M.Sc in Mathematics

Tata Institute of Fundamental Research
In Mumbai

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  • Master
  • Mumbai
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Starts Location
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Mumbai
Homi Bhabha Road, Mumbai , 400005, Maharashtra, India
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Course programme

M.Sc in Mathematics

Algebra

* Group Theory: Basic notions and examples, subgroups,normal subgroups, quotients, products, semi-direct product, group acting on sets, Lagrange theorem, Cauchy's theorem, Sylow theorems, p-groups; Examples to include symmetric and alternating groups, $GL_n({\Bbb Z}/p)$, $SL_n({\BbbZ}/p)$.

References: [Herstein, Ch. 2], [Dummit & Foote, Ch. 4-5].
* Ring Theory: Elementary notions of rings and modules; basic examples and constructions. Notions of ideals, prime ideals, maximal ideals, quotients, integral domains, ring of fractions, PID, UFD.

References: [Herstein, Ch. 3], [Dummit & Foote, Ch. 7-8].
* Linear Algebra: Vector Spaces: Basis, independence of the number of elements in a basis, 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 diagnolisation, Jordan canonical form.

Modules over Principle Ideal Domains.

Elementary notions of quadratic and hermitian forms.

References: [Herstein, Ch. 4 & 6], [Dummit & Foote, Ch. 10-12]
* Field Theory: Elementary notions of algebraic and transcendental extensions, splitting fields, structure theory of finite fields.

References: [Bhattacharya etc., Ch. 15-16], [Artin, Ch. 13]
* Basic Texts
o I.N. Herstein: Topics in Algebra.
o D.S. Dummit and R.M. Foote: Abstract Algebra.
o M. Artin: Algebra.
o P.B. Bhattacharya, S.K. Jain, S. R. Nagpaul: Basic Abstract Algebra.
o S. Lang: Algebra.

Real Analysis

* Countable and uncountable sets; basic notions of metric space and its topology including compactness and connectness.
* Sequences in a metric space, series of complex numbers, completeness, limsup and liminf. Basic convergence tests: comparison, root and ratio; absolute and conditional convergence. Riemann's theorem about re-arrangement of conditionally convergent series; product of series.
* Continuity, uniform continuity, compactness and connectness under continous map, application to existence of maxima, minima, intermediate value; discontinuties of a monotone function.
* Differentiation of a function on ${\Bbb R}$, chain rule, Mean value Theorem, L'Hospital's rule, Taylor's theorem.
* Riemann-Stieltjes integral, upper and lower sums as area under a curve. Interability of continuous functions, Fundamental theorem of Calculus, integration by parts, rectifiable curve.
* Sequence and series of functions, examples and counter-examples. Uniform convergence, 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, radius of convergence, behaviour at the end points. Exponential and trignometric functions. Fourier series, basic convergence theorem, Parseval's theorem.
* Functions of Several variables: Derivative of a function from ${\Bbb R}^n$ to ${\Bbb R}^m$ as a linear map; partial derivative, relation between the two. Chain rule, inverse and implicit function theorems.
* Lebesgue Integration: Construction of Lebesgue measure on ${\Bbb R}$, integration; Lebesgue monotone and dominated convergence theorems; comparison of Lebesgue and Riemann integration, $L^2$-space.
* Basic Texts:
o W. Rudin: Principles of Mathematical Analysis.
o T. Apostol: Mathematical Analysis.
o H.L. Royden: Real Analysis.

Topology and Functional Analysis

* Basic Set Topology:
Notion of a topological space, continuity, compactness, connectness. Heine-Borel Theorem, Tychonoff theorem, sequential compactness, Lebesgue covering lemma, Equi-continuity, Ascoli-Arzela theorem. $T_1$ and $T_2$ spaces, normal space, Urysohn's lemma, Tietze Extension theorem.
* Fundamental Groups: Homotopy of paths, Fundamental group, Convering spaces, Fundamental groups of circle and torus, homotopy lifting, fundamental group of $S^n$.
* Functional Analysis: Normed linear spaces and continuous linear maps between them. Banach spaces and basic theorems about them: Hahn-Banach, Open mapping and Uniform boundedness theorem. Weak* topology on the dual, and the compactness of the sphere under the weak* topology.
* Functional Analysis: Basic notions about Hilbert spaces. Complete Orthonormal bases, Dual space of a Hilbert space, Notion of the adjoint of an operator, Unitary and Normal Operator, Compact operator; Spectral theory of compact self-adjoint operator.
* Funtional Analysis: Basic notions of Banach Algebras, spectrum. Structure of commutative Banach algebras.
* Basic Texts:
o J.R. Munkres: Topology.
o G.F. Simmons: Introduction to topology and modern analysis.
o B.V. Limaye: Functional Analysis.

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


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