hey, I am trying to pick 2 out of these 4 classes. I want to be a well rounded under grad math major. I am in my second year and have done 2 semesters of analysis, 1 abstract algebra, 1 number theory, 2 stats, 1 complex analysis. these are the choices I have narrowed my options down to. I am leaning towards topo and field theory. of course there is no reason not to take the other two later, but these courses aren't offered every semester... Field theory Algebraic and transcendental elements, Galois theory, constructions with straight-edge and compass, solutions of equations of low degrees, insolubility of the quintic equation, Sylow theorems, fundamental theorem of finite Abelian groups. Axiomatic Set theory Development of a system of axiomatic set theory, choice principles, induction principles, ordinal arithmetic including discussion of cancellation laws, divisibility, canonical expansions, cardinal arithmetic including connections with the axiom of choice, Hartog's theorem, Konig's theorem, properties of regular, singular and inaccessible cardinals. Elementary Logic Elementary development of propositional and predicate logic, including semantics and deductive systems and with a discussion of completeness, incompleteness and the decision problem. Introduction to topology Metric spaces, topological spaces, connectedness, compactness (including Heine-Borel and Bolzano-Weierstrass theorems), Cantor sets, continuous maps and homeomorphisms, fundamental group (homotopy, covering spaces, the fundamental theorem of algebra, Brouwer fixed point theorem), surfaces (e.g., Euler characteristic, the index of a vector field, hairy sphere theorem), elements of combinatorial topology (graphs and trees, planarity, coloring problems).