# Graduate Courses

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Code | Course Description |
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Math 201 | Concepts and Techniques in Abstract Algebra
Groups, rings and homomorphism.
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Prerequisite: Math 109/COI Number of units: 3 Number of hours/week: 3 |

Math 202.1 | Analysis I
Real numbers, sequences of real numbers and limits, continuity of functions, derivatives, Riemann integral.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 202.2 | Analysis II
\(n\)-dimensional Euclidean space, functions of several variables, partial derivatives, multiple integrals, complex-valued functions and their derivatives.
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Prerequisite: Math 202.1 Number of units: 3 Number of hours/week: 3 |

Math 203 | Matrices and Applications
Linear systems of equations and matrices, matrix operations, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, applications.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 204 | Classical and Modern Geometry
Finite geometries, euclidean and non-euclidean geometries, projective geometry, geometric transformations.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 205 | Concepts and Methods in Probability and Statistics
Descriptive statistics, probability and probability distributions, sampling theory, estimation and test of hypothesis, linear correlation and regression analysis.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 208 | History and Development of the Fundamental Concepts of Mathematics
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 209.1 | Selected Topics in Applied Mathematics
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 209.2 | Selected Topics in Discrete Mathematics
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Prerequisite: Math 201 Number of units: 3 Number of hours/week: 3 |

Math 210.1 | Modern Algebra I
Semigroups and groups; rings; fields; groups with operators. Selected topics.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 210.2 | Modern Algebra II
A continuation of Mathematics 210.1.
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Prerequisite: Math 210.1 Number of units: 3 Number of hours/week: 3 |

Math 211 | Linear Algebra
Vector spaces, linear mappings; theorem of Hamilton-Cayley; modules over principal ideal domains; Jordan canonical form, rational canonical form; bilinear forms, inner products; law of inertia, spectral theorem; multilinear forms; tensor products.
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Prerequisite: Math 110.2/114/COI Number of units: 3 Number of hours/week: 3 |

Math 214 | Theory of Matrices
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 216 | Lie Groups and Lie Algebras
Classical matrix Lie groups, Lie algebras of Lie groups, nilpotent and solvable algebras, semisimple algebras, representations.
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Prerequisite: Math 210.1 Number of units: 3 Number of hours/week: 3 |

Math 217 | Theory of Numbers
Linear Congruences, Euler’s and Wilson’s Theorems, Quadratic residues, Quadratic Reciprocity Law, Jacobi’s and Kronocker’s symbols, Polian Equation, Positive Binary and Ternary quadratic forms. Theory of the sums of two and three squares.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 218 | Theory of Algebraic Numbers
Algebraic number fields; algebraic integers; basic and discriminant; ideals; fundamental theorem on the decomposition of ideals; ideal classes; Minkowski’s theorem; the class formula; units; Fermat’s last theorem. Selected topics.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 220.1 | Theory of Functions of a Real Variable I
Lebesgue and other integrals; differentiation; measure theory.
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Prerequisite: Math 123.1/COI Number of units: 3 Number of hours/week: 3 |

Math 220.2 | Theory of Functions of a Real Variable II
Continuation of Math 220.1. Selected topics.
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Prerequisite: Math 220.1 Number of units: 3 Number of hours/week: 3 |

Math 221 | Partial Differential Equations
Equations of the first and second order. Green’s function. Boundary value problems.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 222 | Approximation Theory
Taylor’s theorem, Weierstrass approximation theorem, approximation in Hilbert spaces, Fourier Series and Fourier transform, direct and inverse theorems, algebraic and trigonometric interpolation, Whittaker-Shannon sampling theory, wavelet analysis.
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Prerequisite: Math 220.1/COI Number of units: 3 Number of hours/week: 3 |

Math 224 | Control Theory
Elements of the calculus of variations. Naive optimal control theory; Functional analysis; Generalized optimal control theory; The Pontrjagin maximum principle for chattering controls; Research problems.
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Prerequisite: Math 126, 142/COI Number of units: 3 Number of hours/week: 3 |

Math 227 | Calculus of Variation
Euler’s equations. Legendre conditions. Jacobi’s conditions. Isoperimetric problems. Lagrange’s methods. Dirichlet’s principle.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 228 | Theory of Functions of a Complex Variable
Analytic functions; geometric function theory; analytic continuation; Riemann Mapping Theorem.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 229 | Functional Analysis
Linear operators, linear functionals, topological linear spaces, normed spaces, Hilbert spaces, functional equations, Radon measures, distributive and linear partial differential equations, and spectral analysis.
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Prerequisite: Math 220.1 Number of units: 3 Number of hours/week: 3 |

Math 235 | Mathematics in Population Biology
Continuous and discrete population models for single species, models for interacting populations, evolutionary models, dynamics of infectious diseases.
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Prerequisite: Math 121.1/equiv/COI Number of units: 3 Number of hours/week: 3 |

Math 236 | Mathematics in Biological Processes
Biological oscillators and switches, perturbed and coupled oscillators, reaction diffusion, enzyme kinetics, chemotaxis, circadian systems models, coupled cell networks.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 240 | Geometric Crystallography
Isometries, frieze groups, crystallographic groups, lattices and invariant sublattices, finite groups of isometries, geometric and arithmetic crystal classes.
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Prerequisite: Math 210.1/equiv Number of units: 3 Number of hours/week: 3 |

Math 241 | Hyperbolic Geometry
Moebius transformations, hyperbolic plane and hyperbolic metric, geometry of geodesics, hyperbolic trigonometry, groups of isometries on the hyperbolic plane.
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Prerequisite: Math 210.1/equiv Number of units: 3 Number of hours/week: 3 |

Math 242 | General Topology
Topological spaces; metric spaces; theory of convergence; bases; axioms of countability; subspaces; homeomorphisms. Selected topics.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 243 | Algebraic Topology
Homotopy, fundamental group, singular homology, simplicial complexes, degree and fixed point theorems.
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Prerequisite: Math 242 Number of units: 3 Number of hours/week: 3 |

Math 246 | Differential Geometry
Classical theory of curves and surfaces. Mappings of surfaces. Differential structures. Lie groups and frame bundles.
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Prerequisite: Math 123.2/COI Number of units: 3 Number of hours/week: 3 |

Math 247 | Algebraic Geometry
The general projective space. Collineation and correlations in a projective space. Algebraic manifolds. Plane curves. Quadratic transformation of systems of plane curves.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 249 | Selected Topics in Geometry and Topology
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 Additional notes: Topic to be specified for record purposes. |

Math 250 | Probability Theory
Random variables, laws of large numbers, special probability distributions, central limit theorem, Markov chains, Poisson process, martingales.
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Prerequisite: Math 220.1/COI Number of units: 3 Number of hours/week: 3 |

Math 258 | Combinatorial Mathematics
Permutations and combinations. Generating functions. Principle of inclusion and exclusion. Recurrence relations. Occupancy. Matrices of zeros and ones. Partitions. Orthogonal Latin squares. Combinatorial designs.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 260 | Actuarial Theory and Practice
Multiple life theory, multiple decrement theory, applications of multiple decrement theory, risk theory, ruin theory and introduction to credibility theory.
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Prerequisite: Math 164/COI Number of units: 3 Number of hours/week: 3 |

Math 261 | Survival and Loss Models
Hazard rate function, analysis of various survival and loss models, credibility theory.
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Prerequisite: Math 164/COI Number of units: 3 Number of hours/week: 3 |

Math 262.1 | Actuarial Science I
Gross premiums and asset shares, nonforfeiture values, expense analysis, distribution of surplus, valuation of liabilities, product development process, introduction to life insurance accounting.
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Prerequisite: Math 261/COI Number of units: 3 Number of hours/week: 3 |

Math 262.2 | Actuarial Science II
Selection of risks, reinsurance, introduction to investments analysis and finance management, insurance code, actuarial principles in special lines of insurance.
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Prerequisite: Math 262.1/COI Number of units: 3 Number of hours/week: 3 |

Math 265 | Stochastic Calculus
Conditional expectations, martingales, Brownian motion, Ito integral, Ito formula, stochastic differential equation, Girsanov Theorem, applications to mathematical finance.
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Prerequisite: Math 150.1/COI Number of units: 3 Number of hours/week: 3 |

Math 266 | Mathematical Finance
Binomial asset pricing model, vanilla options, exotic options, American options, arbitrage probabilities, profit and loss, stochastic interest rates.
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Prerequisite: Math 265/COI Number of units: 3 Number of hours/week: 3 |

Math 271.1 | Numerical Analysis I
Floating point representation, condition numbers, iterative methods for solving systems of linear and nonlinear equations, numerical integration, numerical linear algebra.
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Prerequisite: Math 171/COI Number of units: 3 Number of hours/week: 3 |

Math 271.2 | Numerical Analysis II
Numerical methods for ordinary differential equations, finite difference methods for partial difference equations, numerical methods for conservation laws, multi-grid methods.
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Prerequisite: Math 271.1/COI Number of units: 3 Number of hours/week: 3 |

Math 280 | Linear Programming
Theories and methods in solving linear programs.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 281 | Nonlinear Programming
Properties of convex sets and functions. Unconstrained optimization. Kuhn- Tucker Theorem. Lagrange Multipliers. Saddle-point Theorems. Algorithms.
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 282 | Integer Programming and Combinatorial Optimization
Applications of integer programming. Converging dual and primal cutting plane algorithms. Branch-bound methods. Total unimodularity and the transportation problem. Applications of graph theory to mathematical programming.
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Prerequisite: Math 280/equiv Number of units: 3 Number of hours/week: 3 |

Math 283 | Applied Dynamic Programming
Deterministic decision problems; Analytical and computational methods; Applications to problems of equipment replacement, resource allocation, scheduling, search and routing.
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Prerequisite: Graduating Standing/COI Number of units: 3 Number of hours/week: 3 |

Math 285 | Introduction to Stochastic Optimization
Probability theory and applications to discrete and continuous time Markov chains; classification of states; algebraic methods, birth and death processes, renewal theory, limit theorems.
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Prerequisite: Math 114, 150.1 Number of units: 3 Number of hours/week: 3 |

Math 286 | Finite Graphs and Networks
Basic graph theory and applications to optimal path problems; flows in network; combinatorial problems.
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Prerequisite: Math 285/COI Number of units: 3 Number of hours/week: 3 |

Math 288 | Numerical Optimization
Deterministic descent type methods, stochastic optimization methods, numerical implementation.
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Prerequisite: Math 271.1/COI Number of units: 3 Number of hours/week: 3 |

Math 290 | Research Paper on College Mathematics
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 |

Math 294 | Independent Study
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Number of units: 3 Additional notes: May be credited once in the M.S. Mathematics/Applied Mathematics programs and twice in the Ph.D. Mathematics program. |

Math 295 | Special Project
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Prerequisite: COI Number of units: 3 |

Math 296 | Graduate Seminar
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Prerequisite: COI Number of units: 1 Number of hours/week: 1 |

Math 297 | Special Topics
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Prerequisite: COI Number of units: 3 Number of hours/week: 3 Additional notes: May be taken at most three times; topic to be specified for record purposes. |

Math 300 | Master’s Thesis
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Number of units: 6 |

Math 400 | PhD Dissertation
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Number of units: 12 |