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GRS MA 614: Statistical Methods 2
Undergraduate Prerequisites: graduate standing in education or in the social sciences.
Second course in statistics, embodying basic statistical methods used in educational and social science research. Reviews all basic concepts covered in a first statistics course (e.g., CAS MA 613) and presents, in detail, more advanced topics such as analysis of variance, covariance, experimental design, correlation, regression, and selected nonparametric techniques. A problem-solving course; students carry out analysis of data taken from educational and other social science sources. This course cannot be taken for credit in addition to the course entitled "Statistical Methods II" that was previously numbered CAS MA 614.
GRS MA 647: Research Methods in Mathematics I
Undergraduate Prerequisites: CAS MA 547 and CAS MA 548; or consent of instructor.
Methods of mathematical research via prolonged study of one selected mathematical topic. Topics are usually chosen from number theory or combinatorics. Written and oral research presentations.
GRS MA 648: Research Methods in Mathematics II
Undergraduate Prerequisites: GRS MA 647; or consent of instructor.
Methods of mathematical reserach via prolonged, directed study of one selected mathematical topic, distinct from that chosen for GRS MA 647. Topics are usually chosen from geometry, number theory, or combinatorics, and may involve open problems. Written and oral research presentation.
GRS MA 665: Introduction to Modeling and Data Analysis in Neuroscience
Undergraduate Prerequisites: CAS MA 122 or CAS MA 124; or equivalent, and graduate standing, or consent of instructor
An introduction to the basic techniques of quantifying neural data and developing mathematical models of neural activity. Major focus on computational methods using computer software and graphical methods for model analysis.
GRS MA 666: Advanced Modeling and Data Analysis in Neuroscience
Undergraduate Prerequisites: CAS MA 226 or CAS MA 231; or equivalent. Graduate standing required, or consent of instructor.
Advanced techniques to characterize neural voltage data and analyze mathematical models of neural activity. Major focus on computational methods using computer software and graphical methods for model analysis.
GRS MA 671: Chaotic Dynamical Systems
Undergraduate Prerequisites: CAS MA 225 or CAS MA 230; or equivalent, and graduate standing.
For graduate students in disciplines outside of mathematics. Iterations of functions of one or several variables. Periodicity, stability, chaos, fractals, bifurcations. Julia sets and the Mandelbrot set. Students are required to perform several experiments on personal computers.
GRS MA 675: Statistics Practicum 1
First of a two-semester sequence aimed at integrating the quantitative training and other skills required for doing statistics in practice. Emphasis on statistical consulting throughout, complemented by modules on speaking, writing, statistical software and programming, and data analysis.
GRS MA 676: Statistics Practicum 2
Undergraduate Prerequisites: admission to the Statistical Practice MS program.
Second of a two-semester sequence aimed at integrating the quantitative training and other skills required for doing statistics in practice. Emphasis on statistical consulting throughout, complemented by modules on speaking, writing, statistical software and programming, and data analysis.
GRS MA 681: Accelerated Introduction to Statistical Methods for Quantitative Research
Undergraduate Prerequisites: CAS MA 225 and CAS MA 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.
GRS MA 684: Applied Multiple Regression and Multivariable Methods
Undergraduate 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. This course cannot be taken for credit in addition to the course with the same title that was previously numbered CAS MA 684.
GRS 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. This course cannot be taken for credit in addition to the course with the same title that was previously numbered CAS MA 685.
GRS MA 703: Statistical Analysis of Network Data
Undergraduate Prerequisites: CAS MA 575 or GRS MA 681; , or consent of instructor.
Methods and models for the statistical analysis of network data, including network mapping and characterization, community detection, network sampling and measurement, and the modeling and inference of network and networked-indexed processes. Balance of theory and concepts, illustrated through various applications.
GRS MA 711: Real Analysis
Graduate Prerequisites: CAS MA 512; 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.
GRS 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.
GRS MA 717: Functional Analysis I
Graduate Prerequisites: GRS MA 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.
GRS MA 721: Differential Topology I
Graduate Prerequisites: CAS MA 511 and CAS MA 512; or equivalent.
Differential manifolds, tangent bundles, transversality, winding numbers, and vector bundles.
GRS MA 722: Differential Topology II
Graduate Prerequisites: GRS MA 721.
Intersection theory, Lefschetz fixed point theory, integration on manifolds, vector fields and flows, and Frobenius' theorem.
GRS MA 725: Differential Geometry I
Graduate Prerequisites: GRS MA 721; or consent of instructor.
Geometry of surfaces in Euclidean space; geodesics and curvature of Riemannian manifolds; topological restrictions on curvature.
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.