{"id":773,"date":"2024-02-28T12:46:45","date_gmt":"2024-02-28T12:46:45","guid":{"rendered":"https:\/\/mysitedemo.in\/iit\/?page_id=773"},"modified":"2026-05-06T15:06:43","modified_gmt":"2026-05-06T09:36:43","slug":"phd","status":"publish","type":"page","link":"\/maths\/?page_id=773","title":{"rendered":"Ph.D."},"content":{"rendered":"\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
<\/div>\n\t\t\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t

Ph.D. in Mathematics<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t
\n\t\t\t\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\tThe department offers Ph.D. programs in recent and emerging research areas in pure and applied mathematics. More than 70 Ph.D. scholars are actively engaged in quality research in a wide range of topics in pure and applied mathematics. Since the inception of the department in 2009, the faculty members are supervising a number of Ph.D. students in all the relevant areas of Mathematics as well as its applications. In an academic year, we organize two rounds of Ph.D. interviews. Students cleared any of the national level examinations such as CSIR \/ UGC NET, NBHM, GATE, INSPIRE, etc are interviewed\/scrutinized for the selection.\n<\/font><\/span><\/span>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t
\n\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t

Ph.D. Courses<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\n\t\t\t\t\t
MA605 Introduction to Nonlinear Dynamics: 3-0-0-6-3 <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t
\n\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nNonlinear Equations:<\/strong> 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.\n

\nNonlinear Oscillations:<\/strong> 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.\n<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA608 Operator Theory: 3-1-0-5-3 <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t
\n\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nOperators 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.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA614 Applied Linear Algebra and Matrix Analysis 4(3-1-2-6-4) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t
\n\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nSpectral theorem with applications:<\/strong> Inner product spaces, Normal, Unitary and self-adjoint operators in finite dimensional spaces, Finite dimensional spectral theorem for normal operators.\n<\/p>\n\n

\nDecompositions:<\/strong> Orthogonal reduction, Range-Null space decomposition, orthogonal decomposition, Singular value decomposition, orthogonal projections, Least-square method.\n<\/p>\n\n

\nPositive and Stochastic matrices:<\/strong> Positive and Stochastic matrices, applications.\n<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA615 Elementary Number Theory 3(3-0-0-6-3)* <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\tContent to update.\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t
\n\t\t\t\t\n\t\t\t\t\t
MA616 Elements of Data Science 3(3-0-2-7-4) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nOverview of Probability and Statistics & statistical learning:<\/strong> Definition, principles and different types of statistical learning, assessing model accuracy, bias-variance tradeoff.\n<\/p>\n\n\n\n

\nRegression models:<\/strong> Simple linear and multiple linear and non-linear.\n<\/p>\n\n

\nResampling methods:<\/strong> Assessing model prediction quality, cross validation, bootstrap.\n<\/p>\n\n

\nModel selection and regularization:<\/strong> Dimensionality reduction, ridge and lasso.\n\n<\/p>\n\n

\nUnsupervised learning:<\/strong> Clustering approaches, K-means and hierarchical clustering\n\n\n<\/p>\n\n

\n\nSupervised learning:<\/strong> classification problem, classification using logistic regression, naive Bayes, classification with Support Vector Machines, neural networks.\n<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA617 Graph Theory 4(3-1-0-5-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nIntroduction to graphs and digraphs: <\/strong> Introduction to graphs, subgraphs, degrees and graphical sequences, Isomorphism, bipartite graphs, directed graphs.\n<\/p>\n\n\n\n

\nConnectivity, Minimum spanning trees, Shortest path problems: <\/strong> Connectivity and edge connectivity; Trees: characterizations, minimum spanning trees, counting the number of spanning trees, cayley\u2019s formula, shortest path problems.\n\n<\/p>\n\n

\nMatchings, Eulerian and Hamiltonian Graphs:<\/strong> Independent sets and covering. Matching in bipartite graphs, Hall\u2019s marriage theorem. Eulerian Graphs: Definition and characterization. Hamiltonian Graphs: Definition, necessary and sufficient conditions.\n\n<\/p>\n\n

\nColoring and Planarity: <\/strong> Graph Coloring: Vertex coloring, edge coloring, chromatic polynomials; Planarity: Planar and non-planar graphs, Euler formula and its consequences, dual of a graph, Kuratowaski\u2019s theorem.\n\n\n<\/p>\n\n

\nNetwork Flows, Triangulated Graphs, Application of Graph Theory:<\/strong> Network Flows; Triangulated Graphs; Applications of graph theory in different real world problems.\n<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA620 Discrete Mathematics 4(3-1-0-5-3)* <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\tContent to update.\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t
\n\t\t\t\t\n\t\t\t\t\t
MA621 Introduction to Calculus of Variations 3(3-0-0-6-3)* <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\tContent to update.\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t
\n\t\t\t\t\n\t\t\t\t\t
MA622 Combinatorics 3(3-0-0-6-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nCounting Principles and Generating Functions, The Method of generating Functions, Recurrence Relations:<\/strong> Linear Recurrence Relation, Binomial coefficients, Binomial theorem, Derangements, Involutions, Fibonacci Numbers, Catalan Numbers, Bell Numbers, Eulerian Numbers.\n<\/p>\n\n\n\n

\nThe Pigeonhole Principle, The Principle of Inclusion and Exclusion:<\/strong> 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\u2019s Lemma, P\u2019olya\u2019s Theory, The Cycle Index, Block Designs: Gaussian Binomial Coefficients, Introduction to Designs, Steiner triple system, Incidence Matrices, Fisher\u2019s inequality, Bruck-Ryser-Chowla Theorem, Coding Theory, Hamming sphere, Reed-Solomon codes.\n\n\n<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA623 Introduction to Knot Theory 3(3-0-0-6-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nIntroduction to Knots and Links:<\/strong> Knots, Knot Projections, Composition of Knots, Reidemeister moves, Links.\n\n<\/p>\n\n

\nInvariants:<\/strong> Classical Knot Invariants \u2013 bridge number, crossing number, unknotting number, linking number, colouring number.\n\n<\/p>\n\n

\nTabulating Knots:<\/strong> Dowker Notation for Knots, Conway\u2019s notation, Knots and Planar Graphs.\n\n\n<\/p>\n\n

\nSeifert Matrices:<\/strong> Seifert Matrices, Invariants from the Seifert Matrices.
\nTorus Knots:<\/strong> Torus knots, Seifert Matrix of a Torus Knot, Invariants of Torus Knots.
\nTangles and 2-Bridge Knots:<\/strong> Tangles, 2-Bridge knots and their properties.\nBraids:<\/strong> The braid group, the braid index, and its properties.
\nPolynomial Invariants:<\/strong> Bracket Polynomial, Alexander Polynomial, Jones Polynomial, Kauffman Polynomial, Amphichirality.
\nVirtual Knots:<\/strong> Introduction to Virtual Knots, Welded Knots and Spatial Graphs.\n\n<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA624 Basics in Coding Theory and Cryptography 3(3-0-0-6-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nElementary number theory:<\/strong> Divisibility, the Euclidean algorithm Congruence and some application.
\nFinite field:<\/strong> Construction of finite field and quadratic residue.
\nIntroduction to coding theory:<\/strong> Codes, hamming codes, hamming bounds, error correction.
\nDifferent codes:<\/strong> Linear codes, cyclic codes, reed solomon codes, and their error correction.
\nCryptosystem:<\/strong> Some simple cryptosystem, Enciphering matrices.
\nPublic key cryptosystem:<\/strong> Public key cryptography, RSA, discrete log, diffie hellmann key exchange, hash function.\n<\/p>\n\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA625 Calculus of Variation and Integral equations, 3 (3-0-0-6-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nCalculus of Variation:<\/strong> Introduction, problem of barchistochrone, isoperimetric problem, concept of extrema of a functional, variation and it\u2019s 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.\nTransversality 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.\n

\nIntegral Equations: <\/strong>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.\nBernstein polynomials and its properties. Solving integral equations by using Bernstein polynomials and general polynomial. Numerical method: Quadrature method for Integral equations. Green\u2019s function in integral equations.\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t

\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\n\t\t\t\t\t
MA626 Problem Solving Methods in Mathematics (1-0-4-4-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t
\n\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nProblems in:<\/strong> Real Analysis, Linear Algebra, Complex Analysis.
\nProblems in:<\/strong> Ordinary Differential Equations, Partial Differential Equations, Calculus of Variations.
\nProblems in:<\/strong> Numerical Analysis, Integral Equations.
\nProblems in:<\/strong> Algebra, Topology.\n<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA627 Theory of Computation 3(3-0-0-6-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t
\n\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nDFA, 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.<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA628 Financial Derivatives Pricing 3(3-0-2-7-4) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t
\n\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\nFinancial 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.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA629 Fuzzy Logic & Applications 3(3-0-0-6-3)* <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\n\nContent to update.\n<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA630 Introduction to Applied Statistical Methods 4(3-1-0-5-3)* <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t

\n\nContent to update.\n\n<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA632 Analysis on Manifolds 3(3-1-0-5-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t\n

\n\n\nReview of differentiability of functions of several real variables, chain rule. Inverse and Implicit function theorems and their applications. Plane and space curves.\n

\nDefinitions 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.\n

\nReview of integral calculus on Rn, Fubini\u2019s 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.
\n

\nDifferential forms and their integrations. Green\u2019s Gauss and Stoke\u2019s theorem and their applications.\n<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA633 Evolutionary PDEs 3(3-1-0-5-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t\n

\n\nBochner Integral:<\/strong> Weakly and strongly measurable functions, Pettis theorem, Bochner integrals, Banach space valued Lebesgue spaces and properties.\n

\nTime-dependent distributions and Sobolev spaces:<\/strong> Calculus on Distributions with values in Hilbert spaces, Sobolev spaces involving time and properties, embedding theorems.\n

\nLinear Parabolic PDEs:<\/strong> Definition and example of parabolic PDEs, notion of weak solution. Existence and uniqueness of weak solutions, maximum principles, Harnack inequality.\n

\nLinear Hyperbolic PDEs:<\/strong> 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.\n\n<\/p>\n\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA634 Financial Risk Management 4(3-1-0-5-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t\n

\nWhat 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.\n\n\n<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA635 Curves and Surfaces 3(3-1-0-5-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t\n

\n\nPlane and space curves:<\/strong> 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.\n

\nDefinition 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.\n

\nFirst and second fundamental forms, surface area and divergence theorem and applications. Brower fixed theorem.\n

\nNormal and principal curvatures, Gaussian curvature and the Gauss map. Ruled and minimal surfaces, Rigid motion and isometries, The Gauss\u2019s 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.\n\n<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA703 Computational Partial Differential Equations: 4(3-1-2-6-4) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t\n\n

\n\nError Analysis:<\/strong>

\n 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\u2019s equivalence theorem, Lax-Wendroff explicit method, CFL conditions, Wendroff implicit approximation. Finite Element Methods.Spectral Methods.\n<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA717 Advanced Partial Differential Equations 3(3-0-0-6-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t\n

\n\nReview of basics of functional analysis and Lebesgue integrals, Lp\u2212spaces and their properties, distribution theory, convolution and Fourier transform.\n

\nSobolev spaces:<\/strong> Definition and examples, approximation and extension properties, Sobolev embedding theorems, Poincar\u00b4e inequality, Fractional order Sobolev spaces and trace theorem.\n

\nElliptic PDEs:<\/strong> Existence of weak solution to elliptic boundary value problems, maximum principle and regularity results.\n\n<\/p>\n\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t

\n\t\t\t\t\n\t\t\t\t\t
MA718 Evolutionary Game theory 3(3-0-0-6-3) <\/div><\/span>\n\t\t\t\t\t\t\t\n\t\t\t<\/path><\/svg><\/span>\n\t\t\t<\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t
\n\t\t\t\t\t
\n\t\t
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t\n

\nIntroduction:<\/strong> Social traps and simple games. Evolutionary stability \u2013 Normal form games \u2013 Evolutionary stable strategies (ESS) \u2013 Characterization of ESS; \u2013 The replicator equation \u2013 Nash equilibrium and evolutionary stable states \u2013 Nash equilibrium strategies \u2013 Perfect equilibrium strategies \u2013 Examples of replicator dynamics and the Lotka-Volterra equation \u2013 The rock-paper-scissors game; \u2013 Other game dynamics \u2013 Imitation dynamics \u2013 General selection dynamics \u2013 Best-response dynamics.\n

\nAdaptive Dynamics:<\/strong> The repeated Prisoner\u2019s Dilemma \u2013 Stochastic strategies for the Prisoner\u2019s Dilemma \u2013 Adaptive Dynamics for the Prisoner\u2019s Dilemma \u2013 Adaptive dynamics and gradients; \u2013 Asymmetric games: Bimatrix games \u2013 A differential equation for asymmetric games \u2013 The case of two players and two strategies; Dynamics for bimatrix games \u2013 Partnership games and zero-sum games \u2013 Conservation of volume \u2013 Nash-Pareto pairs \u2013 Game dynamics and Nash-Pareto pairs.\n

\nThe hypercycle equation \u2013 Permanence: <\/strong> The permanence of the hypercycle \u2013 The competition of disjoint hypercycles; \u2013 Criteria for permanence: Permanence and persistence for replicator equations \u2013 Necessary and Sufficient conditions for permanence; \u2013 Replicator networks \u2013 Cyclic symmetry.\n

\nDiscrete dynamical systems in population genetics:<\/strong> The Hardy-Weinberg law \u2013 The selection model \u2013 The increase in average fitness \u2013 The mutation-selection equation \u2013 The selection-recombination equation \u2013 Fitness under recombination; Continuous selection dynamics: Convergence to a rest point \u2013 The location of stable rest points \u2013 Density dependent fitness \u2013 Mixed strategists and gradient systems; -The selection-mutation model \u2013 Mutation and additive selection.\n<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"

Ph.D. in Mathematics The department offers Ph.D. programs in recent and emerging research areas in pure and applied mathematics. More than 70 Ph.D. scholars are actively engaged in quality research in a wide range of topics in pure and applied mathematics. Since the inception of the department in 2009, the faculty members are supervising a […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"site-sidebar-layout":"no-sidebar","site-content-layout":"page-builder","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"disabled","ast-breadcrumbs-content":"","ast-featured-img":"disabled","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"footnotes":""},"class_list":["post-773","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"\/maths\/index.php?rest_route=\/wp\/v2\/pages\/773","targetHints":{"allow":["GET"]}}],"collection":[{"href":"\/maths\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"\/maths\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=773"}],"version-history":[{"count":774,"href":"\/maths\/index.php?rest_route=\/wp\/v2\/pages\/773\/revisions"}],"predecessor-version":[{"id":17103,"href":"\/maths\/index.php?rest_route=\/wp\/v2\/pages\/773\/revisions\/17103"}],"wp:attachment":[{"href":"\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=773"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}