The selection process for admission in 2020 to the various programs in Mathematics at the TIFR centers – namely, the Ph.D. and Integrated Ph.D. programs at TIFR, Mumbai as well as the programs conducted by TIFR CAM, Bengaluru, and ICTS, Bengaluru – will be held in two stages.

## MSc. Syllabus: TIFR 2020

The pupils enrolled in the Integrated Ph. D. program, that are candidates for receiving an M.Sc. Degree in the School of Mathematics is expected to have mastered the following subjects while satisfying the more complex course prerequisites of the school.

### Algebra:

**Group Theory:**Fundamental notions and examples, subgroups, normal subgroups, quotients, goods, semi-direct Solution, group acting on collections, Lagrange theorem, Cauchy’s theorem, Sylow theorems, p-groups; Cases to include symmetric and alternating groups.**Ring Theory:**Elementary notions of modules and rings; basic examples and structures.**References:**[Herstein, Ch. 7-8].**Linear Algebra:**Vector Spaces: Basis, independence of the number of components in a foundation, direct sums, duals, double dual.**Matrices and Linear transformation:**Linear map as a matrix, rank of a linear map, nullity, Eigenvalues, eigenvectors, minimal and characteristic polynomials, Cayley-Hamilton theorem, Triangulation and diagonalization, Jordan canonical form.**Field Theory:**Elementary notions of algebraic and transcendental extensions, splitting fields, structure theory of finite fields.**References:**[Bhattacharya etc., Ch. 13]

### Real Analysis:

- Countable and uncountable sets; basic notions of metric space and its topologies such as compactness and connectedness.
- Sequences at a metric distance, series of complex numbers, completeness, limp and timing. Standard convergence tests: comparison, origin, and ratio; absolute and conditional convergence. Riemann’s theorem about re-arrangement of conditionally convergent series; the product of series.
- Continuity, uniform continuity, compactness and connectedness under the constant map, application to the existence of maxima, minima, intermediate value; discontinuities of a monotone function.
- Differentiation of a function on, chain rule, Mean value Theorem, L’Hospital’s rule, Taylor’s theorem.
- Riemann-Stieltjes integral, lower and upper sums as the area beneath a curve. Integrability of continuous functions, Fundamental theorem of Calculus, integration by parts, rectifiable curve. Uniform convergence, the limit of continuous and differentiable functions under uniform convergence, integration under uniform convergence. Stone-Weierstrass theorem.
**Power series:**Basic theorems about convergence and continuity of a power series, the radius of convergence, behavior in the endpoints. Fourier series, fundamental convergence theorem, Parseval’s theorem.**Functions of Several factors:**Derivative of a function from to as a linear map; partial derivative, the connection between the two. Chain rule, inverse and implicit function theorems.**Lebesgue Integration:**Construction of Lebesgue measure on, integration; Lebesgue monotone and dominated convergence theorems; the contrast of Lebesgue and Riemann integration, -space.

9- **Basic Texts:**

**W. Rudin:**Principles of Mathematical Analysis.**T. Apostol:**Mathematical Analysis.**H.L. Royden:**Real Analysis.

### Topology and Functional Analysis:

**Basic Set Topology:**The idea of a topological space, continuity, compactness, connectedness. Heine-Borel Theorem, Tychonoff theorem, sequential compactness, Lebesgue covering lemma, Equi-continuity, Ascoli-Arzola theorem. And spaces, normal distance, Urysohn’s lemma, Tietze Extension theorem.**Fundamental Groups:**Homotopy of paths, Fundamental group, Converting spaces, Fundamental groups of circle and torus, homotopy lifting, the fundamental group of.**Functional Evaluation:**Normed linear spaces and continuous linear maps between them. Banach spaces and basic theorems about them: Hahn-Banach, Open mapping, and Uniform boundedness theorem. Terrible * topology on the dual, and the compactness of the world under the weak* topology.**Functional Evaluation:**Basic notions about Hilbert spaces. Total Orthonormal bases, Double space of a Hilbert space, Notion of the adjoint of an operator, Unitary and standard Operator, Compact operator; Spectral theory of compact self-adjoint operator.**Functional Analysis:**Fundamental notions of Banach Algebras, range. Structure of commutative Banach algebras.

6-** Basic Texts:**

**J.R. Munkres:**Topology.**G.F. Simmons:**Introduction to topology and modern analysis.**B.V. Limaye:**Functional Analysis.

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### Complex Analysis:

- Differentiation, Cauchy-Riemann equations, power series and its derivative, Harmonic functions.
- Cauchy’s theorem for a convex domain, Cauchy’s integral formula, Cauchy estimate, power series expansion, Morera’s theorem, Liouville theorem, Fundamental theorem of Algebra.
- Laurent expansion, Singularities, Meromorphic functions.
- Residue calculus, application to some explicit integrals.
- Maximum modulus principle, Phragman-Lindelof theorem.
- Harmonic functions, Poisson integral formula, Harnack’s theorem, mean value property, the Schwartz reflection principle.

7- **Basic Texts:**

**L. Ahlfors:**Complex Analysis.**J.B. Conway:**Functions of one complex variable.

## Ph.d. Syllabus: TIFR 2020

TIFR GS is supervised by the Tata Institute of Fundamental Research for Graduate School Admissions. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in fundamental sciences.

The Institute runs a graduate program leading to the award of Ph.D., Integrated M.Sc.-Ph. D. in Addition to an M.Sc. Degree in certain subjects. With its distinguished faculty, world-class facilities and a stimulating research environment, it’s an ideal place for aspiring scientists to initiate their careers.

### Algebra:

**Groups:**Jordan Holder theorem; solvable groups; symmetric and alternating groups; nilpotent classes; groups acting on sets; Sylow theorems; free groups.**Rings and Modules:**Noetherian and Artinian rings and modules; semisimple rings; Hilbert basis theorem; Principal ideal domains and unique factorization domains; modules over PID; linear algebra and Jordan canonical form; construction theorems for semisimple rings.- Representation theory of finite groups.
**Field theory:**Steinitz theorem; algebraic extensions; Galois theory and applications; only transcendental extensions; Luroth’s theorem.**Homological Algebra:**Categories and functors; adjoint functors Homand Tensor; their exactness properties and derived functors; Tensor, symmetric and exterior algebras.**Commutative Rings:**Integral extensions; Noether normalization theorem; Hilbert’s Nullstellensatz; different valuation rings and Dedekind domain names and some applications to arithmetic; chief decomposition.

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### Topology:

**General and metric topology:**Suitable maps; quotient space construction; illustrations of spheres, real and complex projective spaces, Grassmannians; normal and Hausdorff spaces; paracompact spaces; topological classes and continuous activities; ancient groups*.**Homotopy theory:**Covering spaces; homotopy of maps, homotopy equivalence of spaces, contractible spaces, deformation retractions; fundamental group: universal cover and lifting problem for covering maps; Van Kampen’s theorem, Galois coverings.**Homology theory:**Simplicial complexes; barycentric subdivision; easy approximations; magnificent homology – fundamental attributes – excision, Mayer-Vietoris; mobile homology, and fundamental examples with cellular homology; Kenneth formulation; universal coefficient theorem; singular cohomology; cup product, Poincaré duality; CW complexes; basic truth about topology of CW complexes; CW constructions for standard examples.**Smooth manifolds:**Tough manifolds, tangent and cotangent spaces; Vector fields, integral curves, Frobenius theorem, flows; Immersions and submersions; Implicit and inverse functions theorems; Sard’s theorem.

### Evaluation:

**Measure and integration:**Abstract concept; convergence theorems; merchandise step and Fubini’s theorem; Borel measures on locally compact Hausdorff space, and Riesz representation theorem; Lebesgue measure; regularity properties of Borel measures; Haar steps – theory and examples; complex steps, differentiation, and decomposition of measures; Radon Nikodym theorem; maximal function; Lebesgue differentiation theorem; functions of bounded variation.**Elementary operational evaluation:**Topological vector spaces; Banach spaces; Hilbert spaces; Hahn Banach theorem; open mapping theorem; uniform boundedness principle; bounded linear transformation; linear functionals and dual spaces.**Distances:**Basic theory, Hölder’s inequality, Minkowsky inequality.**Elementary Harmonic analysis:**Analysis; convolutions; approximate identity; approximation theorems; Fourier transform; Fourier inversion formula; Plancherel theorem, Hausdorff-Young inequality*. Operator theory: Spectral theorems for bounded normal operators, compact normal operators; Hilbert-Schmidt operators; Peter-Weyl theorem*. Banach algebras, Gelfand-Naimark theorem*.**Distribution Theory:**The distances, for open in;- As well as their duals; convolution; Fourier transform; Paley-Wiener theorems; basic solutions of constant-coefficient partial differential operators; Sobolev spaces.
**Cauchy-Riemann equation and holomorphic functions:**Fundamental attributes of holomorphic functions; open mapping theorem; maximum modulus theorem; zeros of holomorphic functions, Weierstrass factorization theorem Riemann mapping theorem; meromorphic functions; crucial singularities; Picard’s theorem.

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