Ph.D. in Mathematics
Ph.D. Courses
MA605 Introduction to Nonlinear Dynamics: 3-0-0-6-3
Nonlinear Equations: autonomous and non-autonomous systems, phase portrait, stability of equilibrium points, Lyapunov exponents, periodic solutions, local and global bifurcations, Poincare-Bendixon theorem, Hartmann-G robmann theorem, Center Manifold theorem.
Nonlinear Oscillations: perturbations and the Kolmogorov-Arnold-Moser theorem, limit cycles. Chaos: one-dimensional and two-dimensional Poincare maps, attractors, routes to chaos, intermittency, crisis and quasi periodicity. Synchronization in coupled chaotic oscillators. Applications: Examples from Biology, Chemistry, Physics and Engineering.
MA608 Operator Theory: 3-1-0-5-3
Operators on Hilbert spaces: Bounded linear operator on Hilbert spaces, spectrum of an operator, weak , norm and strong operator topologies, normal, self adjoint, unitary and compact operator and their spectra. Diagonalization, spectral theorem and applications: diagonalization for a compact self adjoint operator, spectral theorem for compact norm operator, spectral calculus, application to strum-Liouville problem. Positive operators : positive linear maps of finite dimensional space and their norms, Schur products, completely positive maps.
MA614 Applied Linear Algebra and Matrix Analysis 4(3-1-2-6-4)
Spectral theorem with applications: Inner product spaces, Normal, Unitary and self-adjoint operators in finite dimensional spaces, Finite dimensional spectral theorem for normal operators.
Decompositions: Orthogonal reduction, Range-Null space decomposition, orthogonal decomposition, Singular value decomposition, orthogonal projections, Least-square method.
Positive and Stochastic matrices: Positive and Stochastic matrices, applications.
MA615 Elementary Number Theory 3(3-0-0-6-3)*
MA616 Elements of Data Science 3(3-0-2-7-4)
Overview of Probability and Statistics & statistical learning: Definition, principles and different types of statistical learning, assessing model accuracy, bias-variance tradeoff.
Regression models: Simple linear and multiple linear and non-linear.
Resampling methods: Assessing model prediction quality, cross validation, bootstrap.
Model selection and regularization: Dimensionality reduction, ridge and lasso.
Unsupervised learning: Clustering approaches, K-means and hierarchical clustering
Supervised learning: classification problem, classification using logistic regression, naive Bayes, classification with Support Vector Machines, neural networks.
MA617 Graph Theory 4(3-1-0-5-3)
Introduction to graphs and digraphs: Introduction to graphs, subgraphs, degrees and graphical sequences, Isomorphism, bipartite graphs, directed graphs.
Connectivity, Minimum spanning trees, Shortest path problems: Connectivity and edge connectivity; Trees: characterizations, minimum spanning trees, counting the number of spanning trees, cayley’s formula, shortest path problems.
Matchings, Eulerian and Hamiltonian Graphs: Independent sets and covering. Matching in bipartite graphs, Hall’s marriage theorem. Eulerian Graphs: Definition and characterization. Hamiltonian Graphs: Definition, necessary and sufficient conditions.
Coloring and Planarity: Graph Coloring: Vertex coloring, edge coloring, chromatic polynomials; Planarity: Planar and non-planar graphs, Euler formula and its consequences, dual of a graph, Kuratowaski’s theorem.
Network Flows, Triangulated Graphs, Application of Graph Theory: Network Flows; Triangulated Graphs; Applications of graph theory in different real world problems.
MA620 Discrete Mathematics 4(3-1-0-5-3)*
MA621 Introduction to Calculus of Variations 3(3-0-0-6-3)*
MA622 Combinatorics 3(3-0-0-6-3)
Counting Principles and Generating Functions, The Method of generating Functions, Recurrence Relations: Linear Recurrence Relation, Binomial coefficients, Binomial theorem, Derangements, Involutions, Fibonacci Numbers, Catalan Numbers, Bell Numbers, Eulerian Numbers.
The Pigeonhole Principle, The Principle of Inclusion and Exclusion: Derangements Revisited, Counting Surjective maps, Stirling Numbers of the first kind, Stirling Numbers of the second kind, Posets and Mobius functions, Lattices, The Classical Mobius function, The Lattice of Partitions, The Orbit-Stabilizer formula, Permutation Groups, Burnside’s Lemma, P’olya’s Theory, The Cycle Index, Block Designs: Gaussian Binomial Coefficients, Introduction to Designs, Steiner triple system, Incidence Matrices, Fisher’s inequality, Bruck-Ryser-Chowla Theorem, Coding Theory, Hamming sphere, Reed-Solomon codes.
MA623 Introduction to Knot Theory 3(3-0-0-6-3)
Introduction to Knots and Links: Knots, Knot Projections, Composition of Knots, Reidemeister moves, Links.
Invariants: Classical Knot Invariants – bridge number, crossing number, unknotting number, linking number, colouring number.
Tabulating Knots: Dowker Notation for Knots, Conway’s notation, Knots and Planar Graphs.
Seifert Matrices: Seifert Matrices, Invariants from the Seifert Matrices.
Torus Knots: Torus knots, Seifert Matrix of a Torus Knot, Invariants of Torus Knots.
Tangles and 2-Bridge Knots: Tangles, 2-Bridge knots and their properties.
Braids: The braid group, the braid index, and its properties.
Polynomial Invariants: Bracket Polynomial, Alexander Polynomial, Jones Polynomial, Kauffman Polynomial, Amphichirality.
Virtual Knots: Introduction to Virtual Knots, Welded Knots and Spatial Graphs.
MA624 Basics in Coding Theory and Cryptography 3(3-0-0-6-3)
Elementary number theory: Divisibility, the Euclidean algorithm Congruence and some application.
Finite field: Construction of finite field and quadratic residue.
Introduction to coding theory: Codes, hamming codes, hamming bounds, error correction.
Different codes: Linear codes, cyclic codes, reed solomon codes, and their error correction.
Cryptosystem: Some simple cryptosystem, Enciphering matrices.
Public key cryptosystem: Public key cryptography, RSA, discrete log, diffie hellmann key exchange, hash function.
MA625 Calculus of Variation and Integral equations, 3 (3-0-0-6-3)
Calculus of Variation: Introduction, problem of barchistochrone, isoperimetric problem, concept of extrema of a functional, variation and it’s properties. Variational problems with fixed boundaries, The Euler equation, The fundamental lemma of calculus of variations. Variational problems with moving boundaries, Reflection and refraction extremals.
Transversality conditions, Sufficient conditions for an extremum, Field of extremals, Jacobi conditions, Legendre Condition. Second variations. Canonical equations and variational principles, Introduction to direct method for variational principle.
Integral Equations: Integral equations, Regular Integral equations: Voltera integral equations, Fredholm integral equations, Volterra and Fredholm equations with regular kernels. Degenerate kernel, Fredholm Thereom, Method of successive approximation.
Bernstein polynomials and its properties. Solving integral equations by using Bernstein polynomials and general polynomial. Numerical method: Quadrature method for Integral equations. Green’s function in integral equations.
MA626 Problem Solving Methods in Mathematics (1-0-4-4-3)
Problems in: Real Analysis, Linear Algebra, Complex Analysis.
Problems in: Ordinary Differential Equations, Partial Differential Equations, Calculus of Variations.
Problems in: Numerical Analysis, Integral Equations.
Problems in: Algebra, Topology.
MA627 Theory of Computation 3(3-0-0-6-3)
DFA, NDFA: Deterministic Finite Automata, Non deterministic finite Automata, Pumping Lemma. Regular Languages, Regular Expressions: Properties of Regular Languages, Equivalence to DFA, NDFA. PDA: Pushdown Automata, Deterministic Pushdown Automata. CFL, CFG: Properties of Context Free Languages, Context Free Grammars, Equivalence to Pushdown Automata, Pumping Lemma for CFG, Chomsky Normal Form. Turing Machines and Undecidability: Turing Machines, Recursive Languages, Recursive Enumerable Languages, Undecidability, Halting Problem, Post Correspondence Problem. Complexity Classes: P, NP, NP-Complete, NP-Hard Problem classes, Intractability.
MA628 Financial Derivatives Pricing 3(3-0-2-7-4)
Financial Securities and Derivatives: In this module, the financial securities like bonds, stocks, swaps, interest rates products, forwards and futures will be introduced. Stochastic Calculus: In this module, the theoretical aspects like conditional expectation, filtration, martingale and Ito calculus will be introduced. Option Pricing: In this part, European style call and put options pricing will be done.
MA629 Fuzzy Logic & Applications 3(3-0-0-6-3)*
Content to update.
MA630 Introduction to Applied Statistical Methods 4(3-1-0-5-3)*
Content to update.
MA632 Analysis on Manifolds 3(3-1-0-5-3)
Review of differentiability of functions of several real variables, chain rule. Inverse and Implicit function theorems and their applications. Plane and space curves.
Definitions and examples of manifolds in Rn, tangent and normal spaces to manifolds in Rn. Definition and examples of domains with smooth boundary in Rn Differentiability of functions defined on manifolds and their derivatives.
Review of integral calculus on Rn, Fubini’s theorem for change of order in multiple integrations, rectifiable sets, partition of unity and change of variable formula and applications. Surface area and volume of manifolds in Rn, integration on manifolds in Rn. Integration by parts formula for domain with smooth boundary.
Differential forms and their integrations. Green’s Gauss and Stoke’s theorem and their applications.
MA633 Evolutionary PDEs 3(3-1-0-5-3)
Bochner Integral: Weakly and strongly measurable functions, Pettis theorem, Bochner integrals, Banach space valued Lebesgue spaces and properties.
Time-dependent distributions and Sobolev spaces: Calculus on Distributions with values in Hilbert spaces, Sobolev spaces involving time and properties, embedding theorems.
Linear Parabolic PDEs: Definition and example of parabolic PDEs, notion of weak solution. Existence and uniqueness of weak solutions, maximum principles, Harnack inequality.
Linear Hyperbolic PDEs: Definition and example of hyperbolic PDEs, notion of weak solutions. Existence, uniqueness and regularity of weak solutions to the initial boundary value problem for 2nd order linear hyperbolic PDEs.
MA634 Financial Risk Management 4(3-1-0-5-3)*
What is risk? Types of financial risk, Why manage financial risk and associated challenges? Risk factors and loss distributions. Different risk measures such as Value at risk (VaR), expected shortfall (ES). Methods to compute VaR and ES. Stylized facts of financial time series, time-series models and estimation. Measuring and monitoring volatility. Multivariate distributions, tests for multivariate normality, Normal mixture distributions, Copulas and measures of dependence. Fitting copulas to data. Modeling credit risk, structural models, reduced form models, credit ratings, credit derivatives. Mean-Variance Markowitz portfolio, Capital Asset Pricing model (CAPM), Active and passive portfolio management, Performance Measures.
MA635 Curves and Surfaces 3(3-1-0-5-3)
Plane and space curves: Definition examples, parametrized curves, regular curves, arc length, convexity and four vertex theorem, curvature, torsion and the Frenet-Serret formula. Fundamental theorems for plane and space curves.
Definition and examples of parametrized surfaces, regular/smooth surfaces, tangent and spaces, change of coordinates and orientability. Differentiability of functions defined between regular surfaces. Diffeomorphic surfaces.
First and second fundamental forms, surface area and divergence theorem and applications. Brower fixed theorem.
Normal and principal curvatures, Gaussian curvature and the Gauss map. Ruled and minimal surfaces, Rigid motion and isometries, The Gauss’s Theorema Egregium, Geodesic and existence of geodesics on surfaces. Geodesic on surfaces of revolutions. The exponential map. The Gauss-Bonnet Theorem for Simple Closed Curves, for Curvilinear Polygons and for Compact Surfaces.
MA703 Computational Partial Differential Equations: 4(3-1-2-6-4)
Error Analysis:
Introduction to Interpolation, differentiation and integration. Finite difference methods for Parabolic Equations: One space dimension, Convergence and stability analysis, two space dimensions. Elliptic Equations: Dirichlet, Neumann and Mixed problems. Hyperbolic equations: One space dimension, two space dimensions, first order equation, system of equations, Lax’s equivalence theorem, Lax-Wendroff explicit method, CFL conditions, Wendroff implicit approximation. Finite Element Methods.Spectral Methods.
MA717 Advanced Partial Differential Equations 3(3-0-0-6-3)
Review of basics of functional analysis and Lebesgue integrals, Lp−spaces and their properties, distribution theory, convolution and Fourier transform.
Sobolev spaces: Definition and examples, approximation and extension properties, Sobolev embedding theorems, Poincar´e inequality, Fractional order Sobolev spaces and trace theorem.
Elliptic PDEs: Existence of weak solution to elliptic boundary value problems, maximum principle and regularity results.
MA718 Evolutionary Game theory 3(3-0-0-6-3)*
Introduction: Social traps and simple games. Evolutionary stability – Normal form games – Evolutionary stable strategies (ESS) – Characterization of ESS; – The replicator equation – Nash equilibrium and evolutionary stable states – Nash equilibrium strategies – Perfect equilibrium strategies – Examples of replicator dynamics and the Lotka-Volterra equation – The rock-paper-scissors game; – Other game dynamics – Imitation dynamics – General selection dynamics – Best-response dynamics.
Adaptive Dynamics: The repeated Prisoner’s Dilemma – Stochastic strategies for the Prisoner’s Dilemma – Adaptive Dynamics for the Prisoner’s Dilemma – Adaptive dynamics and gradients; – Asymmetric games: Bimatrix games – A differential equation for asymmetric games – The case of two players and two strategies; Dynamics for bimatrix games – Partnership games and zero-sum games – Conservation of volume – Nash-Pareto pairs – Game dynamics and Nash-Pareto pairs.
The hypercycle equation – Permanence: The permanence of the hypercycle – The competition of disjoint hypercycles; – Criteria for permanence: Permanence and persistence for replicator equations – Necessary and Sufficient conditions for permanence; – Replicator networks – Cyclic symmetry.
Discrete dynamical systems in population genetics: The Hardy-Weinberg law – The selection model – The increase in average fitness – The mutation-selection equation – The selection-recombination equation – Fitness under recombination; Continuous selection dynamics: Convergence to a rest point – The location of stable rest points – Density dependent fitness – Mixed strategists and gradient systems; -The selection-mutation model – Mutation and additive selection.