Syllabus:
i) Commutative rings with unity: examples, ring homomorphisms, ideals,
quotients, isomorphism theorems with applications to non-trivial
examples. Prime
and maximal ideals, Zorns Lemma and existence of maximal ideals. Product
of rings, ideals in a finite product, Chinese Remainder Theorem. Prime
and
maximal ideals in a quotient ring and a finite product. Field of
fractions of an
integral domain. Irreducible and prime elements; PID and UFD.
ii) Polynomial Ring: universal property; division algorithm; roots of polynomials.
Gausss Theorem (R UFD implies R[X] UFD); irreducibility criteria. Symmetric polynomials: Newtons Theorem. Power Series.
iii) Modules over commutative rings: examples. Basic concepts:
submodules, quotients modules, homomorphisms, isomorphism theorems,
generators,
annihilator, torsion, direct product and sum, direct summand, free
modules, finitely
generated modules, exact and split exact sequences.
iv) Noetherian rings and modules, algebras, finitely generated algebras, Hilbert
Basis Theorem. Tensor product of modules: definition, basic properties and
elementary computations. Time permitting, introduction to projective modules.
Suggested Texts :
(a) D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley (Asian reprint
2003).
(b) N. Jacobson, Basic Algebra Vol. I, W.H. Freeman and Co (1985).
(c) S. Lang, Algebra, GTM (211), Springer (Indian reprint 2004).
(d) N.S. Gopalakrishnan, University Algebra, Wiley Eastern (1986).
(e) N.S. Gopalakrishnan, Commutative Algebra (Chapter 1), Oxonian Press (1984).
https://www.isibang.ac.in/~adean/infsys/database/MMath/C3Alg1.html
- Teacher: Suresh Nayak
- Teacher: G V Krishna Teja
The concept of σ-algebra, Borel subsets of R, Construction of Lebesgue and
Lebesgue-Stieltjes measures on the real line following outer measure.
• Abstract measure theory: definition and examples of measure space, measur-
able functions, Lebesgue integration, convergence theorems (Fatou’s Lemma,
Monotone convergence and dominated convergence theorem).
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• Product measures and Fubini’s theorem.
• Lp-spaces, Riesz-Fischer Theorem, approximation by step functions and con-
tinuous functions.
• Absolute continuity, Hahn-Jordan decomposition, Radon-Nikodym theorem,
Lebesgue decomposition theorem. Functions of bounded variation.
• Complex measures.
If time permits: Vitali covering lemma, differentiation and fundamental theorem of
calculus.
References
(a) H. L. Royden and Patrick Fitzpatrick: Real Analysis, Pearson, 4th edition.
(b) Robert B. Ash and Catherine A. Doleans-Dade, Probability and measure theory,
GTM(211), Academic Press, 2nd edition.
(c) Elias M. Stein, Rami Shakarchi, Real Analysis: Measure Theory, Integration
and Hilbert Spaces , Princeton Lectures in Analysis.
(d) Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications,
Pure and Applied Mathematics, A Wiley Series.
(e) G. de Barra, Measure Theory and Integration.
- Teacher: CRE Raja
• Topological spaces, open and closed sets, basis, closure, interior and bound-
ary. Subspace topology, Hausdorff spaces. Continuous maps: properties and
constructions; Pasting Lemma. Homeomorphisms. Product topology.
• Connected, path-connected and locally connected spaces. Lindel ̈of and Com-
pact spaces, Locally compact spaces, one-point compactification and Tychonoff’s
theorem. Paracompactness and Partitions of unity (if time permits).
• Countability and separation axioms., Urysohn embedding lemma and metriza-
tion theorem for second countable spaces. Urysohn’s lemma, Tietze extension
theorem and applications. Complete metric spaces. Baire Category Theorem
and applications.
• Quotient topology: Quotient of a space by a subspace. Group action, Orbit
spaces under a group action. Examples of Topological Manifolds.
• Topological groups. Examples from subgroups of GLn(R) and GLn(C).
• Homotopy of maps. Homotopy of paths. Fundamental Group.
References
(a) J. R. Munkres, Topology: a first course, Prentice-Hall (1975).
(b) G.F. Simmons, Introduction to Topology and Modern Analysis, TataMcGraw-
Hill (1963).
(c) M.A. Armstrong, Basic Topology, Springer.
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(e) J. Dugundji, Topology, UBS (1999).
(f) I. M. Singer and J. A. Thorpe, Lecture notes on elementary topology and ge-
ometry, UTM, Springer.
- Teacher: Anita Naolekar
• Commutative rings with unity: examples, ring homomorphisms, ideals, quo-
tients, isomorphism theorems with applications to non-trivial examples. Prime
and maximal ideals, Zorn’s Lemma and existence of maximal ideals. Product
of rings, ideals in a finite product, Chinese Remainder Theorem. Prime and
maximal ideals in a quotient ring and a finite product. Field of fractions of an
integral domain. Irreducible and prime elements; PID and UFD.
• Polynomial Ring: universal property; division algorithm; roots of polynomials.
Gauss’s Theorem (R UFD implies R[X] UFD); irreducibility criteria. Symmet-
ric polynomials: Newton’s Theorem. Power Series.
• Modules over commutative rings: examples. Basic concepts: submodules, quo-
tients modules, homomorphisms, isomorphism theorems, generators, annihila-
tor, torsion, direct product and sum, direct summand, free modules, finitely
generated modules, exact and split exact sequences.
9• Noetherian rings and modules, algebras, finitely generated algebras, Hilbert
Basis Theorem. Tensor product of modules: definition, basic properties and
elementary computations. Time permitting, introduction to projective modules.
References
(a) D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley (Asian reprint
2003).
(b) N. Jacobson, Basic Algebra Vol. I, W.H. Freeman and Co (1985).
(c) S. Lang, Algebra, GTM (211), Springer (Indian reprint 2004).
(d) N.S. Gopalakrishnan, University Algebra, Wiley Eastern (1986).
(e) N.S. Gopalakrishnan, Commutative Algebra (Chapter 1), Oxonian Press (1984).
- Teacher: Ramdin Mawia
Metric Topology of Rn. Topology induced by lp norms (p = 1,2,∞) on Rn and
their equivalence. Continuous functions on Rn. Separation properties. Com-
pact subsets of Rn. Path-connectivity. Topological properties of subgroups like
GLn(R), GLn(C), O(n), Hilbert-Schmidt norm and operator norm on Mn(R).
Sequence and series in Mn(R). Exponential of a matrix.
• Differentiation and integration of functions on Rn. Partial derivatives of real-
valued functions on Rn. Differentiability of maps from Rm to Rn and the
derivative as a linear map. Jacobian theorem. Chain Rule. Mean value the-
orem. Higher derivatives and Schwarz theorem, Taylor expansions in several
variables. Inverse function theorem and implicit function theorems. Local max-
ima and minima, Lagrange multiplier method.
• Vector fields on Rn. Integration of vector fields and flows. Picard’s Theorem.
• Riemann integration of bounded real-valued functions on rectangles (product
of intervals). Existence of the Riemann integral for sufficiently well-behaved
functions. Iterated integral and Fubini’s theorem. Brief treatment of multiple
integrals on more general domains. Change of variable and the Jacobian for-
mula.
• Differential forms on Rn. Wedge product of forms. Pullback of differential
forms. Exterior differentiation of forms. Integration of compactly supported
n-forms on Rn. Change of variable formula revisited. Integration of k-forms
along singular k-chains in Rn. Stokes’ theorem on chains. [Special emphasis on
curves and surfaces in R2 and R3. Line integrals, Surface integrals. Gradient,
Curl and Divergence operations, Green’s theorem and Gauss’s (Divergence)
theorem.]
Reference Texts:
(a) M. Spivak: Calculus on manifolds, Benjamin (1965).
(b) T.Apostol: Mathematical Analysis. S. Lang, Algebra, GTM (211), Springer
(Indian reprint 2002).
(c) K. Mukherjea: Differential Calculus in Normed Linear Spaces
- Teacher: Jaydeb Sarkar
Quick review of solutions of a system of linear equations, vector spaces, sub-
spaces, linear independence and span, Zorn’s lemma and existence of basis,
quotient spaces and direct sum of vector spaces, exact sequences and splittings,
linear maps and matrices, matrix of a linear map in a basis, invertibility, rank
and determinant, linear functionals, dual space, annihilator, transpose of a linear
map.
[This part is mostly a review and should be covered quickly with emphasis on
problem solving.]
• Eigenvalues, algebraic and geometric multiplicities, characteristic and minimal
polynomials, upper triangularization, diagonalizability and semisimplicity, de-
composition into nilpotent and semisimple matrices, Cayley-Hamilton Theo-
rem.
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product of linear maps, symmetric and exterior algebra, determinant as a mul-
tilinear map and Laplace expansion.
• Inner-product spaces, orthogonality, Gram-Schmidt orthogonalization, Bessel’s
inequality, projection and orthogonal projection, symmetric and Hermitian op-
erators, orthogonal and unitary diagonalizability, normal operators, spectral the-
orem, bilinear and quadratic forms, positive definite operator, square-root of a
positive operator, polar decomposition, isometry, rigid motions, the rotation
group.
• Structure theory of finitely generated modules over PID and application to
canonical forms.
References
(a) D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley (Asian reprint
2003).
(b) S. Lang, Algebra, GTM (211), Springer (Indian reprint 2002).
(c) K. Hoffman and R. Kunze, Linear Algebra, Prentice-Hall of India (1998).
(d) N.S. Gopalakrishnan, University Algebra, Wiley Eastern (1986).
(e) A. R. Rao and P. Bhimasankaram, Linear Algebra, TRIM(19), Hindustan Book
Agency (2000).
(f) P. R. Halmos, Finite-Dimensional Vector Spaces: Second Edition.
- Teacher: Deepak Pradhan
- Teacher: Maneesh Thakur