Higer Mathematics Program 2017-18

I was the instructor of this course from August 2017 to February 2018.

Lecture Hours:

Friday & Saturday (10.00 PM - 11.30 PM)

Links:

• Course Folder
• Course Page
• Classroom

Grading Scheme (Monthly)

Quizzes and assignments are posted on every lecture day. The assignments are usually due in 7 days. There is also a long test at the end of every month.

• Attendance: 10 points x 8 = 80 points
• Quizzes: 15 points x 8 = 120 points
• Assignments: 25 points x 8 = 200 points
• Test: 100 points x 1 = 100 points

TOTAL: 500 points

This grading scheme is approximate and the individual points structure may be tweaked.

References:

• Bartle, Sherbert - Introduction to Real Analysis
• Rudin - Principles of Mathematical Analysis
• Apostol - Mathematical Analysis
• Apostol - Calculus II
• Hoffman & Kunze - Linear Algebra
• Artin - Algebra
• Dummit & Foote - Abstract Algebra
• Munkres - Topology
• Sarason - Complex Function Theory
• Simmons - Differential Equations with Applications & Historical Notes
• Hoel, Port, Stone - Introduction to Probability Theory
• Bona - A Walk Through Combinatorics

Links:

• Ramsey number
• Simpson's paradox

Syllabus

• M301 Set Theory: Basics, equivalence relations & partitions, order, well-ordering, countable & uncountable sets, axiom of choice, Zorn's lemma.
• Real Analysis:
  • M311 Analysis I (Single Variable Calculus): R, completeness of R, sequences - convergence & divergence, monotone sequences, Cauchy sequences, Bolzano-Weierstrass theorem; series. Functions - limits, continuity, uniform continuity, differentiability, Taylor expansions, Riemann integration.
  • M312 Analysis II (Multi-variable Calculus & Point Set Topology): Limits, continuity, differentiability - partial derivatives, chain rule, directional derivatives, total derivative, Jacobian; integration - line integrals, multiple integrals. Metric spaces, connected sets, compact sets, perfect sets, Heine-Borel theorem, completeness.
  • M313 Analysis III (Vector Calculus): Vector fields, gradient, curl, divergence, Green's theorem, Stokes' theorem, Gauss' divergence theorem. Sequences and series of functions, Weierstrass approximation.
• M321 Linear Algebra: Matrices, systems of linear equations, determinants, vector spaces, subspaces, basis, dimension, direct sum, diagonalisation of matrices, triangular forms, linear transformations, characteristic polynomial, eigenvalues, eigenvectors, nilpotent matrices, dot products, angle, rotations, orthogonal matrices.
• Abstract Algebra:
  • M331 Algebra I (Group Theory): Groups, subgroups, cyclic groups, dihedral groups, homomorphisms, cosets, normal subgroups, product groups, quotient groups, isomorphism theorems. Group actions, conjugacy classes, Sylow theorems.
  • M332 Algebra II (Ring Theory): Rings, ideals, maximal ideals, prime ideals, quotient rings, integral domains, Chinese remainder theorem, polynomial rings, Euclidean domain, PID, UFD, field of fractions, Gauss' lemma, irreducibility criteria.
  • M333 Algebra III (Field Theory): Field extensions, algebraic extensions, finite fields, straight edge and ruler constructions, splitting fields, algebraic closure, separable & inseparable extensions, cyclotomic polynomials & extensions.
• M341 Topology: Topological spaces, basis for a topology, product topology, closed sets and limit points, continuous functions, Hausdorff spaces, metric topology, quotient topology, connectedness, compactness.
• M351 Complex Analysis: Complex numbers, holomorphic functions, Cauchy-Riemann equations, integration, zeroes of analytic functions, Cauchy formulas, maximum modulus theorem, open mapping theorem, Louville’s theorem, poles & singularities, residues & contour integration, conformal maps, Rouche’s theorem, Morera’s theorem.
• M361 Ordinary Differential Equations: First order equations, second order linear equations, Picard's theorem, systems of first order equations, power series solutions.
• M371 Differential Geometry
• M381 Probability Theory (Discrete): Combinatorial probability, independence, conditional probability, Bayes' theorem, expectation, binomial distribution, Poisson distribution.
• Discrete Mathematics:
  • M391 Graph Theory: Basics, planar graphs, Eulerian/Hamiltonian paths/cycles, connectedness, trees, spanning trees - spanning tree algorithms - DFS & BFS, traversals - Dijkstra's algorithm, minimal spanning trees - Kruskal's algorithm & Prim's algorithm, bipartite graphs.
  • M392 Combinatorics: The probabilistic method, double counting.

Italics => completed