The Programme

The Integrated MSc-PhD programme in Mathematics at KSoM is an advanced training program at the postgraduate level leading to doctoral studies. The coursework is very flexible with a focus on grooming students for a research career in Mathematics. Depending on the choice of the electives, a student also has the option to choose a research career in Physics or Theoretical Computer Science as well. The curriculum is among the best that is available in the country at the postgraduate level. The Board of Studies for the Integrated MSc-PhD programme of KSoM will supervise the Integrated MSc-PhD programme.

1.1. Admission

The admission for the Integrated MSc-PhD programme at KSoM will be based on the national level entrance examination conducted by CMI/NBHM/KSoM. Shortlisted candidates will be called for a test/interview thereafter at KSoM. The minimum qualification required for joining the Integrated MSc-PhD programme is B.Sc., B.Math., B.Stat., BE / B.Tech. or an equivalent degree with Mathematics as one of the subjects.

1.2. Duration

The duration of the programme shall be a minimum of ten semesters and a maximum of sixteen semesters. The first four semesters are dedicated to the master's level coursework and the remaining towards doctoral studies leading to a PhD. An academic year has two semesters, one from July/August to November and the other from January to May. There will be a break between two semesters.

1.3. Scholarship

The students selected for the programme are entitled for a scholarship. The scholarship amount is currently Rs. 6000 during the MSc coursework for a maximum period of four semesters. However, a student who does not secure at least 24 credits (excluding Viva) in the first year forfeits her/his scholarship. The scholarship during PhD will be at par with NBHM PhD scholarship.

2.1. Credits for MSc Degree

A total of 76 credits in two years is required to obtain the MSc. degree. Every student has to take the three bridge courses before the commencement of the first semester. The purpose of these bridge courses is to review the fundamentals from a typical undergraduate mathematics programme and to bridge any gaps in the foundations. The bridge course will be undertaken in the first month of the programme.

2.2. Core Course Credits

There will be 11 core courses which contribute to 44 credits of the programme. The number of credits allocated to core and elective courses is equal to the number of contact hours every week including tutorial sessions in that particular course.

2.3. PhD Coursework Requirement

The 36 credits obtained in the second year of the programme will be treated as the coursework requirement for students continuing into the PhD component of the programme.

2.4. Elective Courses

The elective courses for the third and fourth semesters offered at KSoM are listed in subsection 2.3. All the courses listed there may, however, not be offered in an academic year. The list of courses offered every year will be announced at the beginning of the third semester.

2.5. Project and Reading Course Options

Every student has the option to work on a project or a reading course in the fourth semester. The project will comprise a dissertation and Viva at the end of the fourth semester. The project will carry 8 credits. Every student has the option of taking an elective and a reading course of 4 credits in the fourth semester, instead of a project. The reading course involves the study of an advanced topic or a research paper, along with a colloquium at the end of the programme.

2.6. PhD Promotion Criteria

A student may continue into the PhD programme subject to the following conditions:

1. The student has acquired 76 credits at the end the fourth semester.
2. The CGPA of the student is at least 8 (in case of SC/ST candidates the cut-off shall be 7).
3. The student has not obtained a grade of B or C in more than four courses during the MSc coursework.

2.6. PhD Admission Limitations

The admission to the PhD component will be limited based on the availability of the research supervisors.

2.7. MSc Course Structure Summary

The course structure for the MSc is listed below. The detailed syllabi for the courses offered at KSoM are listed in subsections 4.2 and 4.3.

Bridge Courses

Semester 1 & Semester 2

Semester 3 & Semester 4

Every student has to either select (*) or (**) in their final semester. The project carries 8 credits, and all other courses, including the Reading course carry 4 credits. Students continuing into the PhD component of the program should have completed Functional Analysis, Algebraic Topology - I, and Field and Galois Theory as electives in their third semester. Also such students should credit two courses out of Partial Differential Equations, Algebraic Topology-II, and Representation theory as electives in the fourth semester as part of their coursework.

3.1. Evaluation

Evaluation scheme for the core and elective courses offered at KSoM will contain two components: Continuous Assessment and End-semester Examination. The weightage of continuous assessment and end-semester examination in any course offered at KSoM will be as given below.

The weightage allotted to continuous assessment and end semester examination in an elective course will be declared at the beginning of the course by the instructor(s).

3.2. Grading System

Evaluation will be based on seven letter grades (O, A+, A, B+, B, C and F) with numerical values (grade points) of 10, 9, 8, 7, 6, 5 and 0 respectively. In addition to the above, an incomplete grade given by I and an absent grade given by Ab will also be used. The grading scheme involved in any course at KSoM is absolute.

3.3. Continuous Assessment Components

The continuous assessment for the courses offered at KSoM will be based on a predetermined transparent system involving the following components :

The transparent system will be declared at the beginning of the course by the instructor(s). The end-semester examinations for every course will be conducted and evaluated by the instructor(s) of the course.

3.4. Promotion policy

The grades are divided in to the following categories.

3.5. Make-up Examination and CGPA Calculation

A student securing the grade 'F' in the respective course will be treated as having failed in the course. In such a scenario an option of a make-up examination will be provided. The make-up examination will be held in the week before the beginning of the next semester. The Cumulative Grade Point Average (CGPA) out of $10$ is calculated as per the following formula

$$ \text{CGPA} = \frac{\sum_{\text{Courses taken in the programme}} \left(\text{No. of credits of the course} \times \text{Grade Points}\right)}{\text{Net total number of credits obtained in the core and elective courses}} $$

Courses taken in the programme does not include bridge courses. A semester grade point average (SGPA) is calculated similarly every semester.

3.6. MSc Degree Requirement

A student who has obtained 76 credits at the end of the second year, irrespective of satisfying the eligibility criterion for promotion into PhD, will be given an MSc degree.

3.7. Time Limit for MSc Completion and Promotion

The maximum time for completion of the MSc is 4 years during which 76 credits should be acquired. A student needs to acquire a minimum of 20 credits (excluding Viva) to get promoted to the second year of the programme. Any student unable to do so in the first two years will be asked to discontinue the MSc. programme.

4.1. Bridge courses

Three bridge courses will be offered to the students enrolled in the Integrated MSc-PhD programme in the first month after admission. The bridge courses aim at covering the foundational material that will be assumed in all the courses to follow in the MSc programme.

4.1.1. KSMB01: Linear Algebra

Matrix operations, Matrix units, row reduction, The matrix transpose, determinants, Permutation, The other formulas for the determinant, the cofactor matrix.

Suggested texts :

1. Michael Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. (Chapter 1)

4.1.2. KSMB01: Basic Analysis

The Peano axioms, Set theory Fundamentals, Functions, Cardinality of sets, Integers and rationals, The integers, The rationals, Gaps in the rational numbers, The real numbers, Cauchy sequences, The construction of the real numbers, Ordering the reals, The least upper bound property, Real exponentiation, Limits of sequences, real number system, Suprema and Infima of sequences, Limsup, Liminf, and limit points, Rearrangement of series, The root and ratio tests, Infinite sets, Countability, Uncountable sets, The axiom of choice, Ordered sets

Suggested texts :

1. Terence Tao, Analysis. I, third ed., Texts and Readings in Mathematics, vol. 37, Hindustan Book Agency, New Delhi (Chapters 1-8)

4.1.3. KSMB01: Basic Algebra

Laws of Composition, Groups and Subgroups, Subgroups of the Additive Group of Integers, Cyclic Groups, Homomorphism, Isomorphisms, Equivalence Relations and Partitions, Cosets, Modular Arithmetic, The Correspondence Theorem, Product Groups, Quotient Group

Suggested texts :

1. Michael Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. (Chapter 2)

4.2. Core Courses

Every student has to complete 32 credits from core courses compulsorily. The syllabi for the core courses across the first three semesters are detailed below.

Core courses in the first semester:

4.2.1. KSM1C01 : Algebra I

Subspaces of ${\bf R}^n$, Fields, Vector Spaces, Bases and Dimension, Computing with bases, Change of basis, Direct sums, Infinite dimensional spaces, Linear transformations, The dimension formula, Matrix of a linear transformation, Linear operators, Eigenvalues and eigenvectors, The characteristic polynomial, Triangular and Diagonal forms, Jordan form and decomposition, Applications of Linear Operators, Symmetry of Plane, Isometries of the Plane, Finite Groups of Orthogonal Operators on the Plane, Discrete Groups of Isometries, Plane Crystallographic Group, Abstract Symmetry: Group Operations, The Operation on Cosets, The Counting Formula, Operations on Subsets, Permutation Representations, Finite Subgroups of the Rotation Group, Cayley’s Theorem, The Class Equation, p-Groups, Conjugation in the Symmetric Group, Normalizers, The Sylow Theorems, The Free Group, Generators and Relations, Bilinear Forms, Symmetric Forms, Hermitian Form, Orthogonality, Euclidean Spaces and Hermitian Spaces, The Spectral Theorem, Conics and Quadrics, Skew-Symmetric Form.

Suggested texts :

1. Michael Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. (Chapters 3-8)
2. Kenneth Hoffman and Ray Kunze, Linear algebra, Prentice-Hall Mathematics Series, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961.
3. Denis Serre, Matrices, second ed., Graduate Texts in Mathematics, vol. 216, Springer, New York, 2010, Theory and applications.
4. Sheldon Axler, Linear algebra done right, third ed., Undergraduate Texts in Mathematics, Springer

4.2.2. KSM1C02 : Analysis I

Metric spaces, Limits of Functions, Continuous Functions, Continuity and Compactness, Continuity and Connectedness, Discontinuities, Monotonic Functions, Infinite Limits and Limits at Infinity, The Derivative of a Real Function, Mean Value Theorems, The Continuity of Derivatives, L'Hospital's Rule, Derivatives of Higher Order, Taylor's Theorem, Differentiation of Vector-valued Functions, Definition and Existence of the Integral, Properties of the Integral, Integration and Differentiation, Integration of Vector-valued Functions, Sequences and Series of Functions, Uniform Convergence, Equicontinuous Families of Functions, The Stone-Weierstrass Theorem, Functions of Several Variables, Differentiation, The Contraction Principle, The Inverse Function Theorem, The Implicit Function Theorem, The Rank Theorem.

Suggested texts :

1. Walter Rudin, Principles of Mathematical Analysis, McGraw Hill Education; Third edition (Chapters 3,4,5,6,7,9)
2. Terence Tao, Analysis. I & II, third ed., Texts and Readings in Mathematics, vol. 37, Hindustan Book Agency, New Delhi
3. Tom M. Apostol, Mathematical Analysis,

4.2.3. KSM1C03 : Topology

Topological Spaces and Continuous Functions, Connectedness and compactness, Tychonoff's theorem, Countability and Separation Axioms, normal and regular spaces, Urysohn-Tietze theorems, Complete Metric Spaces and Function Spaces, Baire Category Theorem.

Suggested texts :

1. James R. Munkres, Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000, Second edition
2. I. M. Singer and J. A. Thorpe, Lecture notes on elementary topology and geometry, Springer-Verlag, New York-Heidelberg, 1976, Reprint of the 1967 edition, Undergraduate Texts in Mathematics.
3. George F. Simmons, Introduction to topology and modern analysis, McGraw-Hill Book Co., Inc., New York-San Francisco, Calif.-Toronto-London, 1963.

4.2.4. KSM1C04 : Probability Theory

Probability Spaces, Axioms of probability and illustrations, Inclusion-exclusion principle and examples, Bonferroni inequalities, Independence and conditional probability, Bayes’ rule and applications, Discrete probability distributions and examples, Continuos probability distributions and examples, Joint distributions and change of variable formula, Expectation, moments and generating functions, Inequalities (Cauchy-Schwarz, Markov, Chebyshev, Chernoff), Conditional distributions and conditional expectation, Characteristic function and properties, Law of large numbers, Gaussian random variables and properties, Poisson limits, Central limit theorem, Simulation, Simple symmetric random walk.

Suggested texts :

1. William Feller, An introduction to probability theory and its applications. Vol. I, Third edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968.
2. William Feller, An introduction to probability theory and its applications. Vol. II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.
3. Sheldon Ross, A first course in probability, second ed., Macmillan Co., New York; Collier Macmillan Ltd., London, 1984.

Core courses in the second semester:

4.2.5. KSM2C01 : Algebra II

More Group Theory: The Classical Groups, Spheres, The Groups $SU_2$ $SO_3$, One-Parameter Groups, The Lie Algebra, Translation in a Group, Normal Subgroups of $SL_2$, Definition of a Ring, Polynomial Rings, Homomorphisms and Ideals, Quotient Rings, Adjoining Elements, Product Rings, Fractions, Maximal Ideals, Factoring Integers, Unique Factorization Domains, Gauss’s Lemma, Factoring Integer Polynomials, Gauss Primes, Modules, Free Modules, Identities, Diagonalizing Integer Matrices, Generators and Relations, Noetherian Rings, Structure of Abelian Group, Application to Linear Operators, Polynomial Rings in Several Variables.

Suggested texts:

1. Michael Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. (Chapters 9,11,12,14)
2. Serge Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002.
3. David S. Dummit and Richard M. Foote, Abstract algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991.
4. Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York-Berlin, 1980, Reprint of the 1974 original.

4.2.6. KSM2C02 : Analysis II

The problem of measure, Lebesgue measure, The Lebesgue integral, Abstract measure spaces, Modes of convergence, Differentiation theorems, Outer measures, pre-measures, and product measures Measure spaces, Fubini, Radon-Nikodym derivative.

Suggested texts :

1. Terence Tao, An introduction to Measure Theory, AMS publication.
2. Gerald B. Folland, Real analysis, second ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999, Modern techniques and their applications, A Wiley-Interscience Publication.
3. Elias M. Stein and Rami Shakarchi, Real analysis, Princeton Lectures in Analysis, vol. 3, Princeton University Press, Princeton, NJ, 2005, Measure theory, integration, and Hilbert spaces.
4. Walter Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.

4.2.7. KSM2C03 : Complex Analysis

Complex numbers and geometric representation, analytic functions, power series, exponential and logarithmic functions, conformality, Mobius transformations, Complex integration, Cauchy's theorem, Cauchy's integral formula including the homotopy version, singularities, Taylor's theorem, The maximum principle, The residue theorem and applications, Montel's theorem and Riemann mapping theorem.

Suggested texts :

1. Stein & Shakarchi, Complex Analysis, Princeton University press
2. John B. Conway, Functions of one complex variable, second ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978.
3. Ahlfors L, Complex Analysis.
4. Lecture notes of Terence Tao for the course 246A offered at UCLA.

4.2.8. KSM2C04 : Differential Equations

First and second order equations, general and particular solutions, linear and nonlinear systems, linear independence, solution techniques, Existence and Uniqueness Theorems: Peano's and Picard's theorems, Grownwall's inequality, Dependence on initial conditions and associated flows. Linear system: The fundamental matrix, stability of equilibrium points, Phase-plane analysis, Sturm-Liouvile theory, Nonlinear system and their stability: Lyapunov's method, Frobineous's theory.

Suggested texts :

1. Philip Hartman, Ordinary differential equations, Classics in Applied Mathematics, vol. 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002, Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA; MR0658490 (83e:34002)], With a foreword by Peter Bates.
2. Lawrence Perko, Differential equations and dynamical systems, third ed., Texts in Applied Mathematics, vol. 7, Springer-Verlag, New York, 2001.
3. Geral Teschl, Ordinary Differential Equations and Dynamical Systems, AMS publications.
4. Ordinary Differential Equations, Nandakumaran, A. K.; Datti, P. S.; George, Raju K.

4.3. Elective courses

Elective courses are advanced courses intended to form the foundational material for research in mathematics. A student intending to continue into the PhD component of the program has to credit the following three courses in the third semester (and will be treated as core courses of their PhD coursework) among the four electives in their third semester:

1. Functional Analysis
2. Algebraic Topology - I
3. Field and Galois Theory

In addition the following courses may be on offer.

1. Analysis on Manifolds
2. Number Theory
3. Advanced Complex Analysis
4. Commutative Algebra

4.3.1. KSM3E01: Functional Analysis

Normed linear spaces, Banach spaces, Bounded linear functionals, dual spaces and weak topologies, Hahn-Banach theorem, Bounded linear operators, open-mapping theorem, closed graph theorem. The Banach-Steinhaus theorem, Hilbert spaces, Riesz representation theorem, orthogonal complements, bounded operators on a Hilbert space up to (and including) the spectral theorem for compact, self-adjoint operators.

Suggested texts :

1. An Epsilon of Room, I: Real Analysis by Terence Tao, American Mathematical Society 2011
2. Rudin, Functional Anaysis (2nd Ed.), McGraw-Hill, 2006.
3. Yosida, K., Functional Anaysis (4th Edition), Narosa, 1974.
4. Conway, Functional analysis

4.3.2. KSM3E02: Algebraic Topology - I

Cell Complexes, The Fundamental Group, Basic Constructions, Paths and Homotopy, The Fundamental Group of the Circle, Induced Homomorphisms, Van Kampen’s Theorem, Free Products of Groups, The van Kampen Theorem, Applications to Cell Complexes, Covering Spaces, Lifting Properties, The Classification of Covering Spaces, Deck Transformations and Group Actions.

Suggested texts :

1. Algebraic Topology - Alen Hatcher
2. Algebraic Topology : An Introduction - William S. Massey

4.3.3. KSM3E03: Field and Galois Theory

Fields, The characteristic of a field, Extensions, Algebraic and transcendental elements, Transcendental numbers, Constructions with straight-edge and compass, Algebraically closed fields, Homomorphisms from simple extensions, Splitting fields, Homomorphisms of algebraic extensions, Multiplicity of roots, Separable polynomials, Perfect fields, Groups of automorphisms of fields, Separable, normal, and Galois extensions, The fundamental theorem of Galois theory, Constructible numbers, The Galois group of a polynomial, Solvability of equations, Polynomials of degree at most three, Quartic polynomials, Finite fields, Primitive element theorem, Fundamental Theorem of Algebra, Cyclotomic extensions, Dedekind’s theorem on the independence of characters, The normal basis theorem, Cyclic extensions, Kummer theory, Proof of Galois’s solvability theorem.

Suggested texts :

1. Lang, Algebra
2. Artin, E. Galois Theory
3. Joseph Rotman, Galois Theory
4. J.S. Milne, Fields and Galois Theory

4.3.4. KSM3E04: Analysis on Manifolds

The Algebra and Topology of ${\bf R}^n$, Differentiation, Derivative, Continuously Differentiable Functions, The Chain Rule, The Inverse Function Theorem, The Implicit Function Theorem, The Integral over a Rectangle, Existence of the Integral, Evaluation of the Integral, The Integral over a Bounded Set, Rectifiable Sets, Improper Integrals, Partitions of Unity, The Change of Variables Theorem, Diffeomorphisms in ${\bf R}^n$, Proof of the Change of Variables Theorem, Application of Change of Variables, Manifolds, The Volumne of a Parallelopiped, The Volume of a Parametrized-Manifold, Manifolds in ${\bf R}^n$, The Boundary of a Manifold, Integrating a Scalar Function over a Manifold, Differential Forms, Multilinear Algebra, Alternating Tensors, The Wedge Product, Tangent Vectors and Differential Forms, The Differential Operator, Application to Vector and Scalar Fields, The Action of a Differentiable Map, Stokes' Theorem, Integrating Forms over Parametrized-Manifold, Orientable Manifolds, Integrating Forms over Oriented Manifolds.

Suggested texts :

1. Analysis on Manifolds by James Munkres
2. Calculus on Manifolds by Michael Spivak

4.3.5. KSM3E05: Number Theory

The Fundamental Theorem of Arithmetic, Arithmetic Functions and Dirichlet Multiplication, Mean Values of Arithmetic Functions, Characters and Dirichlet Characters, Gauss Sums, Primitive Roots and Quadratic Residues, Primes in Arithmetic Progression, Dirichlet Series and Euler Products, The Riemann Zeta Function $\zeta(s)$, Prime Number Theorem, L-functions, Revisiting the Prime Number Theorem.

Suggested texts :

1. T.M. Apostol, Introduction to Analytic Number Theory, Undergraduate texts in Mathematics, Springer-Verlag, 1976.
2. R. Ayoub, An Introduction to the Analytic Theory of Numbers, American Mathematical Society (AMS), 1963.
3. H. Davenport, Multiplicative Number Theory. Graduate Texts in Mathematics (GTM), Springer-Verlag GTM, 2000.
4. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers (with an appendix by Andrew Wiles and Roger Heath-Brown), Oxford University Press, 2008.

4.3.6. KSM3E06: Advanced Complex Analysis

Automorphisms of Complex plane and Upper half plane, Analytic continuation, meromorphic continuation along a path, monodromy theorem, Conformal Equivalence for Simply Connected Regions, Conformal Equivalence for Finitely Connected Regions, Analytic Covering Maps, De Branges's Proof of the Bieberbach Conjecture, Hardy Spaces on the Disk, Potential Theory in the Plane.

Suggested texts :

1. Functions of One Complex Variable II, Conway John B.,
2. Function theory on planar domains, a second course in complex analysis - D. Fisher
3. Geometric Function Theory Explorations in Complex Analysis - Krantz
4. Potential Theory - Thomas Ransford

4.3.7. KSM3E07: Commutative Algebra

Rings and modules, morphisms, Prime spectrum of a ring, tensor products and change of base rings, Localization, noetherian and artinian rings, associated primes and primary decomposition, integral extensions, Noether normalization and Nullstellensatze, dimension theory of noetherian local rings, Krull principal ideal theorem.

Suggested texts :

1. Introduction to Commutative Algebra - Atiyah M. F. and McDonald I. G.
2. Commutative Algebra - Matsumura
3. Commutative Algebra with a View Toward Algebraic Geometry - Eisenbud, David

4.4. Fourth semester elective courses

Every student has the option to choose between a MSc project (carrying 8 credits) and a Reading course (carrying 4 credits) in the fourth semester. Based on the choice, two or three electives respectively will have to be credited by the students. A student intending to continue into the PhD component of the program has to credit at least two of the following three elective courses in the fourth semester:

1. Partial Differential Equations
2. Algebraic Topology-II
3. Representation theory

4.4.1. KSM4E01: Partial Differential Equations

Order of a PDE, Classification of PDEs into linear, semi-linear, quasilinear, and fully nonlinear equations, Examples of equations from Physics, Geometry, etc. The notion of well-posed PDEs.First Order PDEs: Method of characteristics, existence and uniqueness results of the Cauchy problem for quasilinear and fully nonlinear equations.Second Order Linear PDEs in Two Independent Variables: Classification into hyperbolic, parabolic, and elliptic equations, Canonical forms. The method of Separation of Variables for Laplace, Heat, and Wave equations.Laplace Equation: Definition of Harmonic functions. Mean-value property, Strong maximum principle and Harnack inequality for harmonic functions, Lioville theorem, smoothness of harmonic functions (harmonic functions are real analytic), Fundamental solution, Green's function, Examples for Greens functions like upper half space and ball, Possion's formula. Energy method, Dirichlet Principle, uniqueness using energy method.Heat Equation: Fundamental solution of heat equation, Duhamel's principle and formula for solution of $u_t - \Delta u = f(x,t), u(x,0)= g(x)$. Weak maximum principle, Heat mean value formula, strong maximum principle, Smoothness of solutions of heat equation, ill-posedness of backward heat equation, Energy methods.Wave Equation: Well-posedness of initial and boundary value problem in 1D and d'Alembert's formula, Method of descent in 2D and 3D, Duhamel's principle, domain of dependence, range of influence, and finite speed of propagation, energy method.Real analytic theory: Definition of powerseries in multi dimension, notion of real analytic function and its properties. Cauchy-Kowalevski Theorem, Holmgrem uniqueness theorem.Limitations of classical solutions and weak solutions: Lewy's example of PDE for which no solution exists. Hamilton Jacobi equations, conservation Laws.

Suggested texts :

1. Evans, L. C. Partial Differential Equations, AMS, 2010.
2. Han, Q. A Basic Course in Partial Differential Equations, AMS, 2011.
3. McOwen, R. Partial Differential Equations: Methods and Applications, Pearson, 2002.
4. Pinchover, Y. and Rubinstein, J. An Introduction to Partial Differential Equations, Cambridge, 2005.
5. FritzJohn Partial Differential Equations, springer

4.4.2. KSM4E02: Algebraic Topology-II

Simplicial and Singular Homology, Simplicial Homology, Singular Homology, Homotopy Invariance, Exact Sequences and Excision, The Equivalence of Simplicial and Singular Homology, Computations and Applications, Degree, Cellular Homology, Mayer-Vietoris Sequences, Homology with Coefficients. Cohomology Groups, The Universal Coefficient Theorem, Cohomology of Spaces, Cup Product, The Cohomology Ring, A Kunneth Formula, Spaces with Polynomial Cohomology, Poincar'{e} Duality, Orientations and Homology, The Duality Theorem, Connection with Cup Product, Other Forms of Duality, Universal Coefficients for Homology, The General Kunneth Formula.

Suggested texts :

1. Algebraic Topology - Hathcer
2. An Introduction to Algebraic Topology - Rotman, Joseph
3. Algebraic Topology : An Introduction - William S. Massey

4.4.3. KSM4E03: Representation theory

Generalities on linear representations, Subrepresentations, Irreducible representations, Tensor product of two representations, Symmetric square and alternating square, The character of a representation, Schur's lemma; basic applications, Orthogonality relations for characters, Decomposition of the regular representation, Number of irreducible representations, Canonical decomposition of a representation, Explicit decomposition of a representation, Subgroups, products, induced representations, Abelian subgroups, Product of two groups, Induced representations.

Suggested texts :

1. Representation Theory, A First Course - Fulton and Harris
2. Linear Representations of Finite Groups - Jean-Pierre Serre

The following is a list of elective courses which could be offered in the fourth semester. The list is tentative and all courses may not be on offer in a given semester. In case a student decides to take up an MSc colloquium based reading course instead of an MSc project, one of the following elective courses may be credited.

1. Several Complex Variables
2. Riemann Surfaces
3. Analytic Number Theory
4. Differential geometry
5. Algebraic Geometry
6. Algebraic Number Theory
7. Modular Forms
8. Harmonic Analysis
9. Homological Algebra
10. Introduction to Hyperbolic Geometry
11. Measure Theoretic Probability
12. Theory of Distributions

4.4.4. KSM4E04: Several Complex Variables

The $\overline{\partial}$-operator, holomorphic functions, Cauchy's formula, The Hartog's phenomenon the Dolbeault Lemma, Power series and Reinhardt domains, Domains of holomorphy, Subharmonic and plurisubharmonic functions, Convexity and pseudoconvexity.

Suggested texts :

1. Holomorphic Functions and Integral Representations in Several Complex Variables by Michael Range
2. Function Theory of Several Complex Variables by S. G. Krantz
3. An Introduction to Complex Analysis in Several Variables by Lars H"{o}rmander

4.4.5. KSM4E05: Advanced Functional Analysis

Banach algebra of continuous functions, Banach algebra of operators, Abstract Banach algebras, Space of multiplicative linear functionals, Maximal ideal spaces, Gelfand transform, Gelfand-Mazur theorem, Gelfand theorem for commutative Banach algebras, Stone-Weierstrass theorem, Generalised Stone-Weierstrass theorem, Algebra of bounded measurable functions, Gelfand-Naimark theorem, Spectral theorem for C*-algebras, Functional calculus.

Suggested texts :

1. Banach algebra techniques in operator theory, Ronald G Douglas
2. An invitation to C*-algebras, W. B. Arveson
3. Functional analysis, V. S. Sunder
4. C*-algebras and operator theory, G.J. Murphy
5. Functional Analysis, M. Reed and B. Simon
6. Linear Operator, General Theory N. Dunford, and T. Schwartz

4.4.6. KSM4E06: Riemann surfaces

The Definition of Riemann Surfaces, Elementary Properties of Holomorphic Mappings, Homotopy of Curves. The Fundamental Group, Branched and Unbranched Coverings, The Universal Covering and Covering Transformations, Sheaves, Analytic Continuation, Algebraic Functions, Differential Forms, The Integration of Differential Forms, Compact Riemann Surfaces, Cohomology Groups, Dolbeault's Lemma, A Finiteness Theorem, The Exact Cohomology Sequence, The Riemann-Roch Theorem, The Serre Duality Theorem.

Suggested texts :

1. Lectures on Riemann Surfaces - Otto Forster
2. Riemann Surfaces - Simon Donaldson
3. Riemann Surfaces by Way of Complex Analytic Geometry - Dror Varolin

4.4.7. KSM4E07: Differential Geometry

Vector Bundles, Local and Global Sections of Vector Bundles, Bundle Homomorphisms, Subbundles, Cotangent Bundle The Differential of a Function, Multilinear Algebra, Symmetric and Alternating Tensors, Tensors and Tensor Fields on Manifolds, Differential Forms, The Algebra of Alternating Tensors, Differential Forms on Manifolds, Orientations of Manifolds, Integration on Manifolds, Integration of Differential Forms, Stokes’s Theorem, De Rham Cohomology, The Mayer–Vietoris Theorem, The de Rham Theorem, Singular Cohomology, Smooth Singular Homology, The de Rham Theorem.

Suggested texts :

1. Introduction to Smooth Manifolds, John M Lee, second edition.
2. Foundations of Differentiable Manifolds and Lie Groups, Frank W. Warner, Springer-Verlag New York 1983, first editio

4.4.8. KSM4E08: Algebraic Geometry

Algebraic Sets, The Hilbert basis theorem, The Zariski topology, The Hilbert Nullstellensatz, Regular functions, Noether normalization theorem, Dimension, Affine Algebraic Varieties, Sheaves, Ringed spaces, Morphisms of ringed spaces, Affine algebraic varieties, Subvarieties, Birational equivalence, Noether Normalization Theorem, Local Study, Tangent spaces to plane curves, Tangent cones to plane curves, The local ring at a point on a curve, The differential of a regular map, Tangent spaces to affine algebraic varieties, Algebraic varieties, Products of varieties, Rational maps; birational equivalence, Local study, Smooth maps, Projective Varieties Algebraic subsets of $\mathbb{P}^n$, The homogeneous coordinate ring of a projective variety, Regular functions on a projective variety, Maps from projective varieties, Bezout’s theorem.

Suggested texts :

1. Algebraic Geometry and Commutative Algebra by Siegfried Bosch
2. Basic Algebraic Geometry 1 by Igor R. Shafarevich
3. Algebraic Geometry by Hartshorne

4.4.9. KSM4E09: Algebraic Number Theory

Algebraic Integers, The Theory of Valuations, Riemann-Roch Theory, Abstract Class Field Theory, Local Class Field Theory, Global Class Field Theory.

Suggested texts :

1. Neukirch: Algebraic Number Theory
2. Lang: Algebraic Number Theory
3. Marcus: Number Fields
4. Fr{" o}hlich: Algebraic Number Theory

4.4.10. KSM4E10: Modular Forms

The modular group and congruence subgroups, definition of modular forms and first properties, Eisenstein series, theta series, valence formulas, Hecke operators, Atkin-Lehner-Li theory, L-functions, modular curves, modularity. In the final lecture(s) we will give a global outlook on the use of modular forms (and their associated Galois representations) in the solution of Diophantine equations, in particular Fermat's Last Theorem.

Suggested texts :

1. F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics 228, Springer-Verlag, 2005.
2. Modular forms by Toshitsune Miyake
3. J.S. Milne, Modular Functions and Modular Forms, online course notes.
4. J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer-Verlag, 1973.

4.4.11. KSM4E11: Harmonic Analysis

Topological groups, locally compact groups, Haar measure, Modular function, Convolutions, homogeneous spaces, unitary representations, Gelfand-Raikov Theorem. Functions of positive type, GNS construction, Potrjagin duality, Bochner's theorem, Induced representations, Mackey's impritivity theorem.

Suggested texts :

1. Folland, G. B., A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, 1995.
2. Hewitt, E and Ross, K., Abstract Harmonic Analysis, Vol. 1, Springer 1979.
3. Gaal, S.A., Linear Analysis and Representation Theory, Dover, 2010.

4.4.12. KSM4E12: Homological Algebra

Complexes, Homology sequence, Euler characteristic and the Grothendieck group, Injective modules, Homotopies of morphisms of complexes, Derived functors, Delta-functors, Bifunctors, Spectral sequences, Special complexes, Finite free resolutions, Unimodular polynomial vectors, The Koszul complex

Suggested texts :

1. Homological Algebra - Henri Cartan & Samuel Eilenberg
2. Introduction to Homological Algebra - Rotman

4.4.13. KSM4E13: Introduction to Hyperbolic Geometry

Poincare metric on the upper half space and its geodesics, M"{o}bius Transformations on $\mathbb{R}^n$, Complex M"{o}bius Transformations, Discontinuous Groups, Riemann Surfaces, Hyperbolic Geometry, Fuchsian Groups, Fundamental Domains, Finitely Generated Groups, Universal Constraints On Fuchsian Groups

Suggested texts :

1. The Geometry of Discrete Groups - Beardon, Alan F.
2. Fuchsian Groups, Svetlana Katok

4.4.14. KSM4E14: Measure Theoretic Probability

Probability space, Lebesgue measure, Non-measurable sets, Random variables, Borel Probability measures on Euclidean spaces, Examples of probability measures on the line, A metric on the space of probability measures on $\mathbb{R}^d$, Compact subsets of $\mathcal{P}(\mathbb{R}^d)$, Absolute continuity and singularity, Expectation, Limit theorems for Expectation, Lebesgue integral versus Riemann integral Lebesgue spaces, Some inequalities for expectations, Change of variables, Distribution of the sum, product etc. Mean, variance, moments, Independent random variables, Product measures, Independence Independent sequences of random variables, Some probability estimates, Applications of first and second moment methods, Weak law of large numbers, Applications of weak law of large numbers, Modes of convergence Uniform integrability, Strong law of large numbers, Kolmogorov's zero-one law, The law of iterated logarithm Hoeffding's inequality, Random series with independent terms, Kolmogorov's maximal inequality, Central limit theorem - statement, heuristics and discussion, Central limit theorem - Proof using characteristic functions, CLT for triangular arrays, Limits of sums of random variables Poisson convergence for rare events, Brownian motion, Brownian motion and Winer measure, Some continuity properties of Brownian paths - Negative results, Some continuity properties of Brownian paths - Positive results, L'{e}vy's construction of Brownian motion.

Suggested texts :

1. Rick Durrett Probability: theory and examples.
2. Patrick Billingsley Probability and measure, 3rd ed. Wiley India.
3. Richard Dudley Real analysis and probability, Cambridge university press
4. Leo Breiman Probability, SIAM: Society for Industrial and Applied Mathematics

4.4.15. KSM4E15: Theory of distributions

Theory of Distributions: Introduction, Topology of test functions, Convolutions, Schwartz Space, Tempered distributions, Paley-Wiener theorem, Fourier transform and Sobolev-spaces:Definitions, Extension operators, Continuum and Compact imbeddings, Trace results, Elliptic boundary value problems: Variational formulation, Weak solutions, Maximum Principle, Regularity results.

Suggested texts :

1. Barros-Nato, An Introduction to the Theory of Distributions, Marcel Dekker Inc., New York, 1973.
2. Kesavan S., Topics in Functional Analysis and Applications, Wiley Eastern Ltd., 1989.
3. Evans, L. C., Partial Differential Equations, Univ. of California, Berkeley, 1998.
4. Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis

5.1. Eligibility and Constitution of Doctoral Committee

The promotion of a student to the PhD component is subject to satisfying all conditions listed in 2.5, and on joining a research supervisor at the end of the second year. A doctoral committee will be constituted for each student on promotion to the PhD component. The doctoral committee will conduct the Oral General Comprehensive Examination (OGCE) in the fourth year of the student. The syllabus for the OGCE (four advanced courses in the area of specialization) will be designed by the research supervisor and approved by the doctoral committee.

5.2. Senior Research Fellowship

A student is promoted to Senior Research Fellowship on clearing the OGCE.

5.3. Progress Monitoring

The doctoral committee will monitor the yearly progress of the respective student at the end of each year thereafter.