Fall Course Guide

Courses in Mathematics (Division 428)

Fall Term, 1998 (September 8-December 21, 1998)

513. Introduction to Linear Algebra. Math. 412 or permission of Honors advisor. Two credits granted to those who have completed Math. 417; one credit granted to those who have completed Math 217 or 419. (3). (Excl). (BS).
This is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis is on concepts and proofs with some calculations to illustrate the theory. For students with only the minimal prerequisite, this is a demanding course; at least one additional proof-oriented course (e.g., Math 451 or 512) is recommended. Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear transformations, bases, and matrices; eigenvalues and eigenvectors; applications to linear and linear differential equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form. Math 419 covers much of the same material using the same text, but there is more stress on computation and applications. Math 217 is similarly proof-oriented but significantly less demanding than Math 513. Math 417 is much less abstract and more concerned with applications. The natural sequel to Math 513 is 593. Math 513 is also prerequisite to several other courses (Math 537, 551, 571, and 575) and may always be substituted for Math 417 or 419. WL:2
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520. Life Contingencies I. Math. 424 and Math. 425. (3). (Excl). (BS).
The goal of this course is to teach the basic actuarial theory of mathematical models for financial uncertainties, mainly the time of death. In addition to actuarial students, this course is appropriate for anyone interested in mathematical modeling outside of the physical sciences. Concepts and calculation are emphasized over proof. The main topics are the development of (1) probability distributions for the future lifetime random variable, (2) probabilistic methods for financial payments depending on death or survival, and (3) mathematical models of actuarial reserving. 523 is a complementary course covering the application of stochastic process models. Math 520 is prerequisite to all succeeding actuarial courses. Math 521 extends the single decrement and single life ideas of 520 to multi-decrement and multiple-life applications directly related to life insurance and pensions. The sequence 520-521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Math 522 applies the models of 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc. Recommended text: Actuarial Mathematics (Second Editions) by Bowles et al. WL:2 (Huntington)
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522. Actuarial Theory of Pensions and Social Security. Math. 520. (3). (Excl). (BS).
Practice and theoretical techniques of pension plan evaluation and analysis; social security projections. WL:2
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523. Risk Theory. Math. 425. (3). (Excl). (BS).
Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in finance, risk management, or insurance and provides background for the professional examinations in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society. Students should have a basic knowledge of common probability distributions (Poisson, exponential, gamma, binomial, etc.) and have at least junior standing. Two major problems will be considered: (1) modeling of payouts of a financial intermediary when the amount and timing vary stochastically over time; and (2) modeling of the ongoing solvency of a financial intermediary subject to stochastically varying capital flow. These topics will be treated historically beginning with classical approaches and proceeding to more dynamic models. Connections with ordinary and partial differential equations will be emphasized. Classical approaches to risk including the insurance principle and the risk-reward tradeoff. Review of probability. Bachelier and Lundberg models of investment and loss aggregation. Fallacy of time diversification and its generalizations. Geometric Brownian motion and the compound Poisson process. Modeling of individual losses which arise in a loss aggregation process. Distributions for modeling size loss, statistical techniques for fitting data, and credibility. Economic rationale for insurance, problems of adverse selection and moral hazard, and utility theory. The three most significant results of modern finance: the Markowitz portfolio selection model, the capital asset pricing model of Sharpe, Lintner and Moissin, and (time permitting) the Black-Scholes option pricing model. WL:2
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525/Stat. 525. Probability Theory. Math. 450 or 451. Students with credit for Math. 425/Stat. 425 can elect Math. 525/Stat. 525 for only one credit. (3). (Excl). (BS).
This course is a thorough and fairly rigorous study of the mathematical theory of probability. There is substantial overlap with 425, but here more sophisticated mathematical tools are used and there is greater emphasis on proofs of major results. Math 451 is preferable to Math 450 as preparation, but either is acceptable. Topics include the basic results and methods of both discrete and continuous probability theory. Different instructors will vary the emphasis between these two theories. EECS 501 also covers some of the same material at a lower level of mathematical rigor. Math 425 is a course for students with substantially weaker background and ability. Math 526, Stat 426, and the sequence Stat 510-511 are natural sequels. WL:2
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528. Topics in Casualty Insurance. Math 217, 417, or 419. (1). (Excl).
The insurance policy is the contract describing the services and protection which the insurance company provides to the insured. This course will develop an understanding of the nature of the coverages provided and the bases of exposure used in the respective product lines. It will explore the basic purpose and principles of the underwriting function, how products are designed and modified and the different marketing systems. It will also look at how claims are settled since this determines losses which are key components for insurance ratemaking and reserving. Finally, the course will explore basic ratemaking principles and concepts of loss reserving. WL:2
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537. Introduction to Differentiable Manifolds. Math. 513 and 590. (3). (Excl). (BS).
This course in intended for students with a strong background in topology, linear algebra, and multivariable advanced calculus equivalent to the courses 590, 513, and 551. Its goal is to introduce the basic concepts and results of Differential Topology and Differential Geometry. Topics may include: Inverse and Implicit function theorem in Rⁿ, differentiable manifolds, tangent and cotangent bundles, exterior differential forms, vector fields, partitions of unity, integration on manifolds, Stokes' Theorem, the divergence theorem. Topics in Riemannian Geometry include Riemannian metrics, covariant differentiation and connections, torsion tensor, Levi-Civita connection, Riemann curvature tensor, Gaussian, sectional, Ricci, scalar and mean curvatures, 2-dimensional case, hypersurface case, Gauss and Codazzi equations, length and energy of curves, geodesics, completeness (Hopf-Rinow Theorem), exponential map, Cartan-Hadamard Theorem. Math 433 is an undergraduate version which covers much less material in a less sophisticated way. WL:2
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555. Introduction to Functions of a Complex Variable with Applications. Math. 450 or 451. (3). (Excl). (BS).
This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. Math 596 covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Math 555 is prerequisite to many advanced courses in science and engineering fields. WL:2
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556. Methods of Applied Mathematics I. Math. 217, 419, or 513; 451 and 555. (3). (Excl). (BS).
This is an introduction to methods of applied analysis with emphasis on Fourier analysis for differential equations. Initial and boundary value problems are covered. Students are expected to master both the proofs and applications of major results. The prerequisites include linear algebra, advanced calculus and complex variables. Topics may vary with the instructor but often include: Fourier series; separation of variables for partial differential equations; heat conduction, wave motion, electrostatic fields; Sturm-Liouville problems; Fourier transform; Green's functions; distributions; Hilbert space, complete orthonormal sets; integral operators; spectral theory for compact self-adjoint operators. Math 454 is an undergraduate course on the same topics. WL:2
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561/SMS 518 (Business Administration)/IOE 510. Linear Programming I. Math. 217, 417, or 419. (3). (Excl). (BS).
Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and use various computer programs. WL:2 (Murty)
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562/IOE 511/Aero. 577/EECS 505/CS 505. Continuous Optimization Methods. Math. 217, 417, or 419. (3). (Excl). (BS).
Survey of continuous optimization problems. Unconstrained optimization problems: unidirectional search techniques, gradient, conjugate direction, quasi-Newtonian methods; introduction to constrained optimization using techniques of unconstrained optimization through penalty transformation, augmented Lagrangians, and others; discussion of computer programs for various algorithms. WL:2 (Saigal)
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565. Combinatorics and Graph Theory. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
This course has two somewhat distinct halves devoted to Graph Theory and Enumerative Combinatorics. Proofs, concepts, and calculations play about an equal role. Students should have taken at least one proof-oriented course. Graph Theory topics include Trees; k -connectivity; Eulerian and Hamiltonian graphs; tournaments; graph coloring; planar graphs, Euler's formula, and the 5-Color Theorem; Kuratowski's Theorem; and the Matrix-Tree Theorem. Enumeration topics include fundamental principles, bijections, generating functions, binomial theorem, Catalan numbers, tableaux, partitions and q -series, linear recurrences and rational generating functions, and Pólya theory. There is a small overlap with Math 566, but these are the only courses in combinatorics. 416 is somewhat related but much more concerned with algorithms. WL:2
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567. Introduction to Coding Theory. One of Math 217, 419, 513 . (3). (Excl). (BS).
This course is an introduction to coding theory, focusing on the mathematical background for linear error-correcting codes. It will begin with a discussion of Shannon's theorem and channel capacity. The definition of linear codes will be given along with a review of necessary tools from linear algebra and an introduction to abstract algebra and finite fields. Basic examples of codes will be studied including the Hamming, BCH, cyclic, Melas, Reed-Muller, and Ree-Solomon codes. An introduction to the problem of decoding will be included, starting with syndrome decoding and covering weight enumerator polynomials and the Mac-Williams Sloane identity. Further topics to be included range from consideration of asymptotic parameters and bounds to a discussion of algebraic geometric codes in their simplest form. WL:2
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571. Numerical Methods for Scientific Computing I. Math. 217, 419, or 513; and 454. (3). (Excl). (BS).
This course is a rigorous introduction to numerical linear algebra with applications to 2-point boundary value problems and the Laplace equation in two dimensions. Both theoretical and computational aspects of the subject are discussed. Some of the homework problems require computer programming. Students should have a strong background in linear algebra and calculus, and some programming experience. The topics covered usually include direct and iterative methods for solving systems of linear equations: Gaussian elimination, Cholesky decomposition, Jacobi iteration, Gauss-Seidel iteration, the SOR method, an introduction to the multigrid method, conjugate gradient method; finite element and difference discretizations of boundary value problems for the Poisson equation in one and two dimensions; numerical methods for computing eigenvalues and eigenvectors. Math 471 is a survey course in numerical methods at a more elementary level. Math 572 covers initial value problems for ordinary and partial differential equations. Math 571 and 572 may be taken in either order. Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor. WL:2
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575. Introduction to Theory of Numbers I. Math. 451 and 513. Students with credit for Math. 475 can elect Math. 575 for 1 credit. (3). (Excl). (BS).
Many of the results of algebra and analysis were invented to solve problems in number theory. This field has long been admired for its beauty and elegance and recently has turned out to be extremely applicable to coding problems. This course is a survey of the basic techniques and results of elementary number theory. Students should have significant experience in writing proofs at the level of Math 451 and should have a basic understanding of groups, rings, and fields, at least at the level of Math 412 and preferably Math 512. Proofs are emphasized, but they are often pleasantly short. A computational laboratory (Math 476, 1 credit) will usually be offered as a supplement to this course. Standard topics which are usually covered include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Diophantine equations, primitive roots, quadratic reciprocity and quadratic fields, application of these ideas to the solution of classical problems such as Fermat's last 'theorem'. Other topics will depend on the instructor and may include continued fractions, p-adic numbers, elliptic curves, Diophantine approximation, fast multiplication and factorization, Public Key Cryptography, and transcendence. Math 475 is a non-Honors version of Math 575 which puts much more emphasis on computation and less on proof. Only the standard topics above are covered, the pace is slower, and the exercises are easier. All of the advanced number theory courses (Math 675, 676, 677, 678, and 679) presuppose the material of Math 575. Each of these is devoted to a special subarea of number theory. WL:2
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590. Introduction to Topology. Math. 451. (3). (Excl). (BS).
This is an introduction to topology with an emphasis on the set-theoretic aspects of the subject. It is quite theoretical and requires extensive construction of proofs. Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, surfaces and manifolds, fundamental theorem of algebra, and other topics. Math 490 is a more gentle introduction that is more concrete, somewhat less rigorous, and covers parts of both Math 590 and 591. Combinatorial and algebraic aspects of the subject are emphasized over the geometrical. Math 591 is a more rigorous course covering much of this material and more. Both Math 591 and 537 use much of the material from Math 590. WL:2
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591. General and Differential Topology. Math. 451. (3). (Excl). (BS).
This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces, vector fields, submanifolds, inverse function theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, classification of surfaces. Math 592 is the natural sequel. WL:2
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593. Algebra I. Math. 513. (3). (Excl). (BS).
This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Students should have had a previous course equivalent to 512. Topics include rings and modules, Euclidean rings, principal ideal domains, classification of modules over a principal ideal domain, Jordan and rational canonical forms of matrices, structure of bilinear forms, tensor products of modules, exterior algebras. WL:2
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596. Analysis I. Math. 451. (3). (Excl). (BS). Students with credit for Math. 555 may elect Math 596 for two credits only.
This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Review of analysis in R² including metric spaces, differentiable maps, Jacobians; analytic functions, Cauchy-Riemann equations, conformal mappings, linear fractional transformations; Cauchy's theorem, Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families, Riemann mapping theorem. Math 595 covers some of the same material with greater emphasis on applications and less attention to proofs. WL:2
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100-299

300-499

500-599

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