Course Details:

Review of commutative rings, ideals and modules. Localisation and Spectrum of a ring, Nullstellensatz, Dimension theory, Associated primes and primary decomposition, Flatness, Completion and Artin-Rees lemma, Integral extensions, Graded rings, The Hilbert and Samuel functions, system of parameters and multiplicity, Regular sequences and Koszul complex, Cohen-Macaulay rings and Gorenstein rings, Regular rings, UFDs and Complete intersections.

Course References:

Text Books:
1. H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1989
2. W. Bruns and J. Herzog, Cohen-Macaulay rings, Revised second edition, Cambridge University Press, 1998

Reference Books:
1. D. Eisenbud, Commutative Algebra with view toward Algebraic Geometry, GTM 150, Springer-Verlag, New York, 1995.
2. N. Bourbaki, Elements of Mathematics - Commutative Algebra Chapters 1 - 7, Springer-Verlag, 1989.

Course Details:

Description:
To study in detail the theoretical aspects of Fourier transforms which is useful in various subjects of Science and Engineering

Course Content:
Fourier transform and tempered distributions: Review of the L1 and L2 theory of Fourier transform, Schwartz class of rapidly decreasing functions S(R), Topology on S(R), the class of tempered distributions, calculus of tempered distributions, convolution and Fourier transform of tempered distributions.
Some primary theorems in Fourier analysis: Qualitative and quantitative uncertainty principle, Hardyâ s theorem, Paley-Wiener theorem, Wienerâ s theorem, Wiener-Tauberian theorem, and spherical harmonics.
Introduction to singular integrals: Maximal function, integral of Marcinkiewicz, decomposition in cubes of open sets in Rn, an interpolation theorem for Lp, Carlderon-Zygmund kernel, Lp boundedness of a convolution operator with singular kernel.

Course References:

Text Books:
Reference Books:
1. Steven G. Krantz, A panorama of Harmonic Analysis, The Mathematical Association of America, 1999.
2. W. B. Rudin, Functional analysis, Tata McGraw Hill, 1991.
3. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, 1986.
4. H. Dym and H. P. McKean, Fourier series and integrals, Academic Press, 1972.

Prerequisite:
Fourier Analysis ( MA 6080) or equivalent

Course Details:

Description:
To introduce the students to the geometry of Riemann surfaces, both from analytic as well as algebraic viewpoints, highlighting the rich interplay between the topological, analytical and algebraic aspects of compact Riemann surfaces and their moduli.

Course Content:
Topology of Riemann Surfaces: Definitions and examples of Riemann surfaces, Euler-Poincare and Riemann-Hurwitz formulae, Galois Theory of Coverings, Uniformisation, Kleinian, Fuchsian and Elementary Groups.
Analytic Theory of Riemann Surfaces: Holomorphic differentials, Integration, Weyl's Lemma, Meromorphic functions.
Algebraic Aspects of Riemann Surfaces: Compact Riemann Surfaces as Algebraic Curves, Riemann-Roch Theorem, Abel's Theorem, Jacobian Variety of a Riemann Surface, Jacobi Inversion

Course References:

Text Books:
1. H. M. Farkas, I. Kra, Riemann Surfaces, Graduate Texts in Mathematics 71, Springer-Verlag, 1980.
2. R. Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics vol.5, American Mathematical Society, 1995.

References:
1. O. Forster, Lectures on Riemann Surfaces, Graduate Texts in Mathematics 81, Springer-Verlag, 1981.
2. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, 1997.
3. E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of Algebraic Curves - I,Grundlehren der mat.Wissen.267, Springer-Verlag, 1985

Course Details:

Description:
This course aims to give students an introduction to Cryptology.

Course Content:
Introduction: Brief introduction to number theory, Euclidean algorithm, Euler's totient function, Fermat's theorem and Euler's generalization, Chinese Remainder Theorem, primitive roots and discrete logarithms, Quadratic residues, Legendre and Jacobi symbols.
Private key cryptography: Stream ciphers, Block ciphers, DES and differential and linear cryptanalysis, Advanced encryption standards, Collision resistant hashing.
Authenticated encryption: security against active attacks
RSA public key cryptosystems: RSA system, primality testing, survey of factoring algorithms.
Other public key cryptosystems: El Gamal public key cryptosystem, algorithms for discrete log problem

Course References:

Text Books:
1. Introduction to Modern Cryptography by J. Katz and Y. Lindell.
2. Cryptography: Theory and Practice by D. Stinson.

Reference Books:
1. Handbook of Applied Cryptography by A. Menezes, P. C. Van Oorschot and S. A. Vanstone.
2. A Course in Number Theory and Cryptography by N. Koblitz.

Course Details:

Description:
This course aims to introduce advanced topics in probability theory which will equip graduate students with tools to start research in this area.

Course Content:
Independent Sums: Weak/Strong Law of Large Numbers, Law of Iterated Logarithm.
Central Limit Theorem: Berry-Essen and Linde berg Theorems, Some Extensions.
Convergence of Measures: Modes of Convergence, Infinite divisibility, Invariance Principle.
Wiener's Measure: Gaussian and Markov Aspects
Conditioning and Martingales: Conditional Expectation, Discrete and Continuous Martingales, Doob's Inequality and stopping time theorem, Diffusions.

Course References:

Text Books:
Stroock, D. W. Probability theory. An analytic view. Second edition. Cambridge University Press, Cambridge, 2011.

Reference Books:
1. Varadhan, S. R. S. Probability theory. Courant Lecture Notes in Mathematics, 7. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001.
2. Dudley, R. M. Real analysis and probability. Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002.
3. Billingsley, P. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.
4. Billingsley, P. Probability and measure. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2012.

Prerequisite:
MA5400 or Equivalent

Course Details:

Description:
The course aims to equip the students with commonly used algebraic and probabilistic tools in Combinatorics, and Graph Theory. It not only builds the fundamentals for students who plan to do PhD in Algebra/ Combinatorics / Graph theory and related topics, but also useful to students and researchers in other areas of science and engineering to which the methods of Algebra, Combinatorics, Graph theory and Probability may be applied.

Course Content:
The Twelve Fold way, Techniques from Probabilistic Method , The Method of Linear Algebra , LGV Lemma and applications, Chip firing games. Discharging method for graphs, Combinatorial Nullstellensatz, Polynomial method. Bijective combinatorics and the Catalan garden, Species of Combinatorial structures.

Course References:

Text Books:
1. N. Alon and J.H. Spencer, The probabilistic method, Third Ed., John Wiley & Sons Inc., New Jeresey, 2008.
2. S. Jukna, Extremal combinatorics: with applications to computer science, Second Ed., Springer Verlag, Heidelberg, 2011.

Reference Books:
1. Richard P Stanley, Enumerative Combinatorics - Volume 1, Springer. 2001
2. Richard P Stanley, Enumerative Combinatorics - Volume 2, Springer, 2001.
3. Introduction to combinatorial Species of Structures (lecture notes) â?? Bergeron et al. - (online copy available)

Prerequisite:
Basic Linear algebra, basic combinatorics, familiarity with probability theory concepts.

Course Details:

Description:
Complex networks are huge real-life networks like Neural, Social, and Communication networks or The World Wide Web. The study of these are hindered by their immense size. The only feasible approaches are studying the local structure or use the idea of random graphs and asymptotes. This course aims to prepare the interested student to delve into the fascinating area which is a current topic of huge interest.

Course Content:
Introduction to graphs and connectivity,
Complex networks with real-world examples: Internet, Social networks, Brain, and collaboration network. Scalefreeness and small-world properties. Random graphs and the need for it. Useful probabilistic methods and results.
Phase transition in random graphs: The evolution of clusters. The giant component and Central limit theorem. Criticality properties and bounds.
Models suitable for Complex networks with related results:
1. Generalized random graphs,
2. Configuration model
3. Preferential attachment models.

Course References:

Text Books:
R. van der Hofstad, Random Graphs and complex networks, Vol:1, Cambridge University Press, 2016.

Reference Books:
R. van der Hofstad, Random Graphs and complex networks, Vol:1 & 2, (Class notes), 2013-2014, e-copy available from the author.

Prerequisite:
Probability basics, familiarity with graphs

Course Details:

Description:
The course aims to reveal the fascinating interplay between Algebra, Combinatorics, and Graphs. It not only builds the fundamentals for students who plan to do PhD in Algebra/ Combinatorics / Graph theory and related topics, but also useful to students and researchers in other areas of science and engineering to which the methods of Algebra, Combinatorics, Graph theory may be applied.

Course Content:
Eigenvalues and Walks on graphs. Radon transform and hypercubes. Sperner property, lattices, and boolean algebra. Enumeration under group actions. Ferrer's diagram, Young tableaux, and Matrix Tree theorem. Applications to Electrical networks, planar graphs. Introduction to combinatorial commutative algebra.

Course References:

Text Books:
Richard P Stanley, Algebraic Combinatorics: Walks - Trees - Tableaux and More, Springer, 2013.

Reference Books:
1. Richard P Stanley, Enumerative Combinatorics - Volume 1, Springer. 2001
2. Richard P Stanley, Enumerative Combinatorics - Volume 2, Springer, 2001.
3. Combinatorial Commutative Algebra. Erza Miller and Bernd Sturmfels. Springer, 2005.
4. Rafael H Villarreal, Monomial Algebras, CRC Press, 2015.
5. R B Bapat, Graphs and Matrices, Springer, 2014.

Prerequisite:
A decent course in group theory and a good knowledge in linear algebra.

Course Details:

Groups: Basics of groups and its representations.
Commutative Rings: Dimension Theory, Regular local ring is a UFD. Hilbert functions.
Modules: The tensor, symmetric and exterior algebras of a module.
Non-commutative Rings: Review of Artin-Wedderburn theory including theorems of Jacobson-Bourbaki, Rieffel and Burnside, Kolchin's theorem.
Homological Algebra: Categories and Functors, Abelian categories, Projective and Injective resolutions, Left and Right derived functors, Ext and Tor, Local and Cech homology.

Course References:

Text Books:
1. S. Lang, Algebra, 3rd Edition, Addison-Wesley, 1999.
2. M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1969.
3. W. Fulton, J. Harris, Representation Theory - A First Course, GTM, Readings in Mathematics 129. Springer Verlag (1991)

Reference:
1. H. Matsumura, Commutative Ring Theory, Second edition,Cambridge Studies in Advanced Mathematics, 8.Cambridge University Press, Cambride 1989.
2. H. Matsumura, Commutative Algebra. Second edition. Mathematics Lecture Notes Series 56, Benjamin / Cummings Publishing,1980.
3. M. Artin, Algebra, Prentice Hall, 1994.

Course Details:

Convexity and Extreme Points: Topologies on linear spaces, linear functionals on topological spaces, weak topology, weak*topology, extreme points, Krein-Milman theorem. functions.

BaKCBh Algebras: BaKCBh algebra, complex homomorphisms on a BaKCBh algebra, properties of spectra, spectra radius formula, Gelfand - Mazur theorem, Gelfand transform, Maximal ideal space, involutions, Gelfand - Naimark theorem.

Spectral Theory: Bounded operators on a Hilbert space, normal, self-adjoint, unitary and projection operators, resolutions of the identity, spectral theorem, and symbolic calculus of normal operators.

Course References:

Text Books:
1. W. Rudin, Functional Analysis, International series in pure and applied Mathematics, Tata-McGraw Hill edition, 2007.
2. S.David Promislow, A first course in Functional Analysis, Pure and applied Mathematics, Wiley-Interscience, 2008.

Reference:
1. M.T. Nair, Functional Analysis: A First Course, Prentice-Hall of India, New Delhi,2002.
2. Peter D.Lax, Functional Analysis, Wiley-Interscience,2002.
3. J.B. Conway, A course in Functional Analysis, GTM 96, Springer, 1985.

Course Details:

Ordinary Differential Equations:
Review of existence and uniqueness of solutions of Initial value problems for a system of first-order differential equations
Existence and Uniqueness theorem for a linear system; homogeneous and inhomogeneous linear systems; linear equations with constant coefficients; Fundamental matrix
Linear differential equations with periodic coefficients: Floquet theory Stability for linear systems. Principle oflinearised stability. Stability for autonomous systems. l.iapunov functions.
Plane autonomous systems
Periodic solutions of plane autonomous systems.

Partial Differential Equations:
Review of the method of characteristics for first-order partial differential equations and classification of second-order partial differential equations Nonlinear first-order partial differential equations. Conservation laws. Lax-Oleinik. formula. Riemann's problem.Long-time behavior Separation of variables. Similarity solutions. Transform methods. Converting nonlinear partial differential equations into linear partial differential equations. Asymptotics. Maximum principles

Course References:

Text Books:
1. R. Grimshaw, Nonlinear ordinary differential equations, Blackwell Scientific publications, 1990.
2. Lawrence C. Evans, Partial Differential Equations, American Mathematical Society, 1991.

Reference:
1. David Betounes, Differential equations: Theory and applications, Springer, 2010.
2. L. Perko, Differential equations and dynamical systems, Springer, 2001.
3. Mathew P. Coleman, An introduction to partial differential equations with Matlab, CRC Press, 2005.
4. Sandro Salsa, Partial differential equations in action: From modelling to theory, Springer, 2008.

Course Details:

Combinatorics: Pigeonhole Principle, Principle of Inclusion and Exclusion, Catalan Number Stirling Number, Ramsey Number.
Graph Theory: Matching, Connectivity, Planar Graphs, Hamiltonian Graphs, Graph coloring.
Logic and Set Theory: First Order Logic, Quantifier Rules, Compactness Theorem, Schroder-Burnstein Theorem, Cantor's Diagonalization, Axiom of Choice, Continuum Hypothesis.
Computability and Complexity: C... computable Functions, Halting Problem, Post Correspondence Theorem, Rice Theorem. P, NP and co-NP, NP-completeness, Cooks Theorem, SAT, Knapsac Problem, Vertex Cover, Independent set, TSP.

Course References:

Text Books:
1. PJ Carneron, Combinatorits: Topics, Techniques.Algorithms, Cambridge University Press, 1994.
2. G Chartand and L Lesniak, Graphs and Digraphs, Chapman & HaII/CRC 2004. .
3. A Singh, Logic for Computer Science, PHI 2004.
4. A Singh, Elements of Compuation Theory, Texts in Computer Science, Springer, 2009.

Reference:
1. P R Halmos, Naive Set Theory, Van Nostrand, Princeton, 1960.
2. G Ausiello P Crescenzi G Gambosi V Kann A Maechetti-Spaccamela and M Protasi, Complexity and Approximation, Springer, 1999.
3. Kamala Krithivasan and R.Rama, Introduction to Automata Theory, Formal Languages and Computation, Pearson Education, 2009.

Course Details:

Description:
This course aims to equip interested PhD students with fundamentals of single and multipopulation game dynamics which is an active area of research with applications in Biology, Economics and Learning. The objective is to make these phd students ready to begin research in game dynamics and related areas.

Course Content:
N-person games: Strategy, payoff function, dominance, best response, Nash equilibrium, refinements of Nash equlibrium, symmetric two-person games.
Evolutionary stability: Evolutionarily stable strategies and characterizations, setwise evolutionary stability criteria, preplay communication, role-dependent behavior.
Replicator dynamics: Derivation and basic properties, stationary points of replicator dynamics and their stability, relations with evolutionarily stable strategy, replicator dynamics of doubly symmetric games, discrete-time version of replicator dynamics.
Other selection dynamics: General selection dynamics, replication by imitation, BNN dynamics.
Multipopulation models: Evolutionary stability criteria, standard and adjusted replicator dynamics.

Course References:

Text Books:
1. J. Weibull, Evolutionary Game Theory, MIT Press, 1995.
2. R. Cressman, Evolutionary Dynamics and Extensive Form Games, MIT Press, 2003.

Reference Books:
1. W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, 2010.
2. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.
3. J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. AMS, 40(4): 479-519, 2003.

Prerequisite:
Nil

Course Details:

Operator Semigroups: Hille-Yosida Theorem, Multivalued Operators, Semigroups on Function Spaces, Approximation and Perturbation Theorems.
Stochastic Processes and Martingales: Local Martingales, Projection Theorem, Doob-Meyer Decomposition; Semigroups of Conditioned Shifts, Martingales indexed by directed sets. Convergence of probability measures, Prohorov’s Theorem, Generators and Markov Processes. Levy Processes, Levy-Khinchine Formulation, Stable Laws, Infinite divisibility.

Course References:

Text Books:
1. S. N..; Kurtz, T. G. Markov processes. Characterization and convergence. John Wiley & Sons, Inc., New York, 1986.
2. Stroock, D. W. Probability theory. An analytic view. Second edition. Cambridge University Press, Cambridge, 2011.

Reference Books:
1. Varadhan, S. R. S. Probability theory. Courant Lecture Notes in Mathematics, 7. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001.
2. Dudley, R. M. Real analysis and probability. Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002.
3. Billingsley, P. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.
4. Billingsley, P. Probability and measure. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2012.

Course Details:

Description:
The Main goal of topology is to study the properties of 'objects'. Now-a-days many techniques from topology are applicable to various subjects for example in biology, physics and differential equations. This proposed course will help the students to grasp the some important connections among topology, algebra and geometry.

Course contents:

Homotopy Theory: (16/17 lectures) Operations on Spaces, Homotopy and criteria for homotopy equivalence, Homotopy lifting property. Paths and path homotopy, Fundamental groups and examples. Homotopy invariants. Free groups and their products, Van Kampen theorem and several applications. Covering spaces and Deck transformations, Classification of covering spaces.
Homology Theory: (16/17 lectures) Cell complexes and Delta-complexes, Simplicial and Singular homology, Relative homology groups, Exact sequences and Excision, Equivalence of Simplicial and Singular homology. Homotopy invariance and degrees, Cellular homology and applications, Mayer-Vietories sequences. More applications.
Cohomology Theory: (17/18 lectures) Cohomology groups of spaces, Cup product and cohomology ring, Some Spaces with polynomial cohomology rings, A Kunneth formula. Orientations and homology of manifolds, Cap product and Poincare Duality. Some applications.

Course References:

Text Books:
1. Allen Hatcher: Algebraic topology. Cambridge University Press, Cambridge, 2002.

Reference:
1. James Munkres: Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, CA, 1984.
2. William Massey: A basic course in algebraic topology. Graduate Texts in Mathematics, 127. Springer-Verlag, New York, 1991.
3. Glen Bredon: Topology and geometry. Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1993.

Prerequisite:
1. Basic general topology, group and ring theory

Course Details:

Description:
The course envisages to train research scholar in presenting simulation / preliminary experimental / analytical verification of prior art in the area of research, in the form of a report and an oral presentation to the DC and invitees and take feedback.

Course Content:
To be suggested by the guide

Course References:

Text Books:
To be prescribed by the guide.

Reference Books:
To be prescribed by the guide.

Prerequisite:
Nil