MATHEMATICS

201–2	Algebra I–II (3–3)	Abrams
	Group theory including symmetric groups, free abelian groups, finitely generated abelian groups, Sylow theorems, solvable groups. Factorization in commutative rings, rings of polynomials, chain conditions, semisimple rings, Wedderburn–Artin theorems, Galois theory.
203	Algebra III (3)	Abrams
	An extension of the material of Math 201–2, including Frobenius' theorem on associative division algebras, the Hurwitz problem on composition of forms, valuation theory, formally real fields, rings without finiteness conditions, elements of homological algebra with applications.
206	Topics in Algebra (3)	Abrams, Schmitt
	Topics chosen from Lie groups and Lie algebras, non-associative algebras, abelian groups, classical groups, algebraic number theory, representation theory, algebraic geometry, and ring theory. Prerequisite: Math 201–2. May be repeated for credit with permission.
211	Complex Analysis (3)	Conway, Junghenn
	Topology of the complex plane; complex differentiation and integration; Cauchy's theorem and its consequences; Taylor and Laurent series; classification of singularities; residue theory; conformal mapping; the Riemann mapping theorem. Prerequisite: Math 139 or equivalent.
214	Measure and Integration Theory (3)	Conway, Robinson, Yi
	Lebesgue measure and integration in abstract spaces. Probability measures. Absolute continuity, the Radon–Nikodym theorem, measures on product spaces, and the Fubini theorem. LP spaces and their properties. Prerequisite: Math 139 or equivalent.
215	Introduction to Functional Analysis (3)	Conway, Junghenn, Robinson
	Topological and metric spaces; Tychonoff theorem; Banach spaces; linear functionals and operators; Hahn–Banach, closed graph, and open-mapping theorems; uniform boundedness; Hilbert spaces; eigenvalues, projections. Prerequisite: Math 214 or equivalent.
216	Topics in Real and Functional Analysis (3)	Conway, Junghenn, Yi
	Possible topics include Banach algebras, function algebras, spectral theory for bounded and unbounded operators, harmonic analysis on topological groups and semigroups, topological vector spaces and operator algebras. Prerequisite: permission of instructor. May be repeated for credit with permission.
217	Ordinary Differential Equations (3)	Robinson
	Existence and uniqueness of solutions, continuity and differentiability of solutions with respect to initial conditions. Properties of linear systems, phase portraits, planar systems and Poincaré–Bendixson theory. Prerequisite: Math 140.
219	Partial Differential Equations (3)	Baginski
	Classical techniques for the solution of linear partial differential equations. Laplace's equation, Poisson's equation, heat equation, and wave equation. Existence and uniqueness of solutions. Maximum principles. Separation of variables, Fourier series, eigen function expansions, and Green's functions. Prerequisite: Math 140 or permission of instructor.
221	Modern Partial Differential Equations (3)	Baginski
	Emphasis on modern theory and analytical techniques applied to the solution of partial differential equations. Topics include Sobolev spaces, generalized solutions, strong solutions and regularity; Sobolev imbedding theorem; Rellich–Kondrachov theorem; Leray–Schauder fixed-point theorems; nonlinear eigenvalue problems. Prerequisite: Math 219 or permission of instructor.
222	Introduction to Numerical Analysis (3)	Gupta
	Computer arithmetic and round-off errors. Solution of linear and nonlinear systems. Interpolation and approximations. Numerical differentiation and integration. Eigenvalues and eigenvectors. Prerequisite: Math 32 and 124 and knowledge of a programming language.
223	Numerical Solution of Ordinary and Partial Differential Equations (3)	Gupta
	Initial and boundary value problems for ordinary differential equations. Error propagation, convergence and stability. Finite difference and finite element methods for partial differential equations. Prerequisite: Math 142 and knowledge of a programming language.
225	Ergodic Theory (3)	Robinson, Yi
	Ergodicity, mixing, the K-property and the Bernoulli property. Poincaré recurrence, the Rohlin lemma, the ergodic theorem, and entropy theory. Additional topics from isomorphism theory, spectral theory, the theory of joinings, and coding theory. Prerequisite: Math 214 or permission of instructor.
226	Dynamical Systems and Chaos (3)	Robinson, Yi
	Linear and nonlinear systems, flows, Poincaré maps, structural stability. Examples of chaotic systems in the physical sciences. Local bifurcations, center manifold theory, normal forms, the averaging theorem. Hyperbolic invariant sets, strange attractors, the Smale horseshoe, symbolic dynamics. Prerequisite: Math 124 and 140 or permission of instructor.
231	Topics in Applied Mathematics (3)	Baginski
	Possible topics include, but are not limited to, the calculus of variations, control theory, nonlinear partial differential equations, and mathematical programming. May be repeated for credit with permission.
232	Topics in Numerical Analysis (3)	Gupta
	Numerical methods and software. Introductions to the methods, tools, and ideas of numerical computation. Problem solving using standard mathematical software. Interpolation; linear and nonlinear equations. Differential equations. Prerequisite: Math 142; knowledge of a programming language.
261	Combinatorics (3)	Bonin, Schmitt
	An introduction to fundamental methods and current research problems in partially ordered sets and enumeration. Prerequisite: undergraduate modern algebra and linear algebra or permission of instructor.
262	Graph Theory (3)	Ullman
	Graphical enumeration, factors, planarity and graph coloring, algebraic graph theory, extremal graph theory, applications. Prerequisite: undergraduate modern algebra and linear algebra or permission of instructor.
263	Topics in Combinatorial Mathematics (3)	Bonin, Ullman, Schmitt
	Topics selected from a wide range of research subjects in combinatorics, its relations with other areas of mathematics, and applications. Recent selections have included matroid theory, topological methods in ordered sets, algebraic methods in combinatorics, fractional graph theory, combinatorics of polytopes, the symmetric group. May be repeated for credit with permission.
271	Mathematical Logic (3)	Harizanov, Moses
	Model theory: the relation between a formal language (syntax) and its interpretations (semantics). Consistency, completeness, and compactness. Tarski's theorem on the inexpressibility of truth. Godel's incompleteness theorem and its impact on mathematics.
272	Topics in Logic (3)	Harizanov, Moses
	Topics selected from a broad spectrum of areas of logic and applications, based on students' suggestions and interests. May be repeated for credit with permission.
281	General Topology (3)	Rong, Przytycki, Shumakovitch
	Topological spaces, bases, open sets and closed sets; continuous maps and homeomorphisms; connectedness and compactness; metric topology, product topology and quotient topology; separation axioms; covering spaces and fundamental groups.
282	Algebraic Topology (3)	Rong, Przytycki
	Fundamental groups and the Van Kampen theorem; simplicial complexes, simplicial homology, and Euler characteristic; singular homology, Mayer–Vietoris sequences. Topics may include cohomology, cup products, and Poincaré duality; classification of surfaces; knots and their fundamental groups. Prerequisite: Math 281 or permission of instructor.
289	Topics in Topology (3)	Rong, Przytycki, Shumakovitch
	Topics may include hyperbolic structures on surfaces and 3-manifolds; knot theory; topology of 3-manifolds; topology of 4-manifolds. Prerequisite: Math 282 or permission of the instructor. May be repeated for credit with permission.
295	Reading and Research (arr.)	Staff
	May be repeated for credit.
398	Advanced Reading and Research (arr.)	Staff
	Limited to students preparing for the Doctor of Philosophy general examination. May be repeated for credit.
399	Dissertation Research (arr.)	Staff
	Limited to Doctor of Philosophy candidates. May be repeated for credit.