David Dorman

Calculus II
A continuation of MATH 0121, may be elected by first-year students who have had an introduction to analytic geometry and calculus in secondary school. Topics include a brief review of natural logarithm and exponential functions, calculus of the elementary transcendental functions, techniques of integration, improper integrals, applications of integrals including problems of finding volumes, infinite series and Taylor's theorem, polar coordinates, ordinary differential equations. MATH 0121 or equivalent, or by placement) 4 hrs. lect/disc.

Linear Algebra
Matrices and systems of linear equations, the Euclidean space of three dimensions and other real vector spaces, independence and dimensions, scalar products and orthogonality, linear transformations and matrix representations, eigenvalues and similarity, determinants, the inverse of a matrix and Cramer's rule. (MATH 0121 or equivalent, or by placement) 3 hrs. lect./disc.

Multivariable Calculus
The calculus of functions of more than one variable. Introductory vector analysis, analytic geometry of three dimensions, partial differentiation, multiple integration, line integrals, elementary vector field theory, and applications. (MATH 0122 or equivalent. Not open to students who have taken MATH 0224.) 3 hrs. lect./disc.

Elementary Number Theory
Divisibility and prime factorization. Congruences; the theorems of Lagrange, Fermat, Wilson, and Euler; residue theory; quadratic reciprocity. Diophantine equations. Arithmetic functions and Mobius inversion. Representation as a sum of squares. (MATH 0122 or equivalent)

Abstract Algebra
Groups, subgroups, Lagrange's theorem, homomorphisms, normal subgroups and quotient groups, rings and ideals, integral domains and fields, the field of quotients of a domain, the ring of polynomials over a domain, Euclidean domains, principal ideal domains, unique factorization, factorization in a polynomial ring. (MATH 0200) 3 hrs. lect./disc.

Complex Analysis
An introduction to functions of a complex variable. Mappings of the complex plane, analytic functions, Cauchy Integral Theorem and related topics. (MATH 0200 and MATH 0223, or MATH 0224) 3 hrs. lect./disc.

Quadratic Number Fields
In this senior seminar we will explore the algebra and arithmetic of quadratic extensions of the rational numbers. We will study the rings of integers in these extensions, the structure of the unit group in these rings and unique factorization of ideals in Dedekind domains. We will investigate fractional ideals, splitting of primes, the class group and the finiteness of the class number. Some of the ideas and topics introduced are methods, p-adic methods, cyclotomic theory, Dirichlet’s Units Theorem, Quadratic and Biquadratic Reciprocity and quadratic forms. Using these ideas as a springboard students will investigate a topic of their choosing and write their thesis.