This course studies efficient algorithms and their application to geometric problems. Topics include: convex hulls; line segment intersection; polygons, triangulations, and visibility; low dimension linear programming; range searching; point location search; arrangements of lines and hyperplanes; Voronoi and Delauney diagrams; and randomized algorithms. Topics may include: visibility graphs, shortest paths, and robot motion planning. Algorithms are implemented on the computer.
Advanced topics in algebra and linear algebra that have broad application are studied. Topics include: linear groups (SO(3), SU(2), etc.); an introduction to group representations; the theory of commutative rings and ideals; and an introduction to algebraic geometry. The course includes a detailed study of one or more of: group representations; Lie groups, Lie algebras and their representations; algebraic varieties and Groebner bases.
This course develops the principles and techniques used to identify the best alternative in a set, based on measures of their performance according to two or more criteria. Topics include: properties of the Pareto order and dominance; dominance cones; screening; the existence, construction and properties of value functions; methods of compromise; satisficing; relation of MCDA to techniques such as Multiple-Objective Decision Analysis; extension of MCDA to stochastic criteria; the theorems of May, Arrow, and Gibbard-Satterthwaite.
Monte Carlo techniques and simulation methods are studied in detail. Applications include mathematical modelling and computation of numerical solutions; evaluation of multi-dimensional integrals through pseudo-random numbers, quasi-random numbers, Sobol sequences and other sequences of lattice points. Topics include: sampling algorithms; simulated annealing; Markov processes; variance reduction techniques; importance sampling; adaptive and recursive Monte Carlo methods. Applications include numerical integration of multivariate functions in high dimensions; approximation algorithms for solving partial differential equations; stochastic lattice approaches and path expansions. Additional topics may include parallel algorithms for Monte Carlo simulations.
This course studies orthogonal systems of functions related to ordinary and partial differential equations. Topics include: inner product spaces; Hilbert spaces; Hermitian and self-adjoint linear operators; Sturm-Liouville theory; generating functions and orthogonal polynomials; Green's functions; completeness and over-completeness; eigenfunction expansions in one and several variables; infinite and semi-infinite orthonormal bases. Applications include solution methods for parabolic partial differential equations, Fourier series, wavelets, wavelet transforms, and other expansion methods. Topics may include integral representations and transforms, spectral theory, point-wise convergence and convergence in the norm.
This course introduces topics from number theory with application to public key cryptography. Topics include: elementary number theory; quadratic residues; quadratic reciprocity; finite field arithmetic; elliptic curve groups; RSA public-key cryptography; elliptic curve cryptography; the discrete logarithm problem for elliptic curves; and algorithms for primality testing and factoring.
An introduction to the modern, coordinate-free, formulation of Lagrangian and Hamiltonian mechanics. This formulation provides a unifying framework for many seemingly disparate physical systems, such as N-particle systems, rigid bodies, fluids and other continua, and quantum systems. Topics comprise variational principles, Lagrangian and Hamiltonian dynamics, canonical transformations, Hamilton-Jacobi equations and control, symmetry, Noether's theorem and reduction, integrability, Poisson structures, Poisson brackets, and constrained systems. Applications may include N-particle problems, quantum models, shallow-water and wave dynamics, rigid bodies.
This course introduces tensor analysis with differential geometry and variational calculus for modelling static and dynamical problems. Topics include: vector spaces; affine tensor algebras of arbitrary rank; covariant and contravariant vectors and tensors; cartesian and non-cartesian tensor algebras; symmetries under linear and nonlinear coordinate transformations; tensor fields and their derivatives; tensor analysis on manifolds; differential forms. Additional topics may include Lie differentiation; generalized Stokes' theorem; Riemannian manifolds. Applications of tensors and manifolds may include: the analysis of invariance properties arising in physical phenomena; geodesic flow; Hamilton-Jacobi theory and classical and quantum field theories.
Fundamental concepts of linear and non-linear optimization theory for developing algorithms and models are discussed. Topics include: duality and problem structure for finding, recognizing and interpreting solutions; network optimization problems; problems with integer constraints; combinatorial optimization problems; the simplex algorithm for linear programming; linear programming duality and complementary slackness; the network simplex algorithm; Newton and gradient methods for unconstrained optimization; Lagrange multipliers; penalty and barrier methods for constrained optimization; and an introduction to interior-point methods for linear and convex programming. Search techniques for hard problems may be included. The output of appropriate computer packages is analyzed.
Combinatorial algorithms especially related to optimization in graphs and networks are developed and analyzed. Topics include: combinatorial design theory and applications to experimental design and coding; optimization in intersection graphs and other specially structured classes of graphs; algorithms on ordered sets; network optimization; matching algorithms; matroid greedy algorithm and matroid intersection algorithm; latin squares, block designs, and experimental design; and an introduction to coding theory. Additional topics may include approximation algorithms for selected NP-hard problems, multicommodity flows, and network design.
This is a course in non-cooperative game theory that begins with decision theory and utility theory and covers equilibrium concepts in strategic-form and extensive-form games, including Nash, correlated, subgame-perfect, Bayesian, trembling-hand, perfect Bayesian, and sequential equilibria. Other topics may include evolutionary stability and evolutionary dynamics, and connections to dynamical systems.
This course presents a rigorous development of: point and interval estimation; sufficiency, efficiency, unbiasedness, and consistency. Topics may include: maximum likelihood and Bayesian estimation; exchangeability; invariance; decision theory; large sample theory; optimality criteria and most powerful tests; likelihood ratio tests; robustness and resampling.
This course is a study in modern real and complex analysis. Topics from real analysis include: measure theory and integration; Banach, Hilbert, Lp -spaces; uniform boundedness principle, the open mapping theorem and closed graph theorem. After a review of analytic functions, harmonic functions, the residue theorem and the maximum modulus principle, additional topics in complex analysis include: Riemann mapping theorem, analytic continuation, Poisson's integral formula and Dirichlet's problem. Applications include partial differential equations.
This course introduces the fundamentals of stochastic calculus. Topics include probability measures and random variables; the Itô integral calculus; Itô's Lemma; Markov chains; random walks; the Wiener process; Brownian and geometric Brownian motion; filtrations; adaptive processes; Martingales and super-Martingales; the Martingale Stopping Time Theorem; Girsanov's Theorem and the Radon-Nikodym derivative; stochastic differential equations for single and multiple random processes; Kolmogorov equations and the Feynman-Kac Theorem. Applications include the modelling of continuous diffusion processes, and the development of solution techniques for stochastic differential equations. Topics may include stochastic optimization and jump processes.
This course is a study in infinite dimensional vector spaces and operators between function spaces. Topics include: normed linear spaces (Hilbert, Lebesgue and Sobolev spaces); locally convex spaces; dual spaces; operator theory; Banach algebras; spectral theory; distribution theory; representation theory; the Hahn Banach theorem for Hilbert spaces; strong, weak and weak* topologies; and fixed point theorems. Included are selected applications such as differential equations, and Fourier and other integral transforms.
The principles and theory of harmonic analysis, using examples to motivate the concepts, are introduced. Topics include: Harmonic analysis on the circle, the real line, and the integers; Haar measure and characters; Calderon-Zygmund theory; the uncertainty principle; Fourier and wavelet transforms; Plancherel theorem; Paley-Weiner theory; interpolation of operators; the Hausdorff-Young theorem; and an introduction to noncommutative harmonic analysis.
This course studies the qualitative and quantitative theory of dynamical systems. Topics include: extensions of Picard's theorems; stability properties of continuous-time systems and of discrete-time iterative maps; linearization; Lyapunov direct method; centre manifold; and bifurcation. Other topics may include: delay/feedback equations, limit cycles, strange attractors and deterministic chaos. Applications to neural networks and complex ecosystems are examined.
An introduction to the use of dynamical systems for the purpose of studying biological systems. Models will be chosen from ecology and epidemiology, including structured populations, as well as genetics and systems biology. Mathematical analysis will include techniques from stability analysis, bifurcation theory and persistence theory applied to ordinary differential equations, partial differential equations, difference equations, delay equations, stochastic equations or integral equations.
This course develops the mathematical framework for option pricing in continuous time for equity and interest rate derivatives. Topics include: asset pricing and interest rate processes; derivation of the Black-Scholes partial differential equation; pricing of standard European, American and multi-asset options under geometric Brownian motions; stochastic asset price models; multi-factor interest rate stochastic modelling; bond pricing and interest rate option pricing and calibration; and path dependent options. Topics may include: transformation techniques for solving parabolic PDEs; Green's functions; path integral methodologies for pricing and hedging options; Monte Carlo simulation and stochastic mesh methods for pricing complex multi-asset derivatives.
This seminar course is designed to develop the capacity to abstract salient features of problems in financial mathematics and the scientific disciplines, and to develop, analyze, and interpret models. Problems from financial mathematics and science, using undergraduate mathematics in modelling and analysis, are studied in detail. Commonality of mathematical methods and structures across disciplines is emphasized. Students work individually and in groups, and produce both written and oral reports on their projects.
Individual study of a special topic at an advanced level, under the supervision of a faculty member or other supervisor approved by the department. The topics and evaluation scheme must be approved by the department.
A detailed examination of a special topic not covered by the department's regular course offerings. The topic and evaluation scheme must be approved by the department.
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