# Mathematics (MATH)

**MATH 4050 Combinatorial Set Theory (4 Credits)**

Beginning with a quick review of ZFC, the standard axioms of set theory, the course covers advanced ordinal and cardinal arithmetic and infinitary combinatorics, including Ramsey theory. Additional axioms such as the Continuum Hypothesis, Martin's Axiom, and combinatorial principles such as Diamond and their consequences for mathematics are studied. Prerequisite: MATH 3050.

**MATH 4060 Descriptive Set Theory (4 Credits)**

Descriptive Set Theory is one of the main branches of modern set theory. Set theory provides techniques for the precise study of real analysis. This course covers trees as tools for analyzing sets of real numbers, Polish spaces, the Borel hierarchy, Baire-measurability, extensions of continuous functions, separation theorems, and more. Prerequisite: MATH 3050.

**MATH 4070 Proof Theory (4 Credits)**

Hilbert-style systems, Natural deduction, (simply typed) lambda calculus, combinatory logic, the Curry-Howard correspondence, normalization, cartesian cloased categories, Sequent calculi, cut elimination and applications, structural rules; logical systems: classical, intuitionistic, relevance, linear; algebraic semantics. Recommended prerequisite: MATH 2200.

**MATH 4080 Algebraic Logic (4 Credits)**

Elements of universal algebra, lattice theory and first-order logic; elements of abstract algebraic logic (deductive systems, algebraization, deduction filters, deduction theorems, matrix semantics); sequent calculi for substructural logics, residuated lattices, structure theory for congruences and deductive filters; subvariety lattices (atomic varieties, axiomatizations of joins, translations); algebraic cut elimination; (un)decidability and finite model property. Prerequisites: MATH 3170 and either MATH 3040 or MATH 3060.

**MATH 4110 Topology (4 Credits)**

Point set topology including topological spaces, connectedness, compactness and separate axioms; preparation for advanced courses in analysis. Prerequisite: MATH 3161. Cross listed with MATH 3110.

**MATH 4120 Algebraic Topology (4 Credits)**

Fundamental groups, simplicial homology, Euler characteristic classification of surfaces, manifolds. Prerequisites: MATH 3170 and MATH 3110/4110.

**MATH 4162 Rings and Modules (4 Credits)**

Ideals, left and right R-modules, simple modules, totally decomposable modules, Wedderburn-Artin theorems, Artinian and Noetherian rings and modules, Hopkins theorem, Hilbert basis theorem, free modules, projective and injective modules, Kaplanski theorem. Prerequisites: MATH 3176 or MATH 4176.

**MATH 4163 Universal Algebra (4 Credits)**

Universal algebras, congruences, lattices, distributive lattices, modular lattices, Boolean algebras, subdirectly irreducible algebras, Mal'cev theorems, varieties, Birkhoff theorem. Prerequisites: MATH 3170 and either MATH 3040 or MATH 3060.

**MATH 4164 Galois Theory (4 Credits)**

The fundamental theorem of algebra, field extensions, ruler and compass constructions, normal and separable extensions, field automorphisms, Galois correspondence, solvability and simplicity, calculating Galois groups. Prerequisite: MATH 3176/MATH 4176 and MATH 3166/MATH 4166.

**MATH 4165 Introduction to Real Analysis II (4 Credits)**

A rigorous introduction to the analysis of functions of a real variable, including differentiation, Riemann integration, and the notions of pointwise and uniform convergence for sequences of functions. Prerequisites: MATH 3161.

**MATH 4166 Group Theory (4 Credits)**

Groups and homomorphisms, isomorphism theorems, symmetric groups and G-sets, the Sylow theorems, normal series, fundamental theorem of finitely generated abelian groups. Cross listed with MATH 3166. Prerequisite: MATH 3170.

**MATH 4168 Lie Groups and Lie Algebras (4 Credits)**

Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semisimple Lie groups; classification of semisimple Lie algebras; representation theory of compact and semisimple Lie algebras and Lie groups. Additional topics as time permits: universal enveloping algebras, symmetric spaces. Prerequisites: MATH 3161 and MATH 3170.

**MATH 4176 Rings and Fields (4 Credits)**

Rings, domains, fields; ideals, quotient rings, polynomials; PIDs, UFDs, Euclidean domains; maximal and prime ideals, chain conditions; extensions of fields, splitting fields, algebraic and transcendental extensions; brief introduction to Galois theory. Cross listed with MATH 3176. Prerequisite: MATH 3170 or equivalent.

**MATH 4181 Loop Theory (4 Credits)**

Quasigroups, loops, latin squares, 3-nets, isotopy, multiplication groups, inner mapping groups, nuclei, commutant, center, associator subloop, inverse properties, power-associative loops, Bruck loops, Bol loops, Moufang loops, octonions. Prerequisites: MATH 3166 or MATH 4166.

**MATH 4260 Metric Spaces (4 Credits)**

Metric spaces and continuous functions; completeness and compactness; examples including norm spaces; pointwise and uniform convergence; Baire Category Theorem. Cross listed with MATH 3260. Prerequisite: MATH 3161 or equivalent.

**MATH 4270 Hilbert Spaces (4 Credits)**

Schwarz and triangle inequalities, Reisz lemma, subspaces and othogonal projections, orthonormal bases, spectrum of bounded linear operators, compact, self-adjoint, normal and unitary operators, spectral theorem and, if time permits, unbounded operators. Also, if time permits, applications to partial differential equations, physics and engineering. Prerequisites: MATH 3260 or MATH 4260 or MATH 3110 or MATH 4110.

**MATH 4280 Measure Theory and Applications (4 Credits)**

Definition of Measure spaces; Lebesgue measure; limit theorems; Raydon-Nikodym Theorem; introduction to L_p spaces.
Prerequisite: (MATH 3260 with a minimum grade of D- or MATH 4260 with a minimum grade of C-) or (MATH 3110 with a minimum grade of D- or MATH 4110 with a minimum grade of C-).

**MATH 4290 Dynamical Systems (4 Credits)**

Topological and measure theoretic dynamical systems; properties and invariants of systems; symbolic dynamics; Ergodic Theorems; applications. Prerequisites: MATH 3110/4110 or MATH 3260/4260.

**MATH 4300 Graduate Seminar (1-4 Credits)**

Students research a topic of their choosing with the aid of a faculty member, and then prepare and present a formal lecture on the subject. Prerequisite: graduate standing or consent of the instructor.

**MATH 4400 Differential Geometry (4 Credits)**

Planar and spatial curves, global properties of curves, surfaces in three dimensions, the first fundamental form, curvature of surfaces, Gaussian curvatures, geodesics, Theorema Egregium, hyperbolic geometry. Prerequisites: MATH 3170 and either MATH 3110/4110 or MATH 3260/4260.

**MATH 4501 Functional Analysis (4 Credits)**

Advanced topics in structure of linear spaces; Banach spaces; Hahn-Banach Theorem and Duality; Uniform Boundedness Theorem; Open Mapping and Closed Graph Theorems; Stone-Weierstrass Theorem; Topics in Hilbert Spaces. Prerequisite: MATH 4280.

**MATH 4700 Special Topics in Mathematics (1-4 Credits)**

**MATH 4701 Combinatorial Algorithms (4 Credits)**

Basic enumeration techniques; representations of combinatorial objects; algorithms for searching, sorting, generating combinatorial objects, graph algorithms. Prerequisites: MATH 3701 or MATH 3710.

**MATH 4705 Special Topics Applied Math (1-5 Credits)**

Varying selected advanced topics in mathematics, depending on student demand. Possible alternatives include of variations, partial differential equations, algebraic topology, differential manifolds, special functions.

**MATH 4991 Independent Study (1-10 Credits)**

Cannot be arranged for any course that appears in course schedule for that particular year.

**MATH 4992 Directed Study (1-10 Credits)**

**MATH 4995 Independent Research (1-10 Credits)**

Research projects undertaken in conjunction with a faculty member.

**MATH 5000 Doctoral Seminar (3 Credits)**

Techniques, methods used in mathematical, computing research. Includes proofs, bibliographic searching, writing styles, what constitutes an acceptable dissertation.

**MATH 5991 Independent Study (1-10 Credits)**

Cannot be arranged for any course that appears in the regular course schedule for that particular year.

**MATH 5995 Independent Research (1-10 Credits)**

Research leading to a dissertation.