AMATH 600s


AMATH 655 Control Theory (0.50) LECCourse ID: 011278
Feedback control with applications. System theory in both time and frequency domain, state-space computations, stability, system uncertainty, loopshaping, linear quadratic regulators and estimation. (Heldwith AMATH 455)
Antirequisite: AMATH 455

AMATH 663 Fluid Mechanics (0.50) LECCourse ID: 011279
Incompressible, irrotational flow. Incompressible viscous flow. Introduction to wave motion and geophysical fluid mechanics. Elements of compressible flow. (Heldwith AMATH 463)
Antirequisite: AMATH 463

AMATH 673 Quantum Mechanics (0.50) LECCourse ID: 011280
Vector space formalism, Schrodinger and Heisenberg pictures, elements of second quantization. Angular momentum, selection rules, symmetry and conservation laws. Approximation methods: variation principle, perturbation theory and WKB approximation. Identical particles, Pauli principle and simple applications to atomic, molecular, solid state, scattering and nuclear problems. (Heldwith AMATH 473) [Offered: F]
Antirequisite: AMATH 473

AMATH 675 General Relativity (0.50) LECCourse ID: 011281
Flat space-time and Lorentz transformations. Relativistic mechanics. Maxwell's equations. Curved space-time and the Einstein field equations. The Schwarzschild solution and some experimental tests of general relativity. The weak-field limit. Introduction to black holes and cosmology. (Heldwith AMATH 675)
Antirequisite: AMATH 475

AMATH 700s


AMATH 731 Applied Functional Analysis (0.50) LECCourse ID: 000116
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 732 Asymptotic Analysis and Perturbation Theory (0.50) LECCourse ID: 011283
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.

AMATH 740 Numerical Analysis (0.50) LECCourse ID: 012670
(Cross-listed with CM 770, CS 670)
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 741 Numerical Solution of Partial Differential Equations (0.50) LECCourse ID: 000724
(Cross-listed with CS 778, CM 750)
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 751 Advanced Ordinary Differential Equations (0.50) LECCourse ID: 000118
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 753 Advanced Partial Differential Equations (0.50) LECCourse ID: 000119
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 777 Stochastic Processes in the Physical Sciences (0.50) LECCourse ID: 011284
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 800s


AMATH 851 Stability Theory and Applications (0.50) LECCourse ID: 011285
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.
1 Stability Theory of ODEs

AMATH 855 Advanced Systems Analysis and Control (0.50) LECCourse ID: 000132
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 863 Hydrodynamic Stability and Turbulence (0.50) LECCourse ID: 000133
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.

AMATH 867 Dispersive and Nonlinear Waves (0.50) LECCourse ID: 000134
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.

AMATH 871 Quantum Information Processing (0.50) LECCourse ID: 011589
(Cross-listed with CO 681, PHYS 767, CS 667, QIC 710)
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 872 Introduction to Quantum Field Theory for Cosmology (0.50) LECCourse ID: 012151
(Cross-listed with PHYS 785)
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.
Prerequisite: AMATH 673 or PHYS 701 or equivalent

AMATH 873 Introduction to Quantum Field Theory (0.50) LECCourse ID: 000135
(Cross-listed with PHYS 703)
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 interactin quantum fields to lowest-order quantum electrodynamic processes. Radiative corrections and renormalization.
Prerequisite: AMATH 673 or PHYS 701 or equivalent

AMATH 874 Advanced techniques in General Relativity and Applications to Black Holes (0.50) LECCourse ID: 010439
(Cross-listed with PHYS 784)
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 875 Introduction to General Relativity with Applications to Cosmology (0.50) LECCourse ID: 011282
(Cross-listed with PHYS 786)
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.

AMATH 876 Open Quantum Systems (0.50) LECCourse ID: 012567
(Cross-listed with QIC 845)
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.
Instructor Consent Required
Prereq: AMATH 473/673

AMATH 881 Introduction to Mathematical Oncology (0.50) LECCourse ID: 013468
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.

AMATH 882 Mathematical Cell Biology (0.50) LECCourse ID: 013469
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.

AMATH 900s


AMATH 900 Topics in Applied Mathematics (0.50) LECCourse ID: 011357
Instructor Consent Required
1 Delay Differential Equations
2 Wavelet and Fractal Analysis
3 Calculus of Variations
4 Impulsive Control&Stabilizat'n
5 Modelling of Smart Materials
6 Computational Fluid Mechanics
7 Quantum Fields in Cosmology
8 Wavelets:Theory & Applications
9 Infinite-Dimensional Systems
10 Robust Control
11 Impulsive DEs with Time Delay
12 Fluid Mechanics
13 Impulsive Control&Applications
14 Bifurcation Theory
15 Dynamical Systems and Imaging
16 Stability of Stochastic System
17 Hybrid Systems with Time Delay
18 Waves in Porous Media
19 Bayesian Mthds - Data Analysis
20 Mathmatical Biology
21 Advanced Linear Systems Theory
22 Stochastic Processes
23 Hybrid Dynamical Systems
24 Nonlinear Systems Theory
25 Nonlinear Systems Control
26 Mathematical Imaging
27 Cellular Mathematical Biology
28 Statcl Anlysis-Microarray Data
29 Topics in Spectral Methods
30 Physical Processes in Lakes
31 Mathematical Imaging
32 Finite Element Methods
33 Numrcl Mthds for Hyprbolc PDEs
34 Topics in Continuum Mechanics
35 Intro to Mathematical Oncology
36 Large-scale Ocean Circulation
37 Finite Element Methods
38 Delay Differential Equations
39 Impulsive Delay Systems
40 Foundations of Quantum Theory
41 Numerical Linear Algebra
42 Nonlinear Water Waves
43 Impulsive Dynamical Systems
44 Fractal Analysis, Math Imaging
45 Advanced Numerical PDEs
46 Environmental Fluid Dynamics
47 Nonlinear Dynamical Systems
48 Compressible Fluid Dynamics
49 Computation of Internal Waves
50 Repetitive Control Theory
51 Special Topics in PDEs