Syllabus:
i) Review of normal subgroups, quotient, isomorphism theorems, Group actions
with examples, class equations and their applications, Sylows Theorems; groups
and symmetry. Direct sum and free Abelian groups. Composition series, exact
sequences, direct product and semidirect product with examples. Results on
finite groups: permutation groups, simple groups, solvable groups, simplicity
of An.
ii) Algebraic and transcendental extensions; algebraic closure; splitting fields and
normal extensions; separable, inseparable and purely inseparable extensions;
finite fields. Galois extensions and Galois groups, Fundamental theorem of Galois
theory, cyclic extensions, solvability by radicals, constructibility of regular
n-gons, cyclotomic extensions.
iii) Time permitting: Topics from Trace and Norms, Hilbert Theorem 90, Artin-
Schreier theorem, Transcendental extensions, Real fields.
https://www.isibang.ac.in/~adean/infsys/database/MMath/C9Alg2.html
- Teacher: Manish Kumar
- Teacher: Archita Mondal
Syllabus:
i) Brief review of Division Algorithm, gcd and lcm, Euclidean algorithm; Linear
Diophantine equations, congruences and residues, the Chinese Remainder
Theorem; The ring Z=nZ and its group of units, The Euler -function, Fermats
little theorem, Eulers theorem, Wilsons Theorem, Sums of two and four
squares.
ii) Pythagorean triplets and their geometric interpretation (rational points on circles);
Rational points on general conics; Fermats method of infinite descent
and application to simple Diophantine equations like x4 + y4 = z2; The Hasse
principle for conics (statement only), Brief discussion on rational points on cubics
and the failure of the Hasse principle (statement only).
iii) Polynomial congruences and Hensels Lemma; Quadratic residues and nonresidues,
Eulers criterion, Detailed study of the structure of the group of units
of Z=nZ, Primitive roots; Definition and properties of the Legendre symbol,
Evaluation of Gauss sums, The law of quadratic reciprocity for Legendre symbols;
Extension to the Jacobi symbols.
iv) Arithmetical functions and their convolutions, multiplicative and completely
multiplicative functions, examples like the divisor function d, the Euler
function
, the Mobius function
etc.; The Mobius inversion formula;
Sieve of Eratosthenes; Notion of order of magnitude and asymptotic formulae;
Euler and Abel summation formulae, Hyperbola method of Dirichlet, Average
order of magnitude of various arithmetical functions such as
; d
etc.;
Statement of the Prime Number Theorem; Elementary estimates due to Chebyshev
and Mertens on primes.
v) Algebraic integers, Arithmetic in Z[i] and Z[!], Primes of the forms x2 + y2
and x2 + xy + y2; Integers in quadratic number fields; Examples of failure of
unique factorization; Units in the ring of integers of a real quadratic field and
application to the Brahmagupta-Pell equation.
vi) One or more topics from the following list can be discussed if time permits:
a) Absolute values on Q, Ostrowskis Theorem, Completions of Q, Qp and
Zp, The p-adic topology.
b) The notion of algebraic and transcendental numbers; Transcendence of e,
Diophantine Approximation, Dirichlets Theorem; Liouvilles Theorem,
Statement of Roths Theorem.
c) Continued fractions; Applications to Diophantine approximation, Application
to the Brahmagupta-Pell equation.
d) Introduction to Geometry of numbers.
e)Application of Number Theory to RSA and other cryptosystems.
Suggested Texts :
(a) T. Apostol, Analytic Number Theory.
(b) D. Burton, Elementary Number Theory.
(c) G.H Hardy and E.M. Wright, An Introduction to The Theory of Numbers.
(d) K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory.
(e) Manin and Panchishkin, Introduction to Modern Number Theory.
(f) S.J. Miller and R. Takloo-Bighash, An Invitation to Modern Number Theory.
(g) I. Niven, H. S. Zuckerman, H. L. Montgomery, The Theory of Numbers.
J.-P. Serre, A Course in Arithmetic.
https://www.isibang.ac.in/~adean/infsys/database/MMath/C10NT.html
- Teacher: Suresh Nayak
Syllabus:
i) Review of normal subgroups, quotient, isomorphism theorems, Group actions
with examples, class equations and their applications, Sylows Theorems; groups
and symmetry. Direct sum and free Abelian groups. Composition series, exact
sequences, direct product and semidirect product with examples. Results on
finite groups: permutation groups, simple groups, solvable groups, simplicity
of An.
ii) Algebraic and transcendental extensions; algebraic closure; splitting fields and
normal extensions; separable, inseparable and purely inseparable extensions;
finite fields. Galois extensions and Galois groups, Fundamental theorem of Galois
theory, cyclic extensions, solvability by radicals, constructibility of regular
n-gons, cyclotomic extensions.
iii) Time permitting: Topics from Trace and Norms, Hilbert Theorem 90, Artin-
Schreier theorem, Transcendental extensions, Real fields.
Suggested Texts :
(a) J.J. Rotman, An Introduction to the theory of groups, GTM (148), Springer-Verlag (2002).
(b) D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley (Asian reprint 2003).
(c) S. Lang, Algebra, GTM (211), Springer (Indian reprint 2004).
(d) N.S. Gopalakrishnan, University Algebra, Wiley Eastern (1986).
(e) N. Jacobson, Basic Algebra, W.H. Freeman and Co (1985).
(f) G. Rotman, Galois theory, Springer (Indian reprint 2005).
(g) TIFR pamphlet on Galois theory.
Patrick Morandi, Field and Galois Theory, GTM(167) Springer-Verlag (1996).
(i) M. Nagata, Field theory, Marcel-Dekker (1977).
https://www.isibang.ac.in/~adean/infsys/database/MMath/C9Alg2.html
- Teacher: Ramdin Mawia
Syllabus:
i) Review of fundamental groups, necessary introduction to free product of groups,
Van Kampens theorem. Covering spaces, lifting properties, Universal cover,
classification of covering spaces, Deck transformations, properly discontinuous
action, covering manifolds, examples.
ii) Categories and functors. Simplicial homology. Singular homology groups, axiomatic
properties, Mayer-Vietoris sequence, homology with coefficients, statement
of universal coefficient theorem for homology, simple computation of homology
groups.
iii) CW-complexes and Cellular homology, Simplicial complex and simplicial homology
as a special case of Cellular homology, Relationship between fundamental
group and first homology group. Computations for projective spaces,
surfaces of genus g.
Suggested Texts :
(a) A. Hatcher, Algebraic Topology, Cambridge University Press (2002).
(b) W. S. Massey, A basic course in algebraic topology, GTM (127), Springer (1991).
(c) J. R. Munkres, Topology: a first course, Prentice-Hall (1975).
(d) J. R. Munkres, Elements of algebraic topology, Addison-Wesley (1984).
(e) M. J. Greenberg, Lectures on algebraic topology, Benjamin (1967).
(f) I. M. Singer and J. A. Thorpe, Lecture notes on elementary topology and geometry, UTM, Springer.
(g) E. Spanier, Algebraic Topology, Springer-Verlag (1982).
https://www.isibang.ac.in/~adean/infsys/database/MMath/C8T2.html
- Teacher: Jishnu Biswas
Syllabus:
i) Quick review of sequences and series of functions, equicontinuity, Arzela-
Ascoli theorem.
ii) Normed linear spaces and Banach spaces. Bounded linear operators. Dual of
a normed linear space. Hahn-Banach theorem, uniform boundedness principle,
open mapping theorem, closed graph theorem. Computing the dual of some
well known Banach spaces. Weak and weak-star topologies, Banach-Alaoglu
Theorem. The double dual. Lp-spaces and their duality, Weierstrass and Stone-
Weierstrass Theorems.
iii) Hilbert spaces, adjoint operators, self-adjoint and normal operators, spectrum,
spectral radius, analysis of the spectrum of a compact operator on a Banach
space, spectral theorem for compact self-adjoint operators on Hilbert spaces.
Basics of complex measures and statement of the Riesz representation theorem
for the space C(X) for a compact Hausdorff space X.
iv) If time permits, some of the following topics may be covered: Sketch of proof of
the Riesz Representation Theorem for C(X), Goldsteins Theorem, reflexivity;
spectral theorem for bounded normal operators.
Suggested Texts :
(a) Real and complex analysis, W. Rudin, McGraw-Hill (1987).
(b) Functional analysis, W. Rudin, McGraw-Hill (1991).
(c) A course in functional analysis, J. B. Conway, GTM (96), Springer-Verlag (1990).
(d) Functional analysis, K. Yosida, Grundlehren der MathematischenWissenschaften
(123), Springer-Verlag (1980).
https://www.isibang.ac.in/~adean/infsys/database/MMath/C7FA.html
- Teacher: Soumyashant Nayak
Syllabus:
i) Review of sequences and series of functions including power series, Complex
differentiation and Cauchy-Riemann equation, Cauchys theorem and Cauchys
integral formula, Power series expansion of holomorphic function, zeroes of
holomorphic functions, Maximum Modulus Principle, Liouvilles Theorem,
Moreras Theorem.
ii) Complex logarithm and winding number, Singularities, Meromorphic functions,
Casorati Weierstrass theorem, Riemann sphere, Laurent series, Residue
Theorem and applications to evaluation of definite integrals, Open Mapping
Theorem, Rouches Theorem.
iii) Conformal maps, Schwarz lemma, Linear fractional transformations, automorphisms
of a disc, Introduction to Gamma function.
iv) Equicontinuity and Arzela-Ascoli Theorem, Normal family, Montels theorem
and Riemann mapping theorem.
v) If time permits then the following topics can also be covered: Mittag-Leffler
Theorem, Infinite product, Weierstrass factorization theorem.
Suggested Texts :
(a) Complex Analysis - L. Ahlfors.
(b) Elementary Theory of Analytic Functions of One or Several Complex Variables - H. Cartan
.
(c) Complex Analysis - E. M. Stein, R. Shakarchi.
(d) Complex Analysis - D. Sarason.
https://www.isibang.ac.in/~adean/infsys/database/MMath/C6CA.html
- Teacher: Ramesh Sreekantan