丁香园AV

Graded on a satisfactory/unsatisfactory basis.

Analysis and computation of classical problems from applied mathematics such as eigenfunction expansions, integral transforms, and stability and bifurcation analyses. Methods include perturbation, boundary layer and multiple-scale analyses, averaging and homogenization, integral asymptotics and complex variable methods as applied to differential equations.

First order non-linear partial differential equations (PDEs) and the method of characteristics. Hamilton-Jacobi equation and hyperbolic conservation laws; weak solutions. Second-order linear PDEs (Laplace, heat and wave equations); Green's functions. Sobolev spaces. Second-order elliptic PDEs; Lax-Milgram theorem.

Infinite dimensional vector spaces, convergence, generalized Fourier series. Operator Theory; the Fredholm alternative. Application to integral equations and Sturm-Liouville systems. Spectral theory.

An intensive study of Lebesque measure, integration and the Lebesque convergence theorems together with the treatment of such topics as absolute continuity, the fundamental theorem of calculus, the Lp-spaces, comparison of types of convergence in function spaces, the Baire category theorem.

Conditioning and stability of numerical methods for the solution of linear systems, direct factorization and iterative methods, least squares, and eigenvalue problems. Applications and mathematical software.

Analysis and application of numerical methods for solving partial differential equations. Potential topics include finite difference methods, spectral methods, finite element methods, and multi-level/multi-grid methods.

Theory and algorithms of non-linear programming with an emphasis on modern computational considerations. Topics may include: optimality conditions for unconstrained and constrained optimization, gradient methods, conjugate direction methods, Newton method, quasi-Newton methods, penalty and barrier methods, augmented Langrangian methods and interior point methods.

Basic equations governing compressible and incompressible fluid mechanics. Finite difference and finite volume schemes for hyperbolic, elliptic, and parabolic partial differential equations. Practical applications in low Reynolds number flow, high-speed gas dynamics, and porous media flow. Software design and use of public-domain codes. Students with credit for MATH 930 may not complete this course for further credit.

Analysis of models from the natural and applied sciences via analytical, asymptotic and numerical studies of ordinary and partial differential equations.

Topics vary depending on faculty availability and student interest. Recent offerings include: geophysical fluid dynamics, adaptive numerical methods for differential equations, learning theory, and stability, pattern formation and chaos.

Fundamental algorithmic techniques used to solve computational problems encountered in molecular biology. This area is usually referred to as Bioinformatics or Computational Biology. Students who have taken CMPT 881 (Bioinformatics) in 2007 or earlier may not take CMPT 711 for further credit.

Convex geometry, the simplex method and duality, pivot rules, degeneracy, decomposition and column generation methods, the complexity of linear programming and the ellipsoid algorithm, interior point methods for linear programming.

Algebraic graph theory, extremal graph theory, coloring problems, path and cycle structure of graphs, application of graphs, hypergraphs, and current research topics.

An introduction to the theory of incidence structures (finite geometries, block designs) and their relation to linear codes. Algebraic techniques - finite group actions, orbit enumeration, generation of orbit representatives. Exact and asymptotic enumeration of labelled and unlabelled structures.

Theory and algorithms for problems in data science with an emphasis on mathematical aspects. Topics may include dimension reduction, supervised learning, including regression and classification, unsupervised learning, including clustering and latent variable modeling, deep learning, algorithms for big data, and foundations of learning.

The statistical theory that supports modern statistical methodologies. Distribution theory, methods for construction of tests, estimators, and confidence intervals with special attention to likelihood and Bayesian methods. Properties of the procedures including large sample theory will be considered. Consistency and asymptotic normality for maximum likelihood and related methods (e.g., estimating equations, quasi-likelihood), as well as hypothesis testing and p-values. Additional topics may include: nonparametric models, the bootstrap, causal inference, and simulation. Prerequisite: STAT 450 or permission of the instructor. Students with credit for STAT 801 may not take this course for further credit.

Advanced mathematical statistics for PhD students. Topics in probability theory including densities, expectation and random vectors and matrices are covered. The theory of point estimation including unbiased and Bayesian estimation, conditional distributions, variance bounds and information. The theoretical framework of hypothesis testing is covered. Additional topics that may be covered include modes of convergence, central limit theorems for averages and medians, large sample theory and empirical processes. Prerequisite: STAT 830 or permission from the instructor.