Courses

• GRS MA 711: Real Analysis
  Measure theory and integration on measure spaces, specialization to integration on locally compact spaces, and the Haar integral. Lp spaces, duality, and representation theorems. Introduction to Banach and Hilbert spaces, open mapping theorem, spectral theorem for Hermitian operators, and compact and Fredholm operators.
• GRS MA 713: Functions of a Complex Variable I
  The theory of analytic functions. Integral theorems, contour integration, conformal mapping, and analytic continuation.
• GRS MA 717: Functional Analysis I
  Theory of Banach and Hilbert spaces, and Hahn-Banach and separation theorems. Dual spaces. Banach contraction mapping theorem. Reflexivity and Krein-Milman theorem. Operator theory. Brouwer-Schauder fixed-point theorems. Applications to probability, dynamical systems, and applied mathematics.
• GRS MA 721: Differential Topology I
  Differential manifolds, tangent bundles, transversality, winding numbers, and vector bundles.
• GRS MA 722: Differential Topology II
  Intersection theory, Lefschetz fixed point theory, integration on manifolds, vector fields and flows, and Frobenius' theorem.
• 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.

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