Mathematics & Statistics

• GRS MA 727: Algebraic Topology I
  Graduate Prerequisites: CAS MA 564; or equivalent.
  Covers singular and simplical homology theory. Cohomology and cup products. Duality on manifolds. Lefschetz and fixed-point formula.
• GRS MA 731: Lie Groups and Lie Algebras
  Graduate Prerequisites: GRS MA 721 and GRS MA 741.
  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
  Graduate Prerequisites: GRS MA 741; or consent of instructor.
  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
  Graduate Prerequisites: GRS MA 741; or consent of instructor.
  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
  Graduate Prerequisites: GRS MA 741; or consent of instructor.
  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
  Graduate Prerequisites: GRS MA 745.
  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
  Graduate Prerequisites: CAS MA 575 and CAS MA 581; or consent of instructor.
  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
  Graduate Prerequisites: CAS MA 575 and CAS MA 581; or consent of instructor.
  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 765: Time Series Analysis for Neuroscience Research
  Undergraduate Prerequisites: CAS MA 213 or GRS MA 681; CAS MA 242; CAS MA665/MA666; or consent of instructor.
  Provides an overview of statistical time-series modeling for neuroscience applications. Topics include regression and generalized linear modeling, state space modeling, and parametric and nonparametric spectral analysis. Special emphasis on reading and discussing applications in recent literature.
• GRS MA 770: Mathematical and Statistical Methods of Bioinformatics
  Graduate Prerequisites: graduate standing or advanced undergraduate math/statistics major, (CASMA225), (CASMA242), and previous work in mathematical analysis and probability.
  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
  Undergraduate Prerequisites: GRS MA 711, equivalent, or consent of instructor.
  Graduate Prerequisites: GRS MA 711; equivalent, or consent of instructor.
  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
  Graduate Prerequisites: CAS MA 581 and CAS MA 583 or equivalent, and CAS MA 226 or CAS MA 231or equivalent.
  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
  Graduate Prerequisites: CAS MA 511; or consent of instructor.
  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
  Undergraduate Prerequisites: GRS MA 711.
  Graduate Prerequisites: GRS MA 711.
  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
  Undergraduate Prerequisites: CAS MA 581, MA 582, or consent of instructor.
  Graduate Prerequisites: CAS MA 581 and CAS MA 582; or consent of instructor.
  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
  Undergraduate Prerequisites: GRS MA 782 or consent of instructor.
  Graduate Prerequisites: GRS MA 781; or consent of instructor.
  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.
• GRS MA 783: Advanced Stochastic Processes
  Undergraduate Prerequisites: GRS MA 779 or GRS MA 780; or consent of instructor.
  Proof-based approach to stochastic processes. Brownian motion. Continuous martingales. Stochastic integration. Ito formula. Girsanov's Theorem. Stochastic differential equations. Feynman-Kac formula. Markov Processes. Local times. Levy processes. Semimartingales and the general stochastic integral. Stable processes. Fractional Brownian motion.