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Upper-Level Courses for Sophomores, Juniors, and SeniorsCourses with overlapping content. Students will receive credit for only one of the courses in each of the following groups. Courses with overlapping content are not necessarily equivalent courses. Students are encouraged to consult a mathematics faculty member when choosing between them.
Consult Is There Life After Calculus? for assistance in selecting an appropriate course. MATH 3040 (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 3110 (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 3210 (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 3230 (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 3320 (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 3360 (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 3560 (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 4000-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 3620 (362) Dynamic Models in Biology (also BIOEE 3620)
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 3840 (384) Foundations of Mathematics (also PHIL 3300)
This will be a course on the set theory of Zermelo and Fraenkel: the basic concepts, set-theoretic construction of the natural, integral, rational and real numbers, cardinality, and time permitting the ordinals. Text: Enderton's "Elements of Set Theory." MATH 4010 (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 4030 (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. In addition to the lecture, a problem session (to be arranged) will meet twice a week. MATH 4080 (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 4130 (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 4140 (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 4180 (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 4220. MATH 4200 (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 4220 (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 4240 (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 4250 (425) Numerical Analysis and Differential Equations (also CS 4210)
Introduction to the fundamentals of numerical analysis: error analysis, approximation, interpolation, 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 4250 (CS 4210) and MATH 4260 (CS 4220) provide a comprehensive introduction to numerical analysis; these classes can be taken independently from each other and in either order. MATH 4260 (426) Numerical Analysis: Linear and Nonlinear Problems (also CS 4220)
Introduction to the fundamentals of numerical linear algebra: direct and iterative methods for linear systems, eigenvalue problems, singular value decomposition. In the second half of the course, the above are used to build iterative methods for nonlinear systems and for multivariate optimization. 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 4250 (CS 4210) and MATH 4260 (CS 4220) provide a comprehensive introduction to numerical analysis; these classes can be taken independently from each other and in either order. MATH 4280 (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 4310 (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 4320 (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 4330 (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 4310. MATH 4340 (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 4320. MATH 4370 (437) Computational Algebra
Introduction to Gröbner bases theory, which is the foundation of many algorithms in computational algebra. In this course, students learn how to compute a Gröbner basis for polynomials in many variables. Covers the following applications: solving systems of polynomial equations in many variables, solving diophantine equations in many variables, 3-colorable graphs, and integer programming. Such applications arise, for example, in computer science, engineering, economics, and physics. MATH 4410 (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, König and Birkhoff, and an introduction to network flow theory. Methods of enumeration (inclusion/exclusion, Möbius inversion and generating functions) are introduced and applied to the problems of counting permutations, partitions and triangulations. MATH 4420 (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 Möbius inversion and its application, as well as the Polya theory of counting in the presence of symmetries. MATH 4500 (450) Matrix Groups
An introduction to a topic that is central to mathematics and important in physics and engineering. The objects of study are certain classes of matrices, such as orthogonal, unitary, or symplectic matrices. These classes have both algebraic structure (groups) and geometric/topological structure (manifolds). Thus the course will be a mixture of algebra and geometry/topology, with a little analysis as well. The topics will include Lie algebras (which are an extension of the notion of vector multiplication in three-dimensional space), the exponential mapping (a generalization of the exponential function of calculus), and representation theory (which studies the different ways in which groups can be represented by matrices). Concrete examples will be emphasized. Background not included in the prerequisites will be developed as needed. MATH 4510 (451) Euclidean and Spherical Geometry
Covers topics from Euclidean and spherical (non-Euclidean) geometry. Nonlecture, seminar-style course organized around student participation. MATH 4520 (452) Classical Geometries
Introduction to hyperbolic and projective geometry — 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 4530 (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 Möbius band), elementary knot theory, or the fundamental group and covering spaces. MATH 4540 (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 4550 (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 4710 (471) Basic Probability
Introduction to probability theory which prepares the student to take MATH 4720. 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 4720 (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 4810 (481) Mathematical Logic (also PHIL 4310)
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 4820 (482) Topics in Logic (also PHIL 4311)
This course will focus on intuitionistic logic, including (1) its relationships to classical logic, some “intermediate logics” between intuitionistic and classical, and a modal logic. We’ll consider (2) both proof-theoretic and model-theoretic characterizations of the consequence relations for these logics, (3) algebraic/topological (and time permitting, categorical) characterizations of intuitionistic consequence. (4) We’ll also look at how certain mathematical theories have been developed on the basis of intuitionistic logic. MATH 4860 (486) Applied Logic (also CS 4860)
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 4900 (490) Supervised Reading and Research
Supervised reading and research by arrangement with individual professors. Not for material currently available in regularly scheduled courses. Last modified:March 21, 2008 |
