PhD CoursesSubmitted by admin on Fri, 12/30/2016 - 10:25
Ordinary differential equations: Phase space, existence and uniqueness theorems, The method of successive approximations, dependence on initial conditions, Boundary value problems, Green’s functions, Sturm-Liouville problems.
Partial differential equations: First order partial differential equation; Cauchy problem and classification of second order equations, Laplace equation; Diffusion equation; Wave equation; Methods of solutions (variable separable method, integral transform method).
Real analysis: Metric Spaces, Completeness, Connectedness, Complete Review of Riemann Integral, Lebesgue measure, Lebesgue and convergence, LP spaces.
Complex analysis: complex-analytic functions, Cauchy’s integral formula, power series, Liouville’s theorem, maximum-modulus theorem, Isolated singularities, residue theorem, the Argument Principle, real integrals via contour integration. Mobius transformations, conformal mappings.
Introduction to Error: its sources, propagation and analysis. Root finding of non-linear equations.Systems of linear and non-linear equations.
Finite difference operators, Polynomial interpolation and error analysis, Hermite interpolation, Spline interpolation, B-Splines, Numerical differentiation, Numerical integration: Trapezoidal and Simpson’s rules, Newton-Cotes formula, Method of undermind coefficients and Gaussian Quadrature, Richardson Extrapolation.
Difference equations. Numerical solution of ordinary differential equations: Initial value problems: Numerical stability, Taylor series method, Euler and modified Euler methods and stability analysis, Runge-Kutta methods, Multistep methods, Predictor-Corrector method, convergence and stability. System of ordinary differential equations. Boundary Value Problems: Shooting and direct methods.
Kinematics of Fluids in Motion: Real Fluids and Ideal Fluids, Velocity of fluid at a point, Streamlines and Pathlines, Steady and Unsteady Flows, the Velocity Potential, Vorticity Vector, The Equation of continuity. Equation of motion of a fluid: Pressure at a point in fluid at rest, Pressure at a point in a moving fluid, Euler’s equation of motion, Bernoulli’s equation. Viscous and inviscid fluid, the Navier-Stokes equation of motion, rotational and irrational flows.
Theory of surface wave: Equation of Motion, Wave Terminology, Analytical solution of the wave problem, Dispersion relation of the wave motion, Classification of water waves, Particle motion and Pressure, Superposition of waves, Wave reflection and standing wave, Wave energy and group velocity, Wave Refraction, Wave Diffraction. Finite amplitude waves: Mathematical formulation, Perturbation method of solution. Linear and Nonlinear diffraction theory.
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.
Galois Theory: Field Extensions, Automorphisms, self Normal Extensions, Separable and Inseparable Extensions, The Fundamental Theorem of Galois Theory; Some Galois Extensions: Finite Fields, Cyclotomic Extensions, Norms and Traces, Cyclic Extensions, Hilbert Theorem 90 and Group Cohomology, Kummer Extensions; Applications of Galois Theory: Discriminants, Polynomials of Degree 3 and 4, Ruler and Compass Constructions, Solvability by Radicals; Infinite Algebraic Extensions, Infinite Galois Extensions, Krull topology, The Fundamental Theorem of Infinite Galois theory; Transcendental Extensions: Transcendence Bases, Linear Disjointness, Affine Algebraic Varieties, Algebraic Function Fields, Derivations and Differentials.
Rings and Ideals ;Rings and ring homomorphism , Ideals, quotient rings, Zero divisors, Nil potent Element ,Units, Prime Ideals , and maximal ideals, Nilradical and Jacobson radical, Operation on ideals, Extension and contraction ; Modules ; modules and modules homomorphism ,sub module ,Direct sum product of modules, restriction and extension of scalars , Exactness properties of the tensor product ,algebras ,tensor product of algebra ;Rings and modules of fraction,L ocal Properties Extended and contracted ideals in the rings of fractions, Primary decomposition, Integral dependence, The going up theorem Integrally closed integral domains, Valuation Topologies and completion, Filtrations, Graded Rings and modules, The associated graded ring , Artin Ress Theorem , Dimension Theory, Hilbert function , Dimensions theory of Noetherian local rings, Regular local rings, Transcendental dimension, Depth :M-Regular Sequences, Cohen Macaulay Rings.
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.
Introduction to Financial Derivatives: Futures and options Pricing with no-arbitrage principle, Black-Scholes equation for European style option, Quick introduction to stochastic processes, Basic facts about Brownian Motion and Martingales with applications in finance, Stochastic integration, Stochastic Calculus (Itô's formula, Girsanov theorem, Connection with PDEs) Black-Scholes formula, Greeks.
Prerequisite : Basic Knowledge in Single Variable Calculus
Sigma- rings, sigma algebra, measurable space, countability and sub – addivity of a measure, Borel measure, Lebesgue outer measure, measurable sets, construction of non-measureable set, the contor set. Measurable functions and their properties, almost everywhere property, approximation of measurable functions with the simple measurable functions and step functions. Egorov’s theorem and Lusin’s theorem. Lebesgue integral of functions, Lesbgue integral of integral functions and its linearity, Monotone convergence theorem, Fatou’s lemma, Dominated convergence theorem, Applications of convergence theorems. Singed measures, absolute continues functions and their properties, singular measures , Radon – Nikodym theorem with applications.
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, 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.
Mainfolds: Indentification (quotient) spaces and identification (quotient) maps; topology n-manifolds including surfaces, Sn, RPn, CPn. and lens spaces.
Triangulated manifolds: Representation of triangulated, closed 2-manifolds as connected sums of tori of projective planes.
Fundamental group and covering spaces: Fundamental group, functoriality, retract, deformation retract; Van Kampen’s Theorem, classification of surfaces by abelianizing the fundamental group, covering spaces, path lifting, homotopy lifting, uniqueness of lifts, general lifting theorem for maps, covering transformations, regular covers, correspondence between subgroups of the fundamental group and covering spaces, computing the fundamental group of the circle, RPn, lens spaces via covering spaces.
Simplicial homology: Homology groups, functoriality, topological invariance, Mayer-Vietoris sequence; applications, including Euler characteristic, classification of closed trangulated surfaces via gomology and via Euler characteristic and orientability; degree of a map between oriented manifolds, Lefschetz number, Brouwer Fixed Point Theorem.
Composition of knots, Reidemeister moves, links, Invariants of knots, Surfaces and Knots: Genus and Seifert surfaces, Torus knots knots and its properties, Setelite Knots, Hyperbolic Knots, Braid theory, Alexander polynomial, Bracket polynomial, HOMFLY polynomial, Jones polynomial, Vassiliev Invariants, Knot complements and 3-Manifolds.
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.
Basic concepts of stability theory.Evolution equations and formulation of the linear stability problem.Solution of the initial value problem and normal-mode analysis.Temporal stability of viscous incompressible flow.Spatially instability of viscous incompressible flow.Some general properties of Orr-Summerfield problem.Kelvin Helmholtz Instability. Rayleigh-Taylor Instability, Rayleigh-Benard Covection. Saffman-Taylor Instability.
Artinian and Noetherian Ring, Primitive rings, Radicals, Completely reducible modules, Completely reducible ring, Semiprime rings and their properties, Projective and injective modules, Rings of endomorphism of injective modules, Classical ring of quotients, Regular Ring of quotients, Tensor product of modules exact sequence.
Definition of Group Ring, Basic facts in Group Ring, Augmentation ideals, Partial Augmentations, Ideals in Group Ring, Units in Group Ring, Annihilators, Semiprime Group Rings, Prime Group Rings, Chain Condition in Group Rings, Linear identity, The Delta method, Dimension Subgroup, Polynomial identities, Crossed Products, Zero divisor free Group Ring.
Mathematical theory of hyperbolic conservations laws, first order wave equation, method of characteristics, Burger equation: Discontinuous solutions and expansion waves, Solutions to the Cauchy problem, Uniqueness and continuous dependence, Vanishing viscosity approximations, Hype rbol i c sys t em of PDE’s : ent ropy, symmetrizability, constant coefficient linear systems, definition of wave types, truly non linear fields and linearly degenerate fields, Lax criterion, Riemann problem. Finite differences method, stability, consistency and accuracy of numerical schemes, conservative schemes, Lax-Wendroff theorem. Numerical schemes for scalar equations: 1-D Godunov method, finite volume method, Examples from traffic flow, gas-dynamics and magneto hydrodynamics.
Introduction to Modeling and simulation in Ecology; Single Species Population Dynamics: Exponential and Logistic Growth; Structured Population Dynamics; Population Dynamics of Interacting Species: The Lotka-Volterra Predator-Prey Models and simulation; Qualitative analysis: Stability and phase plane analysis; Modeling of Infectious diseases: SIR models; Qualitative analysis of epidemic models: Computation of R0, stability, equilibria; Spatial Dynamics: Metapopulation models, Diseases in Metapopulation; Adding Stochasticity to models: Sample paths and stochastic differential equations, General stochastic diffusion processes; Key models in Behavioral Ecology: Diet- choice and foraging, Evolutionarily Stable Strategies, Search and predation.
Localization, Integral Dependence, Discrete Valuation Rings and Dedekind Rings, Fractional Ideals and the Class Group, Class numbers, Norms and Traces, Extensions of Dedekind Rings, Discriminant, Ramification, Norms of Ideals, Cyclotomic Fields, Lattices in Real Vector Spaces, The Unit Theorem and Finiteness of the Class Number; Valuations, Completions, Extensions of Nonarchimedean Valuations, Archimedean Valuations, Local Norms and Traces and the Product Formula; Decomposition and Inertia Groups, The Frobenius Automorphism, The Artin Map for Abelian Extensions; Moduli and Ray Classes, Dirichlet Series, Characters of Abelian Groups, L-Series and Product Representations, Frobenius Density Theorem.
Historical introduction, Bernoulli numbers, p-adic norm and p-adic numbers, Hensel's Analogy, Solving Congruences modulo prime power, Absolute values on the field of rational numbers, Completions with respect to p-adic norm, Exploring, Hensel's Lemma, Local and Global principle, p-adic interpolation: A formula for Riemann zeta function at even integers, p-adic L- functions, The p-adic Riemann zeta function as a Mellin-Mazur transform, p-adic distributions, Kummer's Conguences, Bernoulli distributions, Measures and integration, Leopoldt's formula for p-adic L-function, p-adic regulator, p-adic Gamma function and p-adic di-gamma function, p-adic Euler- Lehmer constant and their generalizations.
CLASSICAL BANACH SPACES: Sequence spaces Co,C, ?p, 1= p = 8 , particular properties of ?1 , ?8 , function spaces Lp, for 1= p= 8 , C (K) STRICT COVEXITY AND SMOOTHNESS: Strictly convex and uniform convex Banach spaces, Gateaux differentiability, Frechet differentiability, duality releation between convexity and smoothness.
PROXIMINAL SUBSPACES:M etric projections and their continuity properties, metric projections on Frechet and polihydral spaces, proximinality and strong proximinality their continuity properties, preduality maps,st rong proximinality via finite co-dimension.
Definition of virtual knots and links,de finition and properties of flat virtual knots, Reidemeister moves, virtual isotopic knots, Gauss codes, surface interpretations of virtual knots,lo ng virtual knots, parity and the odd writhe,in variants for virtual knots, Bracket polynomial, parity Bracket polynomial, Z-move, Jones polynomial, Arrow polynomial for virtual and flat virtual knots and links, Virtual braids, categorical structure for the virtual braid groups,A lexander theorem and Markov moves for virtual braids.
- Time-dependent one-dimensional convection-diffusion-reaction equations in finance
- Finite difference methods – theta methods:stability and convergence
- Modelling (local) stochastic volatility: an example of multi-dimensional PDEs with mixed derivative terms.
- Splitting schemes of ADI type: Stability and convergence
- Comparison with other methods: Locally one dimensional (LOD) method, IMEX method.
- Numerical example
- Risk-neural valuation
- Black-Scholes and beyond (Non-constant Volatility, stochastic interest rate, multi-asset options)
- American options: Early exercise and free boundary problem
- Exotic options: Path dependency
- Introduction to interest rate theory: Models and products
- Introduction to FX options: a case study