Master of Science in Applied Mathematics​

Introduction and derivation of real-life equations (vibration, diffusion, flows); solution methods for some linear and nonlinear PDE, separation of variables, Green's functions, Fourier and Laplace transforms.

Analytic functions, Cauchy's theorem and consequences, Mobius transformations, singularities and expansion theorems, maximum modulus principle, residue theorem, and its application, compactness and convergence in the space of analytic and meromorphic functions, elementary conformal mappings, Riemann mapping theorem, elliptic functions, analytic continuation, and Picard's theorem.

Linear transformations. Change of basis, transition matrix, and similarity. Nilpotent linear transformations and matrices. The canonical representation of matrices, Jordan canonical forms. Linear functionals and the dual space. Bilinear forms. Quadratic forms and real symmetric bilinear forms. Complex inner product spaces. Normal operators. Unitary operators. The spectral theorem.

Outer measure, measurable sets, measurable functions, Lebesgue integration, the Lebesgue dominated convergence theorem, Fatou's Lemma, Monotone convergence Theorem, Convergence in Measure, Continuity and differentiability of Monotone functions, The Lebesgue spaces, Duality, Riesz Representation theorem.

The student has to undertake and complete a research topic under the supervision of a faculty member. The thesis work should provide the student with an in-depth perspective of a particular research problem in his chosen field of specialization. It is anticipated that the student is able to carry out his research fairly independently under the direction of his supervisor. The student is required to submit a final thesis documenting his research and defend his work in front of a committee.

Metric and normed spaces. Convergence and completeness. Banach spaces. Linear operators. The dual space. Hilbert spaces and orthogonality. The Riesz representation theorem. Hilbert-adjoint operator. Self-adjoint and compact operators. Fundamental Theorems of Banach spaces include the Hahn-Banach theorem, Uniform boundedness theorem, Open mapping theorem, and Closed graph theorem. Strong, weak, and weak* convergence. Banach fixed point theorem and applications. Basic properties of the spectrum of linear operators.

Sobolev spaces in R, Sobolev spaces in R^n, Lax Milgram Lemma, Hille-Yosida Theorem, linear and elliptic problems, Weak formulation, Existence, regularity, maximum principle Heat equation, Wave equation.

The course presents the advanced analysis of nonlinear systems, with an emphasis on the geometric interpretation of dynamical systems, including linearization, nonlinear feedback control tools, special attention to the averaging technique and the asymptotic tools of perturbation theory, tools for stability analysis of nonlinear systems like Poincare' Stability, Lyapunov's method, and Uniform stability.

This course is designed for specialized topic areas in applied mathematics, which are not covered in the list of courses in the applied mathematics master program.

Convexity of sets and functions. Linear Programming: Theory of the Simplex method, Duality, and the dual Simplex method. Nonlinear Programming: Unconstrained optimization problems, Necessary and sufficient optimality conditions, Line search method, Steepest descent method, Newton's method, Optimization problems with equality and inequality constraints, Method of Lagrange multipliers, Necessary and sufficient KKT optimality conditions, Separable programming, Quadratic programming, Linear combinations method, Game Theory.

Introduction to statistical sciences. Displaying and summarizing Data. Logic, probability, and uncertainty. Discrete random variables and their Bayesian Inference. Continuous random variables and their Bayesian inference. Comparing Bayesian and frequentist inferences for different statistics. Robust Bayesian methods. Bayesian inference for multivariate normal and multiple linear regression model. Computational Bayesian statistics.

Simple linear regression. Residual Analysis, inference for model parameters. Multiple linear regressions with matrix approach Development of linear models. Inference about model parameters. Residuals Analysis. Analysis of variance approach. Model building and variable Selection of the best regression variables. Multicollinearity. Regression with qualitative variables. Using statistical packages to analyse real data sets. Case studies.

This course considers statistical techniques to evaluate processes occurring through time. It introduces students to time series methods and the applications of these methods to different types of data in various contexts (such as actuarial studies, climatology, economics, finance, geography, meteorology, political science, risk management, and sociology). Time series modelling techniques will be considered with reference to their use in forecasting where suitable. While linear models will be examined in some detail, extensions to non-linear models will also be considered. The topics will include: deterministic models; linear time series models, stationary models, homogeneous non-stationary models; the Box-Jenkins approach; intervention models; non-linear models; time-series regression; time-series smoothing; case studies. Statistical software R will be used throughout this course. Heavy emphasis will be given to fundamental concepts and applied work. Since this is a course on applying time series techniques, different examples will be considered whenever appropriate.

Existence and Uniqueness of solutions for Initial Value Problems and BVP's, One-Step and multistep Methods for Non-stiff Initial Value Problems IVPs, Adaptive Control of One-Step Methods, One-Step Methods for Stiff Ordinary Differential Equations ODE, Multistep Methods for ODE and IVPs, Boundary Value Problems for ODEs, Error analysis and stability of methods, Numerical programming, and implementation.

Finite Difference Method for Transport, Wave, Heat, and Poisson equations; Elliptic Partial Differential Equations; Sobolev Spaces; Weak Solutions; Finite Element Method.

The course introduces the generalized linear models that include categorical and discrete responses. It reviews the multiple linear regression models and covers the log-linear models, logistic regression for binary responses, and binomial and Poisson regression. It also includes mixed effects models, model selection and checking, and inference about model parameters of restricted and full data models. The R language, with many packages available that deal with the GLM, is used.

Topics include direct and iterative methods for solving linear systems; vector and matrix norms; condition numbers; least-squares problems.