Courses

• 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 547: Topics in Number Theory
  An exploration of rational arithmetic and its generalizations. Foundations of arithmetic, Euclid's algorithm; the fundamental theorem of arithmetic; arithmetic modulom; continued fractions; Diophantine approximation; Pell's equation; sums of squares; the arithmetic of polynomials over a field; quadratic reciprocity; arithmetic in quadratic number field; lattice point-free regions; Minkowski's theorem on convex bodies
• CAS MA 548: Problem Solving in Number Theory
  Mathematical heuristics, including good use of language and symbolism, and techniques of exploration and discovery. Through intensive work on a large assortment of unusually challenging problems in number theory students practice the art of mathematical discovery--numerical exploration, formulation and critique of conjectures, and techniques of proof and generalization.
• CAS MA 549: Geometry and Symmetry
  Problem-oriented seminar in modern geometry focusing on invariants of transformation groups. Specific topics may include Euclidean and plane geometry, Hilbert's Axioms, conics, tilings, finite, projective, spherical and/or hyperbolic geometry, tessellations, applications to number theory, Platonic Solids.
• 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 557: Mathematical Structures in Physics I
  Relativistic wave equations, quantum equations of motion, Feynman graphs, combinatorics of perturbative expansions and Hopf algebras, renormalization and elimination of divergences, locality of fields, scaling transformations and renormalization group, basic applications to particle physics and condensed matter theory.
• 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.
• CAS MA 573: Qualitative Theory of Ordinary Differential Equations
  Eigenvalues, eigenvectors, Jordan normal forms. Linear systems of differential equations, Phase portrait, Hamiltonian systems, stability theory. Applications to systems arising in mechanics, economics, ecology, electrical circuit theory, etc.
• CAS MA 574: Applied Nonlinear Dynamics
  Attractors and invariant measures for nonlinear dynamical systems. Measures of chaos such as Lyapunov exponents. Time series analysis. Multiple time scales and singular perturbation theory. Synchronization in coupled oscillators. Strong emphasis on applications to realistic biological and mechanical systems.

Note that this information may change at any time.

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