Mathematics & Statistics

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  • 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
    Undergraduate Prerequisites: CAS MA 225 or CAS MA 230.
    Graduate Prerequisites: CAS MA 225 or CAS MA 230.
    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
    Undergraduate Prerequisites: CAS MA 555 and CAS MA 242 or CAS MA 442; or consent of instructor.
    Graduate Prerequisites: CAS MA 555 and CAS MA 242 or CAS MA 442; or consent of instructor.
    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
    Undergraduate Prerequisites: CAS MA 226 or CAS MA 231.
    Graduate Prerequisites: CAS MA 226 or CAS MA 231.
    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
    Undergraduate Prerequisites: CAS MA 561.
    Graduate Prerequisites: CAS MA 561.
    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
    Undergraduate Prerequisites: CAS MA 225 or CAS MA 230; and (CASMA411 or CASMA511); or consent of instructor.
    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
    Undergraduate Prerequisites: CAS MA 225 or CAS MA 230; and (CASMA411 or CASMA511); or consent of instructor.
    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
    Undergraduate Prerequisites: CAS MA 226 or CAS MA 231.
    Graduate Prerequisites: CAS MA 226 or CAS MA 231.
    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 568: Statistical Analysis of Point Process Data
    Undergraduate Prerequisites: CAS MA 213 and CAS MA 214; or consent of instructor.
    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
    Undergraduate Prerequisites: CAS MA 225 or CAS MA 230; , CASMA242 OR CASMA442
    Graduate Prerequisites: CAS MA 225 or CAS MA 230 or CAS MA 242 or CAS MA 442.
    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
    Undergraduate Prerequisites: CAS MA 225 or CAS MA 230 and CAS MA 242 or CAS MA 442.
    Graduate Prerequisites: CAS MA 225 or CAS MA 230 and CAS MA 242 or CAS MA 442.
    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
    Undergraduate Prerequisites: (CASMA226 OR CASMA231) & (CASMA242 OR CASMA442)
    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 575: Linear Models
    Undergraduate Prerequisites: CAS MA 214 ; CAS MA 242 ; CAS MA 581; or consent of instructor.
    Post-introductory course in linear models, with focus on both principles and practice. Simple and multiple linear regression, weighted and generalized least squares, polynomials and factors, transformations, regression diagnostics, variable selection, and a selection from topics on extensions of linear models.
  • CAS MA 576: Generalized Linear Models
    Undergraduate Prerequisites: CAS MA 575; or consent of instructor.
    Covers topics in linear models beyond MA 575: generalized linear models, analysis of binary and polytomous data, log-linear models, multivariate response models, non-linear models, graphical models, and relevant model selection techniques. Additional topics in modern regression as time allows.
  • CAS MA 577: Mathematics of Financial Derivatives
    Undergraduate Prerequisites: CAS MA 581; or consent of instructor.
    Develops the probabilistic tools used in finance and presents the methodologies that are used in the pricing of financial derivatives. No previous knowledge of finance is required.
  • CAS MA 579: Numerical Methods for Biological Sciences
    Undergraduate Prerequisites: CAS MA 226 or CAS MA 231; or equivalent, and elementary knowledge of linear algebra.
    Introduction to the use of numerical methods for studying mathematical models of biological systems. Emphasis on the development of these methods; understanding their accuracy, performance, and stability; and their application to the study of biological systems.
  • CAS MA 581: Probability
    Undergraduate Prerequisites: CAS MA 225 or CAS MA 230; or consent of instructor.
    Graduate Prerequisites: CAS MA 225 or CAS MA 230; or consent of instructor.
    Basic probability, conditional probability, independence. Discrete and continuous random variables, mean and variance, functions of random variables, moment generating function. Jointly distributed random variables, conditional distributions, independent random variables. Methods of transformations, law of large numbers, central limit theorem. Cannot be taken for credit in addition to CAS MA 381.
  • CAS MA 582: Mathematical Statistics
    Undergraduate Prerequisites: CAS MA 581 or CAS MA 381.
    Graduate Prerequisites: CAS MA 581.
    Point estimation including unbiasedness, efficiency, consistency, sufficiency, minimum variance unbiased estimator, Rao-Blackwell theorem, and Rao-Cramer inequality. Maximum likelihood and method of moment estimations; interval estimation; tests of hypothesis, uniformly most powerful tests, uniformly most powerful unbiased tests, likelihood ratio test, and chi-square test.
  • CAS MA 583: Introduction to Stochastic Processes
    Undergraduate Prerequisites: CAS MA 581 or CAS MA 381; or consent of instructor.
    Graduate Prerequisites: CAS MA 581; or consent of instructor.
    Basic concepts and techniques of stochastic process as they are most often used to construct models for a variety of problems of practical interest. Topics include Markov chains, Poisson process, birth and death processes, queuing theory, renewal processes, and reliability.