Mathematics & Statistics

• GRS MA 648: Research Methods in Mathematics II
  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
  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
  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 681: Accelerated Introduction to Statistical Methods for Quantitative Research
  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 699: Teaching College Mathematics I
  The goals, contents, and methods of instruction in mathematics. General teaching-learning issues. Required of all teaching fellows.
• GRS MA 703: Statistical Analysis of Network Data
  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
  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 726: Differential Geometry II
  Topics include connections on vector bundles, moving frames, Hodge theory, spectral geometry, and characteristic classes.
• 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 744: Algebraic Number Theory II
  Advanced topics in number theory. Topics chosen from: Zeta functions of number fields of algebraic varieties; arithmetic of elliptic curves; modular forms and modular curves; class field theory; and Iwasawa theory.

Back to full list of Courses