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2005–2006 Catalog of MATH CoursesUndergraduate Courses
MATH 005: MATH 105 Supplement
MATH 006: MATH 106 Supplement
MATH 011: MATH 111 Supplement
MATH 012: MATH 112 Supplement
MATH 100: Calculus Preparation
Introduces a wide variety of topics of algebra and trigonometry that have applications in various disciplines. Emphasis is on the development of linear, polynomial, rational, trigonometric, exponential, and logarithmic functions. Students will have a better understanding of the behavior of these functions in their application to calculus because of the strong emphasis on graphing. Application of these mathematical ideas is addressed in problem-solving activities. MATH 103: Mathematical Explorations
For students who wish to experience how mathematical ideas naturally evolve. The homework consists of the students actively investigating mathematical ideas. The course emphasizes ideas and imagination as opposed to techniques and calculations. Topics vary depending on the instructor and are announced here several weeks before the semester begins. Some assessment is done through writing assignments. Mathematics and Politics (Topic for Fall 2005): In this course we use tools from mathematics to analyze problems in social sciences and politics. We will discuss different voting procedures, their advantage and disadvantages and will prove that there is no voting which satisfies a number of reasonable conditions and we will apply game theory to analyze political conflicts. MATH 105: Finite Mathematics for the Life and Social Sciences
Introduction to linear algebra, probability, and Markov chains that develops the parts of the theory most relevant for applications. Specific topics include: equations of lines, the method of least squares, solutions of linear systems, matrices; basic concepts of probability, permutations, combinations, binomial distribution, mean and variance, and the normal approximation to the binomial distribution. Examples from biology and the social sciences are used. MATH 106: Calculus for the Life and Social Sciences
Introduction to differential and integral calculus, partial derivatives, elementary differential equations. Examples from biology and the social sciences are used. MATH 109: Precalculus Mathematics
Designed to prepare students for MATH 111. Reviews algebra, trigonometry, logarithms, and exponentials. MATH 111: Calculus I
Topics include functions and graphs, limits and continuity, differentiation and integration of algebraic, trigonometric, inverse trig, logarithmic, and exponential functions; applications of differentiation, including graphing, max-min problems, tangent line approximation, implicit differentiation, and applications to the sciences; the mean value theorem; and antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, substitution in integration, the area under a curve. Graphing calculators are used, and their pitfalls are discussed, as applicable to the above topics. MATH 111 can serve as a one-semester introduction to calculus or as part of a two-semester sequence in which it is followed by MATH 112 or 122. MATH 112: Calculus II
Focuses on integration: applications, including volumes and arc length; techniques of integration, approximate integration with error estimates, improper integrals, differential equations (separation of variables, initial conditions, systems, some applications). Also covers infinite sequences and series: definition and tests for convergence, power series, Taylor series with remainder, and parametric equations. MATH 122: Honors Calculus II
Takes a more theoretical approach to calculus than MATH 112. Topics include differentiation and integration of elementary transcendental functions, techniques of integration, applications, polar coordinates, infinite series, and complex numbers, as well as an introduction to proving theorems. MATH 135: The Art of Secret Writing
Examines classical and modern methods of message encryption, decryption, and cryptoanalysis. Mathematical tools are developed to describe these methods (modular arithmetic, probability, matrix arithmetic, number theory), and some of the fascinating history of the methods and people involved is presented. MATH 171: Statistical Theory and Application In The Real World
Introductory statistics course discussing techniques for analyzing data occurring in the real world and the mathematical and philosophical justification for these techniques. Topics include population and sample distributions, central limit theorem, statistical theories of point estimation, confidence intervals, testing hypotheses, the linear model, and the least squares estimator. The course concludes with a discussion of tests and estimates for regression and analysis of variance (if time permits). The computer is used to demonstrate some aspects of the theory, such as sampling distributions and the Central Limit Theorem. In the lab portion of the course, students learn and use computer-based methods for implementing the statistical methodology presented in the lectures. MATH 190: Calculus For Engineers
Covers the same material as MATH 191 but meant for students with less preparation. Essentially a second course in calculus. Topics include techniques of integration, finding areas and volumes by integration, exponential growth, partial fractions, infinite sequences and series, and power series. MATH 191: Calculus For Engineers
Essentially a second course in calculus. Topics include techniques of integration, finding areas and volumes by integration, exponential growth, partial fractions, infinite sequences and series, and power series. MATH 192: Calculus For Engineers
Introduction to multivariable calculus. Topics include partial derivatives, double and triple integrals, line integrals, vector fields, Greenís theorem, Stokesí theorem, and the divergence theorem. MATH 201: Invitation to Higher Math: Algebra and Geometry
Provides a preview of some of the more advanced parts of mathematics that do not involve calculus. Topics are chosen for their intrinsic interest and beauty rather than practical utility. One theme is to see some of the strange and surprising mathematical universes that can be constructed when one is not confined to the everyday real world. Another theme is the rich interplay between algebra and geometry, how each illuminates the other. A high point is a geometric proof that there is no general formula for solving polynomial equations of degree five and greater like the well-known quadratic formula. Intended for students who may be considering a math major, or who just like math and are good at it. MATH 213: Calculus III
Designed for students who wish to master the basic techniques of multivariable calculus, but whose major will not require a substantial amount of mathematics. Topics include vectors and vector-valued functions; multivariable and vector calculus including multiple and line integrals; first- and second-order differential equations with applications; systems of differential equations; and elementary partial differential equations. The course may emphasize different topics in the syllabus in different semesters. MATH 221: Linear Algebra
Topics include vector algebra, linear transformations, matrices, determinants, orthogonality, eigenvalues, and eigenvectors. Applications are made to linear differential equations. MATH 222: Multivariable Calculus
Differential and integral calculus of functions in several variables, line and surface integrals as well as the theorems of Green, Stokes and Gauss. MATH 223: Theoretical Linear Algebra and Calculus
MATH 223-224 provides an integrated treatment of linear algebra and multivariable calculus designed for students who have been highly successful in their previous calculus courses. The material is presented at a higher level than in 221-222. Topics in 223 include vectors, matrices, and linear transformations; differential calculus of functions of several variables; inverse and implicit function theorems; quadratic forms, extrema, and manifolds; multiple and iterated integrals. MATH 224: Theoretical Linear Algebra and Calculus
Topics include: vector fields; line integrals; differential forms and exterior derivative; work, flux, and density forms; integration of forms over parametrized domains; and Green's, Stokes', and divergence theorems. MATH 231: Linear Algebra with Applications
Introduction to linear algebra for students who wish to focus on the practical applications of the subject. A wide range of applications are discussed and computer software may be used. The main topics are systems of linear equations, matrices, determinants, vector spaces, orthogonality, and eigenvalues. Typical applications are population models, input/output models, least squares, and difference equations. MATH 275: Living in a Random World
Concentrates on applications of probability in the physical, biological, and social sciences, and to understanding the world around us: games, lotteries, option pricing, opinion polls, etc. MATH 281: Deductive Logic (also PHIL 331)
The syntax and model-theory of classical propositional logic and classical predicate logic, including proofs of the soundness and completeness of Natural Deduction formalizations of these logics, with some attention to related material. MATH 293: Engineering Mathematics
Introduction to ordinary and partial differential equations. Topics include first order equations (separable, linear, homogeneous, exact); mathematical modeling (e.g., population growth, terminal velocity); qualitative methods (slope fields, phase plots, equilibria and stability); numerical methods; second order equations (method of undetermined coefficients, application to oscillations and resonance, boundary value problems and eigenvalues); Fourier series; linear partial differential equations (heat flow, waves, Laplace equation); linear systems of ordinary differential equations. MATH 294: Engineering Mathematics
Linear algebra and its applications. Topics include matrices, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and inner product spaces; applications include brief introductions to difference equations, Markov chains, and systems of linear ordinary differential equations. May include computer use in solving problems. MATH 304: Prove It!
In mathematics the methodology of proof provides a central tool for confirming the validity of mathematical assertions, functioning much as the experimental method does in the physical sciences. In this course, students learn various methods of mathematical proof, starting with basic techniques in propositional and predicate calculus and in set theory and combinatorics, and then moving to applications and illustrations of these via topics in one or more of the three main pillars of mathematics: algebra, analysis, and geometry. Since cogent communication of mathematical ideas is important in the presentation of proofs, the course emphasizes clear, concise exposition. This course is useful for all students who wish to improve their skills in mathematical proof and exposition, or who intend to study more advanced topics in mathematics. MATH 311: Introduction to Analysis
Provides a transition from calculus to real analysis. Topics include rigorous treatment of fundamental concepts in calculus: including limits and convergence of sequences and series, compact sets; continuity, uniform continuity and differentiability of functions. Emphasis will be placed upon understanding and constructing mathematical proofs. MATH 321: Manifolds and Differential Forms
A manifold is a type of subset of Euclidean space that has a well-defined tangent space at every point. Such a set is amenable to the methods of multivariable calculus. After a review of some relevant calculus, this course investigates manifolds and the structures that they are endowed with, such as tangent vectors, boundaries, orientations, and differential forms. The notion of a differential form encompasses such ideas as surface and volume forms, the work exerted by a force, the flow of a fluid, and the curvature of a surface, space or hyperspace. Re-examines the integral theorems of vector calculus (Green, Gauss and Stokes) in the light of differential forms and apply them to problems in partial differential equations, topology, fluid mechanics and electromagnetism. MATH 323: Introduction to Differential Equations
Intended for students who want a brief one-semester introduction to the theory and techniques of both ordinary and partial differential equations. Topics for ordinary differential equations may include initial-value and two-point boundary value problems, the basic existence and uniqueness theorems, continuous dependence on data, stability of fix-points, numerical methods, special functions. Topics for partial differential equations may include the Poisson, heat and wave equations, boundary and initial-boundary value problems, maximum principles, continuous dependence on data, separation of variables, Fourier series, Green's functions, numerical methods, transform methods. MATH 332: Algebra and Number Theory
Covers various topics from number theory and modern algebra. Usually includes most of the following: primes and factorization, Diophantine equations, congruences, quadratic reciprocity, continued fractions, rings and fields, finite groups, and an introduction to the arithmetic of the Gaussian integers and quadratic fields. Motivation and examples for the concepts of abstract algebra are derived primarily from number theory and geometry. MATH 335: Introduction to Cryptology (also COM S 480)
Introduction to the algorithmic and mathematical concepts of cryptanalysis. Topics include security vs. feasibility and different types of cryptographic attack, elementary probability, number theory, cryptographic hash functions, secret and public key cryptography. MATH 336: Applicable Algebra
Introduction to the concepts and methods of abstract algebra and number theory that are of interest in applications. Covers the basic theory of groups, rings and fields and their applications to such areas as public-key cryptography, error-correcting codes, parallel computing, and experimental designs. Applications include the RSA cryptosystem and use of finite fields to construct error-correcting codes and Latin squares. Topics include elementary number theory, Euclidean algorithm, prime factorization, congruences, theorems of Fermat and Euler, elementary group theory, Chinese remainder theorem, factorization in the ring of polynomials, and classification of finite fields. MATH 356: Groups and Geometry
A geometric introduction to the algebraic theory of groups, through the study of symmetries of planar patterns and 3-dimensional regular polyhedra. Besides studying these algebraic and geometric objects themselves, the course also provides an introduction to abstract mathematical thinking and mathematical proofs, serving as a bridge to the more advanced 400-level courses. Abstract concepts covered include: axioms for groups; subgroups and quotient groups; isomorphisms and homomorphisms; conjugacy; group actions, orbits, and stabilizers. These are all illustrated concretely through the visual medium of geometry. MATH 362: Dynamic Models in Biology (also BIOEE 362)
Introductory survey of the development, computer implementation, and applications of dynamic models in biology and ecology. Case study format, covering a broad range of current application areas such as regulatory networks, neurobiology, cardiology, infectious disease management, and conservation of endangered species. Students will also learn how to construct and study biological systems models on the computer using a scripting and graphics environment. MATH 384: Foundations of Mathematics (also PHIL 330)
Topic for spring 2006: set theory as a foundation for mathematics, with some attention to its philosophical motivations. This class will cover the ZF axioms, functions, relations and orderings in the set-theoretic context, ordinal numbers, cardinal numbers, and the construction of the standard number systems. MATH 401: Honors Seminar: Topics In Modern Mathematics
Participatory seminar aimed primarily at introducing senior and junior mathematics majors to some of the challenging problems and areas of modern mathematics. Helps students develop research and expository skills in mathematics, which is important for careers in any field that makes significant use of the mathematical sciences (i.e., pure or applied mathematics, physical or biological sciences, business and industry, medicine). Content varies from year to year. MATH 403: History of Mathematics
Survey of the development of mathematics from antiquity to the present, with an emphasis on the achievements, problems, and mathematical viewpoints of each historical period and the evolution of such basic concepts as number, geometry, construction, and proof. Readings from original sources in translation. Students are required to give oral and written reports. MATH 408: Mathematics In Perspective
Examines several basic topics in mathematics, topics that are usually introduced in high school, from the perspective gained through a completed or nearly completed Cornell math major. Emphasizes the connections between branches of mathematics and the role of careful definitions and proofs in both deepening our understanding of mathematics and generating new mathematical ideas. In addition, the course relates these basic subjects to topics of current mathematical interest. Specific topics may include induction and recursion, synthetic and analytic geometry, number systems, the geometry of complex numbers, angle measurement and trigonometry, and the so-called elementary functions. MATH 413: Honors Introduction to Analysis I
Introduction to the rigorous theory underlying calculus, covering the real number system and functions of one variable. Based entirely on proofs. The student is expected to know how to read and, to some extent, construct proofs before taking this course. Topics typically include construction of the real number system, properties of the real number system, continuous functions, differential and integral calculus of functions of one variable, sequences and series of functions. MATH 414: Honors Introduction to Analysis II
Proof-based introduction to further topics in analysis. Topics may include: the Lebesgue measure and integration, functions of several variables, differential Calculus, implicit function theorem, infinite dimensional normed and metric spaces, Fourier series, ordinary differential equations. MATH 418: Introduction To The Theory of Functions of One Complex Variable
Theoretical and rigorous introduction to complex variable theory. Topics include complex numbers, differential and integral calculus for functions of a complex variable including Cauchy's theorem and the calculus of residues, elements of conformal mapping. Students interested in the applications of complex analysis should consider MATH 422. MATH 420: Differential Equations and Dynamical Systems
Covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations. MATH 422: Applied Complex Analysis
Covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions. MATH 424: Wavelets and Fourier Series
Both Fourier series and wavelets provide methods to represent or approximate general functions in terms of simple building blocks. Such representations have important consequences, both for pure mathematics and for applications. Fourier series use natural sinusoidal building blocks and may be used to help solve differential equations. Wavelets use artificial building blocks that have the advantage of localization in space. A full understanding of both topics requires a background involving Lebesgue integration theory and functional analysis. This course presents as much as possible on both topics without such formidable prerequisites. The emphasis is on clear statements of results and key ideas of proofs, working out examples, and applications. Related topics that may be included in the course: Fourier transforms, Heisenberg uncertainty principle, Shannon sampling theorem, and Poisson summation formula. MATH 425: Numerical Analysis and Differential Equations
Introduction to the fundamentals of numerical analysis: error analysis, interpolation, direct and iterative methods for systems of equations, numerical integration. In the second half of the course, the above are used to build approximate solvers for ordinary and partial differential equations. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is required to test the theoretical concepts throughout the course. MATH 428: Introduction To Partial Differential Equations
Topics are selected from first-order quasilinear equations, classification of second-order equations, with emphasis on maximum principles, existence, uniqueness, stability, Fourier series methods, approximation methods. MATH 431: Linear Algebra
Introduction to linear algebra, including the study of vector spaces, linear transformations, matrices, and systems of linear equations. Additional topics are quadratic forms and inner product spaces, canonical forms for various classes of matrices and linear transformations. MATH 432: Introduction to Algebra
Introduction to various topics in abstract algebra, including groups, rings, fields, factorization of polynomials and integers, congruences, and the structure of finitely generated abelian groups. Optional topics are modules over Euclidean domains and Sylow theorems. MATH 433: Honors Linear Algebra
Honors version of a course in advanced linear algebra, which treats the subject from an abstract and axiomatic viewpoint. Topics include vector spaces, linear transformations, polynomials, determinants, tensor and wedge products, canonical forms, inner product spaces and bilinear forms. Emphasis is on understanding the theory of linear algebra; homework and exams include at least as many proofs as computational problems. For a less theoretical course that covers approximately the same subject matter, see MATH 431. MATH 434: Honors Introduction to Algebra
Honors version of a course in abstract algebra, which treats the subject from an abstract and axiomatic viewpoint, including universal mapping properties. Topics include groups, groups acting on sets, Sylow theorems; rings, factorization: Euclidean rings, principal ideal domains, the structure of finitely generated modules over a principal ideal domain, fields, and Galois theory. The course emphasizes understanding the theory with proofs in both homework and exams. An optional computational component using the computer language GAP is available. For a less theoretical course that covers similar subject matter, see MATH 432. MATH 441: Introduction to Combinatorics I
Combinatorics is the study of discrete structures that arise in a variety of areas, in particular in other areas of mathematics, computer science and many areas of application. Central concerns are often to count objects having a particular property (for example, trees) or to prove that certain structures exist (for example, matchings of all vertices in a graph). The first semester of this sequence covers some basic questions in graph theory, including extremal graph theory (how large must a graph be before one is guaranteed to have a certain subgraph) and Ramsey theory (which shows that large enough objects are forced to have structure). Variations on matching theory are discussed, including theorems of Dilworth, Hall, Kˆnig and Birkhoff, and an introduction to network flow theory. Methods of enumeration (inclusion/exclusion, Mˆbius inversion and generating functions) are introduced and applied to the problems of counting permutations, partitions and triangulations. MATH 442: Introduction to Combinatorics II
Continuation of the first semester, although formally independent of the material covered there. The emphasis here is the study of certain combinatorial structures, such as Latin squares and combinatorial designs (which are of use in statistical experimental design), classical finite geometries and combinatorial geometries (also known as matroids, which arise in many areas from algebra and geometry through discrete optimization theory). There is an introduction to partially ordered sets and lattices, including general Mˆbius inversion and its application, as well as the Polya theory of counting in the presence of symmetries. MATH 451: Euclidean and Spherical Geometry
Covers topics from Euclidean and spherical (non-Euclidean) geometry. Nonlecture, seminar-style course organized around student participation. MATH 452: Classical Geometries
Introduction to hyperbolic and projective geometry ó the classical geometries that developed as Euclidean geometry was better understood. For example, the historical problem of the independence of Euclid's fifth postulate is understood when the existence of the hyperbolic plane is realized. Straightedge (and compass) constructions and stereographic projection in Euclidean geometry can be understood within the structure of projective geometry. Topics in hyperbolic geometry include models of the hyperbolic plane and relations to spherical geometry. Topics in projective geometry include homogeneous coordinates and the classical theorems about conics and configurations of points and lines. Optional topics include principles of perspective drawing, finite projective planes, orthogonal Latin squares, and the cross ratio. MATH 453: Introduction to Topology
Topology may be described briefly as qualitative geometry. This course begins with basic point-set topology, including connectedness, compactness, and metric spaces. Later topics may include the classification of surfaces (such as the Klein bottle and Mˆbius band), elementary knot theory, or the fundamental group and covering spaces. MATH 454: Introduction to Differential Geometry
Differential geometry involves using calculus to study geometric concepts such as curvature and geodesics. This introductory course focuses on the differential geometry of curves and surfaces. It may also touch upon the higher-dimensional generalizations, Riemannian manifolds, which underlie the study of general relativity. MATH 455: Applicable Geometry
Introduction to the theory of n-dimensional convex polytopes and polyhedra and some of its applications, with an in-depth treatment of the case of 3-dimensions. Discusses both combinatorial properties (such as face counts) as well as metric properties (such as rigidity). Covers theorems of Euler, Cauchy, and Steinitz, Voronoi diagrams and triangulations, convex hulls, cyclic polytopes, shellability and the upper-bound theorem. Relates these ideas to applications in tiling, linear inequalities and linear programming, structural rigidity, computational geometry, hyperplane arrangements and zonotopes. MATH 457: Computational Homology (also MATH 657)
Undergraduate version of MATH 657. For description, see MATH 657. MATH 471: Basic Probability
Introduction to probability theory which prepares the student to take MATH 472. Begins with basics: combinatorial probability, mean and variance, independence, conditional probability and Bayes formula. Density and distribution functions and their properties are introduced. The law of large numbers and central limit theorem are stated and their implications for statistics are discussed. MATH 472: Statistics
Statistics have proved to be an important research tool in nearly all of the physical, biological, and social sciences. This course will serve as an introduction to statistics for students who already have some background in calculus, linear algebra, and probability theory. Topics include parameter estimation, hypothesis testing, and linear regression. The course will emphasize both the mathematical theory of statistics as well as techniques for data analysis that are useful in solving scientific problems. MATH 481: Mathematical Logic (also PHIL 431)
First course in mathematical logic providing precise definitions of the language of mathematics and the notion of proof (propositional and predicate logic). The completeness theorem says that we have all the rules of proof we could ever have. The Gödel incompleteness theorem says that they are not enough to decide all statements even about arithmetic. The compactness theorem exploits the finiteness of proofs to show that theories have unintended (nonstandard) models. Possible additional topics: the mathematical definition of an algorithm and the existence of noncomputable functions; the basics of set theory to cardinality and the uncountability of the real numbers. MATH 482: Topics In Logic (also PHIL 432)
Topic: proof-theoretic and algebraic aspects of logic. First-order logic: axiomatization, natural deduction, sequent calculi; generalizing natural deduction and sequent calculi; lambda calculi and the Curry/Howard isomorphism; cut-elimination, normalization, and strong normalization. Second-order logic: strong normalization. Arithmetic: the limits of cut-elimination and normalization. Cartesian-closed categories and the Lambek isomorphism. Time permitting, other topics. MATH 483: Intensional Logic (also PHIL 436)
In this course we will investigate various logics of necessity and possibility ("modal logic"). We will study formal proof procedures as well as possible-worlds semantics. We will also prove various "meta" results, including completeness theorems, rendering this course a good introduction to mathematical as well as philosophical logic. The techniques learned in this part of the course will then be applied to the study of conditionals. Further topics will be among the following: quantified modal logic, two-dimensional modal logic, counterpart theory, and epistemic logic. MATH 486: Applied Logic (also COM S 486)
Covers propositional and predicate logic; compactness and completeness by tableaux, natural deduction, and resolution. Other possible topics include: equational logic; Herbrand Universes and unification; rewrite rules and equational logic, Knuth-Bendix method and the congruence-closure algorithm and lambda-calculus reduction strategies; topics in Prolog, LISP, ML, or Nuprl; and applications to expert systems and program verification. MATH 490: Supervised Reading and Research
Supervised reading and research by arrangement with individual professors. Not for material currently available in regularly scheduled courses. Professional Level and Mathematics Education CoursesMATH 505: Educational Issues In Undergraduate Mathematics
Examines various educational issues in undergraduate mathematics and the relationship of these issues to the mathematics itself. The precise choice of topics varies, but the intent is that a balance of different views be presented and discussed. There are extensive readings in the course and occasional guest lectures. Possible topics include: nature of proof and how and when to teach it, calculus "reform," teaching mathematics to school teachers, using writing, using history, alternative assessments, alternatives to lecturing, equity issues, effective uses of technology, what is mathematical understanding and how do we recognize it, what should every mathematics major know, and research in undergraduate mathematics. MATH 507: Teaching Secondary Mathematics: Theory and Practices
Provides direct experience of new approaches, curricula and standards in mathematics education. Discussion of articles, activities for the secondary classroom and videotape of classroom teaching is tied to in-class exploration of math problems. Experience in the computer lab, examining software environments and their use in the mathematics classroom is included. Participants are expected to write short papers, share ideas in class and present their opinions on issues. MATH 508: Mathematics For Secondary School Teachers
Examination of the principles underlying the content of the secondary school mathematics curriculum, including connections with the history of mathematics and current mathematics research. Graduate CoursesMany of our graduate courses are topics courses for which descriptions are not included here; however, during each pre-enrollment period a schedule of graduate courses to be offered the following semester (Fall 2005 coming soon) is posted on a web site that includes more detailed course descriptions, as well as a means for interested students to participate in the process of selecting meeting times. MATH 611: Real Analysis
MATH 611-612 are the core analysis courses in the mathematics graduate program. 611 covers measure and integration and functional analysis. MATH 612: Complex Analysis
MATH 611-612 are the core analysis courses in the mathematics graduate program. 612 covers complex analysis, Fourier analysis, and distribution theory. MATH 613: Topics In Analysis
MATH 614: Topics In Analysis
MATH 615: Mathematical Methods In Physics
Designed to give a working knowledge of the principal mathematical methods used in advanced physics. Covers Hilbert space, generalized functions, Fourier transform, Sturm-Liouville problem in ODE, Green's functions, and asymptotic expansions. MATH 617: Dynamical Systems
Topics include existence and uniqueness theorems for ODEs; Poincaré-Bendixon theorem and global properties of two dimensional flows; limit sets, nonwandering sets, chain recurrence, pseudo-orbits and structural stability; linearization at equilibrium points: stable manifold theorem and the Hartman-Grobman theorem; and generic properties: transversality theorem and the Kupka-Smale theorem. Examples include expanding maps and Anosov diffeomorphisms; hyperbolicity: the horseshoe and the Birkhoff-Smale theorem on transversal homoclinic orbits; rotation numbers; Herman's theorem; and characterization of structurally stable systems. MATH 618: Smooth Ergodic Theory
Topics include invariant measures; entropy; Hausdorff dimension and related concepts; hyperbolic invariant sets: stable manifolds, Markov partitions and symbolic dynamics; equilibrium measures of hyperbolic attractors; ergodic theorems; Pesin theory: stable manifolds of nonhyperbolic systems; Liapunov exponents; and relations between entropy, exponents, and dimensions. MATH 619: Partial Differential Equations
Covers basic theory of partial differential equations. MATH 620: Partial Differential Equations
Covers basic theory of partial differential equations. MATH 621: Measure Theory and Lebesgue Integration
Covers measure theory, integration, and Lp spaces. MATH 622: Applied Functional Analysis
Covers basic theory of Hilbert and Banach spaces and operations on them. Applications. MATH 628: Complex Dynamical Systems
Various topics in the dynamics of analytic mappings in one complex variable, such as: Julia and Fatou sets, the Mandelbrot set, MaÒÈ-Sad-Sullivan's theorem on structural stability. Also covers: local theory, including repulsive cycles and the Yoccoz inequality, parabolic points and Ecalle-Voronin invarients, Siegel disks and Yoccoz's proof of the Siegel Brjuno theorem; quasi-conformal mappings and surgery: Sullivan's theorem on non-wandering domains, polynomial-like mappings and renormalization, Shishikura's constructin of Hermann rings; puzzles, tableaux and local connectivity problems; and Thurston's topological characterization of rational functions, the spider algorithm, and mating of polynomials. MATH 631: Algebra
MATH 631-632 are the core algebra courses in the mathematics graduate program. 631 covers group theory, especially finite groups; rings and modules; ideal theory in commutative rings; arithmetic and factorization in principal ideal domains and unique factorization domains; introduction to field theory; tensor products and multilinear algebra. (Optional topic: introduction to affine algebraic geometry.) MATH 632: Algebra
MATH 631-632 are the core algebra courses in the mathematics graduate program. MATH 632 covers Galois theory, representation theory of finite groups, introduction to homological algebra. Familiarity with the material of a standard undergraduate course in abstract algebra will be assumed. MATH 633: Noncommutative Algebra
Covers Wedderburn structure theorem, Brauer group, and group cohomology. MATH 634: Commutative Algebra
Covers Dedekind domains, primary decomposition, Hilbert basis theorem, and local rings. MATH 649: Lie Algebras
Topics include nilpotent, solvable and reductive Lie algebras; enveloping algebras; root systems; Coxeter groups; and classification of simple algebras. MATH 650: Lie Groups
Topics include topological groups, Lie groups; relation between Lie groups and Lie algebras; exponential map, homogeneous manifolds; and invariant differential operators. MATH 651: Algebraic Topology I
One of the core topology courses in the mathematics graduate program. An introductory study of certain geometric processes for associating algebraic objects such as groups to topological spaces. The most important of these are homology groups and homotopy groups, especially the first homotopy group or fundamental group, with the related notions of covering spaces and group actions. The development of homology theory focuses on verification of the Eilenberg-Steenrod axioms and on effective methods of calculation such as simplicial and cellular homology and Mayer-Vietoris sequences. If time permits, the cohomology ring of a space may be introduced. MATH 652: Differentiable Manifolds I
One of the core topology courses in the mathematics graduate program. It is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle and a section of the tangent bundle is a vector field. Alternatively vector fields can be viewed as first-order differential operators. This course studies flows of vector fields and prove the Frobenius integrability theorem. In the presence of a Riemannian metric, develops the notions of parallel transport, curvature and geodesics. Examines the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits,de Rham cohomology, Morse theory, or other optional topics are introduced. MATH 653: Differentiable Manifolds II
Advanced topics from differential geometry and differential topology selected by instructor. Examples of eligible topics include: transversality, cobordism, Morse theory, classification of vector bundles and principal bundles, characteristic classes, microlocal analysis, conformal geometry, geometric analysis and partial differential equations, and Atiyah-Singer index theorem. MATH 657: Computational Homology (also MATH 457)
Introduction to homology theory in the setting of cubical complexes. Homology is one of the fundamental tools in topology with applications to problems in areas including dynamical systems, pattern formation and classification, and the analysis of high-dimensional data sets. With such problems serving as motivation, the course covers basic concepts in homology theory from a computational and algorithmic point of view. MATH 661: Geometric Topology
Introduction to some of the more geometric aspects of topology and its connections with group theory. Possible topics include: surface theory, 3-manifolds, knot theory, geometric and combinatorial group theory, hyperbolic groups, and hyperbolic manifolds. MATH 662: Riemannian Geometry
Topics include linear connections, Riemannian metrics and parallel translation; covariant differentiation and curvature tensors; the exponential map, the Gauss Lemma and completeness of the metric; isometries and space forms, Jacobi fields and the theorem of Cartan-Hadamard; the first and second variation formulas; the index form of Morse and the theorem of Bonnet-Myers; the Rauch, Hessian, and Laplacian comparison theorems; the Morse index theorem; the conjugate and cut loci; and submanifolds and the Second Fundamental form. MATH 671: Probability Theory I
Conditional expectation, martingales, Brownian motion. Other topics such as random walks and ergodic theory, depending on time and interest of the students and the instructor. MATH 672: Probability Theory II
Content will vary from year to year. Course may be taken more than once for credit. Previously, topics have been chosen from stochastic calculus, diffusion processes, martingale problems, weak convergence, and Markov processes in continuous time. MATH 674: Introduction To Mathematical Statistics
Topics include an introduction to the theory of point estimation, hypothesis testing and confidence intervals, consistency, efficiency, and the method of maximum likelihood. Basic concepts of decision theory are discussed; the key role of the sufficiency principle is highlighted and applications are given for finding Bayesian, minimax and unbiased optimal decisions. Modern computer-intensive methods like the bootstrap receive some attention, as well as simulation methods involving Markov chains. The parallel development of some concepts of machine learning is exemplified by classification algorithms. An optional section may include nonparametric curve estimation and elements of large sample asymptotics. MATH 675: Statistical Theories Applicable to Genomics
Focuses on statistical concepts useful in genomics (e.g., microarray data analysis) that involve a large number of populations. Topics include multiple testing and closed testing (the cornerstone of multiple testing), family-wise error rate, false discovery rate (FDR) of Benjamini and Hochberg, and Storey's papers relating to pFDR. Also discusses the shrinkage technique or the Empirical Bayes approach, equivalent to the BLUP in a random effect model, which is a powerful technique, taking advantage of a large number of populations. A related technique, which allows use of the same data to select and make inferences for the selected populations (or genes), is discussed. If time permits, there may be some lectures about permutation tests, bootstrapping, and QTL identification. MATH 681: Logic
Covers basic topics in mathematical logic, including propositional and predicate calculus; formal number theory and recursive functions; completeness and incompleteness theorems, compactness and Skolem-Loewenheim theorems. Other topics as time permits. MATH 703: Topics in the History of Mathematics
Topics in the history of modern mathematics at the level of Evolution of Mathematics in the 19th Century by Klein, Abrege D'Histoire Des Mathematiques 1700-1900 by Dieudonne, and Source Book of Classical Analysis by Birkhoff. MATH 711: Seminar In Analysis
MATH 712: Seminar In Analysis
MATH 713: Functional Analysis
Covers topological vector spaces, Banach and Hilbert spaces, and Banach algebras. Additional topics selected by instructor. MATH 715: Fourier Analysis
MATH 717: Applied Dynamical Systems (also T&AM 776)
Topics include review of planar (single-degree-of-freedom) systems; local and global analysis; structural stability and bifurcations in planar systems; center manifolds and normal forms; the averaging theorem and perturbation methods; Melnikov's method; discrete dynamical systems, maps and difference equations, homoclinic and heteroclinic motions, the Smale Horseshoe and other complex invariant sets; global bifurcations, strange attractors, and chaos in free and forced oscillator equations; and applications to problems in solid and fluid mechanics. MATH 722: Topics In Complex Analysis
Selections of advanced topics from complex analysis, such as Riemann surfaces, complex dynamics, and conformal and quasiconformal mapping. Course content varies. MATH 728: Seminar In Partial Differential Equations
MATH 731: Seminar In Algebra
MATH 732: Seminar In Algebra
MATH 735: Topics In Algebra
Selection of advanced topics from algebra, algebraic number theory, and algebraic geometry. Course content varies. MATH 737: Algebraic Number Theory
MATH 739: Topics In Algebra
Selection of advanced topics from algebra, algebraic number theory, and algebraic geometry. Course content varies. MATH 740: Homological Algebra
MATH 751: Seminar In Topology
MATH 752: Seminar In Topology
MATH 753: Algebraic Topology II
Continuation of 651. The standard topics most years are cohomology, cup products, Poincaré duality, and homotopy groups. Other possible topics include fiber bundles, fibrations, vector bundles, and characteristic classes. May sometimes be taught from a differential forms viewpoint. MATH 755: Topology and Geometric Group Theory Seminar
MATH 756: Topology and Geometric Group Theory Seminar
MATH 757: Topics In Topology
Selection of advanced topics from modern algebraic, differential, and geometric topology. Content varies. MATH 758: Topics In Topology
Selection of advanced topics from modern algebraic, differential, and geometric topology. Content varies. MATH 761: Seminar In Geometry
MATH 762: Seminar In Geometry
MATH 767: Algebraic Geometry
MATH 771: Seminar In Probability and Statistics
MATH 772: Seminar In Probability and Statistics
MATH 774: Topics in Statistics
Continuation of MATH 674. Selection of advanced topics from mathematical statistics and empirical processes. Applications are emphasized, such as hidden Markov models, pattern recognition, neural networks, decision trees, model selection and other computationally intensive procedures. Content varies. MATH 777: Stochastic Processes
MATH 778: Stochastic Processes
MATH 781: Seminar In Logic
MATH 782: Seminar In Logic
MATH 783: Model Theory
Introduction to model theory at the level of the books by Hodges or Chang and Keisler. MATH 784: Recursion Theory
Covers theory of effectively computable functions; classification of recursively enumerable sets; degrees of recursive unsolvability; applications to logic; hierarchies; recursive functions of ordinals and higher type objects; generalized recursion theory. MATH 787: Set Theory
First course in axiomatic set theory at the level of the book by Kunen. MATH 788: Topics In Applied Logic
Covers applications of the results and methods of mathematical logic to other areas of mathematics and science. Topics vary each year; some recent examples are: automatic theorem proving, formal semantics of programming and specification languages, linear logic, constructivism (intuitionism), nonstandard analysis, and automata theory. This year's topic is automatic structures, i.e. those with presentations given by various types of automata. Students are expected to be familiar with the standard results in graduate level mathematical logic. MATH 790: Supervised Reading and Research
Last modified: July 20, 2005 |
