# Mathematics (MATH)

## Courses

**MATH 075. Beginning Algebra. 2 hours.**

Linear equations and inequalities, functions, linear functions, slope, exponents, polynomials, quadratic equations, rational expressions, rational equations, and applications. Course Information: Satisfactory/Unsatisfactory grading only. Not open to students with credit in Math 070, 090, or a mathematics course at or above the 100-level. No graduation credit. Prerequisite(s): Appropriate score on the department placement test. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Thursday of finals week from 1 to 3 p.m.

**MATH 088. Intermediate Algebra Workshop. 1 hour.**

Individualized lesson plans including: order of operations, properties of real numbers, linear equations, problem solving, graphing linear equations. Course Information: Satisfactory/Unsatisfactory grading only. No graduation credit. Extensive computer use required. Corequisites: Requires concurrent registration in MATH 090.

**MATH 090. Intermediate Algebra. 4 hours.**

Linear equations and inequalities, absolute values, linear graphs and modeling, systems of equations, functions, quadratic equations, exponents and polynomials, factoring, radicals and rational exponents. Course Information: Satisfactory/Unsatisfactory grading only. Not open to students with credit in a mathematics course at or above the 100 level. No graduation credit. Extensive computer use required. Prerequisite(s): MATH 075; or credit or concurrent registration in MATH 088; or appropriate score on the department placement test. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Wednesday of finals week from 6 to 8 p.m. To be properly registered, students must enroll in one Laboratory-Discussion and one Lecture.

**MATH 110. College Algebra. 4 hours.**

Functions, composition and inverses; graphs and transformations, polynomial and rational functions, exponential functions, logarithms and applications; circles and introduction to trigonometry. Course Information: Credit is not given for MATH 110 if the student has credit in MATH 121 or MATH 165 or MATH 175 or MATH 180. Extensive computer use required. Prerequisite(s): MATH 090; or appropriate score on the department placement test. To be properly registered, students must enroll in one Lecture and one Laboratory-Discussion.

**MATH 118. Mathematical Reasoning. 5 hours.**

Elementary topics from algebra applied to descriptive statistics of data, scatter plots, correlation, linear regression, probability, random samples, sampling distributions, experimental designs. Graphing calculator used. Course Information: No credit given if the student has credit in MATH 150 or 160 or 165 or 180, or the equivalent. No credit given if the student has credit in MATH 121 with a grade of C or better. No graduation credit for architecture, business administration, or engineering students. The only mathematics department course for which MATH 118 serves as a prerequisite is MATH 123. It may serve as a preprequisite for statistics courses in the social sciences. It does not replace MATH 090 as a prerequisite for any other mathematics department course. Prerequisite(s): MATH 070, or MATH 075, or appropriate performance on the UIC mathematics placement test. Class Schedule Information: To be properly registered, students must enroll in one Discussion/Recitation and one Lecture.

**MATH 121. Precalculus Mathematics. 5 hours.**

Functions, graphs, exponentials and logarithms, radicals, complex numbers, trigonometry (circle and triangle approaches), trigonometric graphs and inverses, introduction to polar coordinates and vectors Course Information: No credit will be given for MATH 121 if students have credit in MATH 165 or MATH 170 or MATH 180. Extensive computer use required. Prerequisite(s): Grade of C or better in MATH 110; or appropriate score on the department placement test. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Thursday of finals week from 6 to 8 p.m. To be properly registered, students must enroll in one Laboratory-Discussion and one Lecture.

**MATH 122. Emerging Scholars Workshop for Precalculus Mathematics. 1 hour.**

Intensive math workshop for students enrolled in MATH 121. Students work together in groups to solve challenging problems. Course Information: Satisfactory/Unsatisfactory grading only. Prerequisite(s): Admission to the Emerging Scholars Program. Must enroll concurrently in MATH 121.

**MATH 125. Elementary Linear Algebra. 5 hours.**

Introduction to systems of linear equations, matrices and vector spaces, with emphasis on business applications. Course Information: Prerequisite(s): MATH 090; or MATH 110; or appropriate score on the department placement test. Class Schedule Information: To be properly registered, students must enroll in one Lecture and one Discussion. During the fall and spring terms, combined section final exam will be held on Thursday of finals week from 6 to 8 p.m. To be properly registered, students must enroll in one Discussion/Recitation and one Lecture.* Natural World - No Lab course*.

**MATH 140. Arithmetic and Algebraic Structures. 4 hours.**

Problem solving; algebraic thinking; number systems; numeration; number theory; mathematical operations over natural, integer, and rational numbers; and proportional reasoning. Course Information: Prerequisite(s): MATH 090; or appropriate score on the department placement test. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Monday of finals week from 6 to 8 p.m.

**MATH 141. Algebraic and Geometric Structures. 4 hours.**

Area, perimeter, volume, surface area of plane and solid figures; integers, real and rational numbers; trigonometry and extended solution of general polygons; probability. Full purpose calculators used. Course Information: Designed for students in the B.A. in Elementary Education program. Prerequisite(s): Grade of C or better in MATH 140. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Tuesday of finals week from 6 to 8 p.m.

**MATH 160. Finite Mathematics for Business. 5 hours.**

Introduction to probability, statistics, and matrices, with emphasis on business applications. Course Information: MATH 090; or Grade of C or better in MATH 110; or appropriate score on the department placement test. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Thursday of finals week from 6 to 8 p.m. To be properly registered, students must enroll in one Discussion/Recitation and one Lecture.* Natural World - No Lab course*.

**MATH 165. Calculus for Business. 5 hours.**

Introduction to differential and integral calculus of algebraic, exponential and logarithmic functions and techniques of partial derivatives and optimization. Emphasis on business applications. Course Information: Credit is not given for MATH 165 if the student has credit for MATH 180. Prerequisite(s): Grade of C or better in MATH 110; or appropriate score on the department placement test. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Wednesday of finals week from 6 to 8 p.m. To be properly registered, students must enroll in one Discussion/Recitation and one Lecture.* Natural World - No Lab course*.

**MATH 170. Calculus for the Life Sciences. 4 hours.**

Introduction to calculus with applications to the life sciences, mathematical modeling, differentiation, integration and applications. Course Information: Credit is not given for MATH 170 if the student has credit for MATH 165 or MATH 180. Prerequisite(s): Grade of C or better in MATH 110 or Grade of C or better in MATH 121; or appropriate score on the department placement test. Class Schedule Information: To be properly registered, students must enroll in one Lecture and one Discussion.* Natural World - No Lab course*.

**MATH 179. Emerging Scholars Workshop for Calculus I. 1 hour.**

Intensive math workshop for students enrolled in MATH 180. Students work together in groups to solve challenging problems. Course Information: Satisfactory/Unsatisfactory grading only. Prerequisite(s): Admission to the Emerging Scholars Program. Must enroll concurrently in MATH 180.

**MATH 180. Calculus I. 4 hours.**

Differentiation, curve sketching, maximum-minimum problems, related rates, mean-value theorem, antiderivative, Riemann integral, logarithm, and exponential functions. Course Information: Credit is not given for MATH 180 if the student has credit for MATH 165 or MATH 170. Prerequisite(s): Grade of C or better in MATH 121 or appropriate performance on the department placement test. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Thursday of finals week from 1 to 3 p.m. To be properly registered, students must enroll in one Discussion/Recitation and one Lecture.* Natural World - No Lab course*.

**MATH 181. Calculus II. 4 hours.**

Techniques of integration, arc length, solids of revolution, applications, polar coordinates, parametric equations, infinite sequences and series, power series. Course Information: Prerequisite(s): Grade of C or better in MATH 180. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Thursday of finals week from 3:30 to 5:30 p.m. To be properly registered, students must enroll in one Discussion/Recitation and one Lecture.* Natural World - No Lab course*.

**MATH 182. Emerging Scholars Workshop for Calculus II. 1 hour.**

Intensive math workshop for students enrolled in MATH 181. Students work together in groups to solve challenging problems. Course Information: Satisfactory/Unsatisfactory grading only. Prerequisite(s): Admission to the Emerging Scholars Program. Must enroll concurrently in MATH 181.

**MATH 194. Special Topics in Mathematics. 1-4 hours.**

Course content is announced prior to each term in which it is given. Course Information: May be repeated. Prerequisite(s): Approval of the department.

**MATH 210. Calculus III. 3 hours.**

Vectors in space, functions of several variables, partial differential and optimization, multiple integrals, vector fields, Green?s Theorem, Stokes Theorem. Course Information: Prerequisite(s): Grade of C or better in MATH 181. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Thursday of finals week from 1 to 3 p.m. To be properly registered, students must enroll in one Discussion and one Lecture.* Natural World - No Lab course*.

**MATH 211. Emerging Scholars Workshop for Calculus III. 1 hour.**

Intensive math workshop for students enrolled in MATH 210. Students work together in groups to solve challenging problems. Course Information: Satisfactory/Unsatisfactory grading only. Prerequisite(s): Admission to the Emerging Scholars Program. Must enroll concurrently in MATH 210.

**MATH 215. Introduction to Advanced Mathematics. 3 hours.**

Introduction to methods of proofs used in different fields in mathematics. Course Information: Prerequisite(s): Grade of C or better in MATH 181 and approval of the department.

**MATH 220. Introduction to Differential Equations. 3 hours.**

Techniques and applications of differential equations, first and second order equations, Laplace transforms, series solutions, graphical and numerical methods, and partial differential equations. Course Information: Prerequisite(s): Grade of C or better in MATH 210. Class Schedule Information: During the fall and spring terms, combined section final exam will be held on Thursday of finals week from 6 to 8 p.m. To be properly registered, students must enroll in one Laboratory-Discussion and one Lecture.

**MATH 294. Special Topics in Mathematics. 1-4 hours.**

Course content is announced prior to each term in which it is given. Course Information: May be repeated. Prerequisite(s): Approval of the department.

**MATH 300. Writing for Mathematics. 1 hour.**

Fulfills Writing-in-the-Discipline requirement. Course Information: Prerequisite(s): ENGL 161 or the equivalent, and a grade of C or better in MATH 210. Students must have declared a major in the Mathematics, Statistics, and Computer Science Department.

**MATH 310. Applied Linear Algebra. 3 hours.**

Matrices, Gaussian elimination, vector spaces, LU-decomposition, orthogonality, Gram-Schmidt process, determinants, inner products, eigenvalue problems, applications to differential equations and Markov processes. Course Information: Credit is not given for MATH 310 if the student has credit for MATH 320. Prerequisite(s): Grade of C or better in MATH 210.

**MATH 313. Analysis I. 3 hours.**

The real number system, limits, continuous functions, differentiability, the Riemann integral. Course Information: Prerequisite(s): Grade of C or better in MATH 215 or consent of the instructor.

**MATH 320. Linear Algebra I. 3 hours.**

Linear equations, Gaussian elimination, matrices, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors. Course Information: Credit is not given for MATH 320 if the student has credit for MATH 310. Prerequisite(s): A grade of C or better in MATH 215.

**MATH 330. Abstract Algebra I. 3 hours.**

Sets, properties of integers, groups, rings, fields. Course Information: Prerequisite(s): Grade of C or better in MATH 215.

**MATH 394. Special Topics in Mathematics. 2-4 hours.**

Course content is announced prior to each term in which it is given. Course Information: May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department.

**MATH 410. Advanced Calculus I. 3 or 4 hours.**

Functions of several variables, differentials, theorems of partial differentiation. Calculus of vector fields, line and surface integrals, conservative fields, Stokes's and divergence theorems. Cartesian tensors. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 210.

**MATH 411. Advanced Calculus II. 3 or 4 hours.**

Implicit and inverse function theorems, transformations, Jacobians. Point-set theory. Sequences, infinite series, convergence tests, uniform convergence. Improper integrals, gamma and beta functions, Laplace transform. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 410.

**MATH 414. Analysis II. 3 or 4 hours.**

Sequences and series of functions. Uniform convergence. Taylor's theorem. Topology of metric spaces, with emphasis on the real numbers. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 313.

**MATH 417. Complex Analysis with Applications. 3 or 4 hours.**

Complex numbers, analytic functions, complex integration, Taylor and Laurent series, residue calculus, branch cuts, conformal mapping, argument principle, Rouche's theorem, Poisson integral formula, analytic continuation. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade C or better in MATH 210.

**MATH 419. Models in Applied Mathematics. 3 or 4 hours.**

Introduction to mathematical modeling; scaling, graphical methods, optimization, computer simulation, stability, differential equation models, elementary numerical methods, applications in biology, chemistry, engineering and physics. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 220 and grade of C or better in MCS 260.

**MATH 425. Linear Algebra II. 3 or 4 hours.**

Canonical forms of a linear transformation, inner product spaces, spectral theorem, principal axis theorem, quadratic forms, special topics such as linear programming. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 320.

**MATH 430. Formal Logic I. 3 or 4 hours.**

First order logic, syntax and semantics, completeness-incompleteness. Course Information: 3 undergraduate hours. 4 graduate hours. Credit is not given for MATH 430 if the student has credit for PHIL 416. Prerequisite(s): Grade of C or better in CS 202 or grade of C or better in MCS 261 or grade of C or better in MATH 215.

**MATH 431. Abstract Algebra II. 3 or 4 hours.**

Further topics in abstract algebra: Sylow Theorems, Galois Theory, finitely generated modules over a principal ideal domain. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 320 and grade of C or better in MATH 330.

**MATH 435. Foundations of Number Theory. 3 or 4 hours.**

Primes, divisibility, congruences, Chinese remainder theorem, primitive roots, quadratic residues, quadratic reciprocity, and Jacobi symbols. The Euclidean algorithm and strategies of computer programming. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 215.

**MATH 436. Number Theory for Applications. 3 or 4 hours.**

Primality testing methods of Lehmer, Rumely, Cohen-Lenstra, Atkin. Factorization methods of Gauss, Pollard, Shanks, Lenstra, and quadratic sieve. Computer algorithms involving libraries and nested subroutines. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 435.

**MATH 442. Differential Geometry of Curves and Surfaces. 3 or 4 hours.**

Frenet formulas, isoperimetric inequality, local theory of surfaces, Gaussian and mean curvature, geodesics, parallelism, and the Guass-Bonnet theorem. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 320.

**MATH 445. Introduction to Topology I. 3 or 4 hours.**

Elements of metric spaces and topological spaces including product and quotient spaces, compactness, connectedness, and completeness. Examples from Euclidean space and function spaces. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 313.

**MATH 446. Introduction to Topology II. 3 or 4 hours.**

Topics in topology chosen from the following: advanced point set topology, piecewise linear topology, fundamental group and knots, differential topology, applications to physics and biology. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 445.

**MATH 480. Applied Differential Equations. 3 or 4 hours.**

Linear first-order systems. Numerical methods. Nonlinear differential equations and stability. Introduction to partial differential equations. Sturm-Liouville theory. Boundary value problems and Green's functions. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 220.

**MATH 481. Applied Partial Differential Equations. 3 or 4 hours.**

Initial value and boundary value problems for second order linear equations. Eiqenfunction expansions and Sturm-Liouville theory. Green's functions. Fourier transform. Characteristics. Laplace transform. Course Information: 3 undergraduate hours. 4 graduate hours. Prerequisite(s): Grade of C or better in MATH 220.

**MATH 494. Special Topics in Mathematics. 3 or 4 hours.**

Course content is announced prior to each term in which it is given. Course Information: 3 undergraduate hours. 4 graduate hours. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department.

**MATH 496. Independent Study. 1-4 hours.**

Reading course supervised by a faculty member. Course Information: May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the instructor and the department. Class Schedule Information: This course counts toward the limited number of independent study hours accepted toward the degree and the major.

**MATH 502. Mathematical Logic. 4 hours.**

First order logic, completeness and incompleteness theorems, introduction to model theory and computability theory. Course Information: Same as PHIL 562. Prerequisite(s): MATH 430 or consent of the instructor.

**MATH 504. Set Theory. 4 hours.**

Naive and axiomatic set theory. Independence of the continuum hypothesis and the axiom of choice. Course Information: Same as PHIL 565. Prerequisite(s): MATH 430 or MATH 502 or PHIL 562.

**MATH 506. Model Theory I. 4 hours.**

Elementary embeddings, quantifier elimination, types, saturated and prime models, indiscernibles, Morley's Categoricity Theorem. Course Information: Same as PHIL 567. Prerequisite(s): MATH 502 or PHIL 562.

**MATH 507. Model Theory II. 4 hours.**

Stability theory: forking and indpendence, stable groups, geometric stability. Course Information: Same as PHIL 568. Prerequisite(s): MATH 506 or PHIL 567.

**MATH 511. Descriptive Set Theory. 4 hours.**

Polish spaces and Baire category; Borel, analytic and coanalytic sets; infinite games and determinacy; coanalytic ranks and scales; dichotomy theorems. Course Information: Recommended background: MATH 445 or MATH 504 or MATH 533 or MATH 539.

**MATH 512. Advanced Topics in Logic. 4 hours.**

Advanced topics in modern logic; e.g. large cardinals, infinitary logic, model theory of fields, o-minimality, Borel equivalence relations. Course Information: Same as PHIL 569. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department.

**MATH 514. Number Theory I. 4 hours.**

Introduction to classical, algebraic, and analytic, number theory. Euclid's algorithm, unique factorization, quadratic reciprocity, and Gauss sums, quadratic forms, real approximations, arithmetic functions, Diophantine equations.

**MATH 515. Number Theory II. 4 hours.**

Introduction to classical, algebraic, and analytic number theory. Algebraic number fields, units, ideals, and P-adic theory. Riemann Zeta-function, Dirichlet's theorem, prime number theorem. Course Information: Prerequisite(s): MATH 514.

**MATH 516. Second Course in Abstract Algebra I. 4 hours.**

Structure of groups, Sylow theorems, solvable groups; structure of rings, polynomial rings, projective and injective modules, finitely generated modules over a PID. Course Information: Prerequisite(s): MATH 330 and MATH 425.

**MATH 517. Second Course in Abstract Algebra II. 4 hours.**

Rings and algebras, polynomials in several variables, power series rings, tensor products, field extensions, Galois theory, Wedderburn theorems. Course Information: Prerequisite(s): MATH 516.

**MATH 518. Representation Theory. 4 hours.**

Major areas of representation theory, including structure of group algebras, Wedderburn theorems, characters and orthogonality relations, idempotents and blocks. Course Information: Prerequisite(s): MATH 517.

**MATH 519. Algebraic Groups. 4 hours.**

Classical groups as examples; necessary results from algebraic geometry; structure and classification of semisimple algebraic groups. Course Information: Prerequisite(s): MATH 517.

**MATH 520. Commutative and Homological Algebra. 4 hours.**

Commutative rings; primary decomposition; integral closure; valuations; dimension theory; regular sequences; projective and injective dimension; chain complexes and homology; Ext and Tor; Koszul complex; homological study of regular rings. Course Information: Prerequisite(s): MATH 516 and MATH 517; or consent of the instructor.

**MATH 525. Advanced Topics in Number Theory. 4 hours.**

Introduction to topics at the forefront of research in number theory. Topics will vary and may include elliptic curves, automorphic forms, diophantine geometry or sieve methods. Course Information: May be repeated. Prerequisite(s): MATH 515; or consent of the instructor.

**MATH 531. Advanced Topics in Algebra. 4 hours.**

Researchlevel topics such as groups and geometries, equivalencies of module categories, representations of Lie-type groups. Course Information: May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department.

**MATH 533. Real Analysis I. 4 hours.**

Introduction to real analysis. Lebesgue measure and integration, differ entiation, L-p classes, abstract integration. Course Information: Prerequisite(s): MATH 411 or MATH 414 or the equivalent.

**MATH 534. Real Analysis II. 4 hours.**

Continuation of MATH 533. Course Information: Prerequisite(s): MATH 417.

**MATH 535. Complex Analysis I. 4 hours.**

Analytic functions as mappings. Cauchy theory. Power Series. Partial fractions. Infinite products. Course Information: Prerequisite(s): MATH 411.

**MATH 536. Complex Analysis II. 4 hours.**

Normal families, Riemann mapping theorem. Analytic continuation, Harmonic and subharmonic functions, Picard theorem, selected topics. Course Information: Prerequisite(s): MATH 535.

**MATH 537. Introduction to Harmonic Analysis I. 4 hours.**

Fourier transform on L(p) spaces, Wiener's Tauberian theorem, Hilbert transform, Paley Wiener theory. Course Information: Prerequisite(s): MATH 533; and MATH 417 or MATH 535.

**MATH 539. Functional Analysis I. 4 hours.**

Topological vector spaces, Hilbert spaces, Hahn-Banach theorem, open mapping, uniform boundedness principle, linear operators in a Banach space, compact operators. Course Information: Prerequisite(s): MATH 533.

**MATH 541. Partial Differential Equations I. 4 hours.**

Theory of distributions; fundamental solutions of the heat equation, wave equation, and Laplace equation. Harmonic functions. Cauchy problem for the wave equation. Course Information: Prerequisite(s): MATH 417.

**MATH 542. Partial Differential Equations II. 4 hours.**

Cauchy problem for hyperbolic equations. Propagation of singularities. Boundary value problems for elliptic equations. Course Information: Prerequisite(s): MATH 541.

**MATH 546. Advanced Topics in Analysis. 4 hours.**

Subject may vary from semester to semester. Topics include partial differential equations, several complex variables, harmonic analysis and ergodic theory. Course Information: May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department.

**MATH 547. Algebraic Topology I. 4 hours.**

The fundamental group and its applications, covering spaces, classification of compact surfaces, introduction to homology, development of singular homology theory, applications of homology. Course Information: Prerequisite(s): MATH 330 and MATH 445.

**MATH 548. Algebraic Topology II. 4 hours.**

Cohomology theory, universal coefficient theorems, cohomology products and their applications, orientation and duality for manifolds, homotopy groups and fibrations, the Hurewicz theorem, selected topics. Course Information: Prerequisite(s): MATH 547.

**MATH 549. Differentiable Manifolds I. 4 hours.**

Smooth manifolds and maps, tangent and normal bundles, Sard's theorem and transversality, embedding, differential forms, Stokes's theorem, degree theory, vector fields. Course Information: Prerequisite(s): MATH 445; and MATH 310 or MATH 320 or the equivalent.

**MATH 550. Differentiable Manifolds II. 4 hours.**

Vector bundles and classifying spaces, lie groups and lie algbras, tensors, Hodge theory, Poincare duality. Topics from elliptic operators, Morse theory, cobordism theory, deRahm theory, characteristic classes. Course Information: Prerequisite(s): MATH 549.

**MATH 551. Riemannian Geometry. 4 hours.**

Riemannian metrics and Levi-Civita connections, geodesics and completeness, curvature, first and second variation of arc length, comparison theorems. Course Information: Prerequisite(s): MATH 442 and MATH 549.

**MATH 552. Algebraic Geometry I. 4 hours.**

Basic commutative algebra, affine and projective varieties, regular and rational maps, function fields, dimension and smoothness, projective curves, schemes, sheaves, and cohomology, posiive characteristic.

**MATH 553. Algebraic Geometry II. 4 hours.**

Divisors and linear systems, differentials, Riemann-Roch theorem for curves, elliptic curves, geometry of curves and surfaces. Course Information: Prerequisite(s): MATH 552.

**MATH 554. Complex Manifolds I. 4 hours.**

Holomorphic functions in several variables, Riemann surfaces, Sheaf theory, vector bundles, Stein manifolds, Cartan theorem A and B, Grauert direct image theorem. Course Information: Prerequisite(s): MATH 517 and MATH 535.

**MATH 555. Complex Manifolds II. 4 hours.**

Dolbeault Cohomology, Serre duality, Hodge theory, Kadaira vanishing and embedding theorem, Lefschitz theorem, Complex Tori, Kahler manifolds. Course Information: Prerequisite(s): MATH 517 and MATH 535.

**MATH 568. Topics in Algebraic Topology. 4 hours.**

Homotopy groups and fibrations. The Serre spectral sequence and its applications. Classifying spaces of classical groups. Characteristic classes of vector bundles. Course Information: May be repeated. Students may register in more than one section per term. Prerequisite(s): MATH 548 or consent of the instructor.

**MATH 569. Advanced Topics in Geometric and Differential Topology. 4 hours.**

Topics from areas such as index theory, Lefschetz theory, cyclic theory, KK theory, non-commutative geometry, 3-manifold topology, hyperbolic manifolds, geometric group theory, and knot theory. Course Information: Prerequisite(s): Approval of the department.

**MATH 570. Advanced Topics in Differential Geometry. 4 hours.**

Subject may vary from semester to semester. Topics may include eigenvalues in Riemannian geometry, curvature and homology, partial differential relations, harmonic mappings between Riemannian manifolds hyperbolic geometry, arrangement of hyperplanes. Course Information: May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department.

**MATH 571. Advanced Topics in Algebraic Geometry. 4 hours.**

Various topics such as algebraic curves, surfaces, higher dimensional geometry, singularities theory, moduli problems, vector bundles, intersection theory, arithematical algebraic geometry, and topologies of algebraic varieties. Course Information: May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department.

**MATH 574. Applied Optimal Control. 4 hours.**

Introduction to optimal control theory; calculus of variations, maximum principle, dynamic programming, feedback control, linear systems with quadratic criteria, singular control, optimal filtering, stochastic control. Course Information: Prerequisite(s): MATH 411 or consent of the instructor.

**MATH 575. Integral Equations and Applications. 4 hours.**

Fredholm and Volterra equations, Fredholm determinants, separable and symmetric kernels, Neumann series, transform methods, Wiener-Hopf method, Cauchy kernels, nonlinear equations, perturbation methods. Course Information: Prerequisite(s): MATH 411 and MATH 417 and MATH 481; or consent of instructor.

**MATH 576. Classical Methods of Partial Differential Equations. 4 hours.**

First and second order equations, method of characteristics, weak solutions, distributions, wave, Laplace, Poisson, heat equations, energy methods, regularity problems, Green functions, maximum principles, Sobolev spaces, imbedding theorems. Course Information: Prerequisite(s): MATH 410 and MATH 481 and MATH 533; or consent of instructor.

**MATH 577. Advanced Partial Differential Equations. 4 hours.**

Linear elliptic theory, maximum principles, fixed point methods, semigroups and nonlinear dynamics, systems of conservation laws, shocks and waves, parabolic equations, bifurcation, nonlinear elliptic theory. Course Information: Prerequisite(s): MATH 533 and MATH 576 or consent of the instructor.

**MATH 578. Asymptotic Methods. 4 hours.**

Asymptotic series, Laplace's method, stationary phase, steepest descent method, Stokes phenomena, uniform expansions, multi-dimensional Laplace integrals, Euler-MacLaurin formula, irregular singular points, WKBJ method. Course Information: Prerequisite(s): MATH 417 and MATH 481; or consent of instructor.

**MATH 579. Singular Perturbations. 4 hours.**

Algebraic and transcendental equations, regular perturbation expansions of differential equations, matched asymptotic expansions, boundary layer theory, Poincare-Lindstedt, multiple scales, bifurcation theory, homogenization. Course Information: Prerequisite(s): MATH 481 or consent of the instructor.

**MATH 580. Mathematics of Fluid Mechanics. 4 hours.**

Development of concepts and techniques used in mathematical models of fluid motions. Euler and Navier Stokes equations. Vorticity and vortex motion. Waves and instabilities. Viscous fluids and boundary layers. Asymptotic methods. Course Information: Prerequisite(s): Grade of C or better in MATH 410 and grade of C or better in MATH 417 and grade of C or better in MATH 481.

**MATH 581. Special Topics in Fluid Mechanics. 4 hours.**

Geophysical fluids with applications to oceanography and meteorology, astrophysical fluids, magnetohydrodynamics and plasmas. Course Information: Prerequisite(s): Grade of C or better in MATH 580.

**MATH 582. Linear and Nonlinear Waves. 4 hours.**

Derivation and analysis of models for linear and nonlinear wave propagation, including acoustic, hydrodynamic, and eletromagnetic waves. Analytical techniques include exact formulas and asymptotic methods. Course Information: Prerequisite(s): MATH 480 and MATH 481; or consent of the instructor.

**MATH 583. Topics in Wave Propagation. 4 hours.**

Rigorous, asymptotic, and numerical analysis of mathematical models for linear and nonlinear waves. Techniques include inverse scattering, asymptotic analysis, and finite-difference and spectral methods. Course Information: Prerequisite(s): MATH 480 and MATH 481; consent of the instructor.

**MATH 584. Applied Stochastic Models. 4 hours.**

Applications of stochastic models in chemistry, physics, biology, queueing, filtering, and stochastic control, diffusion approximations, Brownian motion, stochastic calculus, stochastically perturbed dynamical systems, first passage times. Course Information: Prerequisite(s): MATH 417 and MATH 481 and STAT 401, or consent of the instructor.

**MATH 585. Ordinary Differential Equations. 4 hours.**

Introduction to ordinary differential equations, existence, uniqueness of solutions, dependence on parameters, autonomous and non-autonomous systems, linear systems, nonlinear systems, periodic solutions, bifurcations, conservative systems. Course Information: Prerequisite(s): MATH 313 or MATH 480 or approval of the department.

**MATH 586. Computational Finance. 4 hours.**

Introduction to the mathematics of financial derivatives; options, asset price random walks, Black-Scholes model; partial differential techniques for option valuation, binomial models, numerical methods; exotic options, interest-rate derivatives. Course Information: Prerequisite(s): Grade of C or better in MATH 220 and grade of C or better in STAT 381; or consent of the instructor.

**MATH 587. Nonlinear Dynamics, Chaos and Applications. 4 hours.**

Introduction to nonlinear dynamics, bifurcations, chaotic dynamics, and strange attractors. Linear response to small external fluctuations. Related numerical methods. Course Information: Prerequisite(s): Grade of C or better in MATH 480 and Grade of C or better in MCS 471; or consent of the instructor.

**MATH 589. Teaching and Presentation of Mathematics. 2 hours.**

Strategies and techniques for effective teaching in college and for mathematical consulting. Observation and evaluation, classroom management, presenting mathematics in multidisciplinary research teams. Required for teaching assistants in MSCS. Course Information: No graduation credit awarded for students enrolled in the Master of Science in the Teaching of Mathematics degree program.

**MATH 590. Advanced Topics in Applied Mathematics. 4 hours.**

Topics from areas such as: elastic scattering, nonlinear problems in chemistry and physics, mathematical biology, stochastic optimal control, geophysical fluid dynamics, stability theory, queueing theory. Course Information: Prerequisite(s): Approval of the department.

**MATH 591. Seminar on Mathematics Curricula. 4 hours.**

Examination of research and reports on mathematics curricula. Analysis of research in teaching and learning mathematics. Developments in using technology in mathematics teaching. Course Information: Prerequisite(s): Enrollment in the Doctor of Arts program in mathematics or consent of the instructor.

**MATH 592. Seminar on Mathematics: Philosophy and Methodology. 4 hours.**

Problems related to teaching and learning mathematics. Analysis of work of Piaget, Gagne, Bruner, Ausabel, Freudenthal, and others and their relation to mathematics teaching. Course Information: Prerequisite(s): Enrollment in the Doctor of Arts program in mathematics or consent of instructor.

**MATH 593. Graduate Student Seminar. 1 hour.**

For graduate students who wish to receive credit for participating in a learning seminar whose weekly time commitment is not sufficient for a reading course. This seminar must be sponsored by a faculty member. Course Information: Satisfactory/Unsatisfactory grading only. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department.

**MATH 594. Internship in Mathematics. 0-8 hours.**

Under the direction of a faculty adviser, students work in government or industry on problems related to their major field of interest. At the end of internship, the student must present a seminar on the internship experiences. Course Information: Satisfactory/Unsatisfactory grading only. May be repeated to a maximum of 8 hours. Only 4 credit hours count toward the 32 credit hours required for the M.S. in MISI degree. Does not count toward the 12 credit hours of 500-level courses requirement. Prerequisite(s): Completion of the core courses in the degree program in which the student is enrolled and approval of the internship program by the graduate adviser and the graduate studies committee.

**MATH 595. Research Seminar. 1 hour.**

Current developments in research with presentations by faculty, students, and visitors. Course Information: Satisfactory/Unsatisfactory grading only. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department.

**MATH 596. Independent Study. 1-4 hours.**

Reading course supervised by a faculty member. Course Information: May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the instructor and the department.

**MATH 598. Master's Thesis. 0-16 hours.**

Research work under the supervision of a faculty member leading to the completion of a master's thesis. Course Information: Satisfactory/Unsatisfactory grading only. Prerequisite(s): Approval of the department.

**MATH 599. Thesis Research. 0-16 hours.**

Research work under the supervision of a faculty member. Course Information: Satisfactory/Unsatisfactory grading only. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department.