UNDERGRADUATE CURRICULUM
FIRST YEAR
First Semester
MATH 111 Fundamentals of Math. (3-0)3
MATH 115 Analytic Geometry (3-0)3 
MATH 153 Calculus for MATH. Students I (4-2)5
PHYS 111 Physics I (4-2)5
ENG 101 Development of Reading and Writing Skills I (4-0)4
 
 Second Semester
MATH 112 Introductory Discrete Math. (3-0)3
MATH 116 Basic Algebraic Structures (3-0)3
MATH 154 Calculus for MATH. Students II (4-2)5
PHYS 112 Physics II (Electricity and Magnetism) (4-2)5
ENG 102  Development of Reading and Writing Skills II (4-0)4
IS 100 Introduction to Information
  Systems and Applications     NC 
 
SECOND YEAR
Third Semester
MATH 251 Advanced Calculus I (4-0)4
MATH 261 Linear Algebra I (4-0)4 
CENG 230 Introduction to C Programming (2-2)3
ENG 211 Advanced Reading and Oral Communication (3-0)3
HIST 2201 Principles of Kemal Atatürk I  NC 
 
 Fourth Semester
MATH 252 Advanced Calculus II (3-2)4
MATH 254 Introduction to Differential Equations I (4-0)4
MATH 262 Linear Algebra II (4-0)4
HIST 2202 Principles of Kemal Atat?rk  II NC 
A non-departmental elective  (3-0)3
THIRD YEAR 
Fifth Semester
MATH 349 Int. to Math. Analysis (4-0)4 
MATH 353 Complex Calculus (4-0)4 
MATH 367 Abstract Algebra (3-2)4
A departmental elective (3-0)3
TURK 303 Turkish I NC
 
 Sixth Semester
MATH 358 Partial Diff. Equations (4-0)4
MATH 371 Differential Geometry (3-0)3
A departmental elective (3-0)3
TURK 304 Turkish II NC
Free Elective (3-0)3
FOURTH YEAR 
Seventh Semester
A departmental elective (3-0)3 
A departmental elective (3-0)3
A departmental elective (3-0)3
A free elective (3-0)3
A non-departmental elective (3-0)3
 
 Eighth Semester
A departmental elective (3-0)3 
A departmental elective (3-0)3
A departmental elective (3-0)3
A free elective (3-0)3
A free elective (3-0)3
Departmental Elective: Courses offered by the mathematics department except Math 223, Math 321, Math 387, Math 388, Math 395, Math 396, Math 470, Math 486 are defined as departmental electives.
DOUBLE MAJOR/MINOR PROGRAM IN MATHEMATICS
See http://www.math.metu.edu.tr/~doublemajor for details.


DESCRIPTION OF UNDERGRADUATE COURSES

 
 
MATH 111 Fundamentals of Mathematics (3-0)3 Symbolic logic. Set theory. Cartesian product. Relations. Functions. Injective, surjective and bijective functions. Composition of functions equipotent sets. Countability of sets. More about relations: Equivalence relations. Equivalence classes and partitions. Quotient set. Order relations: partial order, total order, well ordering. Mathematical Induction and recursive definitions of functions.
MATH 112 Introductory Discrete Mathematics (3-0)3 Basic counting: The sum and product rules, the pigeonhole principle, generalized permutations and combinations. The binomial theorem. Discrete probability. Inclusion-Exclusion. Recurrence relations. Introduction to graphs and trees. 
MATH 115 Analytic Geometry (3-0)3 Fundamental principle of Analytic Geometry. Cartesian coordinates in plane and space. Lines in the plane. Review of trigonometry and polar coordinates. Rotation and translation in the plane. Vectors in plane and space. Lines and planes in 3-space. Basics about conics. Basic surfaces in space, cylinders, surface of revolutions, quadratic surfaces. Cylindrical and spherical coordinates. 
MATH 116 Basic Algebraic Structures (3-0)3 Binary operations. Groups. The symmetric group. Subgroups. The order of an element. Cyclic groups. Rings. Integral domains. Subrings. Ideals. Fields: Q, R, C, Zp The concept of an isomorphism. The ring of integers and the ring of polynomials over a field: Division and Euclidean algorithms. GCD and LCM. Prime factorization. Quotient structures. 
MATH 125 Basic Mathematics I (3-2)4 Logic. Relations and Functions. Matrices and determinants. Inverse of a matrix, matrix polynomials, Cayley-Hamilton theorem. Systems of linear equations, parametric solutions. Counting: principle of inclusion exclusion, pigeonhole principle. Mathematical induction, recursive relations. Permutations, combinations. Discrete probability. Graphs. 
MATH 126 Basic Mathematics II (3-2)4 Analytic Geometry in R2 , R3. Functions of one and several variables: Limit, continuity and differentiation. Chain rule, implicit differen-tiation. Differential calculus, optimization, Lagrange multipliers. The definite integral. The indefinite integral. Logarithmic and exponential functions. Techniques of integration: Integration by substitution, integration by parts, by partial fractions. 
MATH 153 Calculus for Mathematics Students I (4-2)5 Functions, limit and derivative of a function of a single variable, A thorough discussion of the basic theorems of differential calculus: Intermediate value, extreme value, and the Mean Value Theorems, applications: Graph sketching and problems of extrema. 
MATH 154 Calculus for Mathematics Students II (4-2)5 The Riemann Integral, Mean Value Theorem for integrals, Fundamental Theorem of Calculus, Techniques to evaluate anti-derivative, families, various geometric and physical applications. Sequences, improper Integrals, infinite series of constants, power series and Taylor's series with applications. 
Prerequisite: MATH 153.
MATH 155 Calculus I (Accelerated) (4-2)5 Limits and derivatives. The Mean Value Theorem. Definite and indefinite integral. The logarithmic, exponential, inverse trigonometric and hyperbolic functions. L'Hospital rule. Techniques of integration. Numerical methods of integration. Applications to geometry and physics. Area in polar coordinates. Improper integrals. Sequences. Infinite series, power series and Taylor's series.
MATH 156 Calculus II (Accelerated) (4-2)5 Complex numbers. Vectors, lines and planes in space, scalar and vector products. Vector valued functions. Space curves. Functions of several variables: Limit, continuity, partial derivative, directional derivative. Tangent plane. Extreme values. Method of Lagrange multipliers. Multiple integrals. Cylindrical and spherical coordinates. Line, surface integrals. Green's Theorem. Gauss' and Stokes' Theorems. 
Prerequisite:Math 155 
MATH 157 Basic Calculus I (3-2) 4 Functions, Limits, continuity and derivatives. Applications. Extreme values, the Mean Value Theorem and its applications. Graphing. The definite integral. Area and volume as integrals. The indefinite integral. Transedental functions and their derivatives. L'Hopital's rule. Techniques of integration. Improper integrals. Applications. Parametric curves. Polar coordinates. 
MATH 158 Basic Calculus II (3-2) 4 Infinite series, power series, Taylor series. Vectors, lines and planes in space. Functions of several variables: Limit, continuity, partial derivatives, the chain rule, directional derivatives, tangent plane approximation and differentials, extreme values, Lagrange multipliers. Double and triple integrals with applicatons. The line integral. 
Prerequisite: MATH 157 
MATH 201 Elementary Geometry (3-0)3 
(Only for students of EME 413)
Introduces the axiomatic structures in geometry; Euclidean and non-Euclidean geometries. Provides study in geometry and trigonometry including polygones,similar figures, geometric solids, properties of circles, constructions, right triangles, angle measurement in radians and degrees, trigonometric functions and their applications to right triangles, Phytagorean theorem, laws of sine and cosine, graph of trigonometric functions, trigonometric identities, vectors and coordinate conversions. 
MATH 219 Introduction to Differential Equations First order equations and various applications. Higher order linear differential equations. Power series solutions: The Laplace transform: solution of initial value problems. Systems of linear differential equations: Introduction Partial Differential Equations.
Prerequisite: Math 120
MATH 223 Introduction to Object Oriented Programming and C++ (3-0)3 Programming paradigms, object orientation from C to C++, reference variables, iostream methods, default function arguments, function overloadings and template, dynamic memory allocation, classes,constructors, destructors and manipulators, operator overloading, single and multiple inheritance advanced I/O. 
Prerequisite: CENG 230
MATH 250 Advanced Calculus in Statistics (4-2)5 Review of Multidiemsional Calculus.Derivatives of multivariable functions, continuity of multivariable functions. Fundamental Lemma for differentiability. Chain rule and Taylor's Theorem for multivariable functions. Jacobian. Inverse and implicit function theorems. Topology of R2 and R3. Riemann-Stieltjes integral, integrability. Integrability of continuous functions, sequence of integrable functions. Bounded convergence and Riesz representation theorems. Theorems of integral calculus: Integration in cartesian space. Improper and infinite integrals. Series of functions. 
Prerequisite: MATH 152
MATH 251 Advanced Calculus I (4-0)4 Topology of R, R2 and R 3 . Functions of several variables; limits and continuity. Partial derivatives, directional derivatives, gradients. Differentials and the tangent plane: the Fundamental Lemma, approximations. The Mean Value, implicite and Inverse function theorems. Extreme values. Introduction to vector differential calculus: the gradient, divergence and curl. Curvilinear coordinates. 
Prerequisite: MATH 154
MATH 252 Advanced Calculus II (3-2)4 Double Integrals, polar coordinates. Improper double integrals. Change of variables in double integrals. Triple Integrals: Cylindrical and sphe-rical coordinates Applications. Line integrals: Parametrisation of curves, Green's Theorem, Independence of path, exact differentials. Surface Integrals: Parametrisation, and orien-tation of surfaces. Surface Integrals. Divergence and Stokes' Theorems, applications. 
Prerequisite: MATH 251. 
MATH 254 Introduction to Differential Equations (4-0)4 First order equations and various applications. Higher order linear differential equations. Power series solutions: ordinary and regular singular points. The Laplace transform: solution of initial value problems. Systems of linear differential equations: solutions by operator method, by Laplace transform. 
Prerequisite: MATH 152, MATH 154, MATH 156 or MATH 158.
MATH 258 Differential Equations II (3-0)3 Existence and uniqueness theorems for IVP; first order equations, systems and higher order equations. Structure of linear problems. Boundary value problems and eigenvalue problems. Oscillation and comparison theorems. 
Prerequisite: MATH 254 
MATH 260 Basic Linear Algebra (3-0)3 Matrices, determinants and systems of linear equations. Vector spaces, the Euclidian space, linear transformations. Eigenvalues, diagonalization. 
MATH 261 Linear Algebra I (4-0)4 Matrices and systems of linear equations. Vector spaces; subspaces, sums and direct sums of subspaces. Linear dependence, bases, dimension, quotient spaces. Linear transformations, kernel, range, isomorphism. Spaces of linear transformations, Hom (V,W),V*, V** transpose. Representations of linear transformations by matrices, similarity. Determinants. 
MATH 262 Linear Algebra II (4-0)4 Characteristic and minimal polynomials of an operator, eigenvalues, diagonalizability, canonical forms, Smith normal form, Jordan and rational forms of matrices. Inner product spaces, norm and orthogonality, projections. Linear operators on inner product spaces, adjoint of an operator, normal, self adjoint, unitary and positive operators. Bilinear and quadratic forms. 
Prerequisite: MATH 261 
MATH 301 Introduction to Probability Theory (3-0)3 Events and probability. Combinatorial problems. Independence and conditional probability. Random variables and distribution functions. Marginal distributions and conditional distributions. Moments and characteristic functions. Convergence of random variables. Law of large numbers.
Prerequisite: MATH 252 
MATH 303 History of Mathematical Concepts I (3-0)3 Mathematics in Egypt and Mesopotamia, Ionia and Pythagoreans, paradoxes of Zeno and the heroic age. Mathematical works of Plato, Aristotle, Euclid of Alexandria, Archimedes, Appolonius and Diophantus. Mathematics in China and India. 
Prerequisite: Consent of the instructor. 
MATH 304 History of Mathematical Concepts II (3-0)3 Mathematics of the Renaissance, Islamic contributions. Solution of the cubic equation and consequences. Invention of logarithms. Time of Fermat and Descartes. Development of the limit concept. Newton and Leibniz. The age of Euler. Contributions of Gauss and Cauchy. Non-Euclidean geometries. The arithmetization of analysis. The rise of abstract algebra. Aspects of the twentieth century. 
Prerequisite: Consent of the instructor. 
MATH 319 Lebesgue Integral (3-0)3 Review of Riemann integration. Sets of (Lebesgue) measure zero in Rn and charac-terization of Riemann integrable functions. Lebesgue integrable functions and the Lebesgue integral in Rn. Convergence theorems, theorems of Lusin and Egorov. Fubini's theorem, convolutions. Differentiability properties of functions and integrals. Selected applications of Lebesgue theory. 
Prerequisite: MATH 252. 
MATH 320 Set Theory (3-0)3 Language and axioms of set theory. Ordered pairs, relations and functions. Order relation and well ordered sets. Ordinal numbers, transfinite induction, arithmetic of ordinal numbers. Cardinality and arithmetic of cardinal numbers. Axiom of choice, generalized continuum hypothesis. 
Prerequisite: Consent of the instructor.
MATH 321 Automata and Languages (3-0)3 Automata, finite state automata. Minimal and reduced automatas, transformation monoid. Languages, phrase structure grammers, regular and rational languages, context free languages. Varieties, F-varieties, star free languages and aperiodic monoids. 
Prerequisites: Consent of the instructor 
MATH 341 Graph Theory I (3-0)3 Graphs, varieties of graphs, connectedness, extremal graphs, blocks, trees, partitions, line graphs, planarity, Kuratowsky's theorem, colorability, chromatice numbers, five color theorem, four color conjecture. 
Prerequisite: Consent of the instructor 
MATH 344 Introduction to Universal Algebra (3-0)3 Lattices: distributive and modular lattices, complete and algebraic lattices, Boolean lattices. Semigroups: Green's equivalence, semilattice decomposition, completely simple semigroups. Universal Algebra: algebraic lattices and subinverse, congurence and quotients algebras, free algebras. 
Prerequisites: Consent of the instructor. 
MATH 349 Introduction to Mathematical Analysis (4-0)4 LUB Property of real numbers, metric spaces. Review of sequences and series of scalars with emphasis on rigorous proofs. Sequences and series of functions, uniform convergence, applications. 
Prerequisite: MATH 252. 
MATH 353 Complex Calculus (4-0)4 Algebra of complex numbers. Polar representation. Analyticity. Cauchy-Riemann equations. Power series. Elementary functions. Mapping by elementary functions. Linear fractional transformations. Line integral. Cauchy-Theorem. Cauchy integral formula. Taylor's Series. Laurent series. Residues, Residue theorem. Improper integrals. 
Prerequisite: MATH 252 
MATH 355 Operational Calculus (3-0)3 Fourier series. The Fourier transform, inverse Fourier transform. The Laplace transform. The inversion integral for the Laplace transform (complex contour integration). Applications of Laplace transform to linear ordinary, partial differential and integral equations. The z-transform. The inversion integral for the z-transform. Applications of z-transform to difference equations and linear networks. 
Prerequisite: Consent of the instructor. 
MATH 358 Partial Differential Equations (4-0)4 First order equations; linear, quasilinear and nonlinear equations. Classification of second order linear partial differential equations, canonical forms. The Cauchy problem for the wave equation. Dirichlet and Neumann problems for the Laplace equation, maximum principle. Heat equation on the strip. 
Prerequisite: MATH 252, MATH 254 
MATH 365 Elementary Number Theory I (3-0)3 Divisibility, congruences, Euler, Chinese Remainder and Wilson's Theorems. Arithmetical functions. Primitive roots. Quadratic residues and quadratic reciprocity. Diophantine equations. 
Prerequisite: Consent of the instructor. 
MATH 366 Elementary Number Theory II (3-0)3 Arithmetic in quadratic fields. Factorization theory. Continued fractions, periodicity. Transcendental numbers. 
Prerequisite: Consent of the instructor. 
MATH 367 Abstract Algebra (3-2)4 Groups. Isomorphism theorems, direct pro-ducts. Groups acting on sets, Class equation. Statements of Sylow theorems and the F.T on finite abelian groups. Rings, isomorphism theorems. Prime and maximal ideals. Integral domains, field of fractions. Euclidean domains, PIDs, UFDs. Polynomials, polynomials in several variables. Field extensions.Impossibility of certain geometric constructions. Finite fields. 
Prerequisite:Math 116 or Consent of the department.
MATH 368 Field Extensions and Galois Theory (3-0)3 Field extensions, splitting field of a polynomial, multiple roots, Galois group, criteria for solvability by radicals, Galois group as permutation groups of the roots of polynomials of degree n, constructible n-gons, transcendence of e, finite fields. 
Prerequisite: Math 367 or consent of the instructor.
MATH 371 Differential Geometry (3-0)3 Curves in R3: Frenet formulas and Fundamen-tal Theorem. Regular surfaces. Inverse image of regular values. Differentiable functions on surfaces. Tangent plane; the differential of a map, vector fields, the first fundamental form. Gauss map, second fundamental form, normal, principal curvatures, principal and, asymptotic directions. Gauss map in local coordinates. Covariant derivative, geodesics. 
Prerequisite: MATH 251 and MATH 261. 
MATH 373 Geometries I (3-0)3 Foundations: The parallel axiom, models, Hilbert's theorem. Triangles: Theorems of Menelaus and Ceva, classical remarkable points. Circles: Power of a point with respect to a circle, coaxal systems of circles, inversive geometry. Conic sections: Focus and directrix, reflection property, theorems of Poncelet. 
Prerequisite: Consent of the instructor. 
MATH 374 Geometries II (3-0)3 Projective spaces over division rings. Theorems of Desargues and Pappus. Harmonic ranges and pencils, collineations, correlations, involutions, polarities. Affine geometry via "the line at infinity". Euclidean geometry with "circular points at infinity". Conic sections and quadratic surfaces. 
Prerequisite: MATH 373 
MATH 375 Periodic Distributions and Fourier Series (3-0)3 Properties of periodic functions, convolution, approximation, Weierstrass approximation theorem. Periodic distributions, operations on periodic distributions. Hilbert spaces, L2, orthogonal expansions, Fourier series. Applications of Fourier series. 
Prerequisite: MATH 349. 
MATH 381 Numerical Analysis I (3-0)3 Convergence, stability, error analysis and conditioning. Solving systems of linear equations: The LU and Cholosky factorization, pivoting,error analysis in Gaussian elimination. Matrix eigenvalue problem, power method, orthogonal factorizations and least squares problems. Solutions of nonlinear equations. Bisection, Newton's, secant and fixed point iteration methods. 
Prerequisite: Ceng 230 and Math 262/Ceng 230 and Math 260.
MATH 382 Numerical Analysis II (3-0)3 Approximating functions: polynomial interpolation, divided differences, Hermite interpolation, spline interpolation, The B-splines, Taylor Series, least square app-roximation. Numerical differentiation and integration based on interpolation. Richardson extrapolation, Gaussian quadrature, Romberg integration, adaptive quadrature, Bernoulli polynomials and Euler-Maclaurin formula. 
Prerequisite:MATH 381 
MATH 385 Special Functions of Applied Mathematics I (3-0)3 Gamma and Beta functions. Pochhammer's symbol. Hypergeometric series. Hypergeomet-ric differential equation; ordinary and con-fluent hypergeometric functions. Generalized hypergeometric functions; the contiguous function relations. Bessel function; the functional relationships, Bessel's differential equation. Orthogonality of Bessel functions. 
Prerequisites: MATH 254 or MATH 253 or MATH 254 or consent of the Instructor. 
MATH 386 Special Functions of Applied Mathematics II (3-0)3 Legendre functions: Generating Function of Legendre polynomials. Recurrence relations, series of Legendre polynomials. Legendre differential equation. Associated Legendre functions. Hermite polynomials. recurence relations. Hermite's differential equations. Laguerre functions, Laguerre differential equations, associated Laguerre polynomials. 
Prerequisites: Consent of the Instructor. 
MATH 387 Advanced Object Oriented Programming (3-0)3 Further features of the C++ Language and principles of object oriented programming, including dynamic data-structures, stream input and output, encapsulation, inheritance, polymorphism, and further C ++ language constructs. Some object-oriented design and analysis models and methodology. Covers, case studies of OOP in Mathematics. 
Prerequisite: MATH 223 
MATH 388 Data Structures (3-0)3 Review of stacks, queues, linked lists. Algorithms for searching and sorting. More complicated data structures such as multi-linked lists, trees, graphs. 
Prerequisite: MATH 387, or consent of the instructor. 
MATH 390 Computer Algebra     (3-0)3Introductory  information about  reduce. Structure of programs Built-in prefixs operators.  Procedures. A Computers Algebra system.  How to use a Computers algebra  systems.  Representations of  polynomials, rational  functions, algebraic functions,matrices and series. Simlifications of  polynomial equations and real polynomial systems.Advanced algorithms.   g.c.d.  in several variables. Other applications of  modular methods. P-adic methods. Formal integration and differential equations.
Prerequisites: Consent of the Instructor. 
MATH 395 Symbolic Programing Languages (Prolog) (3-0)3
An overview of Prolog. Syntax and meaningof Prolog programs. Lists, 
operators and arithmetic structures. 
Controlling, backtracking,unification.
Input and output.More bultin procedures Programming sytle and techniques.Recursion 
.Operators on data structures. Advanced tree representatitons. 
Prerequisite: CENG 230
MATH 396 Artificial Intelligence and Applications (3-0)3 Basic problem-solving strategies. A heuristic search principle. Problem reduction and AND/OR graphs. Expert systems and knowledge representation. An expert system shell. Planning. Language processing with grammer rules. Machine learning. Game playing. Logic and uncertainty. Meta pragramming. 
Prerequisites: Consent of the Instructor.
MATH 400 Basic Distribution Theory (3-0)3 Test functions. Spaces E,D,S and their duals. Differentiation, localization, convolution. Fourier transform. Supports and singular supports. Kernels and tensor products. Homogeneous distributions. Fundamental solutions of differential operators. Outline of distribution theory on manifolds. 
Prerequisite: MATH 319. 
MATH  401 Probabilty Theory  (3-0)4 
Events  and  probability. Combinatorial problems and equally likely events.  Probability  spaces.  Independence and finite product spaces.Random variables and distribution functions. Integration of  random variables  Lp-spaces. Convergence of random variables. Conditional expectation. Canonical space of a stochastic process.  Markov chains.  Martingales.
Prerequisites: Consent of the Instructor. 
MATH 402 Introduction of Optimization (3-0)3 The importance of optimization, basic definition and facts on convex analysis. Theory of linear programming and convex prog-ramming, simplex method and its applications, nonlinear programming, search methods, basic ideas of classical variational calculus, optimal control theory. Pontraygin's maximum principle and dynamic programming linear theory of optimal control. 
Prerequisite: Consent of the instructor. 
MATH 403 Foundations of Mathematics (3-0)3 Apriosits programme, mathematical empiricism, scientific and mathematical change, formalists, intuitionists, constructivist and logicist views. 
Prerequisite: Consent of the instructor. 
MATH 404 Introduction to Vector Lattices and Applications (3-0)3 Riesz spaces (vector lattics). Riesz subspaces, ideals and bands. Normed Riezs spaces. Order convergence, relatively uniform convergence and norm convergence. Operators on Riezs spaces. 
Prerequisites: Consent of the Instructor. 
MATH 405 Combinatorics Basic counting: Permutations, r-permutations, combinations, multinomial coefficients, occupancy problems, good algorithms. Generating functions: Power series, operating on generating functions, applications to counting, binomial theorem, exponential generating functions, probability generating functions. Recurrence relations: Simple recurrences, linear recurrence relations, characteristic equations, solving recurrences using generating functions, simultaneous equations, recurrences involving convolutions. Divide and conquer algorithms. Experimental design: Blockdesign, balanced incomplete blockdesign. Applications: coding theory, Hadamard designs.
Prerequisites: Consent of the Instructor. 
MATH 406 Introduction to Mathematical Logic and Model Theory (3-0)3 First order language, structures and satisfaction. Completeness and compactness theorems. Isomorphism, elementary equivalence and elementary imbedding. Lowenheim-Skolem theorem. Interpolation and definability. Atomic, universal and saturated models and their characterisation. Extensions of first order logic.
Prerequisites: Consent of the Instructor. 
MATH 410 Modeling Mathematical Methods and Scientific Computing (2-2)3 Introduction to numerical and symbolical computational tools. Balance equations, continuous system models and partial differential equations. Introduction to numerical methods for ordinary and partial differential equations. Case studies from mechanics, fluid dynamics, heat and mass transfer, electrical engineering. Introduction to stochastic process and differential equations. Models from mathematical finance. 
Prerequisite courses: Consent of the Department
Math 420 Elementary Point Set Topology (3-0)3 Topological Spaces; basis, subbasis, subspaces. Closed sets, limit points. Hausdorff Spaces. Continuous functions, homeomorphisms. Product topology. Connected spaces, compo-nents, path connectedness, path components. Compactness, sequential compactness, compactness in metric spaces. Definition of regular and normal spaces. Urysohn's Lemma, Tietsze Extension Theorem. 
Prerequisites: Math 251 
MATH 422 Elementary Geometric Topology (3-0)3 Topology of subsets of Euclidean space. Topological surfaces. Surfaces in Rn. Surfaces via glueing, connected sum and the classification of compact connected surfaces. Simplicial complexes and simplicial surfaces (simplicial complexes with underlying spaces that are topological surfaces). Euler characteristic. 
Prerequisites: Math 252 or consent of the instructor 
MATH 441 Mechanics I (3-0)3 Statics of rigid bodies, statics of suspended strings and cables. Kinematics of a particle. Translation, rotation of rigid body about an axis and about a fixed point, relative motion. Dynamics of a particle, harmonic oscillators, motion of a simple pendulum, flight of a projectile, motion under the action of central forces. Dynamics of a system of particles, motion of a body with varying mass. 
Prerequisite: Consent of the department.
MATH 442 Mechanics II (3-0)3 Analytical statics; principle of virtual work; Lagrange's equation of first and second-kind; Hamilton's canonical equations; variational principles of mechanics; Poisson brackets, canonical transformations and generating functions; the Hamilton-Jacobi equation; Com-pletely integrable systems; canonical pertur-bation theory; Kolmogorov-Moser-Arnolds the-orems. Fundamentals of continuum mechanics. 
Prerequisites: Consent of the department. 
MATH 444 Topics in Universal Algebra (3-0)3 Primal algebras: Maltsev Conditions. Characterisation of finite primal algebras. Term conditions and commutator. Characterisation of abelian algebras. Generalized term conditions. Finite algebras Tame congruence. Types of minimal algebras.
Prerequisites: Consent of the Instructor. 
MATH 450 Potential Theory in The Complex Plane (3-0)3 Harmonic functions, subharmonic functions. Potential Theory. The Dirichlet Problem, Capacity. Some Applications. 
Prerequisites: Math 353 
MATH 452 Introduction to Functional Analysis (3-0)3 Normed linear spaces, Banach spaces. Hahn-Banach Theorem and consequences. Baire category Theorem. Uniform boundedness principle. Open Mapping and Closed Graph Theorems. Selected Topics and applications. 
Prerequisite: Consent of the instructor.
MATH 453 Introduction to Complex Analysis (3-0)3 Riemann mapping theorem and Schwarz-Christofel transformations zn, z1/n. Elementary Riemann surfaces. Applications of conformal mapping: (flows, heat conduction, electrostatistics,...) Analytic continuation. Argument principle, Rouche's theorem. Mapping properties of analytic functions (inverse function theorem, open mapping theorem, maximum modulus theorem). 
Prerequisite: MATH 353. 
MATH 454 Geometric Complex Analysis (3-0)3 Cauchy theorem; maximum and argument principle; normal families; Riemann mapping theorem; isolated singularities. Riemannian metrics; Poincare metric; Curvature; Liouville theorem; spherical metric; Montel, Picard theorems. Caratheodory; Kobayashi metric; automorphisms of domains; hyperbolicity. Functions of several variables; ball and polydisc; their inequivalence. 
Prerequisites: Math 353 
MATH 456 Fourier Analysis and Wavelets (3-0)3
Orthogonality and modes of convergence. Fourier series, convergence of Fourier series, Fourier transform, Fourier inversion, discrete Fourier transform. Haar and Daubechies wavelets, decomposition and reconstruction, multiresolution analysis. Applications.
Prerequisites: Math 349 or consent of the instructor. 
MATH 457 Calculus on Manifolds (3-0)3 Review of differentiation, inverse and implicit function theorems, integration on subsets of Euclidean space, tensors, differential forms, integration on chains, integration on manifolds. Stokes' theorem. 
Prerequisite: MATH 252 and MATH 262  
MATH 461 Rings and Modules (3-0)3 Categories. Universal algebra. Modules. Basic structure theory of rings.Elements of homological algebra. Commutative ideal theory: General theory and Noetherian rings.
Prerequisite: MATH 116 or consent of the department.
MATH 463 Introduction to Group Theory (3-0)3 Group, subgroup, normal subgroup, cyclic subgroup, coset, quotient group. Commutator subgroup, center, homomorphism and isomorphism theorems (invariant subgroup, wreath products), Abelian groups. Free abelian group, rank of an abelian group. Divisible abelian group, periodic Abelian group. Sylow Theorems and their applications, Soluble groups, Nilpotent groups. 
Prerequisite: MATH 367 or consent of the instructor.
MATH 464 Introduction to Representation Theory (3-0)3 Group representations, FG-Modules, Machke Theorem, irreducible modules and group algebras, Characters, inner products of characters, the number of irreducible characters, character table, induced modules and characters, algebraic integers and real representations. 
Prerequisite:MATH 367 or consent of the instructor.
MATH 466 Groups and Geometry (3-0)3 
Symmetry, Isometries of R n, the Euclidean group, symmetry groups of regular polygons and polyhedra, classification of finite subgroups of the three dimensional rotation group. Frieze groups, crytals, wallpaper groups, groups acting on trees. Reflection groups, root systems, classification of finite reflection groups, crystallographic root systems and Weyl groups.
Prerequisite: MATH 367 
MATH 470 File Structures (3-0)3 Update, merge and sort algorithms for sequential files. Direct access files with hashing. Indexed sequential files. Multikey organization. Operations involving data from several files. 
Prerequisite: MATH 387 or consent of the instructor. 
MATH 471 Hyperbolic Geometry (3-0)3 
Parallel postulate and the need for non-Euclidean geometry, models of the hyperbolic plane, Möbius group, classification of Möbius transformations, classical geometric notions such as length, distance, isometry, parallelism, convexity, area, trigonometry in the hyperbolic plane, group acting on the hyperbolic plane, fundamental domains. 
Prerequisite: MATH 252
MATH 473 Ideals, Varieties and Algorithms (3-0)3 Affine varieties. Groebner bases, monomial ideals and Dickson's Lemma, Hilbert Basis Theorem. Buchberger's algorithm. Ideal mem-bership problem. The problem of Elimination Theory. Unique factorization and resultants. Resultant and extension Theorem. 
Prerequisite: MATH 367 
MATH 474 Introduction to Computational Algebraic Geometry (3-0)3 Hilbert's Nullstellensatz, the ideal-variety correspondence. Zariski closure, irreducible varieties and prime ideals, projective varieties, The projective closure of an affine variety, projective elimination theory. The geometry of quadric hypersurfaces. The dimension of a variety; the variety of a monomial ideal, the complement of a monomial ideal, The Hilbert function. Nonsingularity, the tangent cone. 
Prerequisites: Math 473 or consent of the Instructor 
MATH 476 Algebraic Curves (3-0)3 Affine and projective plane curves, local properties of plane curves, multiple points, intersection numbers, Bezout's theorem, Noether's fundamental theorem. Applications to some enumerative geometry problems. Riemann-Roch theorem. 
Prerequisite: MATH 367 and MATH 353. 
MATH 478 Mathematical Aspects of Cryptogrphy (3-0)3 Time estimates for doing arithmetic, some simple cryptosystems, the idea of public key cryptosystems, RSA, discrete log, knapsac, primality and factoring, the rho method, Fermat factorization.
Prerequisite:Consent of the instructor.
MATH 480 Numerical Methods for Differentia lEquations Initial value problems for ordinary differential equations, Convergnece, stability, Stiffnes, Predictor-corrector methods, Boundary value problems, Hyperbolic and Elliptic differentialy equations, Iterative methods
Prerequisite:Consent of the instructor.
Math 484 Algoritms and Complexity (3-0)3 Recursive algorithms: Quicksort, recursive graph algorithms, fast matrix multiplication, the discrete Fourier transform, Complexity of these algorithms. Network Flow: Algorithms for Network Flow, Ford/Fulkerson Algorithm and its complexity, max-flow min-cut theorem. Theory of numbers: GCD, Euclidean Algorithm, primality test, factoring and cryptography, factoring large integers. NP-completeness: Turing machines, Cook's theorem, NP-complete problems, approximate algprithms for hard problems.
Prerequisites: Consent of the instructor 
MATH 486 Fundamentals of Database Systems (2-2)3
Database concepts. Database management Systems (DBMS).Relation Data Model and relation DBMS: Use of ER-diagrams, in database design. Normalizing relations. Relational algebra and query languages. Structured Query Language (SQL). Oracle and/or Access will be introduced in a laboratory environment. 
Prerequisite:Consent of the instructor.
MATH 487 Applied Mathematics I (3-0)3 Mathematical modelling of boundary value problems of partial differential equations. Formulation of Dirichlet and Neumann problems. Green's function. Asymptotic analysis of solutions. Perturbation techniques. 
Prerequisite: MATH 358 consent of the instructor. 
MATH 488 Applied Mathematics II (3-0)3 Introduction to integral equations. Volterra and Fredholm equations. Solutions by Neumann series. Connection with eigenvalue problems. Essentials of calculus of variations, Euler-Lagrange equations, canonical form of the Euler equation, applications to mechanics and mathematical physics. 
Prerequisite: MATH 254 or MATH 358. 
MATH 489 Dynamical Systems (3-0)3 Second order differential equations in phase plane. Linear systems and exponential operators, canonical forms. Stability of equilibria. Lyapunov functions. The existence of periodic solutions. Applications to various fields. 
Prerequisite: Math 261 or consent of the department.
MATH 490 Difference Equations (3-0)3 The Difference calculus. Linear difference equations: First order equations, high order equations. Systems of difference equations. Basic theory. Linear periodic systems. Stability theory. Linear approximation. Lyapunov's second method. The Z transform. Asymptotic behaviour of difference equations. Sturmian theory. Oscillation. 
Prerequisite: Consent of the instructor.
MATH 492 Numerical Optimization (3-0)3 Unconstrained optimization. Treatment of non-linear systems, conjugate gradient directions, Newton-like methods, quasi-Newton methods. Constrained optimization: Linear prog-ramming, simplex method, convex prog-ramming problems, duality, methods of quadratic problems, active set methods for inequality constrained problems. 
Prerequisite: MATH 261 or consent of the instructor.
MATH 493 Philosophy of Mathematics (3-0)3 Philosophical problems about mathematics, Euclidean and non-Euclidean Geometries. The existence of mathematical objects, mathematical truth, Wittgenstein and Lakatos on mathematics. 
Prerequisite: Consent of the instructor. 
MATH 494 The Design of Mathematical Software (3-0)3 The principals of sotfware engineering with special emphasis on mathematical software. (By way of class examples, laboratories and/or term projects students will see two or three small software packages evolve through the stages of specification, design, implementation and testing.) 
Prerequisite: MATH 387 or consent of the instructor 
MATH 496 Supervised Independent Study and Research (2-0)2 Individualized reading, and Study/research in mathematics for students of high intellectual promise. 
Prerequisite: Consent of the department and the instructor. 
MATH 497 Hilbert Space Techniques (3-0)3 Inner product spaces. Examples of inner product spaces; Hilbert spaces (definition and examples); convergence in Hilbert spaces; orthogonal complements and the projection theorem; linear functionals and the Riesz representation theorem; applications to various branches of mathematics. 
Prerequisite: MATH 349