Mathematics

This course offers a study of complex analysis. Topics include: holomorphic functions; analytic functions-- power series and elementary functions; complex integration-- Cauchy's integral formula and applications; the residue theorem and applications-- evaluation of integrals and series; conformal maps.

Prerequisites: Linear Algebra, Differential Calculus, Integral Calculus, Vector Calculus.

The course discusses integration and differentiation in a mathematically rigorous manner. It also discusses the series of functions and their convergence.

The course covers the basic model theory and proof theory of 1st order languages, the Gödel Completeness Theorem and the Godel Incompleteness Theorems characterizing the non-provability of the consistency of a formal system within that system. These theorems are the foundations of 20th century logic.

This course looks at problems that are associated with discrete rather than continuous situations. So the nature of the problems is quite distinct from those that are considered in a calculus paper because the important underlying set is the integers rather than the sets of real or complex numbers. The curriculum includes a selection from the following topics: combinatorics, counting techniques, logic, graph theory, set theory, relations, number theory. There will be an emphasis on both proof techniques and practical algorithms.

This course introduces and develops some of the basic ideas in the areas of Combinatorics and Number Theory. Topics include: mathematical induction, permutations and combinations, counting arguments, modular arithmetic, Euclidean algorithm, Fermat's and Euler's theorems, fundamental theorem of arithmetic, systems of linear congruences, and the Chinese remainder theorem. Students learn to recognize, read, and use standard mathematical symbols and notation. Students learn to ask pertinent questions, to decide which questions are relevant, answerable, and so on. Students gain an understanding of the reasoning behind any methods or procedures they use and are able to demonstrate that understanding. Students also learn to produce examples themselves, in order to illustrate a definition, show a method, or test boundaries of an idea.