Groups and rings and their structure. Sylow theorems. Modules. History and applications.
Polynomial Noetherian rings and ideals. Fields and Galois theory. Structure of fields. History and applications.
A broad yet deep survey of current topics in combinatorics and graph theory essential for teachers K-16, including applications to probability, coding theory, sorting and matching algorithms and optimization.
Analytic and meromorphic functions in the complex plane. Integration, conformal mapping and advanced topics.
Analysis of functions of several variables, unifying and extending ideas from calculus and linear algebra. Includes the implicit function theorem and Stokes' Theorem.
Abstract spaces, Lebesgue measure, continuity, integration and differentiation theorems, Baire category.
The notion of proof, first order logic, set theory, ordinals, cardinals and an overview of the most important recent results in the field.
A survey of topics in arithmetic and analytic number theory, such as Eulers' function, quadratic reciprocity, continued fractions and the distribution of prime numbers.
Topics from various fields of mathematics, for example, algebraic topology, functional analysis, Lie groups and algebras or nonlinear analysis.
Four hours of credit for doctoral dissertation proposal research must be earned in partial fulfillment of requirements before admission to candidacy.
Doctoral Dissertation. S/U graded.