Mathematics & Statistics

• CAS MA 679: Applied Statistical Machine Learning
  Graduate Prerequisites: admission to the MSSP program. - Continues topics of GRS MA 678 at a more advanced level. Application of supervised and unsupervised statistical machine learning techniques with extensive use of computation. Advanced topics such as analysis of network data, Bayesian nonparametric models are considered.
• CAS MA 681: Accelerated Introduction to Statistical Methods for Quantitative Research
  Undergraduate Prerequisites: (CASMA225 & CASMA242) or their equivalents. - Introduction to statistical methods relevant to research in the computational sciences. Core topics include probability theory, estimation theory, hypothesis testing, linear models, GLMs, and experimental design. Emphasis on developing a firm conceptual understanding of the statistical paradigm through data analyses.
• CAS MA 684: Modern Regression Analysis in R
  Graduate Prerequisites: one year of statistics. - Application of multivariate data analytic techniques. Multiple regression and correlation, confounding and interaction, variable selection, categorical predictors and outcomes, logistic regression, factor analysis, MANOVA, discriminant analysis, regression with longitudinal data, repeated measures, ANOVA.
• CAS MA 685: Advanced Topics in Applied Statistical Analysis
  Continues topics of GRS MA 684 at a more advanced level. Canonical correlation, multivariate analysis of variance, multivariate regressions. Categorical dependent variables techniques; discriminant analysis, logistic regression, log-linear analysis. Factor analysis; principal-axes, rotations, factor scores. Cluster analysis. Power analysis. Extensive use of statistical software.
• CAS MA 711: Real Analysis
  Graduate Prerequisites: (CASMA512) or substantial mathematical experience. - 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.
• CAS MA 713: Functions of a Complex Variable I
  Graduate Prerequisites: advanced calculus or substantial mathematical experience. - The theory of analytic functions. Integral theorems, contour integration, conformal mapping, and analytic continuation.
• CAS MA 717: Functional Analysis I
  Graduate Prerequisites: (GRSMA711) or equivalent. - 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.
• CAS MA 721: Differential Topology 1
  Graduate Prerequisites: (CASMA511 & CASMA512) or equivalent. - Differential manifolds, tangent bundles, transversality, winding numbers, and vector bundles.
• CAS MA 722: Differential Topology 2
  Graduate Prerequisites: (GRSMA721) - Intersection theory, Lefschetz fixed point theory, integration on manifolds, vector fields and flows, and Frobenius' theorem.
• CAS MA 725: Differential Geometry I
  Graduate Prerequisites: (GRSMA721) or consent of instructor. - Geometry of surfaces in Euclidean space; geodesics and curvature of Riemannian manifolds; topological restrictions on curvature.
• CAS MA 726: Differential Geometry 2
  Graduate Prerequisites: (GRSMA725) - Topics include connections on vector bundles, moving frames, Hodge theory, spectral geometry, and characteristic classes.
• CAS MA 727: Algebraic Topology I
  Graduate Prerequisites: (CASMA564) or equivalent. - Covers singular and simplical homology theory. Cohomology and cup products. Duality
  on manifolds. Lefschetz and fixed-point formula.
• CAS MA 731: Lie Groups and Lie Algebras
  Graduate Prerequisites: (GRSMA721 & GRSMA741) - 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.
• CAS MA 741: Algebra 1
  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.
• CAS MA 742: Algebra 2
  Graduate Prerequisites: (GRSMA741) 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.
• CAS MA 743: Algebraic Number Theory 1
  Graduate Prerequisites: (GRSMA741) 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.
• CAS MA 745: Algebraic Geometry 1
  Graduate Prerequisites: (GRSMA741) 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.
• CAS MA 746: Algebraic Geometry II
  Graduate Prerequisites: (GRSMA745) - Continuation of topics in algebraic geometry begun in GRS MA 745, including sheaves, schemes, sheaf cohomology, and further study of algebraic curves and surfaces.
• CAS MA 750: Nonparametric and Semiparametric Data Modeling
  Graduate Prerequisites: (CASMA575 & CASMA581) 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.
• CAS MA 751: Statistical Machine Learning
  Graduate Prerequisites: (CASMA575 & CASMA581) 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.