Graduate Course Descriptions
Click on the following course numbers to be taken directly to that course description:
511 Probability Theory (4) (F)
512 Introduction to Statistical Inference (4) (Sp)
513 Introduction to Mathematical Logic I (3)
514 Introduction to Mathematical Logic II (3)
517 Survey of Advanced Calculus I (3)
518 Survey of Advanced Calculus II (3)
519 Basic Algebra (3)
520 Advanced Linear Algebra (3)
524 Survey of Fourier Series Methods (3)
525-526 Introduction to Complex Variables I, II (3, 3)
527-528 Introduction to Topology I, II (3, 3)
529 Introduction to the Theory of Numbers I (3)
530 Introduction to the Theory of Numbers II (3)
531 Introduction to Real Variables I (3) (F)
532 Introduction to Real Variables II (3) (Sp)
533 Calculus of Variations (3)
534 Basic Measure Theory (3)
535 Introduction to Cryptography (3) (F; Sp)
537 Introduction to Numerical Analysis I (3)
538 Introduction to Numerical Analysis II (3)
539-540 Methods of Applied Mathematics I, II (3, 3)
541 Mathematical Biology (3)
543 Fundamentals of Applied Mathematics I (3)
544 Fundamentals of Applied Mathematics II (3)
545 Introduction to Ordinary Differential Equations (3)
549 Introduction to Partial Differential Equations (3)
558 Mathematical Finance (3)
559 Mathematical Finance 2 (3)
560 Theory of Games (3)
561 Lectures on Algebra (3)
562 Lectures on Analysis (3)
563 Lectures on Applied Mathematics (3)
564 Lectures on Combinatorial Analysis (3)
565 Lectures on Geometry (3)
566 Lectures on Logic and Set Theory (3)
567 Lectures on Number Theory (3)
568 Lectures on Computational Mathematics (3)
569 Lectures on Topology (3)
570 Topics in Mathematics (3)
590 History of Mathematics (3)
591 Project Guidance in Applied Mathematics (1-3)
599 Supervised Teaching (1-4)
613-614 Mathematical Logic (3, 3)
615-616 Axiomatic Set Theory (3, 3)
619-620 Algebra (3, 3)
625-626 Complex Variables (3, 3)
627-628 Topology (3, 3)
629-630 Theory of Numbers (3, 3)
631-632 Analysis (3, 3)
635-636 Differential Geometry (3, 3)
637-638 Advanced Numerical Analysis (3, 3)
639 Fourier Analysis (3)
645-646 Advanced Ordinary Differential Equations (3, 3)
647 Integral Equations (3)
649-650 Partial Differential Equations (3, 3)
713-714 Recursive Functions (3, 3)
715-716 Intuitionistic Mathematics (3, 3)
719-720 Advanced Algebra (3, 3)
725 Topics in Complex Analysis (3)
726 Theory of Functions of Several Complex Variables (3)
727-728 Algebraic Topology (3, 3)
729 Diophantine Approximations (3)
730 Algebraic Number Theory (3)
731-732 Functional Analysis (3, 3)
735-736 Algebraic Geometry (3, 3)
743-744 Topics in Differential Geometry (3, 3)
800 Thesis Guidance (1-12)
801 Reading and Conference (1-6)
805 Colloquium (1-4)
807 Graduate Research (1-12)
813 Selected Topics in Mathematical Logic (3)
814 Seminars in Mathematical Logic (variable)
819 Selected Topics in Algebra (3)
820 Seminars in Algebra (variable)
827 Selected Topics in Topology (3)
828 Seminars in Topology (variable)
829 Selected Topics in Number Theory (3)
830 Seminars in Number Theory (variable)
831 Selected Topics in Analysis (3)
832 Seminars in Analysis (variable)
835 Selected Topics in Geometry (3)
836 Seminars in Geometry (variable)
837 Selected Topics in Numerical Analysis (3)
838 Seminars in Numerical Analysis (variable)
839 Selected Topics in Applied Mathematics (3)
840 Seminars in Applied Mathematics (variable)
511 Probability Theory (4) (F)
Prerequisite: MTH 141-MTH 142 or equivalent
Description: A first course in probability. Introduces the basic concepts of probability theory and addresses many concrete problems. A list of basic concepts includes axioms of probability, conditional probability, independence, random variables (continuous and discrete), distribution functions, expectation, variance, joint distribution functions, limit theorems.
512 Introduction to Statistical Inference (4) (Sp)
Description: Topics include: review of probability, conditional probability, Bayes' Theorem; random variables and distributions; expectation and properties; covariance, correlation, and conditional expectation; special distributions; Central Limit Theorem and applications; estimations, including Bayes; estimators, maximum likelihood estimators, and their properties. Includes use of sufficient statistics to 'improve' estimators, distribution of estimators, unbiasedness, hypothesis testing, linear statistical models, and statistical inference from the Bayesian point of view.
513 Introduction to Mathematical Logic I (3)
Prerequisite: MTH 313 and consent of instructor
Description: Informal and formal development of propositional calculus, predicate calculus and predicate calculus with equality. Completeness theorem and some consequences. Additional reading on selected topics.
514 Introduction to Mathematical Logic II (3)
Prerequisite: MTH 513 or consent of instructor
Description: Godel's incompleteness theorem, decidability and recursiveness. Consistency problems. Additional reading on selected topics.
517 Survey of Advanced Calculus I (3)
Prerequisite: MTH 306 or equivalent
Description: Survey of functions of several variables, differentiation, composite and implicit functions, maxima and minima, differentiation under the integral sign, line integrals, Green's theorem. Vector field theory: gradient, divergence and curl, divergence theorem. Stokes' theorem, applications. Review of general theory of sequences and series. Additional reading on selected topics.
518 Survey of Advanced Calculus II (3)
Prerequisite: MTH 306 or equivalent
Description: Fourier series, sine and cosine series, mean convergence, pointwise convergence, orthogonal functions. Linear ordinary differential equations, solution by Laplace transforms, solution by power series. Green's functions. Partial differential equations: solution of boundary value problems by series, methods of separation of variables, solution of boundary value problems by integral transformations, classification and stability of equations. Additional reading on selected topics.
519 Basic Algebra (3)
Prerequisite: MTH 420 or consent of instructor
Description: Definitions and elementary properties of groups and fields, vector space, linear space, linear dependence, dimension, vector space homomorphisms, kernel and cokernel of a vector space homomorphism. Application to linear equations, duality. Rings and ideals. Quotient rings. Integral domains, field of fractions. Polynomial rings. Principal ideal rings, unique factorization, lemma of Gauss. Eisenstein criterion of irreducible polynomials. (Example of irreducible polynomials.) Extension of commutative fields, finite multiplicative subgroup of a field is cyclic characteristic of a field. Roots of unity. Applications to elementary number theory. (Wilson's theorem, Fermat's theorem, etc.) Additional reading on selected topics.
520 Advanced Linear Algebra (3)
Prerequisite: MTH 309, with MTH 311 recommended; or consent of instructor
Description: Topics in advanced linear algebra.
524 Survey of Fourier Series Methods (3)
Prerequisite: Consent of instructor
Description: Partial differential equations of physics, separation of variables and superposition of solutions; orthonormal sets. Fourier series. Fourier transforms; application to boundary value problems. Additional reading on selected topics.
525-526 Introduction to Complex Variables I, II (3, 3)
Prerequisite: MTH 432 or consent of instructor
Description: The notion of analyticity. Calculus over the complex numbers. Cauchy's theorems, residues, singularities, conformal mapping. Weierstrass convergence theorem, analytic continuation. Additional reading on selected topics.
527-528 Introduction to Topology I, II (3, 3)
Prerequisite: MTH 431 or equivalent and consent of instructor
Description: Elementary set theory, functions and relations, partially ordered sets. Zorn's Lemma, abstract topological spaces, semi-metric and metric spaces, bases and subbases, convergence, filters and nets, separation axioms, continuity and homeomorphisms, connectedness, separability, compactness. Additional reading on selected topics.
529 Introduction to the Theory of Numbers I (3)
Prerequisite: MTH 419 and consent of instructor
Description: The Euclidean Algorithm and unique factorization, arithmetical functions, congruences, reduced residue systems, primitive rotos, magic squares, certain diophantine equations. Additional reading on selected topics.
530 Introduction to the Theory of Numbers II (3)
Prerequisite: MTH 420 or consent of instructor
Description: Irrational numbers, continued fractions from a geometric viewpoint, best rational approximations to real numbers, the Fermat-Pell equation, quadratic fields and integers, applications to diophantine equations. Additional reading on selected topics.
531 Introduction to Real Variables I (3) (F)
Prerequisite: MTH 311
Description: This is a comprehensive and rigorous course in the study of real valued functions of one real variable. Topics include sequences of numbers, limits and the Cauchy criterion, continuous functions, differentiation, inverse function theorem, Riemann integration, sequences and series, uniform convergence. This course is a prerequisite for most advanced courses in analysis.
Note: The MTH 311 prerequisite for this course is strictly enforced. Students who have not completed MTH 311, but who have had an equivalent course, need to obtain a waiver from the director of graduate studies.
532 Introduction to Real Variables II (3) (Sp)
Prerequisite: MTH 431
Description: This is a rigorous course in the study of analysis in dimensions greater than one. Three basic theorems: the inverse function theorem, the implicit function theorem, and the change of variables theorem in multiple integrals are among the subjects studied in detail. Topics in this course include continuously differentiable functions, the chain rule, inverse and implicit function theorems, Riemann integration, partitions of unity, change of variables theorem.
533 Calculus of Variations (3)
Prerequisite: MTH 432 or equivalent
Description: Necessary conditions in the calculus of variations. Sufficient conditions, Hamilton-Jacobi Theory. Basic Existence theorems.
534 Basic Measure Theory (3)
Prerequisite: MTH 432 or consent of instructor
Description: The real numbers, the extended real numbers, sequences, limit superior and limit inferior, topology for the real numbers and continuity of functions. The Lebesgue outer measure, measurable sets and Lebesgue measure, nonmeasurable sets, measurable functions. Egoroff's Theorem and Lusin's Theorem. The Riemann integral, the Lebesgue integral and the convergence theorems. Differentiation of functions of bounded variation, absolute continuity. The Lp spaces.
535 Introduction to Cryptography (3) (F; Sp)
Prerequisite: MTH 419 or MTH 429 or consent of instructor
Description: Cryptosystem definitions and basic types of attack. Substitution ciphers. Hill ciphers. Congruences and modular exponentiation. Digital Encryption Standard. Public key and RSA cryptosystems. Pseudoprimes and primality testing. Pollard rho method. Basic finite field theory. Discrete log. Digital signatures.
537 Introduction to Numerical Analysis I (3)
Prerequisite: MTH 145, MTH 241 and MTH 306
Description: Lagrangian interpolation. Newton-Cotes quadrature formulas, Gaussian quadrature and orthogonal polynomials. Romberg quadrature, difference equations, numerical solution of ordinary differential equations, predictor-corrector methods, Runge-Kutta methods. Additional reading on selected topics.
Note: cross-listed with Computer Science 537
538 Introduction to Numerical Analysis II (3)
Prerequisite: MTH 241, MTH 537 or concurrent registration
Description: Solution of nonlinear equations and simultaneous linear equations, linear least-square approximations. Chebyshev polynomials, minimax approximations, calculation of eigenvalues and eigenvectors. Additional reading on selected topics.
Note: cross-listed with Computer Science 538
539-540 Methods of Applied Mathematics I, II (3, 3)
Prerequisite: MTH 418 or consent of instructor
Description: Matrices, equivalence, quadratic and hermitian forms, eigenvalues, invariants, function spaces and Sturm-Liouville problems. Calculus of variations, Euler-Lagrange equations, constraints, variable endpoints, Sturm-Liouville theory, Rayleigh-Ritz method. Integral equations. Green's functions, Hilbert-Schmidt theory, Fredholm theory, singular integral equations.
Note: cross-listed with Engineering Science 543-544
541 Mathematical Biology (3)
Prerequisite: Differential equations, linear algebra, or consent from instructor
Description: Emphasis is on the application of mathematical techniques to help unravel underlying mechanisms involved in various biological processes. Topics will be chosen from a broad range, among the possibilities being reaction kinetic, biological oscillations, population ecology, developmental biology, neurobiology, epidemiology, physiological fluid dynamics, sensory biology, etc.
543 Fundamentals of Applied Mathematics I (3)
Prerequisite: MTH 306 or equivalent
Description: Mathematical formulation and analysis of models for phenomena in the natural sciences. Includes derivation of relevant differential equations from conservation laws and constitutive relations. Potential topics include diffusion, stationary solutions, traveling waves, linear stability analysis, scaling and dimensional analysis, perturbation methods, variational and phase space methods, kinematics and laws of motion for continuous media. Examples from areas might include, but are not confined to, biology, fluid dynamics, elasticity, chemistry, astrophysics, geophysics.
544 Fundamentals of Applied Mathematics II (3)
Prerequisite: MTH 543
545 Introduction to Ordinary Differential Equations (3)
Prerequisite: MTH 431 and consent of instructor
Description: Existence and uniqueness of solutions, continuation of solutions, dependence on initial conditions and parameters; linear systems of equations with constant and variable coefficients; autonomous systems, phase space and stability. Additional reading on selected topics.
549 Introduction to Partial Differential Equations (3)
Prerequisite: Consent of instructor
Description: A rigorous study of the wave, heat, and potential equations in two dimensions, focusing on fundamental concepts, methods and properties of solutions. General properties of second order linear equations in two dimensions, classification, characteristics, well-posed problems and approximation. Solution of the three types of equations by the method of separation of variables and Fourier series. Poisson representation formulas. Nonhomogeneous problems and Green's function. Formulation and properties of the Tricomi problem. Discussion of a simplied problem in fluid dynamics. Additional reading on selected topics.
558 Mathematical Finance (3)
Prerequisite: MTH 241, MTH 309, MTH 306
Description: This course will introduce the mathematical theory and computation of modern financial products used in the banking and corporate world. Mathematical models for the valuation of derivative products will be derived and analyzed.
559 Mathematical Finance 2 (3)
Prerequisite: MTH 458 or MTH 558
Description: Describes the mathematical development of both the theoretical and the computational techniques used to analyze financial instruments. Specific topics include utility functions; forwards, futures, and swaps; and modeling of derivatives and rigorous mathematical analysis of the models, both theoretically and computationally. Develops, as needed, the required ideas from partial differential equations and numerical analysis.
560 Theory of Games (3)
Prerequisite: Consent of instructor
Description: Introduction to von Neumann's theory of games with applications to optimal strategies, decision theory, and linear programming. Additional reading on selected topics.
561 Lectures on Algebra (3)
Prerequisite: MTH 420 and consent of instructor
Description: A topics course. Treats problems, advanced techniques and recent developments in algebra.
Note: Can be taken more than once for credit.
562 Lectures on Analysis (3)
Prerequisite: MTH 432 and consent of instructor
Description: A topics course. Treats problems, advanced techniques and recent developments in analysis.
Note: Can be taken more than once for credit.
563 Lectures on Applied Mathematics (3)
Prerequisite: Consent of instructor
Description: A topics course. Treats problems, advanced techniques and recent developments in applied mathematics.
Note: Can be taken more than once for credit.
564 Lectures on Combinatorial Analysis (3)
Prerequisite: Consent of instructor
Description: A topics course. Treats problems, advanced techniques and recent developments in combinatorial analysis.
Note: Can be taken more than once for credit.
565 Lectures on Geometry (3)
Prerequisite: MTH 309, MTH 419 or equivalent and consent of instructor
Description: A topics course. Treats problems, advanced techniques and recent developments in geometry.
Note: Can be taken more than once for credit.
566 Lectures on Logic and Set Theory (3)
Prerequisite: Consent of instructor
Description: A topics course. Treats problems, advanced techniques and recent developments in logic and set theory.
Note: Can be taken more than once for credit.
567 Lectures on Number Theory (3)
Prerequisite: Consent of instructor
Description: A topics course. Treats problems, advanced techniques and recent developments in number theory.
Note: Can be taken more than once for credit.
568 Lectures on Computational Mathematics (3)
Prerequisite: Consent of instructor
Description: A topics course. Treats problems, advanced techniques and recent developments in computational mathematics.
Note: Can be taken more than once for credit.
569 Lectures on Topology (3)
Prerequisite: Consent of instructor
Description: A topics course. Treats problems, advanced techniques and recent developments in topology.
Note: Can be taken more than once for credit.
570 Topics in Mathematics (3)
Description: Effective Fall 2006, MTH 570 is the number for the topics course previously known as MTH 595
590 History of Mathematics (3)
591 Project Guidance in Applied Mathematics (1-3)
599 Supervised Teaching (1-4)
Prerequisite: Admission only by consent of the Department Chairman
Description: Teaching assignments within the Department will be delegated to all registrants, whose work will be supervised by a member of the department staff. May be taken more than once for credit, hour-allowance of which will depend upon type and amount of instructional duties.
613-614 Mathematical Logic (3, 3)
Prerequisite: MTH 313 or equivalent
Description: Propositional and predicate logic; consistency and completeness results from each. First order theories, particularly arithmetic. Godel's incompleteness theorem.
615-616 Axiomatic Set Theory (3, 3)
Prerequisite: MTH 313 or equivalent
Description: Development of Godel-Bernays axioms for set theory, ordinal numbers, ordinal arithmetic, cardinal numbers, cardinal arithmetic, constructible sets, large cardinal axioms. Recent consistency and independence results (Godel, Cohen).
619-620 Algebra (3, 3)
Prerequisites: 419-420 or equivalent
Description: Basic aspects of monoid theory, group theory, ring theory (including algebras), module theory, field theory, and category theory. The following is a representative list of topics which may be covered. (Of course, individual instructors may modify this list.)
GROUPS: homomorphism theorems, symmetric groups, linear groups, Sylow theorems, solvable groups, group actions;
RINGS: prime and maximal ideals, the radical, UFDs, PIDs, Noetherian and Artinian rings, the Hilbert Basis theorem, localization, I-adic topologies and completions, algebraic varieties;
MODULES: exact sequences, projective and injective modules, tensor products, exterior and symmetric algebras over a module, finitely generated modules, torsion, modules over a PID, Jordan and rational canonical forms for matrices, Cayley-Hamilton theorem;
FIELDS: transcendental extensions, separable and inseparable extensions, cyclotomic extensions, Kummer extensions, algebraic closure, finite fields, Galois theory;
ALGEBRAS: Morita equivalence, semi-simple rings, Wedderburn-Artin theorem, group algebras, Maschke's theorem, representation theory of groups and algebras;
CATEGORY THEORY: categories, functors, natural transformations, representable functors, adjoint functors, universal properties, limits, colimits, Yoneda's lemma.
625-626 Complex Variables (3, 3)
Prerequisite: MTH 431-MTH 432, or the equivalent
Description: Functions (analytic, entire, meromorphic, etc.) of one complex variables, conformal mappings, singularities, complex integration. Cauchy theorem, Cauchy integral formula, power series, Laurent series, calculus of residues, analytic continuation, monodromy theorem. Riemann surfaces, theorems of Liouville, Weierstrass and Mittag-Leffler. Riemann mapping theorem. Picard theorems, approximation by rational functions and polynomials.
627-628 Topology (3, 3)
Prerequisite: MTH 431-MTH 432 or the equivalent
Description: General topology: topological spaces, continuous maps, connected spaces, compact spaces. Homotopy theory: homotopy classes of maps, fundamental groups, Van Kampen's theorem, covering spaces, classification of covering spaces. Elementary manifold theory: tangent vectors, derivative of maps, transversality, Sard's theorem, differential forms, exterior derivative, de Rham cohomology. Singular homology theory: chain complex, relative homology, long exact sequence, excision, Mayer-Vietoris exact sequence. Homology of manifold: cohomology, cup and cap products, Poincaré duality, Lefschetz fixed point theorem.
629-630 Theory of Numbers (3, 3)
Prerequisite: MTH 419-MTH 420, MTH 431-MTH 432 or the equivalent
Description: Classical number theory, binomial coefficients, combinational problems, prime factorization, arithmetic functions, congruences, residue systems, linear congruences, congruences of higher degree, primitive roots, indices, quadratic reciprocity. Analytic number theory, primes, elementary estimates on sums of primes and functions of primes, estimates for sums of arithmetic functions. Selberg's theorem, prime number theorem.
631-632 Analysis (3, 3)
Prerequisite: MTH 431-MTH 432 or the equivalent
Description: Metric spaces, Baire category argument, Stone-Weierstrass theorem, Daniell integral, theory of measure, measurable functions. Lusin's theorem, Egoroff's theorem. Lebesgue integral, Fatou's lemma, convergence in measure, mean convergence, almost uniform convergence. Dominated Convergence Theorem. Riesz representation theorem, absolute continuity. Radon-Nikodym theorem, bounded variation, Lebesgue's differentiation theorem, F.T.C. for Lebesgue integral, density, approximate continuity. Radon-Nikodym theorem, bounded variation, Lebesgue's differentiation theorem, F.T.C. for Lebesgue integral, density, approximate continuity.
635-636 Differential Geometry (3, 3)
Prerequisite: Linear algebra and undergraduate analysis
Description: Analysis on manifolds, Riemannian geometry, and topics selected by the instructor.
637-638 Advanced Numerical Analysis (3, 3)
Prerequisite: Linear algebra and numerical analysis
Description: Computational problems of linear algebra: linear systems and the eigen-problem. Error analysis. Various algorithms: Givens, Jacobi, Householder for Hermitian matrices and L-R, Q-R for the non-Hermitian case as well as Jacobi-type algorithms.
639 Fourier Analysis (3)
Prerequisite: MTH 632
Description: Fourier series and integrals, convergence and summability, theorems on Fourier coefficients, uniqueness properties.
645-646 Advanced Ordinary Differential Equations (3, 3)
Prerequisite: Introductory differential equations and advanced calculus (or introductory real analysis)
Description: Existence theorems, linear and nonlinear differential equations, regular and singular boundary value problems, stability theory of linear and nonlinear systems. Liapunov's second method. Geometric theory of differential equations in the plane.
647 Integral Equations (3)
Prerequisite: Advanced calculus or introductory real and complex analysis
Description: Fredholm theory; Hilbert-Schmidt theorem; singular integral equations; Wiener-Hopf equation. Applications of integral equations to potential theory.
649-650 Partial Differential Equations (3, 3)
Prerequisite: Advanced calculus or introductory real analysis, or permission of instructor
Description: The Cauchy problem for partial differential equations, classification of second order linear partial differential equations, properties of solutions for elliptic, parabolic and hyperbolic equations, existence of solutions for elliptic partial differential equations. Topics from Fourier and Laplace transforms, potential theory, Green's functions, integral equations, Sobolev spaces, and Schwartz distributions.
713-714 Recursive Functions (3, 3)
Prerequisite: MTH 414 or MTH 514
Description: Primitive recursive functions, general recursive functions and partial recursive functions (Kleene's normal form theorem, enumeration theorem, recursive theorem, etc. ). In the second half more advanced topics will be discussed as decided by the students and the instructor.
715-716 Intuitionistic Mathematics (3, 3)
Prerequisite: MTH 414 or MTH 514
Description: Formal systems for intuitionistic predicate calculus and arithmetic and their metatheory. Various systems for intuitionistic analysis described and compared.
719-720 Advanced Algebra (3, 3)
Prerequisite: MTH 619-620, or permission of instructor
Description: Topics selected by the instructor. These may include: tensor product, exterior product, the existence of determinants, bilinear forms, Witt's theorem, Clifford algebra, special theorems, representations of finite groups, characters, theorems of Brauer, Commutative algebra, finitely generated modules over Dedekind domains (the classical ideal theory), dimensions of rings and modules. Hilbert's theorem on syzygies, the finite dimensionality of regular local rings.
725 Topics in Complex Analysis (3)
Prerequisite: MTH 625-626
Description: Topics to be chosen from: Boundary behavior of analytic functions, founded analytic functions, conformal mapping; Riemann surfaces; Potential theory and Nevanlinna theory.
726 Theory of Functions of Several Complex Variables (3)
Prerequisite: MTH 625-626
Description: Topics include fundamental properties of holomorphic functions, complex analytic manifolds, integral representations, Cousin problems.
727-728 Algebraic Topology (3, 3)
Prerequisite: MTH 519 and MTH 527-528 or the equivalent
Description: Abelian groups, simplex, complex, polyhedron, orientation of simplex, chain complex, integral homology group, structure of homology group, chain homomorphism, exact sequence, relative homology group, chain homotopy cone, simplicial subdivision. Simplicial approximation, homotopy, topological invariance of homology group. Kunneth formula, universal coefficient theorem, product complex, homotopy group, relation between homology and homology, fibre bundles, and some applications.
729 Diophantine Approximations (3)
Prerequisite: MTH 430 or consent of instructor
Description: Continued fractions, Perron's modular function, Minkowski's linear forms theorem, badly approximable numbers, approximations to algebraic numbers, inhomogeneous approximations and Kronecker's theorem, uniform distribution and Weyl's criterion, irregularities of distribution.
730 Algebraic Number Theory (3)
Prerequisite: MTH 420 and MTH 430, or consent of instructor
Description: Principal ideal rings, modules over principal ideal rings, integral rings extensions, algebraic field extension, norm, trace, discriminant, Noetherian rings, Dedekind rings. Algebraic number fields: finiteness of the class number, Dirichlet unit theorem, splitting of prime ideals in an extension field, ramification. Galois extensions of number fields. Topics in quadratic, cubic, and cyclotomic fields.
731-732 Functional Analysis (3, 3)
Prerequisite: MTH 631-632 or equivalent
Description: Banach spaces, summability, Banach limits, uniform boundedness, interior mapping theorem, graphs, Hahn-Banach theorem, Lp spaces, C[a,b], finite dimensional, weak and weak* topology. Alaoglu theorem, reflexivity and weak compactness theory. Hilbert spaces, spectral theorem for self-adjoint operators, linear topological vector spaces.
735-736 Algebraic Geometry (3, 3)
Prerequisite: Consent of instructor
Description: Commutative ring theory (including integral dependence, local rings, valuation rings, formal power series). Algebraic varieties with specialization to curves and surfaces, Riemann-Roch theorem.
743-744 Topics in Differential Geometry (3, 3)
Prerequisite: MTH 635-636 or consent of instructor
800 Thesis Guidance (1-12)
Description: Writing and submission of thesis or dissertation under the supervision of the major professor.
801 Reading and Conference (1-6)
Description: Permission of department and instructor required.
805 Colloquium (1-4)
807 Graduate Research (1-12)
813 Selected Topics in Mathematical Logic (3)
Prerequisite: Consent of instructor.
814 Seminars in Mathematical Logic (variable)
819 Selected Topics in Algebra (3)
Prerequisite: Consent of instructor.
820 Seminars in Algebra (variable)
827 Selected Topics in Topology (3)
Prerequisite: Consent of instructor.
828 Seminars in Topology (variable)
829 Selected Topics in Number Theory (3)
Prerequisite: Consent of instructor.
830 Seminars in Number Theory (variable)
831 Selected Topics in Analysis (3)
Prerequisite: Consent of instructor.
832 Seminars in Analysis (variable)
835 Selected Topics in Geometry (3)
Prerequisite: Consent of instructor.
836 Seminars in Geometry (variable)
837 Selected Topics in Numerical Analysis (3)
Prerequisite: Consent of instructor.
838 Seminars in Numerical Analysis (variable)
839 Selected Topics in Applied Mathematics (3)
Prerequisite: Consent of instructor.
840 Seminars in Applied Mathematics (variable)