Curriculum
Courses
Faculty

Students are advised to begin their study of a field with 400-level courses, unless their preparation in that field has been very good. M.A.T. students in mathematics, or M.A., M.S., or Ph.D. students in other departments, may receive graduate credit for any 400-level course that appears in this bulletin. Candidates for the M.A. or Ph.D. in mathematics should note that some 400-level courses do not satisfy certain degree requirements (see footnotes).  

In the following list, the middle digit of the course number indicates the field of mathematics: x0y, algebra; x1y, analysis; x2y, topology; x3y, geometry; x4y, applied mathematics; x5y, mechanics; x6y, probability and statistics; x7y, numerical analysis; x8y, history and foundations.  

• MATH-M 403 Introduction to Modern Algebra I (3 cr.)  

• MATH-M 404 Introduction to Modern Algebra II (3 cr.)  

• MATH-S 403 Honors Course in Modern Algebra I (3 cr.)  

• MATH-S 404 Honors Course in Modern Algebra II (3 cr.)  

• MATH-T 403 Modern Algebra for Secondary Teachers (3 cr.)This course does not ordinarily carry credit toward the M.A. or Ph.D. in mathematics. It may, however, be taken by M.A.T. students and graduate students in other departments for graduate credit.  

• MATH-M 405 Number Theory (3 cr.)P: M212 (Bloomington campus only) This course does not ordinarily carry credit toward the M.A. or Ph.D. in mathematics. It may, however, be taken by M.A.T. students and graduate students in other departments for graduate credit. 

• MATH-M 409 Linear Transformations (3 cr.)Does not count toward the area requirements for the M.A. and Ph.D. in mathematics. 

• MATH-M 413 Introduction to Analysis I (3 cr.) 

• MATH-M 414 Introduction to Analysis II (3 cr.) 

• MATH-M 415 Elementary Complex Variables with Applications (3 cr.) 

• MATH-M 420 Metric Space Topology (3 cr.)This courses does not ordinarily carry credit toward the M.A. or Ph.D. in mathematics. It may, however, be taken by M.A.T. students and graduate students in other departments for graduate credit.  

• MATH-M 425 Graph Network Theory and Combinatorial Analysis (3 cr.) 

• MATH-M 435 Introduction to Differential Geometry (3 cr.)  

• MATH-M 436 Introduction to Geometries (3 cr.)  

• MATH-M 441 Introduction to Partial Differential Equations with Applications I (3 cr.) 

• MATH-M 442 Introduction to Partial Differential Equations with Applications II (3 cr.) 

• MATH-M 447 Mathematical Models and Applications I (3 cr.)This course does not ordinarily carry credit toward the M.A. or Ph.D. in mathematics. It may, however, be taken by M.A.T. students and graduate students in other departments for graduate credit.  

• MATH-M 448 Mathematical Models and Applications II (3 cr.)This course does not ordinarily carry credit toward the M.A. or Ph.D. in mathematics. It may, however, be taken by M.A.T. students and graduate students in other departments for graduate credit.  

• MATH-M 463 Introduction to Probability Theory I (3 cr.)  

• MATH-M 464 Introduction to Probability Theory II (3 cr.)  

• MATH-M 466 Introduction to Mathematical Statistics (3 cr.)  

• MATH-M 471 Numerical Analysis I (3 cr.)P: M301 or M303, M311, M343, and knowledge of a computer language such as Fortran, C, or C++. (Students with other programming backgrounds should consult the instructor.)  

• MATH-M 472 Numerical Analysis II (3 cr.)P: M301 or M303, M311, M343, and knowledge of a computer language such as Fortran, C, or C++. (Students with other programming backgrounds should consult the instructor.)  

• MATH-M 482 Mathematical Logic (3 cr.) 

• MATH-M 490 Problem Seminar (3 cr.) 

• MATH-T 490 Topics for Elementary Teachers (3 cr.)P: T103 or equivalent. Development and study of a body of mathematics specifically designed for experienced elementary teachers. Examples include probability, statistics, geometry, and algebra. Open only to graduate elementary teachers with consent of the instructor. (Does not count toward the area requirements for the M.A. and Ph.D. degrees in mathematics.)  

• MATH-M 501 Survey of Algebra (3 cr.)P: M403- M404. Groups with operators: Jordan-Holder theorem. Sylow theorems. Rings: localization of rings; Chinese remainder theorem. Modules over principal ideal domains: invariants. Fields: algebraic closure; separable and inseparable algebraic extensions; Galois theory; finite fields.  

• MATH-M 502 Commutative Algebra (3 cr.)P: M501. Field theory: transcendental extensions; separable extensions; derivations. Modules: Noetherian and Artinian modules. Primary modules; primary decomposition; Krull intersection theorem. Commutative rings: height and depth of prime ideals. Integral extensions. Notions of algebraic geometry: algebraic sets; Hilbert Nullstellensatz; local rings.  

• MATH-M 503 Noncommutative Algebra (3 cr.)P: M501. Simple and semisimple modules; density theorem; Wedderburn-Artin theorem. Simple algebras: automorphisms; splitting fields; Brauer groups. Representations of finite groups: characters; induced characters; applications. 

• MATH-M 505 Basic Number Theory I (3 cr.)P: M403-M404. Congruence, units modulo n, lattices and abelian groups, quadratic residues, arithmetic functions, diophantine equations, Farey fractions, continued fractions, partition function, the Sieve method, density of subsets of integers, zeta function, the prime number theorem.  

• MATH-M 506 Basic Number Theory II (3 cr.)P: M403-M404. Congruence, units modulo n, lattices and abelian groups, quadratic residues, arithmetic functions, diophantine equations, Farey fractions, continued fractions, partition function, the Sieve method, density of subsets of integers, zeta function, the prime number theorem.  

• MATH-M 507 Introduction to Lie Algebras and Lie Groups (3 cr.)P: M403-M404, and M409 or M501. Nilpotent, solvable, and semisimple Lie algebras, exponential map, PBW theorem, Killing form, Cartan subalgebras, root systems, Weyl group, classification and representations of complex semisimple Lie algebras, Schur's lemma, maximal weight modules; correspondence between real Lie algebras and Lie groups, compact Lie groups, complex and real semisimple Lie groups, symmetric spaces.  

• MATH-M 508 Introduction to Lie Algebras and Lie Groups (3 cr.)P: M403-M404, and M409 or M501. Nilpotent, solvable, and semisimple Lie algebras, exponential map, PBW theorem, Killing form, Cartan subalgebras, root systems, Weyl group, classification and representations of complex semisimple Lie algebras, Schur's lemma, maximal weight modules; correspondence between real Lie algebras and Lie groups, compact Lie groups, complex and real semisimple Lie groups, symmetric spaces.  

• MATH-M 509 Representations of Finite Groups (3 cr.)P: M409 or equivalent. Groups, subgroups. Homomorphisms, isomorphisms. Transformation groups. The orthogonal and Euclidean groups O(3) and E(3). Symmetry and discrete subgroups of E(3). Crystallographic groups. Group representations. Reducible and irreducible representations. Group characters and character tables. Representations of the symmetric groups. Young tableaux. Symmetry classes of tensors. 

• MATH-M 511 Real Variables I (3 cr.)Sets and functions, cardinal and ordinal numbers, set functions, kinds of measures, integration, absolute continuity, convergence theorems, differentiation and integration. Normed linear spaces, function spaces, linear functionals, Banach spaces, Hilbert spaces, Fourier transforms, Schwartz class.  

• MATH-M 512 Real Variables II (3 cr.)Sets and functions, cardinal and ordinal numbers, set functions, kinds of measures, integration, absolute continuity, convergence theorems, differentiation and integration. Normed linear spaces, function spaces, linear functionals, Banach spaces, Hilbert spaces, Fourier transforms, Schwartz class.  

• MATH-M 513 Complex Variables I (3 cr.)Algebra, topology, and geometry of the complex plane; analytic functions; conformal mapping; Riemann surfaces; Cauchy’s theorem and formula; convergence theorems; infinite series and products; Riemann mapping theorem.  

• MATH-M 514 Complex Variables II (3 cr.)Algebra, topology, and geometry of the complex plane; analytic functions; conformal mapping; Riemann surfaces; Cauchy’s theorem and formula; convergence theorems; infinite series and products; Riemann mapping theorem.  

• MATH-M 518 Fourier Analysis (3 cr.)The 
  course will cover basic facts of Fourier series and orthogonal sets of functions, Fourier transforms, and applications. Different convergence properties of the Fourier, Haar, and Sturm-Liouville expansions will be considered. As time permits, applications to discrete and fast Fourier transforms, and wavelets, will be discussed.  

• MATH-M 521 Topology I (3 cr.)Point-set topology, including connectedness, compactness, separation properties, products, quotients, metrization, function spaces. Elementary homotopy theory including fundamental group and covering spaces. Introduction to homology theory with applications such as the Brouwer Fixed Point Theorem.  

• MATH-M 522 Topology II (3 cr.)Point-set topology, including connectedness, compactness, separation properties, products, quotients, metrization, function spaces. Elementary homotopy theory including fundamental group and covering spaces. Introduction to homology theory with applications such as the Brouwer Fixed Point Theorem.  

• MATH-M 529 Introduction to Differential Topology (3 cr.)P: M303, M413, or equivalent. Derivatives and tangents; Inverse Function Theorem; immersions and submersions; Sard’s Theorem. Manifolds; imbedding manifolds. Applications: intersections and degrees (mod 2); Brouwer Fixed Point Theorem. Orientation of manifolds; Euler characteristic; Hopf Degree Theorem.  

• MATH-M 531 Metric Geometry (3 cr.)P: 413-414 Introduction to the geometric study of metric spaces. Topics include: length spaces, model geometries, notions of curvature, Hadamard-Cartan theorem, convexity, metric-measure spaces. 

• MATH-M 533 Differential Geometry I 
  (3 cr.) Differentiable manifolds, multilinear algebra, and tensor bundles. Vector fields, connections, and general integrability theorems. Riemannian manifolds, curvatures, and topics from the calculus of variations.  

• MATH-M 534 Differential Geometry II 
  (3 cr.)Differentiable manifolds, multilinear algebra, and tensor bundles. Vector fields, connections, and general integrability theorems. Riemannian manifolds, curvatures, and topics from the calculus of variations.  

• MATH-M 540 Partial Differential Equations I 
  (3 cr.)P: M441-M442 or equivalent. Introduction to distributions, Sobolev spaces, and Fourier transforms; elliptic equations, Hilbert space theory, potential theory, maximum principle; parabolic equations and systems, characteristics, representations of solutions, energy methods; applications and examples.  

• MATH-M 541 Partial Differential Equations II 
  (3 cr.)P: M441-M442 or equivalent. Introduction to distributions, Sobolev spaces, and Fourier transforms; elliptic equations, Hilbert space theory, potential theory, maximum principle; parabolic equations and systems, characteristics, representations of solutions, energy methods; applications and examples.  

• MATH-M 542 Nonlinear Partial Differential Equations (3 cr.)P: M441-M442 or equivalent. Introduction to an array of topics in linear and nonlinear PDE including elements of calculus of variations and applications to nonlinear elliptic PDE, systems of conservation laws, semi-group theory, reaction-diffusion equations, Schauder theory, Navier-Stokes equations, bifurcation theory.  

• MATH-M 544 Ordinary Differential Equations 
  I (3 cr.)P: M413-M414 or consent of instructor. Existence, uniqueness, continuous dependence; linear systems, stability theory, Floquet theory; periodic solutions of nonlinear equations; Poincaré- Bendixson theory, direct stability methods; almost periodic motions; spectral theory of nonsingular and singular self-adjoint boundary-value problems; two- dimensional autonomous systems; the saddle-point property; linear systems with isolated singularities.  

• MATH-M 545 Ordinary Differential Equations 
  II (3 cr.)P: M413-M414 or consent of instructor. Existence, uniqueness, continuous dependence; linear systems, stability theory, Floquet theory; periodic solutions of nonlinear equations; Poincaré- Bendixson theory, direct stability methods; almost periodic motions; spectral theory of nonsingular and singular self-adjoint boundary-value problems; two- dimensional autonomous systems; the saddle-point property; linear systems with isolated singularities.  

• MATH-M 546 Control Theory (3 cr.)Examples of control problems; optimal control of deterministic systems: linear and nonlinear. The maximal principle: stochastic control problems.  

• MATH-M 548 Mathematical Methods for Biology (3 cr.)P: M414, M463. Deterministic growth models. Birth-death processes and stochastic models for growth. Mathematical theories for the spread of epidemics. Quantitative population genetics.  

• MATH-M 551 Markets and Multi-Period Asset Pricing (3 cr.)P: M463, M345, or equivalent. The concepts of arbitrage and risk-neutral pricing are introduced within the context of dynamic models of stock prices, bond prices, and currency exchange rates. Specific models include multi-period binomial models, Markov processes, Brownian motion, and martingales.  

• MATH-M 553 Cryptography (3 cr.)P: M301 or M303. ***Does not count toward the 500-level requirements. Covers encryption and decryption 
  in secure codes. Topics include: cryptosystems 
  and their cryptanalysis, Data Encryption Standard, differential cryptanalysis, Euclidean algorithm, Chinese remainder theorem, RSA cryptosystem, primality testing, factoring algorithms, ElGamal cryptosystem, discrete log problem, other public key cryptosystems, signature schemes, hash functions, key distribution, and key agreement. Credit not given for both M553 and M453.  

• MATH-M 555 Quantum Computing I (3 cr.)***Does not count toward the 500-level requirements. Covers the interdisciplinary field of quantum information science for graduate students in computer science, physics, mathematics, philosophy, and chemistry. Quantum information science is the study of storing, processing, and communicating information using quantum systems.  

• MATH-M 556 Quantum Computing II 
  (3 cr.)***Does not count toward the 500-level requirements. Covers the interdisciplinary field of quantum information science for graduate students in computer science, physics, mathematics, philosophy, and chemistry. Quantum information science is the study of storing, processing, and communicating information using quantum systems.  

• MATH-M 557 Introduction to Dynamical Systems and Ergodic Theory (3 cr.)Iteration of mappings, flows. Topological, smooth, measure-theoretic, and symbolic dynamics. Recurrence and chaos. Ergodic theory, spectral theory, notions of entropy. Low- dimensional phenomena; hyperbolicity; structural stability and rigidity. Application to number theory, data storage, Internet search and Ramsey theory.  

• MATH-M 558 Introduction to Dynamical Systems and Ergodic Theory (3 cr.)Iteration of mappings, flows. Topological, smooth, measure-theoretic, and symbolic dynamics. Recurrence and chaos. Ergodic theory, spectral theory, notions of entropy. Low- dimensional phenomena; hyperbolicity; structural stability and rigidity. Application to number theory, data storage, Internet search and Ramsey theory.  

• MATH-M 560 Applied Stochastic Processes 
  (3 cr.)P: M343, M463, or consent of instructor. Simple random walk as approximation of Brownian motion. Discrete-time Markov chains. Continuous- time Markov chains; Poisson, compound Poisson, and birth-and-death chains; Kolmogorov’s backward and forward equations; steady state. Diffusions as limits of birth-and-death processes. Examples drawn from diverse fields of application.  

• MATH-M 563 Theory of Probability I (3 cr.)P: M463 and M512; or consent of instructor. Basic concepts of measure theory and integration, axiomatic foundations of probability theory, distribution functions and characteristic functions, infinitely divisible laws and the central limit problem, modes of convergence of sequences of random variables, ergodic theorems, Markov chains, and stochastic processes.  

• MATH-M 564 Theory of Probability II 
  (3 cr.)P: M463 and M512; or consent of instructor. Basic concepts of measure theory and integration, axiomatic foundations of probability theory, distri- bution functions and characteristic functions, infinitely divisible laws and the central limit problem, modes of convergence of sequences of random variables, ergodic theorems, Markov chains, and stochastic processes.  

• MATH-M 566 Mathematical Statistics I 
  (3 cr.)P: M466, M512; or consent of instructor. Modern statistical inference, including such topics as sufficient statistics with applications to similar tests and point estimates, unbiased and invariant tests, lower bounds for mean square errors of point estimates, interval estimation, linear hypothesis, analysis of variance, sequential analysis, decision functions, and nonparametric inference.  

• MATH-M 567 Mathematical Statistics II 
  (3 cr.)P: M466, M512; or consent of instructor. Modern statistical inference, including such topics as sufficient statistics with applications to similar tests and point estimates, unbiased and invariant tests, lower bounds for mean square errors of point estimates, interval estimation, linear hypothesis, analysis of variance, sequential analysis, decision functions, and nonparametric inference.  

• MATH-M 568 Time Series Analysis (3 cr.)P: M466 or consent of instructor. Trends, linear filters, smoothing. Stationary processes, autocorrelations, partial autocorrelations. Autoregressive, moving average, and ARMA processes. Fitting of ARMA 
  and related models. Forecasting. Seasonal time series. Spectral density of stationary processes. Periodograms and estimation of spectral density. Bivariate time series, cross-correlations, cross- spectrum. Other topics as time permits. Equivalent to STAT S650.  

• MATH-M 571 Analysis of Numerical Methods I 
  (3 cr.)P: M441-M442 and M413-M414. Solution of systems of linear equations, elimination and iterative methods, error analyses, eigenvalue problems; numerical methods for integral equations and ordinary differential equations; finite difference, finite element, and Galerkin methods for partial differential equations; stability of methods.  

• MATH-M 572 Analysis of Numerical Methods II 
  (3 cr.)P: M441-M442 and M413-M414. Solution of systems of linear equations, elimination and iterative methods, error analyses, eigenvalue problems; numerical methods for integral equations and ordinary differential equations; finite difference, finite element, and Galerkin methods for partial differential equations; stability of methods.  

• MATH-M 583 Set Theory (3 cr.)P: M482 or M511 or M521. Zermelo-Fraenkel axioms for set theory, well-foundedness and well-orderings, induction and recursion, ordinals and cardinals, axiom of choice, cardinal exponentiation, generalized continuum hypothesis, infinite combinatorics and large cardinals. Martin’s axiom, applications to analysis and topology. 

• MATH-M 584 Recursion Theory (3 cr.)P: One of M482, M511, M521 or CSCI C452; or consent of instructor. Classes of recursive functions, models of computation, Church’s thesis, normal forms, recursion theorem, recursively enumerable sets, reducibilities, lattice of r.e. sets, jump operator, priority arguments, degrees of unsolvability, and hierarchies.  

• MATH-M 590 Seminar (3 cr.) 

• MATH-M 595 Seminar in the Teaching of College Mathematics I (1 cr.) Methods of teaching undergraduate college mathematics. Does not count toward meeting any of the 500-level requirements toward an M.A. or Ph.D.  

• MATH-M 596 Seminar in the Teaching of College Mathematics II (1 cr.) Methods of teaching undergraduate college mathematics. Does not count toward meeting any of the 500-level requirements toward an M.A. or Ph.D.  

• MATH-M 599 Colloquium (1 cr.)Attendance at Department of Mathematics colloquia required. May not be used in fulfillment of the 36 credit hours of 500-, 600-, or 700-level coursework required for 
  the Ph.D. Also not applicable to 30 credit hours for master’s degree. May be repeated.  

• MATH-M 601 Algebraic Number Theory I (3 cr.)P: M501-M502. Valuations, fields of algebraic functions, cohomology of groups, local and global class field theory.  

• MATH-M 602 Algebraic Number Theory II 
  (3 cr.)P: M501-M502. Valuations, fields of algebraic functions, cohomology of groups, local and global class field theory.  

• MATH-M 607 Group Representations I 
  (3 cr.)P: Consent of instructor. Review of abstract group theory. Representation theory of finite and infinite compact groups. Detailed study of selected classical groups. Lie groups, covering groups, 
  Lie algebras, invariant measure and induced representations. May be taught in alternate years by members of the Departments of Mathematics and Physics; see PHYS P607.  

• MATH-M 608 Group Representations II (3 cr.) P: Consent of instructor. Review of abstract group theory. Representation theory of finite and infinite compact groups. Detailed study of selected classical groups. Lie groups, covering groups, 
  Lie algebras, invariant measure and induced representations. May be taught in alternate years by members of the Departments of Mathematics and Physics; see PHYS P607.  

• MATH-M 611 Functional Analysis I (3 cr.) Fundamentals of the theory of vector spaces; Banach spaces; Hilbert space. Linear functionals and operators in such spaces, spectral resolution of operators. Functional equations: applications 
  to fields of analysis, such as integration and measure, integral equations, ordinary and partial differential equations, ergodic theory. Nonlinear problems. Schauder-Leray fixed-point theorem and its applications to fundamental existence theorems of analysis. 

• MATH-M 612 Functional Analysis II (3 cr.) Fundamentals of the theory of vector spaces; Banach spaces; Hilbert space. Linear functionals and operators in such spaces, spectral resolution of operators. Functional equations: applications 
  to fields of analysis, such as integration and measure, integral equations, ordinary and partial differential equations, ergodic theory. Nonlinear problems. Schauder-Leray fixed-point theorem and its applications to fundamental existence theorems of analysis.  

• MATH-M 621 Algebraic Topology I (3 cr.)Basic concepts of homological algebra, universal coefficient theorems for homology and cohomology, Künneth formula, duality in manifolds. Homotopy theory including Hurewicz and Whitehead theorems, classifying spaces, Postnikov systems, spectral sequences, homotopy groups of spheres. Offered every other year, alternating with M623-M624.  

• MATH-M 622 Algebraic Topology II (3 cr.)Basic concepts of homological algebra, universal coefficient theorems for homology and cohomology, Künneth formula, duality in manifolds. Homotopy theory including Hurewicz and Whitehead theorems, classifying spaces, Postnikov systems, spectral sequences, homotopy groups of spheres. Offered every other year, alternating with M623-M624.  

• MATH-M 623 Geometric Topology I 
  (3 cr.)P: M522. Topics in geometric topology chosen from K-theory, simple homotopy theory, topology of manifolds, fiber bundles, knot theory, and related areas. May be taken more than once. Offered every other year, alternating with M621-M622.  

• MATH-M 624 Geometric Topology II 
  (3 cr.)P: M522. Topics in geometric topology chosen from K-theory, simple homotopy theory, topology of manifolds, fiber bundles, knot theory, and related areas. May be taken more than once. Offered every other year, alternating with M621-M622.  

• MATH-M 630 Algebraic Geometry (3 cr.)P: M522. A study in the plane, based on homogeneous point and line coordinates; a study of algebraic curves and envelopes, including such topics as invariants, singularities, reducibility, genus, polar properties, Pascal and Brainchon theorems, and Jacobian, Hessian, and Plücker formulas.  

• MATH-M 633 Algebraic Varieties (3 cr.) Geometric and cohomological properties of algebraic varieties and schemes.  

• MATH-M 634 Algebraic Varieties II (3 cr.) Geometric and cohomological properties of algebraic varieties and schemes.  

• MATH-M 635 Relativity I (3 cr.) Mathematical foundations of the theory of relativity. Lorentz groups, Michelson-Morley experiment, aberration 
  of stars, Fizeau experiment, kinematic effects, relativistic second law of Newton, relativistic kinetic energy, Maxwell equations, ponderomotive equations. Curvature tensor and its algebraic identities, Bianchi’s identity, gravitation and geodesics. Schwarzschild solution, relativistic orbits, deflection of light.  

• MATH-M 636 Relativity II (3 cr.) Mathematical foundations of the theory of relativity. Lorentz groups, Michelson-Morley experiment, aberration of stars, Fizeau experiment, kinematic effects, relativistic second law of Newton, relativistic kinetic energy, Maxwell equations, ponderomotive equations. Curvature tensor and its algebraic identities, Bianchi’s identity, gravitation and geodesics. Schwarzschild solution, relativistic orbits, deflection of light.  

• MATH-M 637 Theory of Gravitation I 
  (3 cr.)Introduction to the general theory of relativity, stress-energy tensor, parallel transport, geodesics, Einstein’s equation, differential geometry, manifolds, general covariance, bending of light, perihelion advance. Modern cosmology: Robertson-Walker metric, equations of state, Friedmann equations, Hubble’s law, redshift, cosmological constant, inflation, quintessence, cosmic microwave background, Big Bang nucleosynthesis, structure formation. May be taught in alternate years by members of the Department of Physics; see PHYS P637.  

• MATH-M 638 Theory of Gravitation II 
  (3 cr.)Gravitation waves, Schwarzschild geometry and black holes, Kerr metric, Reissner-Nordstrom metric, extremal black holes, Penrose diagrams, Hawking radiation, Lie derivatives, isometries and Killing vectors, variational principle and the Palatini formalism, spinors in general relativity, vierbeins, gravitation as a gauge theory, quantum gravity. May be taught in alternate years by members of the Department of Physics; see PHYS P638.  

• MATH-A 641 Elliptic Differential Equations 
  (3 cr.)P: M511, M513, M540, or consent of instructor. Green’s identity, fundamental solutions, function theoretic methods, partition of unity, 
  weak and strong derivatives, Sobolev inequalities, embedding theorems, Garding’s inequality, Dirichlet problem, existence theory, regularity in the interior, regularity on the boundary, and selected topics.  

• MATH-A 642 Evolution Equations (3 cr.)P: M511, M513, M540, or consent of instructor. Hyperbolic equations and systems, parabolic equations, Cauchy problems in higher dimension, method of descent, fundamental solutions and their construction, strongly continuous semigroups, analytic semigroups, uniqueness theorems in Hilbert space, fractional powers of operators, analyticity of solutions, and selected topics.  

• MATH-A 643 Integral Equations (3 cr.)P: M511, M513, M540, or consent of instructor. Covers the Volterra-Fredholm theory of integral equations and the abstract Riesz theory of compact operators. Other topics include ideals of compact operators, Fredholm operators, convolution equations and their relationship to Toeplitz operators, Wiener-Hopf factorization.  

• MATH-A 647 Mathematical Physics (3 cr.)P: M541 or consent of instructor. Applications of the theory 
  of normed linear spaces, distributions, unbounded operators in Hilbert space, and related topics to problems in mathematical physics. May be taught in alternate years by members of the Department of Physics; see PHYS P647. 

• MATH-A 655 Mathematical Foundations of Quantum Mechanics (3 cr.)P: Consent of instructor. Philosophical and mathematical analysis of the concepts: quantum observable, compatibility, quantum state, superposition principle, symmetry. Axiomatic construction of conventional quantum mechanics. May be taught in alternate years by members of the Department of Physics; see PHYS P655.  

• MATH-A 656 Kinetic Theory and Statistical Mechanics I (3 cr.)Introduction to the classical theory and modern developments. Historical development of kinetic-statistical theories; rigorous equilibrium statistics; kinetic gas dynamics according to Boltzmann equation; kinetic theories of transport processes in liquids. May be taught in alternate years by members of the Departments of Mathematics and Physics; see PHYS P656-P657.  

• MATH-A 657 Kinetic Theory and Statistical Mechanics I (3 cr.)Introduction to the classical theory and modern developments. Historical development of kinetic-statistical theories; rigorous equilibrium statistics; kinetic gas dynamics according to Boltzmann equation; kinetic theories of transport processes in liquids. May be taught in alternate years by members of the Departments of Mathematics and Physics; see PHYS P656-P657.  

• MATH-A 658 Continuum Mechanics I 
  (3 cr.)P: Consent of instructor. Two-semester course dealing with mathematical foundations of continuum mechanics; content varies yearly; topics selected from elasticity, plasticity, or fluid mechanics and related areas.  

• MATH-A 659 Continuum Mechanics II 
  (3 cr.)P: Consent of instructor. Two-semester course dealing with mathematical foundations of continuum mechanics; content varies yearly; topics selected from elasticity, plasticity, or fluid mechanics and related areas.  

• MATH-M 663 Weak Convergence of Probability Measures and Applications (3 cr.)P: M512, M564. Weak convergence of probability measures on metric spaces. Prohorov’s theorem and tightness. Brownian motion. Donsker’s invariance principle. Weak convergence on D [0,1]. Convergence of empirical distributions. Functional central limit theorems under dependence.  

• MATH-M 664 Large Sample Theory of Statistics (3 cr.) P: M563, M566. Asymptotic distributions 
  of sample moments, sample quantiles, and U- statistics; methods of estimation: maximum likelihood estimates, method of moments, L- estimators, Bayes estimators; asymptotic efficiency; likelihood ratio tests, chi-square tests, asymptotic relative efficiencies of tests; weak convergence of the empirical distribution function to a Brownian bridge and application; selection of topics from the following: large deviations, second-order asymptotic efficiency, bootstrap rank tests.  

• MATH-M 671 Numerical Treatment of Differential and Integral Equations I (3 cr.)P: M540 or consent of instructor. Finite difference methods of ordinary and partial differential equations; relaxation methods; discrete kernel functions; methods of Ritz, Galerkin, and Trefftz approximate methods for integral equations.  

• MATH-M 672 Numerical Treatment of Differential and Integral Equations II (3 cr.)P: M540 or consent of instructor. Finite difference methods of ordinary and partial differential equations; relaxation methods; discrete kernel functions; methods of Ritz, Galerkin, and Trefftz approximate methods for integral equations. 

• MATH-M 680 Logic and Decidability 
  (3 cr.)P: M584 and M404; or consent of instructor. Effective syntax and semantics of propositional 
  and first-order logics, theory of decidability and some decidable theories, theory of undecidability and implicit definability, Gödel’s theorems on incompleteness and the unprovability of consistency.  

• MATH-M 682 Model Theory (3 cr.)P: M583, 
  M680, and M502; or consent of instructor. Elementary equivalence, completeness and model- completeness, interpolation, preservation and characterization theorems, elementary classes, types, saturated structures, introduction to categoricity and stability.  

• MATH-M 701 Selected Topics in Algebra I (3 cr.) 

• MATH-M 702 Selected Topics in Algebra II (3 cr.)  

• MATH-M 711 Selected Topics in Analysis I (3 cr.)  

• MATH-M 712 Selected Topics in Analysis II (3 cr.)  

• MATH-M 721 Selected Topics in Topology I (3 cr.)  

• MATH-M 722 Selected Topics in Topology II (3 cr.) 

• MATH-M 731 Selected Topics in Differential Geometry I (3 cr.) 

• MATH-M 732 Selected Topics in Differential Geometry II (3 cr.) 

• MATH-M 733 Selected Topics in Algebraic Geometry I (3 cr.) 

• MATH-M 734 Selected Topics in Algebraic Geometry II (3 cr.) 

• MATH-M 741 Selected Topics in Applied Mathematics I (3 cr.) 

• MATH-M 742 Selected Topics in Applied Mathematics II (3 cr.) 

• MATH-M 743 Selected Topics in Mathematical Physics I (3 cr.) Content varies from year to year. May be taught in alternate years by members of the Department of Physics; see PHYS P743.  

• MATH-M 744 Selected Topics in Mathematical Physics II (3 cr.)Content varies from year to year. May be taught in alternate years by members of the Department of Physics; see PHYS P743.  

• MATH-M 751 Selected Topics in Mechanics I (3 cr.)  

• MATH-M 752 Selected Topics in Mechanics II (3 cr.)  

• MATH-M 761 Selected Topics in Probability I (3 cr.)  

• MATH-M 762 Selected Topics in Probability II (3 cr.)  

• MATH-M 771 Selected Topics in Numerical Analysis I (3 cr.)  

• MATH-M 772 Selected Topics in Numerical Analysis II (3 cr.)  

• MATH-M 781 Selected Topics in Mathematical Logic (3 cr.)  

• MATH-M 782 Selected Topics in Mathematical Logic (3 cr.)  

• MATH-M 800 Mathematical Reading and Research (arr. cr.)**These courses are eligible for a deferred grade.