Mathematics & Statistics

• GRS MA 725: Differential Geometry I
  Geometry of surfaces in Euclidean space; geodesics and curvature of Riemannian manifolds; topological restrictions on curvature.
• GRS MA 727: Algebraic Topology I
  Covers singular and simplical homology theory. Cohomology and cup products. Duality on manifolds. Lefschetz and fixed-point formula.
• GRS MA 728: Algebraic Topology II
  Topics include homotopy theory, theory of characteristic classes and covering spaces, and cobordism theory.
• GRS MA 731: Lie Groups and Lie Algebras
  Classical Lie groups, associated Lie algebras, exponential map, closed subgroups and homogeneous spaces, classification of simple Lie algebras, and elementary representation theory of Lie algebras. Selection of applications to analysis, geometry, or algebra.
• GRS MA 741: Algebra I
  Basic properties of groups, rings, fields, and modules. Specific topics include the Jordan-Holder and Sylow theorems, local rings, theory of localization, modules over PIDs, and Galois theory.
• GRS MA 742: Algebra II
  Advanced topics in algebra. Linear and multilinear algebra, commutative algebra, and an introduction to category theory and homological algebra. Further topics may include representation of groups, completions, real fields, and elementary algebraic number theory and algebraic geometry.
• GRS MA 743: Algebraic Number Theory I
  Algebraic integers, completions, ramification and the discriminant, cyclotomic and quadratic fields, ideal class groups, Dirichlet's unit theorem, ideles, and adeles. Further topics are chosen from analytic number theory, class field theory, and the theory of Diophantine equations.
• GRS MA 745: Algebraic Geometry I
  Affine and projective varieties, morphisms and rational maps, nonsingular varieties, Bezout's theorem, and an introduction to sheaves and schemes. Further topics are chosen from the advanced theory of schemes, algebraic curves, Riemann-Roch theorem, algebraic surfaces, and sheaf cohomology.
• GRS MA 746: Algebraic Geometry II
  Continuation of topics in algebraic geometry begun in GRS MA 745, including sheaves, schemes, sheaf cohomology, and further study of algebraic curves and surfaces.
• GRS MA 750: Nonparametric and Semiparametric Data Modeling
  Introduces theory and methods of non- and semiparametric data analysis. Topics include scatterplot smoothers, bias/variance trade-off, selection of smoothing parameter, generalized additive model, smoothing spline, and Bayesian nonparametric models. Applications in various fields are discussed.
• GRS MA 751: Statistical Machine Learning
  Foundations and applications of statistical machine learning. Supervised and unsupervised learning. Machine classification and regression methods, regularized basis methods, kernel methods, boosting, neural networks, support vector machines, and graphical models.
• GRS MA 770: Mathematical and Statistical Methods of Bioinformatics
  Mathematical and statistical bases of bioinformatics methods and their applications. Hidden Markov models, kernel methods, mathematics of machine learning approaches, probabilistic sequence alignment, Markov chain Monte Carlo and Gibbs sampling, mathematics of phylogenetic trees, and statistical methods in microarray analysis.
• GRS MA 771: Introduction to Dynamical Systems
  Diffeomorphisms and flows; periodic points, nonwandering points, and recurrent points; hyperbolicity, topological conjugacy, and structural stability; stable manifold theorem; symbolic dynamics; Axiom A and chaotic systems.
• GRS MA 775: Ordinary Differential Equations
  Stable and center manifolds theorem, linearization of vector fields, variational equations, Floquet theory and Poincare; maps for periodic orbits, bifurcation of rest points, averaging theory, topics from singular perturbations, Hamiltonian systems, non-linear oscillations, normal forms, and applications.
• GRS MA 776: Partial Differential Equations
  Hyperbolic, elliptic, and parabolic equations. Characteristics and separation of variables. Eigenvalue problems, Fourier techniques, Sobolev spaces, and potential theory. Introduction to pseudodifferential operators.
• GRS MA 777: Multiscale Methods for Stochastic Processes and Differential Equations
  Methods and models for the analysis of systems that possess many characteristic length and time scales. Asymptotic expansions, coarse-graining of multiscale stochastic models, mathematical analysis and statistical inference. Balance of theory and concepts illustrated via various applications.
• GRS MA 779: Probability Theory I
  Introduction to probability with measure theoretic foundations. Fundamentals of measure theory. Probability space. Measurable functions and random variables. Expectation and conditional expectation. Zero-one laws and Borel-Cantelli lemmas. Chracteristic functions. Modes of convergence. Uniform integrability. Skorokhod representation theorem. Basic limit theorems.
• GRS MA 780: Probability Theory II
  Probability topics important in applications and research. Laws of large numbers. Three series theorem. Central limit theorems for independent and non-identically distributed random variables. Speed of convergence. Large deviations. Laws of the iterated logarithm. Stable and infinitely divisible distributions. Discrete time martingales and applications.
• GRS MA 781: Estimation Theory
  Review of probability, populations, samples, sampling distributions, and delta theorems. Parametric point estimation. Rao-Cramer inequality, sufficient statistics, Rao-Blackwell theorem, maximum likelihood estimation, least squares estimation, and general linear model of full rank. Confidence intervals. Bayesian analysis and decision theory.
• GRS MA 782: Hypothesis Testing
  Parametric hypothesis testing, uniformly and locally the most powerful tests, similar tests, invariant tests, likelihood ratio tests, linear model testing, asymptotic theory of likelihood ratio, and chi-squared test. Logit and log-lin analysis of contingency tables.

Back to full list of Courses