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University Bulletin: Undergraduate Programs 2003-2004 The George Washington University  

 
    
 
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MATHEMATICS


 

Professors H.D. Junghenn, M.M. Gupta, E.A. Robinson, F.E. Baginski, D.H. Ullman, J. Przytycki, J. Bonin, V. Harizanov, Y. Rong, J.B. Conway (Chair)

Associate Professors M. Moses, W. Schmitt, L. Abrams, X. Ren

Assistant Professors I. Yi, A. Shumakovitch, H. Wu, M. Musielak

Bachelor of Arts or Bachelor of Science with a major in mathematics—The department offers the Bachelor of Arts and Bachelor of Science with a major in mathematics through three tracks: pure mathematics, applied mathematics, and computational mathematics. Each track is designed to give students a broad background in the theory and practice of modern mathematics. The pure mathematics and applied mathematics tracks are complementary and equally rigorous, differing mainly in their emphasis. The Bachelor of Science in either track provides strong preparation for graduate study in mathematics. The Bachelor of Arts, while providing a strong mathematics background, is designed to permit a wider selection of electives to enable students to plan for careers such as teaching, medicine, or law. The computational mathematics track is designed to prepare students for careers in government and industrial settings in which modeling and computation play a large role; it is intended for students who plan to enter the job market immediately after graduation.

The following requirements must be fulfilled:

1. The general requirements stated under Columbian College of Arts and Sciences.

2. Prerequisite courses—Math 21 or 31 and Math 32, 33, 71, 72, and 84.

3. Required courses for the major—

(a) The track in pure mathematics: Math 121, 139, 140, and Math 122 or 125; 9 credits of additional 100-level math courses for the B.A. or 15 credits of additional 100-level math courses for the B.S.

(b) The track in applied mathematics: Math 139, 142, 143, 153, and 159; 6 credits of additional 100-level math courses for the B.A. or 12 credits of additional 100-level math courses for the B.S.

(c) The track in computational mathematics: Math 142, 143, 153, 159, and one course selected from Stat 157 or CSci 49, 50, 100, or 102; 9 credits of additional 100-level math courses for the B.A. or 15 credits of additional 100-level math courses for the B.S.

Special Honors—To graduate with Special Honors, a student must meet the general requirements stated under University Regulations; maintain a grade-point average of at least 3.5 in courses in the major; enroll in 3 credit hours of Math 195 in addition to the other required courses in the major; and present an oral defense of a senior thesis prepared for Math 195.

Minor in mathematics—18 credits in mathematics courses, including Math 84, and of which at least 9 are at the 100 level or higher, chosen in consultation with a departmental advisor.

With permission, graduate courses in the department may be taken for credit toward an undergraduate degree. See the Graduate Programs Bulletin for course listings.

Note: Math 20 and 21 each cover one-half the material of Math 31. Because Math 21, 31, and 52 are related in their subject matter, credit for only one of the three may be applied toward a degree. For courses that indicate the placement examination as prerequisite, see https://my.gwu.edu/mod/placement/.

7 Mathematics and Politics (3) Staff
  A mathematical treatment of fair representation, voting systems, power, and conflict. The impossibility theorems of Balinsky and Young and of Arrow. The electoral college. The prisoner’s dilemma.
8 History of Mathematics (3) Gupta
  The history of mathematics, with emphasis on its importance in the evolution of human thought. Students learn some useful mathematics from areas such as geometry, number theory, and probability and develop an appreciation of the mathematical endeavor.
9—10 Mathematical Ideas I—II (3—3) Staff
  Math 9: Elementary mathematical models of growth and decay, scaling, chaos, and fractals. Math 10: Elementary graph theory, scheduling, probability theory.
20—21 Calculus with Precalculus I—II (3—3) Staff
  An introduction to single-variable calculus (differentiation and integration of algebraic and trigonometric functions with applications), with the concepts and techniques of precalculus developed as needed. Prerequisite to Math 20: the placement examination or a score of 560 or above on the SAT II in mathematics; Math 20 is prerequisite to Math 21.
31 Single-Variable Calculus I (3) Staff
  Limits and continuity. Differentiation and integration of algebraic and trigonometric functions with applications. Prerequisite: the placement examination or a score of 720 or above on the SAT II in mathematics.
32 Single-Variable Calculus II (3) Staff
  The calculus of exponential and logarithmic functions. L’Hopital’s rule. Techniques of integration. Infinite series and Taylor series. Polar coordinates. Prerequisite: Math 21 or 31.
33 Multivariable Calculus (3) Staff
  Partial derivatives and multiple integrals. Vector-valued functions. Topics in vector calculus, including line and surface integrals and the theorems of Gauss, Green, and Stokes. Prerequisite: Math 32.
51 Finite Mathematics for the Social and Management Sciences (3) Staff
  Systems of linear equations, matrix algebra, linear programming, probability theory, and mathematics of finance. Prerequisite: the placement examination or a score of 560 or above on the SAT II in mathematics.
52 Calculus for the Social and Management Sciences (3) Staff
  Differential and integral calculus of functions of one variable; applications to business and economics. Prerequisite: the placement examination or a score of 560 or above on the SAT II in mathematics.
71 Introduction to Mathematical Reasoning (2) Staff
  An introduction to the fundamental abstract concepts of modern mathematics as well as various proof techniques demonstrated on numerous examples taken from within discrete and continuous mathematics. Prerequisite: Math 21 or 31. Open to majors and to others with permission of instructor or the departmental undergraduate advisor.
72 Introduction to Computing in Mathematics (2) Staff
  An introduction to the use of computers in modern mathematics and a primer in basic programming skills covering Maple, Matlab, and LaTex. Prerequisite: Math 21 or 31. Open to majors and to others with permission of instructor or the departmental undergraduate advisor.
84 Linear Algebra I (3) Staff
  Linear equations, matrices, inverses, and determinants. Vector spaces, rank, eigenvalues, and diagonalization. Applications to geometry and ordinary differential equations. Prerequisite: Math 21 or 31, or 51 and 52, or permission of instructor.
91 Introductory Special Topics (1 to 3) Staff
  Admission by permission of instructor. May be repeated for credit.
101 Introduction to Mathematical Logic (3) Moses
  Symbolic logic as a precise formalization of deductive thought. Logical correctness of reasoning. Formal languages, interpretations, and truth. Propositional logic and first-order quantifier logic suited to deductions encountered in mathematics. Goedel’s completeness theorem; compactness. Prerequisite: Math 71 or permission of instructor.
102 Axiomatic Set Theory (3) Harizanov
  Cantor’s theory of sets. Russell’s paradox. Axiomatization of set theory as a framework for a contradiction-free mathematics. The Zermelo—Fraenkel axioms and the axiom of choice. Finite, countable, and uncountable sets; ordinal and cardinal arithmetic. The continuum hypothesis. Prerequisite: Math 71 or permission of instructor.
103 Computability Theory (3) Harizanov
  The unlimited register machine as a model of an idealized computer. Computable and partial computable functions; Church—Turing thesis. Kleene’s recursion theorem. Algorithmic enumerability. Unsolvability of the halting problem and other theoretical limitations on what computers can do. Discussion of Goedel’s incompleteness theorem. Prerequisite: Math 71 or permission of instructor.
104 Computational Complexity (3) Harizanov
  Automata and languages. Deterministic and nondeterministic Turing machines. Space and time complexity measures and classes. The P versus NP problem. The traveling salesman problem and other NP-complete problems. Intractability. Circuit complexity. Introduction to probabilistic and quantum algorithms. Prerequisite: Math 71 or permission of instructor.
106 Introduction to Topology (3) Przytycki, Rong
  Metric spaces: completeness, compactness, continuity. Topological spaces: continuity, bases, subbases, separation axioms, compactness, local compactness, connectedness, product and quotient spaces. Prerequisite: Math 71 or permission of instructor.
113 Introduction to Combinatorics (3) Bonin
  Introduction to combinatorial enumeration. Basic counting techniques, inclusion—exclusion principle, recurrence relations, generating functions, pigeonhole principle, bijective correspondences. Prerequisite: Math 71 or permission of instructor.
120 Elementary Number Theory (3) Bonin
  Divisibility of integers, prime numbers, greatest common divisor, the Euclidean algorithm, congruence, the Chinese remainder theorem, number theoretic functions, Möbius inversion, Euler’s phi function, and applications to cryptography and primality testing. Prerequisite: Math 71 or permission of instructor.
121 Introduction to Abstract Algebra I (3) Abrams, Schmitt
  Study of groups and associated concepts, including Lagrange’s theorem, Cayley’s theorem, the fundamental theorem of homomorphisms, and applications to counting. Prerequisite: Math 71 and 84 or permission of instructor.
122 Introduction to Abstract Algebra II (3) Abrams
  Study of rings, through maximal and prime ideals, and the study of fields, through Galois theory. Prerequisite: Math 121 or permission of instructor.
125 Linear Algebra II (3) Yi
  Theory of vector spaces, linear transformations, and matrices. Quadratic and bilinear forms. Characteristic polynomials and the Cayley—Hamilton theorem. Similarity and Jordan canonical form. Prerequisite: Math 71 and 84 or permission of instructor.
132 Introduction to Graph Theory (3) Ullman
  Fundamental concepts, techniques, and results of graph theory. Topics include trees, connectivity, traversability, matchings, coverings, colorability, planarity, networks, and Polya enumeration. Prerequisite: Math 71 or permission of instructor.
135 Projective Geometry (3) Bonin
  Projective spaces, projectivities, conics, pairs and pencils of conics, finite planes, coordinates, collineation, Desarguesian planes. Prerequisite: Math 120 or 121 or permission of the instructor.
139 Real Analysis (3) Junghenn
  A rigorous study of differentiation, integration, and convergence. Topics include sequences and series, continuity and differentiability of real-valued functions of a real variable, the Riemann integral, sequences of functions, and power series. Prerequisite: Math 32 and either 71 or 84 or permission of instructor.
140 Real Analysis (3) Ullman
  Continuation of Math 139. Topics include: topology of Rn, derivatives of functions of several variables, inverse and implicit function theorems, multiple integrals, generalized Stokes’s theorem. Prerequisite: Math 33 and 139 or permission of instructor.
142 Ordinary Differential Equations (3) Musielak, Ren
  A first course in ordinary differential equations with an emphasis on mathematical modeling: solution curves, direction fields, existence and uniqueness, approximate solutions, first and second order linear equations, linear systems, phase portraits, and Laplace transforms. Prerequisite: Math 32 and 84 or permission of instructor.
143 Partial Differential Equations (3) Baginski
  A first course in partial differential equations: Fourier series and separation of variables, vibrations of a string, Sturm—Liouville problems, series solutions, Bessel’s equation, linear partial differential equations, wave and heat equations, separation of variables. Prerequisite: Math 33 and 84 or permission of instructor.
148 Differential Geometry (3) Robinson
  Curves in space, regular surfaces, tensors, fundamental forms of a surface. Gauss—Bonnet theory, minimal surfaces. The geometry of the Gauss map. Prerequisite: Math 33 and 84 or permission of instructor.
153 Introduction to Numerical Analysis (3) Gupta
  Accuracy and precision. Linear systems and matrices. Direct and iterative methods for solution of linear equations. Sparse matrices. Solution of nonlinear equations. Interpolation and approximate representation of functions, splines. Prerequisite: Math 33 or permission of instructor. Math 72 and 84 are recommended.
157 Introduction to Complex Variables (3) Conway
  Analytic functions and power series. Contour integration and the calculus of residues. Conformal mapping. Physical applications. Prerequisite: Math 33 and 84 or permission of instructor.
159 Introduction to Mathematical Modeling (3) Musielak
  An introduction to the fundamental modeling ideas of dimensional analysis, scaling, and elementary approximations of curves and functions. Applications to development of models from science and engineering. Prerequisite: Math 72 and 142.
181 Seminar: Topics in Mathematics (3) Staff
  Past topics have included computational mathematics, fractals; network flows and combinatorial optimization; information theory and coding theory; dynamical systems; queuing theory. May be repeated for credit with permission. Prerequisite: Math 33 and 84 or permission of instructor.
191 Special Topics (arr.)  
  Admission by permission of instructor. May be repeated for credit.
195 Reading and Research (arr.)  
  Under the personal direction of an instructor. Limited to majors with demonstrated capability. Prior approval of instructor required. May be repeated for credit.
 

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© 2009 University Bulletin
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Information in this bulletin is generally accurate as of fall 2008. The University reserves the right to change courses, programs, fees, and the academic calendar, or to make other changes deemed necessary or desirable, giving advance notice of change when possible.