Courses

• GRS MA 725: Differential Geometry I
  Geometry of surfaces in Euclidean space; geodesics and curvature of Riemannian manifolds; topological restrictions on curvature.
• GRS MA 726: Differential Geometry II
  Topics include connections on vector bundles, moving frames, Hodge theory, spectral geometry, and characteristic classes.
• 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 744: Algebraic Number Theory II
  Advanced topics in number theory. Topics chosen from: Zeta functions of number fields of algebraic varieties; arithmetic of elliptic curves; modular forms and modular curves; class field theory; and Iwasawa theory.
• 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: Advanced Statistical Methods I
  First course in a two-semester PhD sequence on post-classical statistical methods and their applications. Selection from topics in non- and semi-parametric modeling and inference, such as smoothing, splines, generalized additive models, projection pursuit, and classification and regression trees.
• GRS MA 751: Advanced Statistical Methods II
  Second course in a two-semester PhD sequence on post-classical statistical methods and their applications. Selection from topics in statistical learning, such as 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 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.

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