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Prof. Monica VanDieren Dept of Mathematics Office: John Jay 318 Email: mylastname at rmu dot edu URL: http://academics.rmu.edu/~vandieren/index.htm |
Office hours: other times not listed below by appointment Mondays 3-4 Wednesdays 11-12, 3-4 Fridays 11-12 and 1-2 |
Course Description:
The spring semester course will cover topics in number theory and abstract algebra as well as set theory and metric space theory. The main goals are to understand the foundations and structure of the real line and to gain fluency with abstract mathematical concepts and proof techniques. The course will also provide the student with an introduction to basic set theory which was invented by Cantor in order to understand the relationship between different infinite sets. His work was motivated by questions surrounding the Fourier series.
Cantor defined the notion of different sizes of infinity. He showed, by a method called diaganolization, that the size of the set of the real numbers is strictly larger than the size of the set of the natural numbers. His work led to the question of whether the continuum hypothesis is true. Roughly speaking, the continuum hypothesis says that there are no sizes of infinity strictly between the size of the real numbers and the size of the natural numbers. Paul Cohen in the 1960s developed a new technique so that he could prove that there was no way to prove the continuum hypothesis was true (or false) within the rules and axioms that mathematicians use.
Topics to be covered include basic properties of natural numbers, countable and uncountable sets, construction of the real numbers, some basic facts about the topology of the real line, cardinal numbers and cardinal arithmetic, the continuum hypothesis, well ordered sets, ordinal numbers and transfinite induction, the axiom of choice and Zorn's lemma. Topics covered in number theory include mathematical induction, divisibility algorithms, factorization methods, primes, congruences, and Diophantine equations. Topics covered in abstract algebra include binary and equivalence relations, groups and subgroups, isomorphisms and homorphisms, rings, and ideals.
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