MATH 410-1,2,3 Analysis (1) (1) (1) First and Second Quarters: Real analysis. Topological spaces, metric spaces. Lebesque measure and integration. Function spaces, including Banach and Hilbert spaces. Elementary functional analysis. Weak convergence. Third Quarter: Complex analysis. Holomorphic functions, Cauchy's theorem, power series, harmonic functions, conformal mapping, analytic continuation.

MATH 413-1,2,3 Functions of a Complex Variable (1) (1) (1) Holomorphic functions: theorems of Cauchy, Morera, and Rouché residue and open mapping theorems; harmonic and entire functions; analytic continuation; conformal mapping. Schlicht functions, functions of several complex variables, Hp spaces, and complex manifolds.

MATH 415-1, 2 Functional Analysis (1) (1) Topological groups and topological vector spaces; Banach spaces, linear functionals, and operators; applications to functional equations.

MATH 420-1, 2, 3 Partial Differential Equations (1) (1) (1) Introduction to basic differential equations, with emphasis on the theory of partial differential equations. Prerequisites: advanced calculus and linear algebra or permission of instructor.

MATH 425 - 1, 2, 3 Partial Differential Equations II (1) (1) (1) Nonlinear elliptic differential equations, nonlinear hyperbolic differential equations, pseudodifferential operators, and other topics.

MATH 428 Geometric Measure Theory and Applications (1) General measure theory, Hausdorff measure, area and co-area formulas, Sobolev functions, BV functions and set of finite perimeter, Gauss-Green theorem, differentiability and approximation, applications.

MATH 429 Fourier Analysis (1) A short overview of classical Fourier analysis on the circle. Selected topics about Fourier analysis on the line and in Euclidean space. Prerequisite: permission of instructor.

MATH 430-1,2,3 Dynamical Systems (1) (1) (1) Qualitative theory of differentiable dynamical systems, emphasizing global properties such as structural stability theorems.

MATH 435 Ergodic Theory (1) Introduction to abstract ergodic theory, focusing on the asymptotic behavior of measure preserving transformations. Topics to be covered include: measure preserving transformations and flows, convergence theorems, recurrence properties, isomorphism invariants, and applications to problems in number theory, probability, and combinatorics.  Prerequisites:  MATH 412-1.

MATH 438-1,2,3 Interdisciplinary Nonlinear Dynamics (1) First quarter: Example-oriented survey of nonlinear dynamical systems, including chaos, combining numerical, analytical and geometrical approaches to differential equations. Second and third quarters: Interdisciplinary theoretical, computational and experimental projects involving complex systems in science and engineering directed by cross-disciplinary faculty teams.

MATH 440-1, 2, 3 Differential Geometry (1) (1) (1) MATH 440-1 Manifolds and differentiable maps, the tangent bundle, orientations, integration of differential forms, MATH 440-2 Riemannian geometry, vector bundles and connections, MATH 440-3 Sheaves, Morse theory, characteristic classes.  Prerequisites:  For MATH 440-2:  MATH 440-1; For MATH 440-3:  MATH 440-2.

MATH 444 Hamiltonian Dynamics and Symplectic Geometry (1) Symplectic structure and cotagent bundle.  Hamiltonian flow and their invariants.  Integrable systems and stability.  Lagrangian intersection theory and symplectic fixed points theorems.  Arnold conjecture on n-torus.

MATH 445 Complex Manifolds (1) Hodge theory, complex structures, Kahler manifolds, Hodge decomposition, symplectic manifolds and other topics.

MATH 448-1,2,3 Geometry and Topology (1) (1) (1) This is a working seminar for students with interests in geometry, topology, and related fields. Its primary aim is to introduce students to research subjects of current interest to faculty members in these areas. Potential topics include: topological field theory, noncommutative geometry, derived algebraic geometry.

MATH 450-1,2 Probability (1) (1) Probability spaces, random variables, distribution functions, conditional probability, laws of large numbers, and central limit theorem. Random walk, Markov chains, martingales, and stochastic processes.

MATH 455-1,2,3 Stochastic Analyses (1) (1) (1) Definition and properties of standard Brownian motion. Stochastic Integration and stochastic differential calculus, with applications to diffusion processes.

MATH 460-1,2,3 Algebraic Topology I (1) (1) (1) MATH 460-1 Fundamental group and covering spaces, MATH 460-2 Simplical, singular, and cellular (co-) homology; universal coefficient and Kuenneth theorems, MATH 460-3 Cohomology rings and Poincare duality; Thom Isomorphism and characteristic classes.  Prerequisites:  For MATH 460-2:  MATH 460-1; For MATH 460-3:  MATH 460-2.

MATH 465-1,2,3 Algebraic Topology II (1) (1) (1) Cohomology theories and operations, homotopy and obstruction theory, and CW complexes; spectral sequences. Multiple registrations allowed.

MATH 468 Homological Algebra (1) Exact sequences, Ext and Tor, and homological dimensions.

MATH 470-1,2,3 Algebra (1) (1) (1) Free, permutation, solvable, simple, and linear groups. Actions of groups on sets; Sylow theorems. Rings and modules: polynomials and power series, Euclidean domains, PIDs, UFDs, and free and projective modules. Field and Galois theory. Extensions: algebraic, transcendental, normal, and integral. Splitting fields. Wedderburn theory. Commutative algebra: prime ideals; localization.

MATH 477 Commutative Algebra (1) Research in commutative algebra: theory of depth (regular sequences, Koszul complexes), dimension theory, completions, Hilbert functions, Cohen-Macaulay modules, excellent rings, Hensel rings, and minimal resolutions. Prerequisites: MATH 470-1,2,3 or equivalent.

MATH 478 Representation Theory (1) Topics in the representation theory and cohomology of finite and infinite groups, including compact and non-compact Lie groups.

MATH 482-1,2 Algebraic Number Theory (1) (1) 1. The theory of global and local fields; various special topics. 2. Abelian Galois extensions of algebraic number fields (class field theory). Complex multiplication, other examples, and relations with geometry.

MATH 483-1,2,3 Algebraic Geometry (1) (1) (1) Introduction to classical and scheme theoretic methods of algebraic geometry. Algebraic vector bundles, sheaf cohomology, the Riemann-Roch theorem for curves, and intersection theory.

MATH 484 Lie Theory (1) Topics in the theory of Lie algebras and Lie groups including classification.

MATH 485 - 1, 2 Modular Forms (1) (1) First quarter: introduction to the theory of modular forms.  Congruence subgroups of SL (2,Z), the definitions of modular functions and modular forms, Fourier expansions, Hecke operators, theta functions, modular curves.  Second quarter: possible topics include the connections between modular forms and the representation theory of GL (2), automorphic forms, Galois representations attached to modular forms, and the relations with algebraic geometry and other areas of mathematics.

MATH 486-1,2,3 Algebraic K-Theory (1) (1) (1) Classical algebraic K-theory. Functors K0 and K1; origins in and relations with topology; congruence subgroup problem; techniques of computation: exact sequences, localization, resolution, and devissage; polynomial and related extensions; higher K- theories: Karoubi-Villamayor, Quillen.

MATH 495 Statistical Phenomena in the Theory of Networks (1) This interdisciplinary course combines graph theory and probability theory to develop a rigorous foundation for the study of network-related problems.

MATH 499 Independent Study (1 or 2) Permission of instructor and department required. May be repeated for credit.

Seminars MATH 511-MATH 519 cover topics not regularly offered in other courses.

MATH 511-1,2,3 Topics in Analysis (1) (1) (1)

MATH 512-1,2,3 Topics in Partial Differential Equations (1) (1) (1)

MATH 513-1,2,3 Topics in Dynamical Systems (1) (1) (1)

MATH 514-1,2,3 Topics in Geometry (1) (1) (1)

MATH 516-1,2,3 Topics in Topology (1) (1) (1)

MATH 517-1,2,3 Topics in Algebra (1) (1) (1)

MATH 518-1, 2, 3 Topics in Number Theory (1) (1) (1)

MATH 519 -1, 2, 3 Topics in Representation Theory (1) (1) (1)

MATH 580 Seminar in College Teaching (0) A weekly two-hour seminar introducing the technique, philosophy, and practice of teaching undergraduate mathematics. Student presentations are critiqued by fellow students, as well as a senior faculty member.

MATH 590 Research (1, 2, or 3) Independent investigation of selected problems pertaining to thesis or dissertation. May be repeated for credit.

300 Level Math Courses

MATH 300 Foundations of Higher Mathematics (1) Sets, logic, relations, elementary number theory, cardinality, and real numbers.

MATH 310-1,2,3 Probability and Statistics (1) (1) (1) First Quarter: Discrete probability spaces, random variables, expected value, combinational problems, special distributions, independence, and conditional probability. Second Quarter: Integrating density functions, convolutions, law of large numbers, central limit theorem, random walk, and stochastic processes. Third Quarter: Elementary decision theory, estimation, testing hypotheses, Bayes procedures, linear models, and nonparametric procedures.

MATH 318 Introduction to Optimization (1) Mathematical methods and concepts underlying continuous nonlinear optimization theory (determining the maxima and minima of a function on an allowable set of points). Examples drawn from economics, business, and engineering.

MATH 320-1,2,3 Introduction to Real Analysis (1) (1) (1) Sets, functions, limits, and properties of the real number system. Metric spaces. Foundations of differential and integral calculus: the Riemann integral and infinite series. Lebesque integration. Fourier series. Primarily for undergraduate students; open to graduate students only with permission of department.

MATH 321-1,2,3 MENU: Real Analysis (1) (1) (1) Metric space topology, properties of Euclidean spaces, limits and continuity, differentiation and integration, sequences and series; inverse and implicit function theorems. Lebesgue integration with applications. Credit not granted for both MATH 310-1 and 312-1 or for both 310-2 and 312-2. Prerequisite: permission of department.

MATH 325 Complex Variables for Applications (1) Complex numbers, functions of a complex variable, the theory of analytic functions, series development, analytical continuation, contour integration, and conformal mapping.

MATH 330-1,2,3 Introduction to Modern Algebra (1) (1) (1) First Quarter: Elementary theory of groups, rings, and fields; applications to the ring of integers and polynomial rings. Second and Third Quarters: Linear algebra done abstractly: vector spaces, bilinear forms, canonical forms, and modules.

MATH 331-1,2,3 MENU: Algebra (1) (1) (1) 1. Groups and their structure, including the Sylow theorems; elementary ring theory; polynomial rings. 2. Basic field theory; Galois theory. 3. Module theory, including application to canonical form theorems of linear algebra.

MATH 334 Linear Algebra for Applications (1) Solution of linear equations; number of independent solutions. Vector spaces. Eigenvalues and eigenvectors, diagonalization of a matrix, and minima for quadratic forms. Symmetric, orthogonal, hermitian, and unitary matrices. Inner products. Least squares. Applications to science, engineering, and economics.

MATH 336-1,2 Introduction to the Theory of Numbers (1) (1) First Quarter: Divisibility and primes, congruences, quadratic reciprocity, and diophantine problems. Second Quarter: Additional topics in analytic and algebraic number theory.

MATH 340 Geometry (1) Axiomatics for Euclidean geometry, non-Euclidean geometry. Projective geometry. Introduction of coordinate system from the axioms. Quadrics. Erlangen program. Introduction to plane algebraic curves. Prerequisite: permission of instructor.

MATH 342 Introduction to Differential Geometry (1) Curves and surfaces in three-dimensional space. Prerequisite: permission of instructor.

MATH 344 Introduction to Topology (1) Point-set topology. Prerequisites: MATH 308 and MATH 310-1 (may be corequisite).

MATH 351 Fourier Series and Boundary Value Problems (1) Expansion in orthogonal functions with emphasis on Fourier series. Applications to the solution of partial differential equations arising in physics and engineering.

MATH 353 Differential Equations (1) Intermediate course in differential equations. Linear systems, nonsingular boundary value problems, theory of periodic solutions, stability theory, asymptotic expansions, and special functions of mathematical physics.

MATH 354-1,2 Chaotic Dynamical Systems (1) (1) 1. Chaotic phenomena in deterministic discrete dynamical systems, primarily through iteration of functions of one variable. 2. Iteration of functions of two and more variables, including the study of the horseshoe map, attractors, and the Henon map. Complex analytic dynamics, including the study of the Julia set and Mandelbrot set.

MATH 364 Game Theory (1) Topics in game theory: noncooperative games, matrix games, optimal strategies, and cooperative games.

MATH 366-1,2 Mathematical Models in Finance (1) (1) Basic financial concepts; interest theory; financial derivatives; discreet probability theory; binomial tree model; normal distribution; random walk; Brownian motion; Ito's formula; Black-Scholes formula.

MATH 368-0 Introduction to Optimization (1) Mathematical ideas and theory of continuous optimization.  First and second order derivative conditions.  Kuhn-Tucker Theorem.  Convex structures.  Kakutani Fixed Point Theorem for correspondences with applications to Nash equilibria in game theory.  Dynamic programming.  Examples drawn from economics.  Prerequisites:  MATH 300 or graduate standing.  Math 320-1 or 321-1 is useful.

MATH 370 Mathematical Logic (1) Mathematical formulation and rigorous discussion of logical systems, particularly the propositional calculus and the functional calculi of first and second order Well-informed formulae, formal languages, proofs, tautologies, effective procedures, deduction theorems, and axiom schemata.

MATH 374 Theory of Computability and Turing Machines (1) Algorithms, computability, decidability, and enumerability; formal replacements and Church's thesis. Turing machines, primitive recursive functions, mu-recursive functions, and recursive functions. Undecidable predicates; the undecidability and incompleteness of arithmetic.

MATH 395 Statistical Phenomena in the Theory of Networks (1) This interdisciplinary course combines graph theory and probability theory to develop a rigorous foundation for the study of network-related problems.

Related Course in the Department of Chemical Engineering CHEM ENG 438-1,2,3 Interdisciplinary Nonlinear Dynamics (1) (1) (1)

Related Course in the Department of Electrical and Computer Engineering ECE 438-1,2,3 Interdisciplinary Nonlinear Dynamics (1) (1) (1)

Related Course in the Department of Engineering Science and Applied Mathematics ES APPM 438-1,2,3 Interdisciplinary Nonlinear Dynamics (1) (1) (1)

Related Course in the Department of Mechanical Engineering MECH ENG 438-1,2,3 Interdisciplinary Nonlinear Dynamics (1) (1) (1)