Mathematics programmes at SPS
M.Sc. in Mathematics:
Overview

Semester I  Semester II 
Algebra I  Algebra II 
Complex Analysis  Measure Theory 
Real Analysis  Functional Analysis 
Basic Topology  Discrete Mathematics 
Semester III  Semester IV 
Probability and Statistics  Partial Differential Equations 
Computational Mathematics  Elective I 
Ordinary Differential Equations  Elective II 
Project  Elective III 
List of Elective Courses  
Number Theory  Differential Topology 
Harmonic Analysis  Analytic Number Theory 
Proofs  Advanced Algebra 
Algebraic Topology  Banach and Operator Algebras 
Ph.D. in Mathematics:
After clearing the entrance examination and interview, a student with an M.Phil degree in Mathematics or with an M.A./M.Sc. degree in Mathematics and a valid UGCCSIR/NBHM JRF can enrol for a Ph.D. programme in Mathematics at SPS, JNU. For details regarding eligibility requirements and dates of examination and interview, please visit JNU's Admissions page.
Selected students who just have an M.A./M.Sc. degree have to do a course work during the first year. An advisor will be assigned only if the student finishes the course work with suitable grades.
Course work in Mathematics (2 semesters)
Students are required to obtain at least 14 credits during the two semesters of course work.
 The core courses PS 641M, PS 642M and PS 643M of 3 credits each are essential for students to expand their knowledge of basic Mathematics for doing research. Typically these are covered in the 1st semester.
 Apart from the three core courses totalling 9 credits, depending upon the courses being offered and the availability of faculty, students may opt for one of the following combinations for the remaining 5 credits in their 2nd semester:
 One research course and one of the three optional courses PS 644M, PS 645M or PS 712M.
 Two research courses (3 credits each).
Core Courses:
PS 641M Algebra (3 credits)
PS 642M Analysis (3 credits)
PS 643M Topology (3 credits)
Optional Courses:
PS 644M Topological Groups (2 credits)
PS 645M Functional Analysis and Operator Theory (2 credits)
PS 712M Algebraic Number Theory (3 credits)
PS 646M Research Course I (3 credits)
PS 647M Research Course II (3 credits)
Details of Core Courses
 Groups: Nilpotent and solvable groups, Sylow’s theorems, free groups.
 Representation theory of finite groups, PeterWeyl theorem.
 Rings and Modules: Commutative rings, Noetherian and Artinian rings and modules, principal ideal domains (PID), unique factorization domain, modules over PID, tensor products.
 Field Theory: Algebraic and transcendental extensions, introduction to Galois theory.
Suggested Texts:
 M. Artin. Algebra.
 I.N. Herstein. Topics in Algebra.
 S. Lang. Algebra.
 Prerequsites: Real Analysis, Basic Measure Theory, Topology and basics of Hilbert and Banach Spaces.
Review of Basic Measure Theory: Outer measures, Lebesgue measure, nonmeasurable sets, general measure spaces, Egorov's Theorem, sigma finite and complete measures, completion of measures, integration on measure spaces, approximation theorems for measurable and integrable functions, product measures, Fubini and Tonelli Theorem.
Signed and Complex measures: Signed and complex measures, different variations of signed measures, Hahn decomposition Theorem, Jordan decomposition Theorem, total variation of complex measures, complex regular measures, discrete and continuous measures, absolute continuity, mutual singularity, continuous and discrete decomposition, Lebesgue decomposition theorem, RadonNikodym Theorem.
L_{p} spaces: Holder and Minkowski inequalities, completeness, approximation by simple and continuous functions, duality, VitaliLuzin theorem and denseness of C_{ c}(X) in L_{ p}(X).
Riesz Representation Theorem (RRT): Original RRT, RRT for positive linear functionals, RRT for complex measures.
Fourier Analysis: Fourier transforms on L_{ 1}(R), boundedness and continuity of Fourier transforms, RiemannLebesgue Lemma, convolution on L_{ 1}(R), approximate identities, inversion formula, Fourier transforms on L_{ 2}(R), Plancheral Theorem, Parseval Formula, Existence and uniqueness of Haar measure on compact abelian group using KakutaniMarkov fixed point Theorem.
 Main text:
Integration and Modern Analysis , by Benedettor and Czaja, Birkhauser Advanced Texts, 2009
General references:  Real and Complex Analysis, W. Rudin, McGrawHill, 2006.
 A course in Abstract Analysis, J. B. Conway, AMS
 Analysis Now, G. K. Pederson, Springer, GTM
General Topology: Introduction, metric topology, separation axioms, compactness, Connectedness, product topology, introduction to manifolds, submanifolds.
Homotopy Theory. Covering spaces, homotopy maps, homotopy equivalence,Contractible spaces, deformation retraction.
Fundamental Groups: Universal cover and lifting problem for covering maps, Fundamental groups of S1 and Sn.
Introduction to Homology Theory.
Suggested texts:
 C.O. Christenson and W.L. Voxman. Aspects of Topology.
 J.R. Munkres. General Topology.
 I.M. Singer and J.A. Thorpe. Lecture Notes in Elementary Topology and Geometry.
Details of Optional Courses
PS 644M: Topological Groups and Lie Groups (2 credits)
Topological Groups: Introduction, integration on locally compact spaces, Haar Measure, Character groups, group action.
Lie groups and lie algebras: Basic theory, linear groups.
Suggested texts:
 K. Chandrasekharan. A Course on Topological Groups.
 W. Fulton and J. Harris. Representation Theory.
 F.W. Warner. Foundations of Differentiable Manifolds and Lie Groups.
PS 645M: Functional Analysis and Operator Theory (2 credits)
 Functional Analysis
Topological vector spaces: Separation properties, Linear Mappings, Finite dimensional spaces, Metrizability, Seminorms and local convexity.
Completeness: Baire Category Theorem, BanachSteinhauss Theorem, Open Mapping Theorem, Closed Graph Theorem.
Convexity: HahnBanach Theorems, Weak topologies, BanachAlaouglu Theorem, Exteme points, KreinMilman Theorem.
Some Applications: StoneWeirstrass Theorem, KakutaniMarkov fixed point Theorem and Haar measure for compact groups  Operator Theory
Compact operators on Banach spaces.
Bounded operators on a Hilbert space: Riesz Representation Theorem, Bounded operators, Adjoints, Normal Operators, Unitary Operators.
Spectral Theorem: Spectrum of an operator, Resolution of Identity, Spectral Theorem, Functional Calculus of normal operators, Spectral theorem for compact normal operators.
Suggested texts:
 W. Rudin, Functional Analysis, ISPAM, McGrawHill, 2006.
 J. B. Conway, A Course in Functional Analysis, GTM, Springer, 1990.
 R. J. Zimmer, Essential Results of Functional Analysis, Chicago Lecture in Mathematics, 1990.
 G. F. Simmons, Introduction to Topology and Modern Analysis, Tata McGrawHill, 2004.
 Functional Analysis
PS 712M: Algebraic Number Theory (2 credits)
Number fields, number rings and their structure as Dedekind domains.
Factorization of prime ideals, quadratic and cyclotomic extensions.
Decomposition group, inertia group.
Group of units, ideal class group, theorems of Dedekind and Minkowski.
Introduction to zeta function, Dirichlet character.Suggested texts:
 D. Marcus. Number Fields
 Borevich and Shafarevich. Number Theory
 Esmonde and Murty. Problems in Algebraic Number theory
 Frohlich and Taylor. Algebraic Number Theory
 Hasse. Number Theory
PS 646M, 647M: Research Courses I & II (3 credits each)
The research courses are advanced courses to prepare students to work in a specific area. The details of these courses are usually decided by the instructors.