Graduate Catalog

2012-2013

MATHEMATICAL SCIENCES

Department website: http://www.math.nmsu.edu

(575) 646-3901

gradcomm@nmsu.edu

J. Lakey, department head, Ph.D. (Maryland) – applied harmonic analysis; P. Baggett, Ph.D. (Colorado) – mathematics education; M. Ballyk, Ph.D. (McMaster) – mathematical biology and ecology; E. Barany, Ph.D. (Ohio State) – mathematical physics; G. Bezhanishvili, Ph.D. (Tokyo Institute of Technology) – logic; M. S. Cohen, Ph.D. (Chicago) – mathematical biology, mathematical physics; R. DeBlassie, Ph.D. (MIT) – Probability; L. Fouli, Ph.D. (Purdue) – commutative algebra; T. Giorgi, Ph.D. (Purdue) – applied mathematics; J. Harding, Ph.D. (McMaster) – logic and foundations; D. S. Kurtz, Ph.D. (Rutgers) – harmonic analysis; G. Lodder, Ph.D. (Stanford) – algebraic topology; P. Morandi, Ph.D. (California-San Diego) – algebra; B. Olberding, Ph.D. (Wesleyan) – commutative algebra, valuation theory and module theory; D. Ramras, Ph.D. (Stanford) – algebraic topology; S. Salamanca-Riba, Ph.D. (M.I.T.) – Lie groups and representation theory; R. Smits, Ph.D. (Purdue) – probability, harmonic analysis; R. Staffeldt, Ph.D. (California-Berkeley) – algebraic topology; T. Stanford, Ph.D. (Columbia) – low dimensional topology; T. Wang, Ph.D. (Windsor) – mathematical statistics. Associated Faculty: Annie Selden, Ph.D. (Clarkson) – mathematics education; John Selden, Ph.D. (Georgia) – mathematics education.

DEGREE: Master of Science
MAJOR: Mathematics

DEGREE: Professional Master of Financial Mathematics

DEGREE: Doctor of Philosophy
MAJOR: Mathematics

MINOR: Mathematics

The Department of Mathematical Sciences offers graduate instruction leading to the Master of Science degree, Doctor of Philosophy degree, and a Professional Master's Degree in Financial Mathematics. Possible areas of study are various topics in pure mathematics and applied mathematics, statistics, and mathematics education. Students may also pursue an interdisciplinary program of study. Our program has 50 to 60 graduate students, most of them supported by a combination of teaching assistantships, research assistantships, fellowships, and job opportunities at nearby teaching or research units.

For more information on our programs and on our working environment, and to learn more about the research interests of the faculty, please see our web site at www.math.nmsu.edu, phone us at (575) 646-3901, or write to Graduate Secretary, Department of Mathematical Sciences, NMSU, Las Cruces, NM 88003-8001, email: gradcomm@nmsu.edu

Students applying for regular admission to graduate study in mathematics are expected to have 24 credits of upper-division courses in mathematics and statistics, including a three-credit course in modern analysis and a three-credit course in modern algebra. Students who do not meet these requirements may be admitted with deficiencies and allowed to complete the requirements at New Mexico State University.

The minimum application to be admitted as a regular graduate student in mathematics includes:

1. a completed Graduate School admission application,
2. complete transcripts of all undergraduate and graduate work,
3. application fee,
4. three letters of recommendation from professors, employers, or others who are qualified to judge potential for graduate work in mathematics,
5. a one-page statement of educational objectives.

Items 1, 2, and 3 should be submitted to the Graduate School by domestic applicants and to the Center for International Programs by international applicants. Items 4 and 5 and copies of items 1 and 2 should be submitted to the Department of Mathematical Sciences.

Although GRE subject test scores are not required for admission, applicants are encouraged to submit them, if available. The test scores may be used to help allocate available teaching assistantships among entering students.

To ensure full consideration for admission, candidates should submit their applications by the following deadlines:

Application Deadlines-Domestic Applicants:

Semester Admission only Admission /Financial Aid

Fall July 1 February 1

Spring/Summer October 1 October 1

Application Deadlines-International Applicants:

Semester Admission only Admission /Financial Aid

Fall February 1 February 1

Spring/Summer October 1 October 1

DEGREE: Master of Science
MAJOR: Mathematics

The Master's degree is designed to increase one's knowledge and understanding of mathematics beyond the Bachelor's degree level. It also prepares a student for future graduate work.

A candidate for a master's degree may select up to two minors in addition to the major. A minimum of 8 credits of graduate work is necessary for a minor.

Minimum Requirements for the Master's Degree

1. In fulfillment of the Graduate School requirement of a minimum of 30 semester credits of course work, the student must take at least 24 credits of mathematics or statistics, numbered above 500.
2. The student must complete, transfer, or challenge MATH 525, MATH 527, MATH 528, and MATH 581.
3. In addition, 6 of the 24 Math credits must be from the following list of courses: Algebra (MATH 582), Complex Analysis (MATH 591, 592), Differential Equations (MATH 531, 532), Logic and Foundations (MATH 557, 585), Probability and Statistics (STAT 562, 571), Real Analysis (MATH 593, 594) and Topology (MATH 541, 542).
4. At most 6 credits of individual study courses such as MATH 540 may be used to fulfill the course requirement.
5. MATH 511 through 516, and MATH 563 through 569 may not be used to fulfill any of these requirements.
6. The student's program of study must be approved by the departmental Graduate Studies Committee.
7. The student must successfully complete a final master's examination.

The Master's Final Examination

The Master's final examination is an oral examination administered by the student's committee and covers the student's coursework. The student's committee consists of at least three departmental members and a Graduate faculty member from another department who serves as the Dean's representative. If the student has a minor area of study, then a member must come from the minor department. The examination is restricted to course work presented in the student's program of studies. When a master's thesis has been written, the master's final exam will be in part an oral defense of the thesis and in part a general examination of the candidate's course work. The oral exam must be completed at least 10 days prior to the end of the semester in which the candidate wishes to receive the degree.

DEGREE: Professional Master of Financial Mathematics

The Professional Master in Financial Mathematics Program prepares students for successful careers in the financial industry or energy sector, including banks, insurance companies, investment and securities firms, energy companies, utilities, and corporations with exposure to exchange rate or commodities risk. The program provides students with a solid mathematics and statistics background complemented by studies in financial management and financial mathematics including sophisticated problems directly originating from the financial industry. Financial Mathematicians are expected to work in financial product development and pricing, risk management, and portfolio management. Course Requirements for the Professional Master's Degree

1. MATH 518, MATH 521, MATH 522, MATH 577
2. STAT 525, STAT 535
3. FIN 511, FIN 535, FIN 545
4. FIN 590, or any additional FIN course numbered 500 and above with consent of advisor, or MATH 523.

The Financial Mathematics Program is not currently accepting applicants.

DEGREE: Doctor of Philosophy
MAJOR: Mathematics

Candidates for the Ph.D. degree in the Department of Mathematical Sciences must pass a qualifying examination, three comprehensive written examinations, a basic mathematical reading knowledge test in a language other than English, a comprehensive oral examination, a series of courses, and a final oral doctoral thesis examination. These are briefly described below. For more information, see the Graduate School requirements in this catalog, and the Mathematics Graduate Student handbook at www.math.nmsu.edu

Qualifying examination: Every student admitted to the Ph.D. program must complete the Ph.D. oral qualifying examination. Its purpose is to determine the areas in which the student shows strength or weakness, as well as the ability to assimilate subject matter presented at the graduate level. Students who complete their mathematics master's degree at NMSU may request, at the time of applying for their master's oral final examination, that the Master's examination also fulfill the Ph.D. qualifying examination requirement. In all other cases, towards the end of the student's first semester in the Ph.D. program, the student and his or her advisor will convene an oral examination with three examiners, the examiners being the advisor and some of the student's current or past instructors. As a result of the Qualifying examination, the department will take one of the following actions: (1) admit the student to further work toward the Ph.D.; (2) recommend that the student's program be limited to a Master's degree; (3) recommend a reevaluation of the student's progress after the lapse of one semester; or (4) recommend a discontinuation of the student's graduate program in mathematics.

Written comprehensive examinations: Candidates for the Ph.D. degree must pass written comprehensive examinations in three of the seven areas of algebra, complex analysis, differential equations, logic and foundations, real analysis, statistics, and topology. To ensure adequate breadth, a combination of three comprehensive examinations must include real analysis, and at least one of algebra and topology.

The seven examinations are based on the following comprehensive examination sequence courses: Algebra (MATH 525, MATH 581, MATH 582), Complex Analysis (MATH 517, MATH 591, MATH 592), Differential Equations (MATH 518, MATH 531, MATH 532), Logic and Foundations (MATH 504, MATH 557, MATH 585), Real Analysis (MATH 527, MATH 528, MATH 593, MATH 594), Probability and Statistics (STAT 562, STAT 571), and Topology (MATH 541, MATH 542).

Full time students should complete the comprehensive written exams in the first two years. Those who have not made substantial progress towards completion of their written exams at the start of the fifth semester may be removed from the program. Students who have not completed the written exams by the start of the sixth semester will normally have any departmental funding revoked.

Exams are offered every August and January. A student must register to take exams in the semester prior to taking the exams. A student has three consecutive examination periods to complete the written comprehensive exam requirements (Example: if s/he starts in August, s/he has the August, January and August examination periods to complete the exams). This does not extend the time limit mentioned above. Students will normally not be given more than two attempts at any one exam.

Course requirements: Before graduation, a student must pass a total of four comprehensive exam sequences, but needs to take the comprehensive examinations in only three of them. In addition, a student must pass four more (one-semester) MATH/STAT courses from the seven comprehensive exam sequences listed above.

A student may pass any of the four comprehensive examination sequences before enrolling as a Ph.D. student, but the four additional courses have to be passed after enrolling as a Ph.D. student.

The following courses will not count towards the course requirements: Any course below MATH 501, 511 through 516, and MATH 563 through 569, MATH/STAT 540, MATH/STAT 598, MATH 599, MATH 600, MATH 700.

Students and advisors are encouraged to consider further courses beyond this minimum.

Foreign language examination: The department requires that each Ph.D. student pass a basic mathematical reading knowledge exam in a language, other than English, relevant to the student's research interests. This exam is coordinated by the student's advisor and consists of the open-dictionary written translation into English of a mathematical text of interest to the student. The language requirement must be fulfilled prior to the oral part of the Ph.D. comprehensive examination.

Oral Comprehensive Exam: The student must take this exam at the end of the semester after completing the written comprehensive exams. The student should present a proposed direction for thesis work.

Final Oral Exam: This should be an exam over the student's thesis and administered by the same committee of the oral comprehensive exam.

MATHEMATICS

MATH 451. Introduction to Differential Geometry 3 cr.

Applies calculus to curves and surfaces in three dimensional Euclidean space. Prerequisites: MATH 280 and MATH 391, or consent of instructor.

MATH 452. Foundations of Geometry 3 cr.

Topics in projective, axiomatic Euclidean or non-Euclidean geometries. Prerequisite(s): C or better in Math 331 or Math 332. Restricted to: Main campus only.

MATH 453. Introduction to Topology 3 cr.

Introduction to topological spaces and metric spaces, with connections to analysis, geometry, and the classification of surfaces. Prerequisite: MATH 332 or consent of instructor.

MATH 454. Mathematical Logic 3 cr.

Propositional calculus and the first order predicate calculus, including Godel's completeness theorem for the latter, and additional topics at the option of the instructor. Prerequisite(s): C or better in Math 331 or Math 332, or consent of instructor.

MATH 455. Elementary Number Theory 3 cr.

Covers primes, congruences and related topics. Prerequisite: grade of C or better in MATH 331 or consent of instructor.

MATH 457. Topics in Algebra 3 cr.

Topics may include coding theory, crytography, algebraic geometry, or symmetry groups. Prerequisites: C or better in Math 331.

MATH 459. Survey of Geometry 3 cr.

Basic concepts of Euclidean geometry, ruler and compass constructions. May include topics in non-Euclidean geometry. For non-math majors. Prerequisite(s): C or better in Math 331 or Math 332. Restricted to: Main campus only.

MATH 466. Lattice Theory 3 cr.

Introduction to partially ordered sets, distributive, modular, and Boolean lattices. Prerequisites: MATH 330 or MATH 331 or MATH 332 or consent of instructor.

MATH 471. Complex Variables 3 cr.

A first course in complex function theory, with emphasis on applications. Prerequisite: MATH 391 or both MATH 392 and MATH 291G.

MATH 472. Fourier Series and Boundary Value Problems 3 cr.

Fourier series and methods of solution of the boundary value problems of applied mathematics. Prerequisite: MATH 392.

MATH 473. Calculus of Variations and Optimal Control 3 cr.

Euler's equations, conditions for extrema, direct methods, dynamic programming, and the Pontryagin maximal principle. Prerequisite: MATH 392.

MATH 475. Business Applications 3 cr.

Taught with MATH 375 with additional work. Does not fulfill requirements for degrees in mathematics. Prerequisite(s): C or better in Math 142G, or in MATH 191G, or in Math 235.

MATH 480. Matrix Theory and Applied Linear Algebra 3 cr.

An application driven course, whose topics include rectangular systems, matrix algebra, vector spaces and linear transformations, inner products, and eigenvalues and eigenvectors. Applications may include LU factorization, least squares, data compression, QR factorization, singular value decomposition, and search engines. Prerequisite(s): C or better in any 300-level course with a MATH or STAT prefix.

MATH 481. Advanced Linear Algebra 3 cr.

Rigorous treatment of vector spaces and linear transformations including canonical forms, spectral theory, inner product spaces and related topics. Prerequisite: grade of C or better in MATH 331.

MATH 491. Introduction to Real Analysis I 3 cr.

Rigorous discussion of the topics introduced in calculus. Sequences, series, limits, continuity, differentiation. Prerequisite: grade of C or better in MATH 332 or consent of instructor.

MATH 492. Introduction to Real Analysis II 3 cr.

Continuation of MATH 491. Integration, metric spaces and selected topics. Prerequisite: MATH 491 or consent of instructor.

MATH 498. Directed Reading 1-6 cr.

May be repeated for a maximum of 6 credits. Graded S/U.

MATH 501. Introduction to Differential Geometry 3 cr.

Same as MATH 451 with additional work for graduate students.

MATH 502. Foundations of Geometry 3 cr.

Same as MATH 452 with additional assignments for graduate students.

MATH 503. Introduction to Topology 3 cr.

Same as MATH 453 with additional work for graduate students.

MATH 504. Mathematical Logic 3 cr.

Same as MATH 454 with additional assignments for graduate students.

MATH 505. Elementary Number Theory 3 cr.

Same as MATH 455 with additional assignments for graduate students.

MATH 506. Lattice Theory 3 cr.

Same as MATH 466 with additional assignments for graduate students.

MATH 507. Topics in Algebra 3 cr.

Topics may include coding theory, cryptography, algebraic geometry, or symmetry groups. Same as Math 457 with additional work for graduate students. Prerequisites: C or better in Math 331.

MATH 509. Information Theory 3 cr.

This class is a study of Shannon's measure of information and discusses mutual information, entropy, and channel capacity, the noiseless source coding theorem, the noisy channel coding theorem, channel coding and random coding bounds, rate-distortion theory, and data compression. Prerequisite(s): EE 571 or Stat 515. Restricted to: Main campus only. Crosslisted with: E E 586

MATH 511. Fundamentals of Elementary Mathematics I 3 cr. (3+1P)

Topics from real numbers, geometry, measurement, and algorithms, incorporating calculator technology. Intended for K-8 teachers. As part of course students mentor MATH 111 undergraduates. Does not fulfill degree requirements for M.S. in mathematics.

MATH 512. Fundamentals of Elementary Mathematics II 3 cr. (3+1P)

Real numbers, geometry, and statistics, incorporating calculator technology. Intended for K-8 teachers. Students serve as mentors to MATH 112 undergraduates. Does not fulfill degree requirements for M.S. in mathematics.

MATH 513. Fundamentals of Algebra and Geometry I 3 cr. (3+1P)

Algebra and metric geometry, incorporating appropriate calculator technology. Intended for K-8 teachers. Students serve as mentors to MATH 313 undergraduates. Does not fulfill degree requirements for M.S. in mathematics.

MATH 516. Calculus with Hands-on Application 3 cr.

This course, primarily for in-service teachers, is taught in an interactive laboratory format. Students design and construct physical objects for which the planning stage requires calculus techniques. All numerical computations are carried out on graphing calculators. Meets simultaneously with Math 316, primarily for prospective teachers. Does not fulfill degree requirements for M.S. in Mathematics. Prerequisite(s): Math 511 and Math 512 or consent of instructor.

MATH 517. Complex Variables 3 cr.

Same as MATH 471 with additional work for graduate students.

MATH 518. Fourier Series and Boundary Value Problems 3 cr.

Same as MATH 472 with additional work for graduate students.

MATH 519. Calculus of Variations and Optimal Control 3 cr.

Same as MATH 473 with additional work for graduate students.

MATH 521. Financial Mathematics I: Portfolio Optimization 3 cr.

Complete and incomplete markets, optimal investment paths, dynamic optimization, the Black-Scholes model, European options, American options. Prerequisite: STAT 515 and either MATH 280 or MATH 480.

MATH 522. Financial Mathematics II 3 cr.

Bonds, Swaps, Exotic options, Barrier options, Asian options, Lookback options, options with transaction costs, Fokker Plank theory: computing expectations, The Heath-Jarrow-Morton theorem, the Ho-Lee model, Stochastic volatility models, Exponential-Affine models, numerical methods. Prerequisite: MATH 521.

MATH 523. Numerical Optimization and Applications to Financial Mathematics 3 cr.

Dynamic optimization of a monopolist, trading off inflation and unemployment, the optimal adjustment of labor demand, infinite planning horizon, the optimal investment path of a firm, the optimal social saving behavior, phase-diagram analysis, optimal control theory, the political business cycle, the dynamics of a revenue-maximizing firm, economic examples of state-space constraints. This course is offered simultaneously with Math 423. Prerequisite: Math 521.

MATH 525. Advanced Linear Algebra 3 cr.

Same as MATH 481 with additional work for graduate students. Prerequisite: grade of C or better in MATH 331.

MATH 527. Introduction to Real Analysis I 3 cr.

Same as MATH 491 with additional work for graduate students.

MATH 528. Introduction to Real Analysis II 3 cr.

Same as MATH 492 with additional work for graduate students.

MATH 530. Special Topics 1-3 cr.

Specific subjects to be announced in the Schedule of Classes. May be for unlimited credit with approval of the department.

MATH 531. Ordinary Differential Equations 3 cr.

Linear algebra and linear ordinary differential equations, existence and uniqueness of solution, smooth dependence on initial conditions, flows, introduction to smooth dynamical systems. Prerequisites: MATH 392 and MATH 527, or consent of instructor.

MATH 532. Partial Differential Equations 3 cr.

The basic equations of mathematical physics. Elliptic, hyperbolic, and parabolic equations. Characteristic surfaces. Well-posed problems. Prerequisite: MATH 518 or consent of instructor.

MATH 534. Nonlinear Programming 3 cr.

Theoretical and computational methods to solve optimization problems in engineering, statistics, economics, and operations research. Topics include convexity, optimality conditions, Newton's method, Lagrange multipliers, search algorithms for unconstrained and constrained problems, as well as barrier and penalty methods.

MATH 541. Topology I 3 cr.

Topological spaces, connectedness, compactness, Tychonoff's theorem, separation axioms, Tietze's extension theorem, Urysohn s metrization theorem, elementary homotopy theory, the fundamental group, the Seifert-van Kampen theorem. Prerequisites: MATH 525 and MATH 528, or consent of instructor.

MATH 542. Topology II 3 cr.

Covering spaces and their classification, CW-complexes, singular and cellular homology, Brouwer's fixed point theorem, and other applications. Prerequisites: MATH 541 or consent of instructor.

MATH 555. Differentiable Manifolds 3 cr.

Differentiable structures, tangent bundles, vector fields and differential equations. Additional topics may include differential forms, De Rham cohomology, Riemannian geometry, and topics chosen by the instructor. May be repeated for a maximum of 9 credits. Consent of instructor required. Prerequisite(s): MATH 525 and MATH 528, or consent of instructor.

MATH 557. Axiomatic Set Theory 3 cr.

A detailed study of Zermelo-Fraenkel and Bernays set theories. Prerequisite: MATH 504 or equivalent.

MATH 561. The Role of History in the Teaching of Mathematics 3 cr.

In-depth study of selected mathematical topics through examination of their historical development, with emphasis on studying original sources. Pedagogical aspects of using history and original sources in teaching mathematics. Research and preparation of classroom materials based on original sources.

MATH 562. History and Theories of Mathematics Education 3 cr.

A study of the history of the mathematics taught in American schools, including an examination of authentic original textbooks and the changes in their content and the approach to the subject over time, together with writings of people who have influenced the development and changes of mathematics education. Theories of learning mathematics, and current issues in mathematics education. Prerequisite(s): Restricted to graduate students.

MATH 563. Algebra with Connections 3 cr.

Connections between Algebra and other K-12 curriculum strands, especially Geometry and Probability / Data Analysis. Apply algebraic modeling and reasoning to a variety of mathematical problem solving situations. Does not fulfill requirements for degrees in mathematics. Consent of instructor required. Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 564. From Number to Algebra 3 cr.

The progression from Number to Algebra in the K-12 curriculum as a concrete-to-abstract progression. Key concepts considered across the grade levels include the different uses of variables, equivalence in different contexts, patterns, and ratios. Does not fulfill requirements for degrees in mathematics. Consent of instructor required. Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 565. Modeling Linear Decisions for Middle School Teachers 3 cr.

Introduction to linear decision-making algorithms. Topics include network models, systems of equations and linear programming. Does not fulfill requirements for degrees in mathematics. Prerequisite: MATH 185 or equivalent.

MATH 566. Data Analysis with Applications 3 cr.

Statistical concepts and terminology in professional uses of data by teachers, such as standardized test score reports and educational research; visual displays of data; measures of variation and central tendency; consideration of how K-12 topics in Data Analysis are developed from one grade level to the next. Does not fulfill requirements for degrees in mathematics. Consent of instructor required. Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 567. From Measurement to Geometry 3 cr.

The progression from Measurement to Geometry in the K-12 curriculum as a concrete-to abstract progression. Important concepts such as angle, length, and area progress from concrete, measurable situations to more abstract problems which require reasoning and proof. Does not fulfill requirements for degrees in mathematics. Consent of instructor required. Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 568. Using Number Throughout the Curriculum 3 cr.

Understand number concepts more deeply by seeing many examples of those concepts applied in other content strands. Develop mathematical knowledge and understanding to build a repertoire of ways for students to practice and review basic number skills and concepts as part of later, more advanced courses. Does not fulfill requirements for degrees in mathematics. Consent of instructor required. Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 569. Geometry with Connections 3 cr.

Connections between Geometry and other K-12 curriculum strands, especially Algebra and Probability / Data Analysis. Address key attributes of geometric concepts by considering their connections within and across grade levels. Does not fulfill requirements for degrees in mathematics. Consent of instructor required. Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 577. Numerical Analysis I 3 cr.

Topics may include interpolation, differential equations, nonlinear equations, optimization. Prerequisites: MATH 480 and 527, or consent of instructor.

MATH 581. Algebra 1 3 cr.

Examines groups, commutative rings, solvability of polynomials, Galois theory, ruler and compass constructions. Prerequisite/corequisite: MATH 525.

MATH 582. Algebra II 3 cr.

Group actions, fundamental theorem of finite Abelian groups, Sylow theorems, solvable groups, noncommutative rings, Noetherian rings, unique factorization domains, modules, tensor products. Prerequisite: MATH 581.

MATH 583. Algebraic Number Theory 3 cr.

Number fields and number rings, prime decomposition in number rings, ideal theory and the ideal class group, and selected other topics. Prerequisite: MATH 581 or consent of instructor.

MATH 584. Representation Theory 3 cr.

Topics from representation theory of finite or infinite groups. Prerequisite: consent of instructor. May be repeated for a maximum of 9 credits.

MATH 585. Universal Algebra 3 cr.

Universal algebra and category theory. Theorems of Birkhoff and Tarski relating equational classes, free algebras and their construction through homomorphisms, subalgebras and products. Topics from model theory, sheaf theory and representation by subdirect products. Prerequisite: consent of instructor. May be repeated for a maximum of 6 credits.

MATH 586. Nonlinear Dynamics I 3 cr.

Same as PHYS 586.

MATH 591. Complex Analysis I 3 cr.

Rigorous treatment of complex differentiation and integration, properties of analytic functions, series and Cauchy's integral representations. Prerequisites: MATH 517 and MATH 528, or consent of instructor.

MATH 592. Complex Analysis II 3 cr.

Harmonic functions, product representations, conformal mappings, Riemann's mapping theorem, Riemann surfaces, and selected other topics. Prerequisite: MATH 591 or consent of instructor.

MATH 593. Measure and Integration 3 cr.

Measure spaces, measurable functions, extension and decomposition theorems for measures, integration on measure spaces, absolute continuity, iterated integrals. Prerequisite: MATH 528 or consent of instructor.

MATH 594. Real Analysis 3 cr.

Differentiation, Lp spaces, Banach spaces, measure and topology, other selected topics. Prerequisite: MATH 593.

MATH 598. Special Research Programs 1-3 cr.

Individual analytical or experimental projects. Maximum of 3 credits per semester. More than 3 credits total requires approval of graduate committee. Six credits maximum.

MATH 599. Master's Thesis 0-88 cr.

Thesis.

MATH 600. Doctoral Research 1-88 cr.

Research.

MATH 601. Special Topics 1-3 cr.

Specific subjects to be announced in the Schedule of Classes. May be repeated for unlimited credit with approval of the department.

MATH 643. Topology III 3 cr.

Topics may include higher homotopy groups, fibrations, cohomology operations and obstruction theory, spectral sequences, or others chosen by instructor. Prerequisites: MATH 542 or consent of instructor. May be repeated for a maximum of 9 credits.

MATH 649. Applications of Tensor Analysis 3 cr.

Same as PHYS 649.

MATH 655. Topics in Differential Geometry 3 cr.

Representation theory of Lie groups, Riemannian geometry, or another topic chosen by instructor. Content varies. Prerequisite: MATH 555 or consent of instructor. May be repeated for a maximum of 9 credits.

MATH 683. Homological Algebra 3 cr.

Basic topics in homological algebra and category theory. Prerequisite: MATH 542 or MATH 582 or consent of instructor. May be repeated for a maximum of 9 credits.

MATH 686. Nonlinear Dynamics II 3 cr.

Same as PHYS 686.

MATH 695. Introduction to Functional Analysis I 3 cr.

Banach spaces. The three basic principles: uniform boundedness principle, closed graph/open mapping theorems, Hahn-Banach theorem. Prerequisites: MATH 541 and MATH 594, or consent of instructor.

MATH 696. Introduction to Functional Analysis II 3 cr.

Continuation of MATH 695. Topics selected from topological vector spaces, Hilbert space, spectral theory, Banach algebras, and distribution theory. Prerequisite: MATH 695 or consent of instructor.

MATH 698. Selected Topics 1-88 cr.

Selected topics.

MATH 700. Doctoral Dissertation 1-88 cr.

Dissertation.

STATISTICS

STAT 470. Probability: Theory and Applications 3 cr.

Basic probability distributions including binomial, normal; random variables, expectation; laws of large numbers; central limit theorem. Prerequisites: MATH 291G and at least one-300 level Math course.

STAT 480. Statistics: Theory and Applications 3 cr.

Point and interval estimation; sufficiency; hypothesis testing; regression; analysis of variance; chi-square tests. Prerequisite: STAT 470.

STAT 515. Probability: Theory and Applications 3 cr.

Same as STAT 470 with additional work for graduate students.

STAT 525. Statistics: Theory and Applications 3 cr.

Same as STAT 480 with additional work for graduate students.

STAT 535. Elementary Stochastic Processes 3 cr.

Markov chains, Poisson processes, Brownian motion, branching processes, and queuing processes, with applications to the physical, biological, and social sciences. Prerequisite: STAT 515 or consent of instructor.

STAT 540. Directed Reading 1-6 cr.

Prerequisite: consent of instructor and graduate committee. May be repeated for a maximum of 6 credits. Graded S/U.

STAT 562. Foundations of Probability 3 cr.

Probability spaces, expectation and conditional expectation, limit theorems and laws of large numbers. Prerequisite: MATH 593.

STAT 571. Continuous Multivariate Analysis 3 cr.

Theory and applications of the multivariate normal distribution. Prerequisites: MATH 480 and STAT 525, or consent of instructor.

STAT 572. Linear Models 3 cr.

Theory of regression, analysis of variance, analysis of covariance in various linear models. Prerequisite: STAT 571.

STAT 581. Advanced Theory of Statistics I 3 cr.

Testing hypotheses, probability and sufficiency, uniformly most powerful tests, unbiasedness, invariance, and minimax principle. Prerequisite: STAT 525 or consent of instructor.

STAT 582. Advanced Theory of Statistics II 3 cr.

Estimation of parameters; unbiased estimators; equivariance; Bayes properties; large sample theory and optimality. Prerequisite: STAT 581 or consent of instructor.

STAT 598. Special Research Problems 1-3 cr.

Individual investigations or consulting programs. Maximum of 3 credits.