Advanced Algebra (M.Sc. Mathematics)
Group theory, ring theory, modules, field extensions and Galois theory.
📋 Course Learning Outcomes
On successful completion of this course, the learner will be able to:
- Apply the Sylow theorems to classify groups of small order and analyse solvable and nilpotent groups [Apply]
- Analyze ideals in rings and characterise unique factorisation, principal ideal and Euclidean domains [Analyze]
- Examine the structure of finitely generated modules over a principal ideal domain [Analyze]
- Distinguish algebraic and transcendental field extensions and compute degrees of composite extensions [Evaluate]
- Apply the fundamental theorem of Galois theory to determine intermediate fields of a finite Galois extension [Apply]
- Construct proofs of insolvability of the general quintic by radicals using Galois-theoretic arguments [Create]
📚 Chapters
- Unit I — Sylow theorems, solvable and nilpotent groups
- Unit II — Rings, ideals, polynomial rings and unique factorisation domains
- Unit III — Modules, submodules and modules over PIDs
- Unit IV — Field extensions, algebraic and transcendental extensions
- Unit V — Galois theory and solvability by radicals
TutorDA LMS
Measure Theory (M.Sc. Mathematics)
Lebesgue measure, measurable functions, Lebesgue integration and L^p spaces.
📋 Course Learning Outcomes
On successful completion of this course, the learner will be able to:
- Construct Lebesgue outer measure on R and identify the sigma-algebra of Lebesgue measurable sets [Create]
- Analyze measurable functions and approximate them by simple functions [Analyze]
- Apply the monotone convergence, Fatou's lemma and dominated convergence theorems to evaluate Lebesgue integrals [Apply]
- Compare Riemann and Lebesgue integrals and identify functions integrable in one sense but not the other [Analyze]
- Examine functions of bounded variation and absolute continuity in connection with differentiation of integrals [Analyze]
- Establish completeness of L^p spaces using Holder's and Minkowski's inequalities [Evaluate]
📚 Chapters
- Unit I — Lebesgue outer measure and measurable sets
- Unit II — Measurable functions and Lebesgue measure on R
- Unit III — Lebesgue integral and convergence theorems
- Unit IV — Differentiation, functions of bounded variation and absolute continuity
- Unit V — L^p spaces, Holder and Minkowski inequalities, completeness
TutorDA LMS
Functional Analysis (M.Sc. Mathematics)
Banach and Hilbert spaces, bounded linear operators and the fundamental theorems of functional analysis.
📋 Course Learning Outcomes
On successful completion of this course, the learner will be able to:
- Identify normed linear spaces and verify completeness to characterise Banach spaces [Understand]
- Analyze bounded linear operators between normed spaces and compute their operator norms [Analyze]
- Apply the Hahn-Banach, open mapping, closed graph and uniform boundedness theorems to prove structural results [Apply]
- Evaluate orthonormal expansions in Hilbert spaces using Bessel's inequality and Parseval's identity [Evaluate]
- Apply the Riesz representation theorem to identify the dual of a Hilbert space [Apply]
- Distinguish between self-adjoint, normal and unitary operators through their spectral and structural properties [Analyze]
📚 Chapters
- Unit I — Normed linear spaces and Banach spaces
- Unit II — Bounded linear operators and dual spaces
- Unit III — Hahn-Banach, open mapping and closed graph theorems
- Unit IV — Hilbert spaces, orthonormal sets and Riesz representation
- Unit V — Adjoint, self-adjoint, normal and unitary operators
TutorDA LMS
Topology (M.Sc. Mathematics)
Topological spaces, continuity, connectedness, compactness and separation axioms.
📋 Course Learning Outcomes
On successful completion of this course, the learner will be able to:
- Define topological spaces using open sets, basis and subbasis and identify the subspace topology [Remember]
- Analyze continuity of functions between topological spaces and construct product and quotient topologies [Analyze]
- Prove that connectedness and path-connectedness are topological invariants and apply them to characterise spaces [Create]
- Apply the Heine-Borel theorem and the Tychonoff theorem to determine compactness of standard spaces [Apply]
- Evaluate separation properties (T0 to T4) of topological spaces and use Urysohn's lemma to construct continuous functions [Evaluate]
- Construct counterexamples that distinguish between countability and separation axioms [Create]
📚 Chapters
- Unit I — Topological spaces, basis and subspace topology
- Unit II — Continuous functions, product and quotient topology
- Unit III — Connectedness and path connectedness
- Unit IV — Compactness, local compactness and Tychonoff theorem
- Unit V — Countability and separation axioms, Urysohn's lemma
TutorDA LMS