Review of topics from Math 3351 including vector spaces, bases, dimension, matrices and linear transformations, diagonalization; the material is covered in greater depth with emphasis on theoretical aspects. The course continues with more advanced topics including Jordan and rational canonical forms of matrices and introduction to bilinear forms. Additional topics such as modules and tensor products may be included.
Prerequisite: MATH 3351.
Structural properties of basic algebraic systems such as groups, rings and fields. A special emphasis is made on polynomials in one and several variables, including irreducible polynomials, unique factorization and symmetric polynomials. Time permitting, such topics as group representations or algebras over a field may be included. Prerequisite: MATH 3351 or 4651, or instructor permission.
The study of the integers and related number systems. Includes polynomial congruences, rings of congruence classes and their groups of units, quadratic reciprocity, diophantine equations, and number-theoretic functions. Additional topics such as the distribution of prime numbers may be included. Prerequisite: MATH 3354.
This course will cover the representation theory of finite groups and other interactions between linear and abstract algebra. Topics include: bilinear and sesquilinear forms and inner product spaces; important classes of linear operators on inner product spaces; the notion of group representations; complete reducibility of complex representations of finite groups; character theory; some applications of representation theory.
Prerequisite: MATH 3351 (or 4651) and MATH 3354 (or 4652)
Solutions of polynomials, algebraic field extensions, field automorphisms, and the fundamental theorem of Galois theory. Applications include the unsolvability of the quintic, as well as ruler and compass constructions.
Prerequisite: MATH 3351 (or 4651) and MATH 4652
Geometric study of curves/surfaces/their higher-dimensional analogues. Topics vary and may include curvature/vector fields and the Euler characteristic/the Frenet theory of curves in 3-space/geodesics/the Gauss-Bonnet theorem/and/or an introduction to Riemannian geometry on manifolds. Prerequisites: MATH 2310 and MATH 3351 or instructor permission.
Examines the knotting and linking of curves in space. Studies equivalence of knots via knot diagrams and Reidemeister moves in order to define certain invariants for distinguishing among knots. Also considers knots as boundaries of surfaces and via algebraic structures arising from knots. Prerequisite: MATH 3354 or instructor permission.
Introduces abstract topological spaces. Topics include topological spaces & continuous functions/connectedness/compactness/countability/separation axioms. Rigorous proofs emphasized. Covers myriad examples, i.e., function spaces/projective spaces/quotient spaces/Cantor sets/compactifications. May include intro to aspects of algebraic topology, i.e., the fundamental group. Prerequisites: MATH 2310, MATH 3351, and MATH 3310.
This course will introduce students to mathematical research. Students will independently work with mathematical literature on a topic assigned by the instructor and present their findings in various formats (presentation, paper etc.).
Prerequisite: MATH 3310, MATH 3351 (or 4651), MATH 3354 (or 4652) and at least two of the following: APMA 3340, MATH 3340, MATH 4310, MATH 5330, MATH 4651, MATH 4652, MATH 4653, MATH 4720, MATH 4770.
This course provides a framework for the completion of a Distinguished Major Thesis, a treatise containing an exposition of a chosen mathematical topic. A faculty advisor guides a student through the beginning phases of the process of research and writing. Prerequisite: Acceptance into the Distinguished Major Program.
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