A broad introduction to Mathematical Logic. The notions of logical consequence and derivation are introduced in the settings of propositional and...

Topology of Euclidean spaces, continuity, norms, completeness. Contraction mapping principle. Fourier series. Various applications, for example, to...

Complex numbers, Cauchy-Riemann equations, analytic functions, conformal maps and applications to the solution of Laplace's equation, contour...

Rings, ideals, factor rings, homomorphisms, finite and infinite fields, polynomials and roots, field extensions, algebraic numbers, and applications,...

Groups, permutation groups, subgroups, homomorphisms, symmetry groups in 2 and 3 dimensions, direct products, Polya-Burnside enumeration. [Note: PMATH...

An elementary approach to the theory of numbers; the Euclidean algorithm, congruence equations, multiplicative functions, solutions to Diophantine...

Elementary properties of rings, polynomial rings, Gaussian integers, integral domains and fields of fractions, homomorphisms and ideals, maximal...

Elementary properties of groups, cyclic groups, permutation groups, Lagrange's theorem, normal subgroups, homomorphisms, isomorphism theorems and...

Normed and metric spaces, open sets, continuous mappings, sequence and function spaces, completeness, contraction mappings, compactness of metric...

Analytic functions, Cauchy-Riemann equations, Goursat's theorem, Cauchy's theorems, Morera's theorem, Liouville's theorem, maximum modulus principle,...

An introduction to affine, projective and non-Euclidean forms of geometry. Conic sections in the projective plane. Inversion in circles. Theorems of...

An introduction to local differential geometry, laying the groundwork for both global differential geometry and general relativity. Submanifolds of...

Relations, functions, well-orderings, Schroder-Bernstein theorem, recursion, axiom of choice and equivalents, ordinals, cardinals, continuum...

The mathematics of iterated functions, properties of discrete dynamical systems, Mandelbrot and Julia sets. [Note: Programming experience on one...

Prereq: Not open to General Mathematics students

The concepts of formal provability and logical consequence in first order logic are introduced, and their equivalence is proved in the soundness and...

Model theory: the semantics of first order logic including the compactness theorem and its consequences, elementary embeddings and equivalence, the...

An introduction to elementary and analytic number theory; primitive roots, law of quadratic reciprocity, Gaussian sums, Riemann zeta-function,...

An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine...

Normal series, elementary properties of solvable groups and simple groups, algebraic and transcendental extensions of fields, adjoining roots,...