Mathematics & Statistics

• CAS MA 505: History of Mathematics
  Patterns of mathematical thought from antiquity to the seventeenth century. Emphasis throughout on the background and origins of the mathematical revolution of the seventeenth century in which Descartes, Newton, and Leibniz played key roles.
• CAS MA 511: Introduction to Analysis I
  Fundamental concepts of mathematical reasoning. Properties of the real-number system, elementary point-set theory, metric spaces. Limits, sequences, series, convergence, uniform convergence, continuity. Differentiability for functions of a single variable, Riemann-Stieltjes integration.
• CAS MA 512: Introduction to Analysis II
  Background of CAS MA 511 used to develop further topics of calculus. Exponential and logarithmic functions, Taylor series, power series, real analytic functions. Differential and integral calculus for functions of several variables. Line and surface integrals, divergence theorem, Stokes's theorem, inverse and implicit function theorems, change of variables. Fourier analysis.
• CAS MA 528: Introduction to Modern Geometry
  The foundations of Euclidean geometry. Transformation and symmetries in the plane, inversive and projective planes, coordinates, conics and quadrics, the Golden Section, intermediary and Dedekind's axiom, models for non-Euclidean geometries.
• CAS MA 531: Mathematical Logic
  The syntax and semantics of sentential and quantificational logic, culminating in the Gödel Completeness Theorem. The Gödel Incompleteness Theorem and its ramifications for computability and philosophy. Also offered as CAS PH 461.
• CAS MA 532: Foundations of Mathematics
  Axiomatic set theory as a foundation for mathematics and as a field of mathematics: Axiom of Choice, the Continuum Hypothesis, and consistency results.
• CAS MA 539: Methods of Scientific Computing
  (Meets with CAS CS 539.) An introduction to topics including computational linear algebra, solutions of linear equations, numerical integration and solution of differential equations, finite element methods, and methods of stochastic simulation (i.e., Monte Carlo methods).
• CAS MA 541: Modern Algebra I
  Basic properties of groups, Sylow theorems, basic properties of rings and ideals, Euclidean rings, polynomial rings.
• CAS MA 542: Modern Algebra II
  Vector spaces and modules, Galois theory, linear transformations and matrices, canonical forms, bilinear and quadratic forms.
• CAS MA 555: Numerical Analysis I
  Numerical solutions of equations, iterative methods, analysis of sequences. Theory of interpolation and functional approximation, divided differences. Numerical differentiation and integration. Polynomial theory. Ordinary differential equations.
• CAS MA 556: Numerical Analysis II
  Numerical linear algebra; norms, elimination methods, error analysis, conditioning, eigenvalues, iterative methods, least squares and nonlinear functional minimization. Partial differentiation equation boundary value and initial value problems. Finite element methods. Legendre and Chebyshev polynomials. Treatment in greater depth of selected topics from CAS MA 555.
• CAS MA 561: Methods of Applied Mathematics I
  Derivation and analysis of the classical equations of mathematical physics; heat equation, wave equation, and potential equation. Initial boundary value problems, method of separation of variables, eigenvalue problems, eigenfunction expansions. Fourier analysis. Existence and uniqueness of solution.
• CAS MA 562: Methods of Applied Mathematics II
  Calculus of variations, first-order non-linear partial differential equations, Hamilton-Jacobi theory, Rayleigh-Ritz procedure, perturbation methods.
• CAS MA 563: Introduction to Differential Geometry
  Study of local properties of curves and surfaces in the three-dimensional Euclidean space; curvature, torsion, Frenet equations, tangent and normal planes; first and second fundamental form; developable surfaces, principal, mean and Gaussian curvature; vector fields, covariant differentiation, geodesics, surfaces of constant curvature.
• CAS MA 564: Introduction to Topology
  Introduction to point set and algebraic topology. Topological spaces and continuity. Compactness and connectedness. Metrizable topological spaces. Product topology and Tychonoff's theorem. The fundamental group and van Kampen's theorem. Covering spaces and the universal cover.
• CAS MA 565: Mathematical Models in the Life Sciences
  An introduction to mathematical modeling, using applications in the biological sciences. Mathematics includes linear difference and differential equations, and an introduction to nonlinear phenomena and qualitative methods. An elementary knowledge of differential equations and linear algebra is assumed.
• CAS MA 566: Geometric Methods in Mechanics
  Modern geometric theories applied to motion of physical objects. Differential forms. Symplectic manifolds. Lie groups and their Lie algebras. Hamiltonian and Lagrangian systems; Liouville's theorem. Poincaré’s return theorem, Noether's theorem. Additional topics according to instructor.
• CAS MA 568: Statistical Analysis of Point Process Data
  Introduces the theory of point processes and develops practical problem-solving skills to construct models, assess goodness-of-fit, and perform estimation from point process data. Applications to neural data, earthquake analysis, financial modeling, and queuing theory.
• CAS MA 569: Optimization Methods of Operations Research
  Optimization of linear functions: linear programming, simplex method; transportation, assignment, and network problems. Optimization of non-linear functions: unconstrained optima, constrained optima and Lagrange multipliers, Kuhn-Tucker conditions, calculus of variations, and Euler's equation.
• CAS MA 570: Stochastic Methods of Operations Research
  Poisson processes, Markov chains, queuing theory. Matrix differential equations, differential-difference equations, probability-generating functions, single- and multiple-channel queues, steady-state and transient distributions.

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