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 Graduate Courses 501 | 512 | 513 | 520 | 521 | 523 | 525 | 526 | 528 | 531 | 537 | 547 | 548 | 550 | 555 | 556 | 557 | 558 | 559 | 561 | 562 | 563 | 565 | 566 | 567 | 571 | 572 | 575 | 582 | 590 | 591 | 592 | 593 | 594 | 596 | 597 | 602 | 604 | 605 | 612 | 614 | 615 | 619 | 623 | 625 | 626 | 631 | 632 | 635 | 636 | 637 | 650 | 651 | 654 | 656 | 657 | 658 | 660 | 663 | 664 | 665 | 669 | 671 | 675 | 676 | 677 | 678 | 679 | 681 | 682 | 684 | 694 | 695 | 696 | 697 | 703 | 704 | 709 | 710 | 711 | 715 | 731 | 732 | 756 | 775 | 776 | 781 | 797 | 990 | 993 | 995 | Math 501: Applied and Interdisciplinary Mathematics (AIM) Student Seminar Prerequisites: Graduate Standing Frequency: Fall (I), Winter (II) Credit: 1 credit Recent Texts: N/A Past instructors: P. Smereka, P. Miller Student Body: Graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: During their first three years of study, students in the AIM graduate program are required to enroll in Math 501 in both the Fall and Winter terms. In part, this seminar course is coordinated with the Applied and Interdisciplinary Mathematics Research Seminar. The AIM Student Seminar will (i) present the background to the research to be discussed at a more advanced level in the week's AIM Research Seminar, (ii) put the work in context and enable discussion of the importance of the results, and (iii) generally provide an introduction to the topic of the research seminar. Thus students gain meaningful exposure to a broad range of problems. Through direct speaking opportunities in class, the AIM Student Seminar also teaches students to give presentations to an interdisciplinary audience. Both aspects of Math 501, listening and speaking, are vital to general interdisciplinary training, and hence Math 501 is an important part of the AIM graduate program. In Math 501, students will learn both what other students are doing and also what the current of modern research is, and in this way the course will foster interactions and camaraderie among AIM students and faculty. Content: This is a student-focused seminar series directed by core AIM faculty, that features a variety of speakers on interdisciplinary topics. In addition, students present short talks on topics of their choice. These talks must be understandable to a general interdisciplinary audience, and this exercise is thus excellent training for those who will interact with other disciplines throughout their careers. Other speakers in the seminar series will be invited speakers from the University of Michigan or elsewhere. Alternatives: None Subsequent Courses: N/A Math 512: Algebraic Structures Prerequisites: Math 296, 412, 451, or permission of instructo Frequency: Fall (I) Credit: 3 Credits. Recent Texts: Algebra by Artin Past instructors: K. Smith Student Body: Mainly undergraduate mathematics concentrators with a few graduate students from other fields Background and Goals:Math 512-513 is one of the more abstract and difficult sequences in the undergraduate program. It is frequently elected by students who have completed the 295-296 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, fields, and modules) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Excercises tend to be quite challenging. Students must have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard. Students from the Math 185-286 and Math 156-256 sequences are strongly advised to take both Math 217 and some 400-500 level course (for examples, Math 451) prior to attempting Math 512. Content:The course covers basic definitions and properties of groups, fields, and vetor spaces including homomorphisms, isomorphisms, subgroups, and bilinear forms. Further topics are selected from: Sylow theorems; structure theorem for finitely-generated abelian groups; permutation representation; the symmetric and alternating groups; vector spaces over arbitrary fields; spectral theorem; and linear groups. Alternatives: Math 412 (Introduction to Modern Algebra) is a substantially lower level course which covers about half of the material of Math 512. The sequence Math 593-594 covers about twice as much Group and Field Theory as well as several other topics and presupposes that students have had a previous introduction to these concepts at least at the level of Math 412. Subsequent Course: Math 513 (Introduction to Linear Algebra) Math 513: Introduction to Linear Algebra Prerequisites: Math 512 Frequency: Winter (II) Credit: 3 credits Recent Texts: Algebra by Artin Past instructors: K.E. Smith Student Body: Mainly undergraduate mathematics concentrators with a few graduate students from other fields Background and Goals: Math 512-513 is one of the more abstract and difficult sequences in the undergraduate program. It is frequently elected by students who have completed the 295-296 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, fields, and modules) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Excercises tend to be quite challenging. Students must have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard. Students from the Math 185-286 and Math 156-256 sequences are strongly advised to take both Math 217 and some 400-500 level course (for examples, Math 451) prior to attempting Math 512. Content: This course is a continuation of Math 512. It covers basic definitions and properties of rings and modules including quotients, ideals, factorization, and field extensions. Further topics are selected from: representation theory; structure theory of modules over a PID; Jordan canonical form; Galois theory, Nullstellensatz; finite fields; Euclidean, Principal ideal, and unique factorization domains; polynomial rings in one and several variables; and algebraic varieties. Alternatives: None Subsequent Courses: The natural sequel to Math 513 is Math 593 Math 520: Life Contingencies I Prerequisites: Math 424 and 425 or permission Frequency: Fall (I) Credit: 3 credits Recent Texts: Actuarial Mathematics (N.L. Bowers et al.) Past instructors: C. Huntington Student Body: Undergraduate students of actuarial mathematics Background and Goals: 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.  Content: 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. This corresponds to Chapters 3--6 and part of 7 of Bowers.  Alternatives: Math 523 (Risk Theory) is a complementary course covering the application of stochastic process models.  Subsequent Courses: Math 520 is prerequisite to all succeeding actuarial courses. Math 521 (Life Contingencies II) 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 (Act. Theory of Pensions and Soc. Sec) applies the models of 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc.  Math 521: Life Contingencies II Prerequisites: Math 520 or permission Frequency: Winter (II) Credit: 3 credits Recent Texts: Actuarial Mathematics (N.L. Bowers et al.) Past instructors: C. Huntington Student Body: Undergraduate students of actuarial mathematics Background and Goals: This course extends the single decrement and single life ideas of Math 520 to multi-decrement and multiple-life applications directly related to life insurance. The sequence 520--521 covers covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Concepts and calculation are emphasized over proof.  Content: Topics include multiple life models--joint life, last survivor, contingent insurance; multiple decrement models---disability, withdrawal, retirement, etc.; and reserving models for life insurance. This corresponds to chapters 7--10, 14, and 15 of Bowers et al.  Alternatives: Math 522 (Act. Theory of Pensions and Soc. Sec) is a parallel course covering mathematical models for prefunded retirement benefit programs.  Subsequent Courses: none  Math 523: Risk Theory Prerequisites: Math 425 Frequency: Fall (I), Winter (II) Credit: Recent Texts: Loss Models - From Data to Decisions (Klugman, Panjer, et al.) Past instructors: J. Conlon Student Body: Undergraduate students of financial and actuarial mathematics Background and Goals: 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.  Content: 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.  Alternatives: none  Subsequent Courses: none    Math 525 (Stat. 525): Probability Theory Prerequisites: Math 450 or 451 Frequency: Fall (I), Winter (II) Credit: 3 credits Recent Texts: Grimmet and Stirzaker, Probability and Random Processes (required);  Ross, Introduction to Probability Models (optional) Past instructors: J. Marker, M. Rudelson, A. Barvinok  Student Body: A mix of undergraduate and graduate students, drawn largely from mathematics, statistics, and engineering, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This course is a thorough and fairly rigorous study of the mathematical theory of probability. There is substantial overlap with Math 425 (Intro. to Probability), but here more sophisticated mathematical tools are used and there is greater emphasis on proofs of major results. Math 451 is the required prerequisite. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program.  Content: Topics include the basic results and methods of both discrete and continuous probability theory. Different instructors will vary the emphasis between these two theories.  Alternatives: EECS 501 also covers some of the same material at a lower level of mathematical rigor. Math 425 (Intro. to Probability) is a course for students with substantially weaker background and ability.  Subsequent Courses: Math 526 (Discr. State Stoch. Proc.), Stat 426 (Intro. to Math Stat.), and the sequence Stat 510 (Mathematical Statistics I)--Stat 511 (Mathematical Statistics II) are natural sequels.  Math 526 (Stat. 526): Discrete State Stochastic Processes Prerequisites: Required: Math 525 or EECS 501 or basic probability theory including: Random variables, expectation, independence, conditional probability. Recommended: Good understanding of advanced calculus covering limits, series, the notion of continuity, differentiation and the Riemann integral ; Linear algebra including eigenvalues and eigenfunctions. Frequency: Varies Credit: 3 credits Required textbook:A First Course in Stochastic Processes, 2nd ed. (Karlin and Taylor) Background and Goals: The theory of stochastic processes is concerned with systems which change in accordance with probability laws. It can be regarded as the 'dynamic' part of statistic theory. Many applications occur in physics, engineering, computer sciences, economics, financial mathematics and biological sciences, as well as in other branches of mathematical analysis such as partial differential equations. The purpose of this course is to provide an introduction to the many specialized treatise on stochastic processes. Most of this course is on discrete state spaces. It is a second course in probability which should be of interest to students of mathematics and statistics as well as students from other disciplines in which stochastic processes have found significant applications. Special efforts will be made to attract and interest students in the rich diversity of applications of stochastic processes and to make them aware of the relevance and importance of the mathematical subtleties underlying stochastic processes. Content: The material is divided between discrete and continuous time processes. In both, a general theory is developed and detailed study is made of some special classes of processes and their applications. Some specific topics include generating functions; recurrent events and the renewal theorem; random walks; Markov chains; limit theorems; Markov chains in continuous time with emphasis on birth and death processes and queueing theory; an introduction to Brownian motion; stationary processes and martingales. Significant applications will be an important feature of the course. Coursework: weekly or biweekly problem sets and a midterm exam will each count for 30% of the grade. The final will count for 40%. Additional information: Those wishing to discuss the course should contact taoluo@umich.edu. Math 528: Topics in Casualty Insurance Prerequisites: Math 217, 417, or 419, or permission Frequency: Sporadically Credit: 3 credits Recent Texts: Past instructors: C. Huntington Student Body: Undergraduate students of actuarial mathematics and insurance majors in Business Background and Goals: Historically the Actuarial Program has emphasized life, health, and pension topics. This course will provide background in casualty topics for the many students who take employment in this field. Guest lecturers from the industry will provide some of the instruction. Students are encouraged to take the Casualty Actuarial Society's Part 3B examination at the completion of the course.  Content: The insurance policy is a 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, the bases of exposure 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.  Alternatives: none  Subsequent Courses: none  Math 531: Transformation Groups in Geometry Prerequisites: Math 215, 255, or 285 Frequency: Winter (II) Credit: 3 credits Recent Texts: Groups and Symmetry (Armstrong); Notes on Geometry (Rees) Past instructors: R. Spatzier Student Body: Undergraduate and graduate mathematics students Background and Goals: This course gives a rigorous treatment of a selection of topics involving the interaction of group theory and geometry. Most students have substantial preparation beyond the formal prerequisite (e.g. Math 512) and are taking concurrently other advanced courses (e.g. Math 490)  Content: The content will vary significantly with the instructor. One version includes subgroups of the group of Euclidian motions of R, crystallographic groups, hyperbolic and projective geometry, and Fuchsian groups. Other possible topics are tilings of the plane, affine geometries, and regular polytopes.  Alternatives: none  Subsequent Courses: This course is not prerequisite for any later course but provides good general background for any course in Topology (590, 591, 592) or Geometry (537, 635, 636).  Math 537: Introduction to Differentiable Manifolds **See Math 437 Math 547: Biological Sequence Analysis Prerequisites: Flexible. Basic probability (level of Math/Stat 425) or molecular biology (level of Biology 427) or biochemistry (level of Chem/BioChem 451) or basic programming skills desirable; or permission of instructor. Frequency: Annually; check for semester Credit: 3 credits Recent Texts: Biological Sequence Analysis (R. Durbin, et al.) Past instructors: D. Burns Student Body: Interdisciplinary: mainly Math, Statistics, Biostatistics and Bioinformatics students; also Biology, Biomedical and Engineering students. Background and Goals: Content: Probabilistic models of proteins and nucleic acids. Anaylsis of DNA/RNA and protein sequence data. Algorithms for sequence alignment, statistical analysis of similarity scores, hidden markov models, neural networks, training, gene finding, protein family proviles, multiple sequence alignment, sequence comparison and structure prediction. Analysis of expression array data. Alternatives: Bioinformatics 526 Subsequent Courses: Bioinformatics 551 (Preteome Informatics) Math 548: Computations in Probabilistic Modeling in Bioinformatics Prerequisites: Math 215, 255, or 285; Math 217, and Math 425 Frequency: Sporadically Credit: 1 credit Student Body: graduate and undergraduate students from many disciplines Background and Goals: This course is a computational laboratory course designed in parallel with Math/Stat. 547 Prob. mod. Bioinformatics. Content: weekly hand on problems with be presented on the algorithms presented in the course, the use of public sequence data basis, the design of hidden Markov models. Concrete examples of homology, gene finding, structure analysis. Alternatives: None Subsequent Courses: None  Math 550: Intro to Adaptive Systems Prerequisites: Math 215, 255, or 285; Math 217, and Math 425 Frequency: Sporadically Credit: 3 credits Recent Texts: Past instructors: C. Simon Student Body: graduate and undergraduate students from many disciplines Background and Goals: This course centers on the construction and use of agent-based adaptive models study phenomena which are prototypical in the social, biological and decision sciences. These models are "agent-based" or "bottom-up" in that t he structure placed at the level of the individuals as basic components; they are "adaptive" in that individuals often adapt to their environment through evolution or learning. The goal of these models is to understand how the structure at the individual or micro level leads to emergent behavior at the macro or aggregate level. Often the individuals are grouped into subpopulations or interesting hierarchies, and the researcher may want to understand how the structure of development of these populations affects macroscopic outcomes.  Content: The course will start with classical differential equation and game theory approaches. It will then focus on the theory and application of particular models of adaptive systems such as models of neural systems, genetic algorithms, classifier system and cellular automata. Time permitting, we will discuss more recent developments such as sugarscape and echo.  Alternatives: Complex Systems 510 is the same course.  Subsequent Courses: none  Math 555: Intro to Complex Variables Prerequisites: Math 450 or 451 Frequency: Fall (I), Winter (II), Spring (IIIa) Credit: 3 credits Recent Texts: Complex Variables and Applications, 6th ed. (Churchill and Brown); Past instructors: B. Stensones, C. Doering, J. Fornaess Student Body: largely engineering and physics graduate students with some math and engineering undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: 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 princip les to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program.  Content: Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. This corresponds to Chapters 1--9 of Churchill.  Alternatives: Math 596 (Analysis I (Complex)) covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications.  Subsequent Courses: Math 555 is prerequisite to many advanced courses in science and engineering fields.  Math 556: Methods of Applied Math I: Applied Functional Analysis Prerequisites: Math 217, 419, or 513; 451 and 555 Frequency: Fall (I) Credit: 3 credits Recent Texts: Applied Functional Analysis (Griffel) Past instructors: P Miller, J Schotland Student Body: Graduate students in matehematics, science, and engineering, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This is an introduction to methods of applied functional analysis. Students are expected to master both the proofs and applications of major results. The prerequisites include linear algebra,  undergraduate analysis, advanced calculus and complex variables. This course is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program. Content: Topics may vary with the instructor but often include Fourier transform, distributions, Hilbert space, Banach spaces, fixed point theorems, integral equations, spectral theory for compact self-adjoint operators. Alternatives: Math 602 is a more theoretical course covering many of the same topics Subsequent Courses: Math 557 (Methods of Applied Math II), Math 558 (Ordinary Diff. Eq.), Math 656 (Partial Differential Equations) and 658 (Ordinary Differential Equations.)  Math 557: Methods of Applied Math II Prerequisites: Math 217, 419, or 513; 451 and 555 Frequency: Winter (II) Credit: 3 credits Recent Texts: Asymptotic Analysis (Murray) Past instructors: C. Doering, V. Booth Student Body: Graduate students in mathematics, science and engineering, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This is an introduction to methods of asymptotic analysis including asymptotic expansions for integrals and solutions of ordinary and partial differential equations. The prerequisites include linear algebra, advanced calculus and complex variables. Math 556 is not a prerequisite. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Content: Topics include stationary phase, steepest descent, characterization of singularities in terms of the Fourier transform, regular and singular perturbation problems, boundary layers, multiple scales, WKB method. Additional topics depend on the instructor but may include non-linear stability theory, bifurcations, applications in fluid dynamics (Rayleigh-Benard convection), combustion (flame speed). Alternatives: none  Subsequent Courses: Math 656 (Partial Differential Equations) and 658 (Ordinary Differential Equations.)  Math 558: Applied Nonlinear Dynamics Prerequisites: Math 451 Frequency: Fall (I) Credit: 3 credits Recent Texts: Nonlinear Ordinary Differential Equations (Jordan and Smith) Past instructors: R. Krasny, C. Doering Student Body: grad students in math, science, and engineering, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This course is an introduction to dynamical systems (differential equations and iterated maps).  The aim is to survey a broad range of topics in the theory of dynamical systems with emphasis on techniques and results that are useful in applications.  Chaotic dynamics will be discussed. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Content: Topics may include: bifurcation theory, phase plane analysis for linear systems, Floquet theory, nonlinear stability theory, dissipative and conservative systems, Poincare-Bendixson theorem, Lagrangian and Hamiltonian mechanics, nonlinear oscillations, forced systems, resonance, chaotic dynamics, logistic map, period doubling, Feigenbaum sequence, renormalization, complex dynamics, fractals, Mandelbrot set, Lorenz model, homoclinic orbits, Melnikov's method, Smale horseshoe, symbolic dynamics, KAM theory, homoclinic chaos Alternatives: Math 404 (Intermediate Diff. Eq.) is an undergraduate course on similar topics Subsequent Courses: Math 658 (Ordinary Differential Equations)  Math 559: Topics in Applied Mathematics Prerequisites: Vary by topic, check with instructor Frequency: Sporadically Credit: 3 credits Recent Texts: Varies Past instructors: Aaron King, Victoria Booth Student Body: undergraduate and graduate students in mathematics or science Background and Goals: This is an advanced topics course intended for students with strong interests in the intersection of mathematics and the sciences, but not necessarily experience with both applied mathematics and the application field. Mathematical concepts, as well as intuitions arising from the field of application will be stressed. Content: This course will focus on particular topics in emerging areas of applied mathematics for which the application field has been strongly influenced by mathematical ideas. It is intended for students with interests in the mathematical, computational, and/or modeling aspects of interdisciplinary concepts. The applications considered will vary with the instructor and may come from physics, biology, economics, electrical engineering, and other fields. Recent examples have been: Nonlinear Waves, Mathematical Ecology, and Computational Neuroscience.  Alternatives: none  Subsequent Courses: Other courses in applied mathematics  Math 561 (Bus. Adm. Stat. 518, IOE 510): Linear Programming I Prerequisites: Math 217, 417, or 419 Frequency: Fall (I), Winter (II), and Spring (IIIa) Credit: 3 credits Recent Texts: Linear Optimizations and Extensions: Theory and Algorithms(Fang and Puthenpura) Past instructors: J. Goldberg Student Body: Graduate and undergraduate students from many fields Background and Goals: A fundamental problem is the allocation of constrained resources such as funds among investment possibilities or personnel among production facilities. Each such problem has as it’s goal the maximization of some positive objective such as investment return or the minimization of some negative objective such as cost or risk. Such problems are called Optimization Problems. Linear Programming deals with optimization problems in which both the objective and constraint functions are linear (the word "programming" is historical and means "planning" rather that necessarily computer programming). In practice, such problems involve thousands of decision variables and constraints, so a primary focus is the development and implementation of efficient algorithms. However, the subject also has deep connections with higher-dimensional convex geometry. A recent survey showed that most Fortune 500 companies regularly use linear programming in their decision making. This course will present both the classical and modern approaches to the subject and discuss numerous applications of current interest. Content: 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; a lgorithmic complexity; the elipsoid method; scaling algorithms; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to form ulate and solve models developed from more complex case studies and use various computer programs.  Alternatives: Cross-listed as IOE 510.  Subsequent Courses: IOE 610 (Linear Programming II) and IOE 611 (Nonlinear Programming) Math 562 (IOE 511, Aero Eng. 577): Continuous Optimization Meth. Prerequisites: Math 217, 417, or 419 Frequency: Fall (I) Credit: 3 credits Recent Texts: Past instructors: Student Body: Background and Goals: Not Available  Content: 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.  Alternatives: Cross-listed as IOE 511.  Subsequent Courses: This is not a prerequisite for any other course. Math 563: Advanced Mathematical Methods For the Biological Sciences Prerequisites: Math 217, 417, or 419 and Math 216, 286, or 316 and to have a basic familiarity with partial differential equations as would be gained by the completion of Math 450 or 454 or to have permission of the instructor. Frequency: Winter (II) Student Body: Graduate Students, Math, Science, Engineering and Medical School (Both the Department of Mathematics and the Program in Bioinformatics have approved this course for cross-listing. Further approval is in process). Credit: 3 Credits. Recent Texts: Math Biology, J. D. Murray Past Instructors: T. Jackson Background and Goals: Mathematical biology is a fast growing and exciting modem application which has gained worldwide recognition. This course will focus on the devrivation, analysis, and simulation of partial differential equations (PDEs) which model specific phenomena in molecular, cellular, and population biology. A goal of this course is to understand how the underlying spatial variability in natural systems influences motion and behavior. Content: Mathematical topics covered include derivation of relevant PDEs from first principle; reduction of PDEs to ODEs under steady state, quasi-state and traveling wave assumptions; solution techniques for PDEs and concepts of spatial stability and instability. These concepts will be introduced within the setting of classical and current problems in biology and the biomedical sciences such as cell motion, transport of biological substances and, biological pattern formation. Above all, this course aims to enhance the interdisciplinary training of advanced undergraduate and graduate students from mathematics and other disciplines by introducing fundamental properties of partial differential equations in the context of interesting biological phenomena. Alternatives: None Subsequent Courses:   Math 565: Combinations and Graph Theory Prerequisites: Math 412 or 451 or equivalent experience with abstract mathematics Frequency: Fall (I) Credit: 3 credits Recent Texts: A Course in Combinatorics (van Lint and Wilson) Past instructors: N. Reading, A. Blass, S. Fomin Student Body: Largely math and EECS grad students with a few math undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program. Background and Goals: This course has two somewhat distinct halves devoted to Graph Theory and Combinatorics. Proofs, concepts, and applications play about an equal role. Students should have taken at least one proof-oriented course. This course is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program.   Content: Eulerian and Hamiltonian graphs; tournaments; network flows; graph coloring; the 5-Color Theorem; Kuratowski's Theorem; the Matrix-Tree Theorem; fundamental enumeration principles, bijections, and generating functions; inclusion-exclusion; partially ordered sets; matroids; Ramsey's Theorem. Alternatives: There are small overlaps with Math 566 (Introduction to Algebraic Combinatorics) and Math 416 (Theory of Algorithms). Math 465 has similar content but is significantly less demanding than Math 565. Subsequent Courses: Math 566 (Introduction to Algebraic Combinatorics) Math 566: Combinatorial Theory Prerequisites: Math 512, or equivalent experience with abstract algebra Frequency: Winter (II) Credit: 3 credits Recent Texts: Enumerative Combinatorics (Stanley) Past Instructors: M. Skandera, J. Stembridge, S. Fomin Student Body: Undergraduates and graduates from Math, EECS, or IOE Background and Goals: This course is a rigorous introduction to modern algebraic combinatorics, primarily focused on enumeration.  Content: varies considerably with instructor. Topics may include: generating functions (ordinary and exponential); sieve methods; Lagrange inversion; perfect matchings; words and formal languages; group-theoretic enumeration methods; partitions and tableaux; algebraic graph theory. Alternatives: Math 664 (Combinatorial Theory I) occasionally covers similar material in greater depth at a faster pace. Subsequent Courses: Sequels include Math 665 and Math 669.  Math 567: Introduction to Coding Theory Prerequisites: Math 217, 417, or 419 Frequency: Winter (II) Credit: 3 credits Recent Texts: Introductin to Coding Theory (van Lint) Past instructors: T. Wooley Student Body: Undergraduate math majors and EECS graduate students Background and Goals: This course is designed to introduce math majors to an important area of applications in the communications industry. From a background in linear algebra it will cover the foundations of the theory of error-correcting codes and prepare a student to take further EECS courses or gain employment in this area. For EECS students it will provide a mathematical setting for their study of communications technology. Content: Introduction to coding theory focusing on the mathematical background for error-correcting codes. Shannon's Theorem and channel capacity. Review of tools from linear algebra and an introduction to abstract algebra and finite fields. Basic examples of codes such and Hamming, BCH, cyclic, Melas, Reed-Muller, and Reed-Solomon. Introduction to decoding starting with syndrome decoding and covering weight enumerator polynomials and the Mac-Williams Sloane identity. Further topics range from asymptotic parameters and bounds to a discussion of algebraic geometric codes in their simplest form. Alternatives: none Subsequent Courses: Math 565 (Combinatorics and Graph Theory) and Math 556 (Methods of Applied Math I) are natural sequels or predecessors. This course also complements Math 312 (Applied Modern Algebra) in presenting direct applications of finite fields and linear algebra. Math 571: Numerical Methods for Scientific Computing I Prerequisites: Math 217, 417, 419, or 513 and Math 450, 451, or 454 or permission Frequency: Fall (I) and Winter (II) Credit: 3 credits Recent Texts: A Multigrid Tutorial (Briggs), Introduction to Numerical Linear Algebra and Optimization (Ciarlet) Past instructors: R. Krasny, S. Karni, J. Rauch Student Body: math and engineering grads, strong undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: 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. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program.  Content: 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.  Alternatives: Math 471 (Intro to Numerical Methods) is a survey course in numerical methods at a more elementary level.  Subsequent Courses: Math 572 (Numer Meth for Sci Comput II) 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.  Math 572: Numerical Methods for Scientific Computing II Prerequisites: Math 217, 417, 419, or 513 and one of Math 450, 451, or 454 or permission Frequency: Winter (II) Credit: 3 credits Recent Texts: Numerical Solutions of PDE's (Morton and Mayer) Past instructors: S. Karni, P. Smereka Student Body: math and engineering grads, strong undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. Graduate students from engineering and science departments and strong undergraduates are also welcome. The course is an introduction to numerical methods for solving ordinary differential equations and hyperbolic and parabolic partial differential equations. Fundamental concepts and methods of analysis are emphasized. Students should have a strong background in linear algebra and analysis, and some experience with computer programming. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program.  Content: Content varies somewhat with the instructor. Numerical methods for ordinary differential equations; Lax's equivalence theorem; finite difference and spectral methods for linear time dependent PDEs: diffusion equations, scalar first order hyperbolic equations, symmetric hyberbolic systems.  Alternatives: There is no real alternative; Math 471 (Intro to Numerical Methods) covers a small part of the same material at a lower level. Math 571 and 572 may be taken in either order.  Subsequent Courses: Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor.  Math 575: Intro to Theory of Numbers Prerequisites: Math 451 and 513 or permission Frequency: Fall (I) Credit: 3 credits; 1 credit after Math 475 Recent Texts: An introduction to the Theory of Numbers (Niven, Zuckerman, and Montgomery) Past instructors: T. Wooley, H. Montgomery/li> Student Body: Roughly half honors math undergrads and half graduate students Background and Goals: 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. Content: 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'(proved recently by A. Wiles). 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. This material corresponds to Chapters 1--3 and selected parts of Chapters 4, 5, 7, 8, and 9 of Niven, Zuckerman, and Montgomery.  Alternatives: Math 475 (Elementary Number Theory) 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.  Subsequent Courses: 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.  Math 582: Intro to Set Theory Prerequisites: Math 412 or 451 or equivalent experience with abstract mathematics Frequency: Winter (II) Credit: 3 credits Recent Texts: Elements of Set Theory (H. Enderton) Past instructors: A. Blass, P. Hinman Student Body: undergraduate math (often honors) majors and some grad students Background and Goals: One of the great discoveries of modern mathematics was that essentially every mathematical concept may be defined in terms of sets and membership. Thus Set Theory plays a special role as a foundation for the whole of mathematics. One of the goals of this course is to develop some understanding of how Set Theory plays this role. The analysis of common mathematical concepts (e.g. function, ordering, infinity) in set-theoretic terms leads to a deeper understanding of these concepts. At the same time, the student will be introduced to many new concepts (e.g. transfinite ordinal and cardinal numbers, the Axiom of Choice) which play a major role in many branches of mathematics. The development of set theory will be largely axiomatic with the emphasis on proving the main results from the axioms. Students should have substantial experience with theorem-proof mathematics; the listed prerequisites are minimal and stronger preparation is recommended. No course in mathematical logic is presupposed.  Content: The main topics covered are set algebra (union, intersection), relations and functions, orderings (partial, linear, well), the natural numbers, finite and denumerable sets, the Axiom of Choice, and ordinal and cardinal numbers. Alternatives: Some elementary set theory is typically covered in a number of advanced courses, but Math 582 is the only course which presents a thorough development of the subject.  Subsequent Courses: Math 582 is not an explicit prerequisite for any later course, but it is excellent background for many of the advanced courses numbered 590 and above.  Math 590: Intro to Topology Prerequisites: Math 451 Frequency: Fall (I) Credit: 3 credits Recent Texts: An Introduction to Topology and Homotopy (Sieradski) Past instructors: M. Brown, A. Wasserman Student Body: math grads, some non-math grads, math undergrads Background and Goals: 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.  Content: Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, surfaces and manifolds, fundamental theorem of algebra, and other topics.  Alternatives: Math 490 (Introduction to Topology) is a more gentle introduction that is more concrete, somewhat less rigorous, and covers parts of both Math 591 and Math 592 (General and Differential Topology). Combinatorial and algebraic aspects of the subject are emphasized over the geometrical. Math 591 (General and Differential Topology) is a more rigorous course covering much of this material and more.  Subsequent Courses: Both Math 591 (General and Differential Topology) and Math 537 (Intro to Differentiable Manifolds) use much of the material from Math 590.  Math 591: General and Differential Topology Prerequisites: Math 451 Frequency: Fall (I) Credit: 3 credits Recent Texts: Topology (Munkres); Differential Topology (Guillemin and Pollack) Past instructors: P. Scott, R. Canary, J. Lott Student Body: mainly math grads, a few math undergrads and non-math grads Background and Goals: 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.  Content: 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.  Alternatives: none  Subsequent Courses: Math 592 (An Introduction to Algebraic Topology) is the natural sequel.  Math 592: An Introduction to Algebraic Topology Prerequisites: Math 591 Frequency: Winter (II) Credit: 3 credits Recent Texts: Elements of Algebraic Topology (Munkres) Past instructors: I. Kriz, P. Scott, R. Canary Student Body: largely math graduate students Background and Goals: 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.  Content: Fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, Eilenberg-Steenrod axioms, Brouwer's and Lefschetz' fixed-point theorems, and other topics.  Alternatives: none  Subsequent Courses: Math 695 (Algebraic Topology I)  Math 593: Algebra I Prerequisites: Math 513 Frequency: Fall (I) Credit: 3 credits Recent Texts: Algebra (Artin) Past instructors: A. Moy, P.J. Hanlon, R.L. Griess, Jr. Student Body: largely math graduate students Background and Goals: 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 Math 512 (Algebraic Structures).  Content: 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.  Alternatives: none  Subsequent Courses: Math 594 (Algebra II) and Math 614 (Commutative Algebra I).  Math 594: Algebra II Prerequisites: Math 593 Frequency: Winter (II) Credit: 3 credits Recent Texts: Algebra, A Graduate Course (Isaacs) Past instructors: I.V. Dolgachev, R. Lazarsfeld, R.L. Griess, Jr. Student Body: largely math graduate students Background and Goals: 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.  Content: Topics include group theory, permutation representations, simplicity of alternating groups for n>4, Sylow theorems, series in groups, solvable and nilpotent groups, Jordan-Holder Theorem for groups with operators, free groups and presentations, fields and field extensions, norm and trace, algebraic closure, Galois theory, transcendence degree.  Alternatives: none  Subsequent Courses: Math 612 (Algebra III), Math 613 (Homological Algebra), Math 614 (Commutative Algebra I) and various topics courses in algebra.  Math 596: Analysis I (Complex) Prerequisites: Math 451 Frequency: Fall (I) Credit: 3 credits; 2 credits after Math 555 Recent Texts: Complex Analysis, 3rd ed. (L. Ahlfors) Past instructors: D.M. Burns, Jr., P. Duren Student Body: largely math grad students Background and Goals: 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.  Content: Review of analysis in R^2 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.  Alternatives: Math 555 (Intro to Complex Variables) covers some of the same material with greater emphasis on applications and less attention to proofs.  Subsequent Courses: Math 597 (Analysis II (Real)), Math 604 (Complex Analysis II), and Math 605 (Several Complex Variables).  Math 597: Analysis II (Real) Prerequisites: Math 451 and 513 Frequency: Winter (II) Credit: 3 credits Recent Texts: Real Analysis (Bruckert et. al.) Past instructors: D. Barrett, J. Heinonoen, L. Ji, B. Stensones Student Body: largely math graduate students Background and Goals: 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.  Content: Topics include Lebesgue measure on the real line; measurable functions and integration on R; differentiation theory, fundamental theorem of calculus; function spaces, L^p(R), C(K), Holder and Minkowski inequalities, duality; general measure spaces, product measures, Fubini's Theorem; Radon-Nikodym Theorem, conditional expectation, signed measures, introduction to Fourier transforms.  Alternatives: none  Subsequent Courses: Math 602 (Real Analysis II).  Prerequisite: Math 590 and 597. Introduction to functional analysis; metric spaces, completion, Banach spaces, Hilbert spaces, L^p spaces; linear functionals, dual spaces, Riesz representation theorems; principle of uniform boundedness, closed graph theorem, Hahn-Banach theorem, B aire category theorem, applications to classical analysis. Prerequisite: Math 596. Selected topics such as potential theory, geometric function theory, analytic continuation, Riemann surfaces, uniformization and analytic varieties. Prerequisite: Math 604 or consent of instructor. Power series in several complex variables, domains of holomorphy, pseudo convexity, plurisubharmonic functions, the Levi problem. Domains with smooth boundary, tangential Cauchy-Riemann equations, the Lewy and Bochner extension theorems. The $\overlin e {\partial }$-operator and Hartog's Theorem, Dol beault-Grothendieck lemma, theorems of Runge, Mittag-Leffler and Weierstrass. Analytic continuation, monodromy theorem, uniformization and Koebe's theorem, discontinuous groups. Prerequisite: Math 593 and 594 or consent of instructor. Representation Theory of semisimple Lie algebras over the complex numbers. Weyl's Theorem, root systems, Harish Chandra's Theorem, Weyl's formulae and Kostant's Multiplicity Theorem. Lie groups, their Lie algebras and further examples of representatio ns. Prerequisite: Math 593. Review of commutative rings and modules. Local rings and localization. Noetherian and Artinian rings. Integral independence. Valuation rings, Dedekind domains, completions, graded rings. Dimension theory. Prerequisite: Math 614 or permissions of instructor. This is a continuation of Math 614: structure of complete local rings, regular, Cohen-Macaulay, and Gorenstein rings, excellent rings, Henselian rings, etale maps, equations over local rings. Prerequisite: Math 593. Selected topics. Prerequisites: Math 316 and 425 or 525 Frequency: Winter (II) Credit: 3 credits Recent Texts: Mathematics of Financial Derivatives (Wilmot et.al) Past instructors: J. Conlon, E. Bayraktar Student Body: graduate students in Math, Physics, Engineering, and Finance/Business Background and Goals: The field of computational finance is rising rapidly in academic and industry. There is a growing need for students with such skills. This course will fill this demand. Documented computer projects will be required in addition to a final examination.  Content: This is a course in computational methods in finance and financial modeling. Particular emphasis will be put on interest rate models and interest rate derivatives. Specific topics include Black-Scholes theory, no-arbitrage and complete markets theory, term structure models, Hull and White models, Heath-Jarrow-Morton models, the stochastic differential equations and martingale approach, multinomial tree and Monte Carlo methods, the partial differential equations approach, finite difference methods.  Alternatives: none  Subsequent Courses: none Prerequisite: Math 597. Axiomatics; measures and integration in abstract spaces. Fourier analysis, characteristic functions. Conditional expectation, Kolmogoroff extension theorem. Stochastic processes; Wiener-Levy, infinitely divisible, stable. Limit theorems, law of the it erated logarithm. Prerequisite: Math 625. Selected topics from among: diffusion theory and partial differential equations; spectral analysis; stationary processes, and ergodic theory; information theory; martingales and gambling systems; theory of partial sums. Prerequisite: Math 594 or 614 or permission of instructor). Theory of algebraic varieties: affine and projective varieties, dimension of varieties and subvarieties, singular points, divisors, differentials, intersections. Schemes, cohomology, curves and surfaces, varieties over the complex numbers. Prerequisite: Math 631). Continuation of Math 631. Prerequisite: Math 537 or permission of instructor. Second fundamental form, Hadamard manifolds, spaces of constant curvature, first and second variational formulas, Rauch comparision theorem, and other topics chosen by the instructor Prerequisite: Math 635.  Prerequisite: Some familiarity with the theory of algebraic groups. Prerequisite: Math 602 and 596. General properties of orthogonal systems. Convergence criteria for Fourier series. The Fourier integral, Fourier transform and Plancherel theorem. Wiener's Tauberian theorem. Elements of harmonic analysis. Applications. Prerequisite: Math 451, 555 and one other 500-level course in analysis or differential equations. Topics such as celestial mechanics, continuum mechanics, control theory, general relativity, nonlinear waves, optimization, statistical mechanics. Prerequisite: Math 555, 556 Texts: A Mathematical Introduction to Fluid Mechanics (Chorin and Marsden) Instructors: R. Krasny Student Body: Graduate students in math, science, and engineering. Background and Goals: This is an introductory survey of mathematical fluid dynamics. Content: Derivation of the Euler and Navier-Stokes equations, compressible and incompressible flow, conservation laws for mass, momentum, and energy, stream function, flow map, vorticity, Biot-Savart law, circulation, Kelvin theorem, Helmholtz thoerem, potential flow past a bluff body, Bernoulli principle, viscous flow, lift and drag, Prandtl boundary layer equations, point vortices, vortex sheets, Kelvin-Helmholtz instability. Subsequent Courses: Math 655 (Topics in Fluid Dynamics) Prerequisite: Math 558, 596 and 597 or permission of instructor. Characteristics, heat, wave and Laplace's equation, energy methods, maximum principles, distribution theory. Prerequisite: Math 656 or permission of instructor. A survey of ideas and methods arising in the study of nonlinear partial differential equations, nonlinear variational problems, bifurcation theory, nonlinear semigroups, shock waves, dispersive equations. This course will discuss geometric aspects of the modern theory of ordinary differential equations and dynamical systems, with applications to various mechanical and physical systems. Topics will include: the qualitative theory of ODE's on manifolds, symplectic and Poisson geometry, nonlinear stability theory, Lagrangian and Hamiltonian mechanics, integrable systems, reduction and symmetries, mechanical systems with constraints including nonholonomically constrained systems, and mechanical systems with forces and controls Prerequisite: Math 561. Primal-dual algorithm. Resolution of degeneracy, upper bounding. Variants of simplex method. Geometry of the simplex method, application of adjacent vertex methods in nonlinear programs, fractional linear programming under uncertainty. Ranking algorit hms, fixed charge problem. Integer programming. Combinatorial problems. Prerequisite: Math 561. Modeling, theorems of alternatives, convex sets, convex and generalized convex functions, convex inequality systems, necessary and sufficient optimality conditions, duality theory, algorithms for quadratic programming, linear complementarity problems and fixed point computing. Methods of direct search, Newton and quasi-Newton, gradient projection, feasible direction, reduced gradient; solution methods for nonlinear equations. An introduction to the techniques of enumeration. Basic material for first half of this course is found in Stanley's `Enumerative Combinatorics, Vol. I''. The second half consists of topics such as ordinary and exponential generating functions, Sieve methods, partitions and $q$-series, Polya Theory and other optional topics as time permits. Prerequisite: Math 565 or 566 or 664 or permission of instructor. Selected topics from the foundations of combinatorics, including the analysis of general partially ordered sets, combinatorial designs in loops and structures in abstract systems, enumeration under group action, combinatorial aspects of finite simple groups. Prerequisite: Math 571, 572, or permission of instructor This is a course on special topics in numerical analysis and scientific computing. Subjects of current research interest will be included. Recent topics have been: Finite difference methods for hyperbolic problems, Multigrid methods for elliptic bound ary value problems. Students can take this class for credit repeatedly. Prerequisite: Math 575, 596. Theory of the Riemann zeta-function and the L-functions, distribution of primes, Dirichlet's theorem on primes in a progression, quadratic forms, transcendental numbers. Prerequisite: Math 575, 594. Arithmetic of algebraic extensions, the basis theorems for units, valuation and ideal theory. Prerequisite: Math 575. Topics in diophantine approximation, diophantine equations and transcendence. Prerequisite: Math 596 and 575. A basic introduction to modular functions, modular forms, modular groups. Hecke operators, Selberg trace formula. Applications to theory of partitions, quadratic forms, class field theory and elliptic curves. Topics in the theory of elliptic curves. Prerequisite: Mathematical maturity appropriate to a 600-level course. (No previous knowledge of mathematical logic is needed.) Syntax and semantics of first-order logic. Formal deductive systems. Soundness and completeness theorems. Compactness principle and applications. Decision problems for formal theories. Additional topics may include non-standard models and logical syst ems other than classical first-order logic. Prerequisite: Math 681 or Equivalent. Axiomatic development of set theory including cardinal and ordinal numbers. Constructible sets and the consistency of the axiom of choice and the generalized continuum hypothesis. Forcing and the independence of choice and the continuum hypothesis. Ad ditional topics may include combinatorial set theory, descriptive set theory, or further independence results. Prerequisite: Math 681 or equivalent. Elementary theory of recursive functions, sets, and relations and recursively enumerable sets and relations. Definability and incompleteness in arithmetic. Godel's incompleteness theorems. Properties of r.e. sets. Relative recursiveness, degrees of un solvability and the jump operator. Oracle constructions. The Friedberg-Muchnik Theorem and the priority method. Prerequisite: Math 537 and 591 or permission of instructor. Transversality, embedding theorems, vector bundles and selected topics from the theories of cobordism, surgery, and characteristic classes. Prerequisite: Math 591 or permission of instructor. Cohomology Theory, the Universal Coefficient Theorems, Kunneth Theorems (product spaces and their homology and cohomology), fiber bundles, higher homotopy groups, Hurewicz' Theorem, Poincar{\accent 19 e} and Alexander duality. Prerequisite: Math 695 or permission of instructor. Further topics in algebraic topology typically taken from: obstruction theory, cohomology operations, homotopy theory, spectral sequences and computations, cohomology of groups, characteristic classes. An intermediate level topics course. Prerequisite: Math 604. Selected advanced topics in function theory. May be taken for credit more than once, as the content will vary from year to year. Prerequisite: Math 604. Selected advanced topics in function theory. May be taken for credit more than once, as the content will vary from year to year. Prerequisite: Varies Selected advanced topics in analysis. Prerequisite: Math 597. Selected advanced topics in analysis. Prerequisite: Math 594 and 612 or permission of instructor. Topics of current research interest, such as groups, rings, lattices, etc., including a thorough study of one such topic. May be taken more than once for credit. Selected topics in algebraic geometry. Prerequisite: Math 631 or 731. Selected topics in algebraic geometry. May be taken more than once for credit. Prerequisite: Math 675. Selected topics in analytic number theory. Prerequisite: Math 676. Selected topics in algebraic number theory. Prerequisite: Varies according to content. Advanced topics in mathematical logic. Content will vary from year to year. May be repeated for credit. Prerequisite: Permission of instructor. Prerequisites: Appointment as GSI in Mathematics Department.

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