Mathematics & Statistics

• CAS MA 679: Applied Statistical Machine Learning
  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
  Prerequisites: (CASMA 225) AND (CASMA 242) 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: Applied Multiple Regression and Multivariable Methods
  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
  Prerequisites: (CASMA 684 or CASMA 416 or CASMA 575) or consent of instructor. Continues topics of CAS 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
  Prerequisites: (CASMA 512) 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
  Prerequisites: (CASMA 511) AND (CASMA512) or equivalent, or consent of instructor. - The theory of analytic functions. Integral theorems, contour integration, conformal mapping, and analytic continuation.
• CAS MA 717: Functional Analysis 1
  Prerequisites: (CASMA 711) 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
  Prerequisites: (CASMA 511) AND (CASMA512) or equivalent. - Differential manifolds, tangent bundles, transversality, winding numbers, and vector bundles.
• CAS MA 722: Differential Topology 2
  Prerequisites: (CASMA 721) - Intersection theory, Lefschetz fixed point theory, integration on manifolds, vector fields and flows, and Frobenius' theorem.
• CAS MA 725: Differential Geometry 1
  Prerequisites: (CASMA 721) 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
  Prerequisites: (CASMA 725) - Topics include connections on vector bundles, moving frames, Hodge theory, spectral geometry, and characteristic classes.
• CAS MA 727: Algebraic Topology I
  Prerequisites: (CASMA 564) or equivalent. - Covers singular and simplical homology theory. Cohomology and cup products. Duality on manifolds. Lefschetz and fixed-point formula.
• CAS MA 728: Algebraic Topology 2
  Prerequisites: (CASMA 727) - Topics include homotopy theory, theory of characteristic classes and covering spaces, and cobordism theory.
• CAS MA 731: Lie Groups and Lie Algebras
  Prerequisites: (CASMA 721) AND (CASMA741) - 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
  Prerequisites: (CASMA541) AND (CASMA542) or equivalent, or consent of instructor. – 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
  Prerequisites: (CASMA 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.
• CAS MA 743: Algebraic Number Theory 1
  Prerequisites: (CASMA 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.
• CAS MA 745: Algebraic Geometry 1
  Prerequisites: (CASMA 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.
• CAS MA 746: Algebraic Geometry 2
  Prerequisites: (CASMA 745) - Continuation of topics in algebraic geometry begun in CASMA 745, including sheaves, schemes, sheaf cohomology, and further study of algebraic curves and surfaces.
• CAS MA 750: Nonparametric and Semiparametric Data Modeling
  Prerequisites: (CASMA 575) AND (CASMA 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.