological processes, with emphasis on population biology. Topics include ecology, epidemiology, microbiology, and physiology. Techniques include difference equations, ordinary differential equations, partial differential equations, stability analysis, phase plane analysis, travelling wave solutions, mathematical software. Includes collaborative projects and computer labs.
An introduction to some of the deep connections between mathematics and music. Topics covered include acoustics, including pitch and harmonics, basic Fourier analysis, the mathematics behind the differing pitch and timbre of string, wind and percussion instruments, scales and temperaments, digital music, musical synthesis.
AMATH 391Data Analysis with Fourier and Wavelet Methods0.50 units
Fourier and wavelet methods are fundamental tools in data analysis ranging from time-series data to image and audio compression. These methods are increasingly finding application in areas such as machine learning and quantum computing alongside traditional application areas such as signal processing for images and audio. This course covers the mathematical development of Fourier and wavelet analysis, with an emphasis on both theory and practical application. Students will learn the necessary theory to apply Fourier and wavelet methods to the analysis of a variety of practical problems with hands on coding experience through a number of assignments.
AMATH 442Computational Methods for Partial Differential Equations0.50 units
This course studies several classes of methods for the numerical solution of partial differential equations in multiple dimensions on structured and unstructured grids. Finite volume methods for hyperbolic conservation laws: linear and nonlinear hyperbolic systems; stability; numerical conservation. Finite element methods for elliptic and parabolic equations: weak forms; existence of solutions; optimal convergence; higher-order methods. Examples from fluid and solid mechanics. Additional topics as time permits.
Prerequisites
Must have completed the following:
AMATH342 - Computational Methods for Differential Equations (0.50)
The course provides an in-depth exploration of how deep learning techniques are applied in various scientific and medical domains. The course introduces basic concepts of deep learning, explores different deep learning architectures and algorithms, and focuses on their applications in scientific and biomedical research. The integration of scientific knowledge with machine learning techniques is emphasized throughout the course. Students will gain hands-on experience by executing the acquired concepts using Python.
Prerequisites
Must have completed at least 1 of the following:
AMATH250 - Introduction to Differential Equations (0.50)
AMATH251 - Introduction to Differential Equations (Advanced Level) (0.50)
AMATH350 - Differential Equations for Business and Economics (0.50)
ECE205 - Advanced Calculus 1 for Electrical and Computer Engineers (0.50)
MATH211 - Advanced Calculus 1 for Electrical and Computer Engineers (0.50)
MATH218 - Differential Equations for Engineers (0.50)
MATH228 - Differential Equations for Physics and Chemistry (0.50)
An introduction to neural network methods, with some discussion of their relation to neuroscience. Simple neuron models and networks of neurons. Training feedforward networks. Backpropagation of errors and stochastic gradient descent. Unsupervised learning methods. Recurrent neural networks. Convolutional neural networks. Continuous time models. Adversarial attacks. Advanced topics may include neural engineering, biologically plausible learning methods.
AMATH 451Introduction to Dynamical Systems0.50 units
A unified view of linear and nonlinear systems of ordinary differential equations in Rn. Flow operators and their classification: contractions, expansions, hyperbolic flows. Stable and unstable manifolds. Phase-space analysis. Nonlinear systems, stability of equilibria, and Lyapunov functions. The special case of flows in the plane, Poincare-Bendixson theorem, and limit cycles. Applications to physical problems will be a motivating influence.
Prerequisites
Must have completed at least 1 of the following:
AMATH250 - Introduction to Differential Equations (0.50)
AMATH251 - Introduction to Differential Equations (Advanced Level) (0.50)
AMATH 453Partial Differential Equations 20.50 units
A thorough discussion of the class of second order linear partial differential equations with constant coefficients, in two independent variables. Laplace's equation, the wave equation and the heat equation in higher dimensions. Theoretical/qualitative aspects: well-posed problems, maximum principles for elliptic and parabolic equations, continuous dependence results, uniqueness results (including consideration of unbounded domains), domain of dependence for hyperbolic equations. Solution procedures: elliptic equations -- Green functions, conformal mapping; hyperbolic equations -- generalized d'Alembert solution, spherical means, method of descent; transform methods -- Fourier, multiple Fourier, Laplace, Hankel (for all three types of partial differential equations); Duhamel's method for inhomogeneous hyperbolic and parabolic equations.
Feedback control with applications. System theory in both time and frequency domain, state-space computations, stability, system uncertainty, loopshaping, linear quadratic regulators, and estimation.
Prerequisites
Complete all of the following
Must have completed at least 1 of the following:
AMATH250 - Introduction to Differential Equations (0.50)
AMATH251 - Introduction to Differential Equations (Advanced Level) (0.50)
Concept of functional and its variations. The solution of problems using variational methods - the Euler-Lagrange equations. Applications include an introduction to Hamilton's principle and optimal control.
Prerequisites
Complete all of the following
Must have completed at least 1 of the following:
MATH237 - Calculus 3 for Honours Mathematics (0.50)
Incompressible, irrotational flow. Incompressible viscous flow. Introduction to wave motion and geophysical fluid mechanics. Elements of compressible flow.
The Hilbert space of states, observables, and time evolution. Feynman path integral and Greens functions. Approximation methods. Co-ordinate transformations, angular momentum, and spin. The relation between symmetries and conservation laws. Density matrix, Ehrenfest theorem, and decoherence. Multiparticle quantum mechanics. Bell inequality and basics of quantum computing.
AMATH 474Quantum Theory 3: Quantum Information and Foundations0.50 units
Theory of correlations and entanglement; theory of quantum channels, detectors; the measurement problem, in quantum mechanics; phase space formulation of quantum mechanics; entanglement in infinite dimensional quantum systems; introduction to open quantum systems; and exploration of current research directions in quantum information.
AMATH 475Introduction to General Relativity0.50 units
Tensor analysis. Curved space-time and the Einstein field equations. The Schwarzschild solution and applications. The Friedmann-Robertson-Walker cosmological models.
AMATH 477Stochastic Processes for Applied Mathematics0.50 units
Random variables, expectations, conditional probabilities, conditional expectations, convergence of a sequence of random variables, limit theorems, minimum mean square error estimation, the orthogonality principle, random process, discrete-time and continuous-time Markov chains and applications, forward and backward equation, invariant distribution, Gaussian process and Brownian motion, expectation maximization algorithm, linear discrete stochastic equations, linear innovation sequences, Kalman filter, various applications.
Prerequisites
Complete all of the following
Must have completed at least 1 of the following:
AMATH250 - Introduction to Differential Equations (0.50)
AMATH251 - Introduction to Differential Equations (Advanced Level) (0.50)
AMATH350 - Differential Equations for Business and Economics (0.50)
ECE205 - Advanced Calculus 1 for Electrical and Computer Engineers (0.50)
MATH211 - Advanced Calculus 1 for Electrical and Computer Engineers (0.50)
MATH218 - Differential Equations for Engineers (0.50)
MATH228 - Differential Equations for Physics and Chemistry (0.50)
A research-based course that allows students with an interest in applied mathematics or mathematical physics to participate in a research project with a faculty member. The student is required to approach a faculty member of the Department of Applied Mathematics for supervision. The enrolled student is expected to hand in a written report of their research to the supervisor and present a summary of the project orally in front of a subset of the applied mathematics undergraduate committee as a final assessment.
AMATH 642Computational Methods for Partial Differential Equations0.50 units
This course studies several classes of methods for the numerical solution of partial differential equations in multiple dimensions on structured and unstructured grids. Finite volume methods for hyperbolic conservation laws: linear and nonlinear hyperbolic systems; stability; numerical conservation. Finite element methods for elliptic and parabolic equations: weak forms; existence of solutions; optimal convergence; higher-order methods. Examples from fluid and solid mechanics. Additional topics as time permits. (Held with AMATH 442. Students in AMATH 642 will be expected to meet some additional learning objectives.)
Antirequisites
Complete all of the following
Not completed nor concurrently enrolled in any of the following:
AMATH741 - Numerical Solution of Partial Differential Equations (0.50)
The course provides an in-depth exploration of how deep learning techniques are applied in various scientific and medical domains. The course introduces basic concepts of deep learning, explores different deep learning architectures and algorithms, and focuses on their applications in scientific and biomedical research. The integration of scientific knowledge with machine learning techniques is emphasized throughout the course. Students will gain hands-on experience by executing the acquired concepts using Python. (Held with AMATH 445. Students in AMATH 645 will be expected to meet some additional learning objectives.)
AMATH 646Introduction to Mathematics of Deep Learning0.50 units
This course introduces the basic concepts of deep learning in a mathematically rigorous fashion. Topics include mathematical definitions of deep neural networks, approximation theory, gradient-based optimization, and generalization analysis. (Held with AMATH 446. Students in AMATH 646 will be expected to meet some additional learning objectives.)
An introduction to neural network methods, with some discussion of their relevance to neuroscience. Simple neuron models and networks of neurons. Training feedforward networks for classification or regression. Learning using the backpropagation of errors. Unsupervised learning methods. Optimal linear decoding. Recurrent neural networks. Convolutional neural networks. Advanced topics, including adversarial inputs and biologically plausible learning methods.
AMATH 651Introduction to Dynamical Systems0.50 units
A unified view of linear and nonlinear systems of ordinary differential equations in Rn. Flow operators and their classification: contractions, expansions, hyperbolic flows. Stable and unstable manifolds. Phase-space analysis. Nonlinear systems, stability of equilibria and Lyapunov functions. The special case of flows in the plane, Poincare-Bendixson theorem and limit cycles. Applications to physical problems will be a motivating influence. (Held with AMATH 451. Students in AMATH 651 will be expected to meet some additional learning objectives.)
AMATH 653Partial Differential Equations 20.50 units
A thorough discussion of the class of second order linear partial differential equations with constant coefficients, in two independent variables. Laplace's equation, the wave equation and the heat equation in higher dimensions. Theoretical/qualitative aspects: well-posed problems, maximum principles for elliptic and parabolic equations, continuous dependence results, uniqueness results (including consideration of unbounded domains), domain of dependence for hyperbolic equations. Solution procedures: elliptic equations -- Green functions, conformal mapping; hyperbolic equations -- generalized d'Alembert solution, spherical means, method of descent; transform methods -- Fourier, multiple Fourier, Laplace, Hankel (for all three types of partial differential equations); Duhamel's method for inhomogeneous hyperbolic and parabolic equations. (Held with AMATH 453. Students in AMATH 653 will be expected to meet some additional learning objectives.)
Feedback control with applications. System theory in both time and frequency domain, state-space computations, stability, system uncertainty, loopshaping, linear quadratic regulators and estimation. (Held with AMATH 455. Students in AMATH 655 will be expected to meet some additional learning objectives.)
Concept of functional and its variations. The solution of problems using variational methods - the Euler- Lagrange equations. Applications include an introduction to Hamilton's principle and optimal control. (Held with AMATH 456. Students in AMATH 656 will be expected to meet some additional learning objectives.)
AMATH 662Introduction to Mathematics of Climate Modelling0.50 units
This course will introduce mathematical techniques, including analytical, computational, and machine learning methods, used to study climate change. Course material will examine both historical evidence of climate change and future predictions related to climatological and societal impacts based on current climate models. The course will have a strong computational component to study the governing geophysical fluid dynamics equations in climate models and their properties. The course will also examine the application of learned principles to climate modeling through in-depth discussions of journal readings. (Held with AMATH 462. Students in AMATH 662 will be expected to meet some additional learning objectives.)
Incompressible, irrotational flow. Incompressible viscous flow. Introduction to wave motion and geophysical fluid mechanics. Elements of compressible flow. (Held with AMATH 463. Students in AMATH 663 will be expected to meet some additional learning objectives.)
The Hilbert space of states, observables, and time evolution. Feynman path integral and Greens functions. Approximation methods. Co-ordinate transformations, angular momentum, and spin. The relation between symmetries and conservation laws. Density matrix, Ehrenfest theorem, and decoherence. Multiparticle quantum mechanics. Bell inequality and basics of quantum computing. (Held with AMATH 473. Students in AMATH 673 will be expected to meet some additional learning objectives.)
AMATH 674Quantum Theory 3: Quantum Information and Foundations0.50 units
Theory of correlations and entanglement, theory of quantum channels, detectors and the measurement problem in quantum mechanics, phase space formulation of quantum mechanics and entanglement in infinite dimensional quantum systems, introduction to open quantum systems, exploration of current research directions in quantum information. (Held with AMATH 474. Students in AMATH 674 will be expected to meet some additional learning objectives.)
AMATH 675Introduction to General Relativity0.50 units
Tensor analysis. Curved space-time and the Einstein field equations. The Schwarzschild solution and applications. The Friedmann-Robertson-Walker cosmological models. (Held with AMATH 475. Students in AMATH 675 will be expected to meet some additional learning objectives.)
AMATH 677Stochastic Processes for Applied Mathematics0.50 units
Random variables, expectations, conditional probabilities, conditional expectations, convergence of a sequence of random variables, limit theorems, minimum mean square error estimation, the orthogonality principle, random process, discrete-time and continuous-time Markov chains and applications, forward and backward equation, invariant distribution, Gaussian process and Brownian motion, expectation maximization algorithm, linear discrete stochastic equations, linear innovation sequences, Kalman filter, various applications. (Held with AMATH 477. Students in AMATH 677 will be expected to meet some additional learning objectives.)
Basic concepts of functional analysis. Topics include: theory of linear operators, nonlinear operators and the Frechet derivative, fixed point theorems, approximate solution of operator equations, Hilbert space, spectral theory. Applications from various areas will be used to motivate and illustrate the theory. A previous undergraduate course in real analysis is strongly recommended.
AMATH 732Asymptotic Analysis and Perturbation Theory0.50 units
Elements of asymptotic analysis. Techniques of perturbation theory such as Poincare-Lindstedt, matched asymptotic expansions and multiple scales. Applications to various areas form an essential aspect of the course. Previous courses in real analysis and differential equations at the undergraduate level are strongly recommended.
Introduction to basic algorithms and techniques for numerical computing. Error analysis, interpolation (including splines), numerical differentiation and integration, numerical linear algebra (including methods for linear systems, eigenvalue problems, and the singular value decomposition), root finding for nonlinear equations and systems, numerical ordinary differential equations, and approximation methods (including least squares, orthogonal polynomials, and Fourier transforms).
AMATH 741Numerical Solution of Partial Differential Equations0.50 units
Discretization methods for partial differential equations, including finite difference, finite volume and finite element methods. Application to elliptic, hyperbolic and parabolic equations. Convergence and stability issues, properties of discrete equations, and treatment of non-linearities. Stiffness matrix assembly and use of sparse matric software. Students should have completed a course in numerical computation at the undergraduate level.
AMATH 751Advanced Ordinary Differential Equations0.50 units
Qualitative theory of systems of ODEs. Topics include: existence/uniqueness of solutions, comparison principle, iterative techniques, stability and boundedness, Lyapunov method, periodic solutions, Floquet theory and Poincare maps, hyperbolicity, stable, unstable and center manifolds, structural stability and bifurcation. Applications from various areas will be used to motivate and illustrate the theory. A previous course in ordinary differential equations at the undergraduate level is strongly recommended.
AMATH 753Advanced Partial Differential Equations0.50 units
The main themes are well-posedness of problems, Hilbert space methods, variational principles and integral equation methods. Topics include: first-order nonlinear partial differential equations, quasilinear hyperbolic systems, potential theory, eigenfunctions and eigenvalues, semi-groups, and power series solutions. Applications from various areas will be used to motivate and illustrate the theory. A previous course in partial differential equations at the undergraduate level is strongly recommended.
AMATH 777Stochastic Processes in the Physical Sciences0.50 units
Basic concepts and classification of stochastic processes. Stochastic differentiation and integration, Markov processes, Chapman-Kolmogorov equation, Fokker-Planck equation, Master equations: mesoscopic vs. macroscopic description. Spectral representation of stationary processes. Correlation function theory. A previous course in probability theory at the undergraduate level is strongly recommended.
AMATH 840Advanced Numerical Methods for Computational and Data Sciences0.50 units
Theory and practice of a selection of advanced numerical methods for computational and data sciences. Algorithms for eigenvalues and singular value decomposition. Multigrid methods for linear and nonlinear systems. Sparse optimization and compressed sensing. Low-rank tensor and matrix decomposition. Nonlinear convergence acceleration. Randomized numerical linear algebra. Adjoint methods and automatic differentiation for neural networks and optimal control. Stochastic gradient descent and variants. Efficient computer implementation of the algorithms and applications with real-world data. Students should have completed an introductory course on numerical methods.
Mathematical theory of finite element methods, with error analysis and stability. Continuous and discontinuous Galerkin methods and mixed finite element methods, applied to elliptic, parabolic, and hyperbolic partial differential equations including compressible and/or incompressible Euler and Navier-Stokes equations. Implementation in two and three dimensions. Students should have completed an introductory course on numerical methods for partial differential equations.
AMATH 851Stability Theory and Applications0.50 units
Concepts of stability and boundedness, basic stability criteria, comparison methods, large scale systems, method of decomposition and aggregation, method of several Lyapunov functions, method of vector Lyapunov functions, method of higher derivatives. Stability problems in ecology, mechanics, neural networks and control systems. Students should have completed AM751 or equivalent.
AMATH 855Advanced Systems Analysis and Control0.50 units
The main theme is the extension of control theory beyond systems modelled by linear ordinary differential equations. Topics include: advanced systems theory, control of nonlinear systems, control of partial differential equations and delay equations. Students should have completed an introductory undergraduate course in control theory.
AMATH 863Hydrodynamic Stability and Turbulence0.50 units
Mathematical methods, stability of parellel flows for unstratified and stratified fluids, Rayleigh-Taylor instability, centrifugal instability, barotropic and baroclinic instabilities, the effects of viscosity and the Orr-Sommerfeld equation, transition to turbulence, averaged equations, closure problem, homogeneous isotropic turbulence, turbulent boundary layers, effects of stratification. Students should have completed an introductory undergraduate course in fluid mechanics.
Dispersive waves, propagation of dispersive waves in an inhomogeneous medium (WKB theory). Nonlinear resonant interactions. Solitons: completely integrable nonlinear wave equations (e.g., the KdV equation, nonlinear Schrodinger equations) and the inverse Scattering Transform. Applications to water waves and nonlinear optics. Introducation to weakly nonlocal solitary waves and beyond-all-orders asymptotics. Completion of an upper year course in partial differential equations is strongly recommended.
Review of basics of quantum information and computational complexity; Simple quantum algorithms; Quantum Fourier transform and Shor factoring algorithm: Amplitude amplification, Grover search algorithm and its optimality; Completely positive trace-preserving maps and Kraus representation; Non-locality and communication complexity; Physical realizations of quantum computation: requirements and examples; Quantum error-correction, including CSS codes, and elements of fault-tolerant computation; Quantum cryptography; Security proofs of quantum key distribution protocols; Quantum proof systems. Familiarity with theoretical computer science or quantum mechanics will also be an asset, though most students will not be familiar with both.
AMATH 872Introduction to Quantum Field Theory for Cosmology0.50 units
Introduction to scalar field theory and its canonical quantization in flat and curved spacetimes. The flat space effects of Casimir and Unruh. Quantum fluctuations of scalar fields and of the metric on curved space-times and application to inflationary cosmology. Hawking radiation.
AMATH 873Introduction to Quantum Field Theory0.50 units
Review of relativistic quantum mechanics and classical field theory. Quantization of free quantum fields (the particle interpretation of field quanta). Canonical quantization of interacting fields (Feynman rules). Application of the formalism of interacting quantum fields to lowest-order quantum electrodynamic processes. Radiative corrections and renormalization.
AMATH 874Advanced techniques in General Relativity and Applications to Black Holes0.50 units
Review of elementary general relativity. Timelike and null geodesic congruences. Hypersurfaces and junction conditions. Lagrangian and Hamiltonian formulations of general relativity. Mass and angular momentum of a gravitating body. The laws of black-hole mechanics.
AMATH 875Introduction to General Relativity with Applications to Cosmology0.50 units
Introduction to the differential geometry of Lorentzian manifolds. The priniciples of general relativity. Causal structure and cosmological singularities. Cosmological space-times with Killing vector fields. Friedmann-Lemaitre cosmologies, scalar, vector and tensor perburbations in the linear and nonlinear regimes. De Sitter space-times and inflationary models.
Review of the axioms of quantum theory and derivation of generalized axioms by considering states, transformations, and measurements in an extended Hilbert space. Master equations and the Markov approximation. Standard models of system-environment interactions and the phenomenology of decoherence. Introduction to quantum control with applications in NMR, quantum optics, and quantum computing.
Review mathematical formulation of operational quantum theory; theory of measurements and decoherence; quantum-classical contrast; review of historical perspectives on interpretation, including EPR paradox; Bell's theorem, non-locality and contextuality; PBR theorem; selected topics including overviews of current interpretations of quantum mechanics and critical experiments in quantum foundations.
AMATH 881Introduction to Mathematical Oncology0.50 units
Biological and clinical aspects of cancer. Overview of recent mathematical models developed to examine different stages of cancer growth and therapeutic strategies, including ordinary and partial differential equation models, discrete models and models based on continuum mechanics. Various analytical and numerical methods will be used in the analysis of these models.
Dynamic mathematical modelling of biological process at the cellular level. Intracellular networks: metabolism, signal transduction, and genetic regulatory networks. Neural networks: from biophysical modelling of single neurons to the analysis of network behaviour. Modelling will be carried out primarily through ordinary differential equations; analysis will involve application of dynamical systems tools and simulations. Other relevant modelling frameworks (PDEs, delay equations, stochastic methods) will be touched on as time allows.