Semester I

Algebra I :

Group Theory: the Jordan-Holder Theorem; solvable groups; symmetric and alternating groups; nilpotent groups; groups acting on sets; the Sylow Theorems; free groups.

Category theory: categories and functors; equivalence of categories; inductive and projective limits.

Rings and Modules: Noetherian and Artinian conditions; the Hilbert Basis Theorem; principal ideal domains; unique factorization domains; inductive and projective limits of rings and modules; bilinear maps and forms; the tensor product.

Field Theory: the Steinitz Theorem on algebraic closures; algebraic extensions; finite fields; Galois theory and applications.

Textbooks:
1. S. Lang, Algebra.

2. M. Artin, Algebra.

3. D.S. Dummit and R. M. Foote, Abstract Algebra.

4. P.M. Cohn, Basic Algebra

5. T.W. Hungerford, Algebra.

6. M.P. Murthy et. al., Galois Theory, TIFR Pamphlet No. 3. 7. N. Jacobson, Basic Algebra, Vols. 1 and 2.

Analysis I :

Calculus : summary of calculus of several real variables; the Stone-Weierstrass Theorem; Ascoli’s Theorem.

Measure theory: Measure spaces; convergence theorems; product measure and Fubini’s Theorem; Borel measures on locally compact spaces, and the Riesz Representation Theorem; the Lebesgue measure; regularity properties of Borel measures; complex measures, differentiation and decomposition of measures; the Radon-Nikodym Theorem.

Functional analysis: Topological vector spaces; Banach spaces; Hilbert spaces; the Hahn-Banach Theorem; the Open Mapping Theorem; The Banach-Steinhaus Theorem; bounded linear maps; linear functionals and dual spaces; Lp spaces; Ho ̈lder’s inequality, Minkowski’s inequality.

Textbooks:

1. W. Rudin, Real and complex analysis.

2. S. Lang, Real analysis.

3. W. Rudin, Functional analysis.

4. E. Stein and R. Shakarchi, Real Analysis.

5. E. Stein and R. Shakarchi, Functional Analysis.

 

Topology I :

Topological spaces: Topologies; bases; continuous maps; subspaces; quotient spaces; products; connectedness and compactness; proper maps.

Convergence: Nets; filters; limits of nets and filter; the relations between convergence and countability and separation axioms; relations with compactness and proper maps.

Topological groups: Topological groups; uniform structures; products of compact spaces; compactifications; actions; orbit spaces; proper actions; homogeneous spaces.

Metrisability: metrisability and paracompactness; complete metric spaces; function spaces. Inductive and projective limits: Inductive and projective limits of topological spaces.

Homotopy theory: Homotopy; retraction and deformation; suspension; mapping cylinder; fundamental group; the Van Kampen Theorem; etale spaces; covering spaces; homotopy lifting property; relations with the fundamental group; lifting of maps; universal coverings; automorphisms of a covering; Galois coverings; the basic definitions regarding higher homotopy groups.

Textbooks:

1. N. Bourbaki, General topology, vol. 1.

2. E. Spanier, Algebraic topology (1966).

3. W. S. Massey, Algebraic topology: an introduction.

4. A. Hatcher, Algebraic Topology.

5. W. Fulton, Algebraic Topology.

6. J. Munkres, Elements of Algebraic Topology

Differentiable Manifolds:

Differentiable manifolds: basic notions; the effects of second countability and Hausdorffness; tangent and cotangent spaces; submanifolds; consequences of the Inverse Function Theorem; vector fields and their flows; the Frobenius Theorem; Sard’s theorem.

Differential forms: recapitulation of multilinear algebra; tensors; differential forms; the de Rham complex and its behaviour under differentiable maps; the Lie derivative; differential ideals.

Lie groups: Lie groups; Lie algebras; homomorphisms; Lie subgroups; coverings of Lie groups; the exponential map; closed subgroups; the adjoint representation; homogeneous manifolds.

Integration on manifolds: orientation; the integral of differential forms on differentiable singular chains; integration of differential forms of top degree on an oriented differentiable manifold; the theorems of Stokes; the volume form on an oriented Riemannian manifold; the divergence theorem; integration on a Lie group. de Rham cohomology: definition; real differentiable singular cohomology; statement of the de Rham theorem; the Poincare ́ lemma.

Textbooks:

1. F. W. Warner, Foundations of differentiable manifolds and Lie groups.

2. I. Madsen and J. Tornehave, From calculus to cohomology

3. John M. Lee, Introduction to smooth manifolds.

Research methodology:

Philosophy and Ethics, Scienti c conduct, publication ethics, open access publishing, publication misconduct, database and research metrics

In this course, the student is also expected to get familiarised with the process of writing academic reports, planning of a research project, writing mathematical texts using Latex software, and oral presentation, apart from the scientific conduct and ethics mentioned above. • Detailed syllabus:

Semester II

Algebra II : Multilinear algebra: tensor, symmetric, and exterior algebras; right exactness of tensoring, and flat and faithfully flat modules; left exactness of Hom, and injective and projective modules.

Commutative algebra: rings and modules of fractions; local rings; integral extensions; transcendence degree; Noether’s Normalization Theorem; Hilbert’s Nullstellensatz; discrete valuation rings; Dedekind domains; primary decomposition.

Linear algebra: modules over principal ideal domains; the minimal polynomial of an endomorphism; the Jordan canonical form; the characteristic poly- nomial of an endomorphism; the Cayley-Hamilton Theorem.

Textbooks:

1. S. Lang, Algebra.
2. M.F. Atiyah and I.G. McDonald, Introduction to Commutative Algebra .

3. P.M. Cohn, Basic Algebra.
4. N. Bourbaki, Algebra, Chapters 2, 3 and 7.
5. N. Bourbaki, Commutative Algebra, Chapters 1 to 6.
6. N. Jacobson, Basic Algebra, Vol. 2.

Analysis II:

Distributions: The spaces D(U ) and E (U ) for an open subset U of R ; basic operations on distributions; the support of a distribution; convolution; approximate identities; the Fourier transform on L1(Rn); the Schwartz space of Rn; the Inversion Theorem; Plancherel theorem; tempered distributions.

Functional analysis: Banach algebras; the Gelfand-Naimark Theorem; bounded operators on a Hilbert space; the Spectral Theorem for bounded normal operators on a Hilbert space; compact operators; Fredholm operators and the index.

Complex analysis: Basic properties of holomorphic functions; relations with the fundamental group and covering spaces; the Open Mapping Theorem; the Maximum Modulus Theorem; zeros of holomorphic functions; classification of singularities; meromorphic functions; the Weierstrass Factorization Theorem; brief account of the Riemann Mapping Theorem; the Little Picard Theorem.

Textbooks:

1. W. Rudin, Real and complex analysis.

2. S. Lang, Real analysis.

3. W. Rudin, Functional analysis.

4. E. Stein and R. Shakarchi, Complex Analysis.

5. E. Stein and R. Shakarchi, Functional Analysis.

6. V.S. Sunder, Functional Analysis : Spectral Theory.

7. J.B. Conway, Functions of One Complex Variable 1.

8. . H.A. Priestley, Introduction to Complex Analysis.

9. F. Treves, Topological Vector Spaces, Distributions and Kernels.

Topology II: Simplicial topology: Simplicial complexes; triangulations; polyhedra; barycentric subdivision; the Simplicial Approximation Theorem with applications.

Homology: simplicial homology; singular homology; the Mayer-Vietoris sequence; The Jordan-Brouwer Separation Theorem; the Universal Coefcient Theorem; the Kunneth Formula; CW complexes; cellular homology and computations for projective spaces; the Lefschetz Fixed Point Theorem.

Cohomology: singular cohomology; the Universal Coefcient Theorem; the Kunneth Formula; cup and cap products; Poincare ́ duality for a topological manifold.

Textbooks:

1. E. Spanier, Algebraic topology

2. M. J. Greenberg and J. R. Harper, Algebraic topology

3. .A. Hatcher, Algebraic Topology

4. G. E. Bredon, Topology and Geometry

5. J. W. Vick, Homology Theory : An Introduction to Algebraic Topology.

 

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