semester 4 math courses at IISc
| course name | course details | course grade |
|---|---|---|
| Introduction to Basic Analysis (UM 204) | Basic notions from set theory, countable and uncountable sets. Metric spaces: definition and examples, basic topological notions. The topology of R n: topology induced by norms, the Heine-Borel theorem, connected sets. Sequences and series: essential definitions, absolute versus conditional convergence of series, some tests of convergence of series. Continuous functions: properties, the sequential and the open- set characterizations of continuity, uniform continuity. Differentiation in one variable. The Riemann integral: formal definitions and properties, continuous functions and integration, the Fundamental Theorem of Calculus. Uniform convergence: definition, motivations and examples, uniform convergence and integration, the Weierstrass Approximation Theorem. | B+ |
| Introduction to algebraic structures (UM 205) |
Set theory: equivalence classes, partitions, posets, axiom of choice/Zorn’s lemma, countable and uncountable sets. Combinatorics: induction, pigeonhole principle, inclusion-exclusion, Möbius inversion formula, recurrence relations. Number theory: Divisibility and Euclids algorithm, Pythagorean triples, solving cubics, Infinitude of primes, arithmetic functions, Fun- damental theorem of arithmetic, Congruences, Fermat’s little theorem and Euler’s theorem, ring of integers modulo n, factorisation of poly- nomials, algebraic and transcendental numbers. Graph theory: Basic definitions, trees, Eulerian tours, matchings, matrices associated to graphs. Algebra: groups, permutations, group actions, Cayley’s theorem, di- hedral groups, introduction to rings and fields. |
B+ |