These are the Graduate Courses of the Institute of Mathematics.

 

COURSE NO. DESCRIPTION SYLLABUS
Math 201

Concepts and Techniques in Abstract Algebra
Groups, rings and homomorphisms.

Prerequisite: Math 109 or COI

Credit: 3 u.

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Math 202.1

Analysis I
Real numbers; sequences of real numbers and limits; continuity of functions; derivatives; Riemann integral.

Prerequisite: COI

Credit: 3 u.

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Math 202.2

Analysis II
n-dimensional Euclidean space; functions of several variables; partial derivatives; multiple integrals;  complex-valued functions and their derivatives

Prerequisite: Math 202.1

Credit: 3 u.

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Math 203

Matrices and Applications
Linear systems of equations and matrices; matrix operations; determinants; vector spaces,linear transformations; eigenvalues, eigenvectors, applications.

Prerequisite: COI

Credit: 3 u.

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Math 204

Classical and Modern Geometry
Finite geometries, euclidean and non-euclidean geometries,projective geometry, geometric transformations

Prerequisite: COI

Credit: 3 u.

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Math 205

Concepts and Ms in Probability and Statistics
Descriptive statistics, probability and probability distributions, sampling theory, estimation and test of hypothesis, linear correlation and regression analysis.

Prerequisite: COI

Credit: 3 u.

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Math 208

History and Development of the Fundamental Concepts of Mathematics
The Development of Mathematics: a historical over; The Nature of Mathematics, Issues and Aspects of Mathematics.

Prerequisite: COI

Credit: 3 u.

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Math 209.1

Selected Topics in Applied Mathematics

Prerequisite: COI

Credit: 3 u.

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Math 209.2

Selected Topics in Discrete Mathematics

Prerequisite: Math 201

Credit: 3 u.

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Math 210.1

Modern Algebra I
Semi-groups and groups; rings; fields; groups with operators; selected topics.

Prerequisite: COI

Credit: 3 u.

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Math 210.2

Modern Algebra II
Continuation of Math 210.1

Prerequisite: Math 210.1

Credit: 3 u.

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Math 211

Linear Algebra
Vector spaces; linear mappings, theorem o Hamilton-Cayley; modules over principal ideal domains; Jordan canonical forms; rational canonical forms; bilinear forms; inner products; law of inertia; spectral theorem; multi-linear forms; tensors products.

Prerequisite: Math 110.2 or Math 114 or COI

Credit: 3 u.

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Math 212

Theory of Groups
Definitions and examples; normal subgroups and homomorphisms; Abelian groups; Sylow theorems; composition series and solvable groups.

Prerequisite: COI

Credit: 3 u.

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Math 213

Theory of Rings
Representation and structure of rings; Ideal Theory.

Prerequisite: COI

Credit: 3 u.

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Math 214

Theory of Matrices

Prerequisite: COI

Credit: 3 u.

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Math 216

Lie Groups and Lie Algebras
Classical matrix Lie groups, Lie algebras of Lie groups, nilpotent and solvable algebras, semi-simple algebras, representations.

Prerequisite: Math 210.1

Credit: 3 u.

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Math 217

Theory of Numbers
Linear Congruences, Euler's and Wilson 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.

Prerequisite: COI

Credit: 3 u.

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Math 218

Theory of Algebraic Numbers
Algebraic number fields; algebraic integrals; basic and discriminants ideals, fundamental theorem on the decomposition of ideals; ideal classes; Minckowski's theorem; the class formula; units; Fermat's last theorem.

Prerequisite: COI

Credit: 3 u.

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Math 220.1

Theory of Functions of a Real Variable I
Lebesque and other integrals; differentiation; measure theory.

Prerequisite: Math 123.1 or COI

Credit: 3 u.

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Math 220.2

Theory of Functions of a Real Variable II
Continuation of Math 220.1; selected topics

Prerequisite: COI

Credit: 3 u.

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Math 221

Partial Differential Equations
Equations of the firsts and second order; Green's functions; boundary value problems.

Prerequisite: Math 220.1

Credit: 3 u.

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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.

Prerequisite: Math 220.1 or COI

Credit: 3 u.

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Math 224

Control Theory
Elements of the calculus of variations. Naïve optimal control theory; Functional analysis; Generalized optimal control theory; the Pontrajgin maximum principle for chattering controls; Research problems.

Prerequisite: Math 126 and Math 142 or equivalent

Credit: 3 u.

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Math 227

Calculus of Variations
Euler's equation; Legendre conditions; Jacobi's conditions; Isoperimitic problems; Lagrange method; Dirichlet's principle.

Prerequisite: COI

Credit: 3 u.

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Math 228

Theory of Functions of a Complex Variable
Analytic functions; geometric functions theory; analytic continuation; Reimann mapping theorem

Prerequisite: COI

Credit: 3 u.

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Math 229

Functional Analysis
Linear operators; linear functionals; topological linear spaces; normed spaces; Hilbert spaces; functional equations; Randon measures; distributive and linear partial differential equations and spectral analysis.

Prerequisite: Math 220.1

Credit: 3 u.

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Math 235

Mathematics in Population Biology
Continuous and discrete population models for single species, models for interacting populations, evolutionary models, dynamics of infectious diseases.

Prerequisite: Math 121.1 or equivalent, or COI

Credit: 3 u.

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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.

Prerequisite: COI

Credit: 3 u.

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Math 240

Geometric Crystallography
Isometries, frieze groups, crystallographic groups, lattices and invariant sublattices, finite groups of isometries, geometric and arithmetic crystal classes.

Prerequisite: Math 210.1 or equivalent

Credit: 3 u.

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Math 241

Hyperbolic Geometry
Moebius transformations, hyperbolic plane and hyperbolic metric, geometry of geodesics, hyperbolic geometry, groups of isometries on the hyperbolic plane.

Prerequisite: Math 210.1 or equivalent

Credit: 3 u.

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Math 242

General Topology
Topological spaces; metric spaces; theory of convergence; bases; axioms of countability; subspaces; homomorphisms; selected topics.

Prerequisite: COI

Credit: 3 u.

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Math 243

Algebraic Topology
Homotopy, fundamental group, singular homology, simplicial complexes, degree and fixed point theorems.

Prerequisite: Math 242

Credit: 3 u.

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Math 246

Differential Geometry
Classsical Theory of curves and surfaces; mappings of surfaces; differential structures; Lie groups and frame bundles.

Prerequisite: Math 123.2 of COI

Credit: 3 u.

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Math 247

Algebraic Geometry
The general projective space; collineation and correlation in a projective space; algebraic manifolds; plane curves; quadratic transformation of systems of plane curves.

Prerequisite: COI

Credit: 3 u.

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Math 249

Selected Topics in Geometry and Topology
topic to be specified for record purposes

Prerequisite: COI

Credit: 3 u.

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Math 250

Probability Theory
Random variables, laws of large numbers, special probability distributions, central limit theorem, Markov chains, Poisson process, martingales.

Prerequisite: Math 220.1 or COI

Credit: 3 u.

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Math 255

Mathematics of Decision Making
Some applications of Bayesian statistics; use of experiments in decision problems; group decision making and risk sharing.

Prerequisite: Math 155

Credit: 3 u.

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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.

Prerequisite: Math COI

Credit: 3 u.

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Math 260

Actuarial Theory and Practice
Multiple life theory, multiple decrement theory, applications of multiple decrement theory, risk theory and introduction to credibility theory.

Prerequisite: Math 164 or equivalent

Credit: 3 u.

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Math 261

Survival and Loss Models.
Hazard rate function, analysis of various survival and loss models, credibility theory.

Prerequisite: Math 164 or equivalent

Credit: 3 u.

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Math 262.1

Actuarial Science I
Gross premiums and asset shares, non forfeiture values, expense analysis, distribution of surplus, valuation of liabilities, product development process, introduction to life insurance accounting.

Prerequisite: Math 261 or COI

Credit: 3 u.

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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.

Prerequisite: Math 262.1 or COI

Credit: 3 u.

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Math 265

Stochastic Calculus
Conditional expectations, martingales, Brownian motion, Ito integral, Ito formula, stochastic differential equation, Girsanov Theorem, applications to mathematical finance.

Prerequisite: Math 150.1 or COI

Credit: 3 u.

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Math 266

Mathematical Finance
Binomial asset pricing model, vanilla options, exotic options, American options, arbitrage probabilities, profit and loss, stochastic interest rates.

Prerequisite: Math 265 or COI

Credit: 3 u.

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Math 271.1

Numerical Analysis I
Floating point representation, condition numbers, iterative methods for solving systems of linear and non-linear equations, numerical integration, numerical linear algebra.

Prerequisite: Math 171 or COI

Credit: 3 u.

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Math 271.2

Numerical Analysis II
Numerical methods for ordinary differential equations, finite-difference methods for partial differential equations, numerical methods for conservation laws, multi-grid methods.

Prerequisite: Math 271.1 or COI

Credit: 3 u.

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Math 272

Automata Theory
Finite state automata. Regular expressions, decomposition of finite automata and their realization. Turing machines.  Introducton to formal languages.

Prerequisite: COI

Credit: 3 u.

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Math 276

Introduction to Computer Simulation
Introduction to computer simulation of theoretical system and real-time processes.  Examples of simulation for the solution of both theoretical and practical problems in various fields of applications.

Prerequisite: COI

Credit: 3 u.

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Math 280

Linear Programming
Simplex method, duality, geometry of linear programs, parametric programming, decomposition and upper-bounded variables.

Prerequisite: Math 114 and Math 180.2

Credit: 3 u.

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Math 281

Nonlinear Programming
Properties of convex sets and functions; unconstrained optimization; Kuhn - Tucker theorem.  Lagrange multipliers.  Saddle-point Theorems; algorithms.

Prerequisite: Math 280 or equivalent

Credit: 3 u.

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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.

Prerequisite: Math 280 or equivalent

Credit: 3 u.

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Math 283

Applied Dynamic Programming
Deterministic decision problems; Analytical and computational methods; Applicatons to problems of equipment replacement, resource allocation, scheduling, search and routing.

Prerequisite: Graduating status or COI

Credit: 3 u.

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Math 285

Introduction to Stochastic Optimization
Probability thoery and applications to discrete and continuous Markov chains; classification of states; algebraic methods, birth and death processes, renewal theory, limit theorems.

Prerequisite: Math 114 and Math 150.1

Credit: 3 u.

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Math 286

Finite graphs and Networks
Basic graph theory and applications to optimal path problems; flows in network; combinatorial problems.

Prerequisite: Math 285 or COI

Credit: 3 u.

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Math 288

Numerical Optimization.
Deterministic descent type methods, stochastic optimization methods, numerical implementation.

Prerequisite: Math 271.1 or COI

Credit: 3 u.

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Math 290

Research Paper on College Mathematics

Prerequisite: COI

Credit: 3 u.

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Math 294

Independent Study

Credit: 3 u. May be credited once in the M.S. Mathematics/ Applied Mathematics programs and twice in the Ph.D. Mathematics program.

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Math 295

Special Project

Prerequisite: COI

Credit: 3 u.

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Math 296

Graduate Seminar

Credit: 1 u.

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Math 297

Special Topics

Credit: 3 u. ; topic to be specified for record purposes.

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Math 300

Master's Thesis

Credit: 6 u.

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Math 400

Ph.D. Dissertation

Credit: 12 u.

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