College Graduate Courses

Math | Engineering

Graduate Math Courses

A typical graduate course work covers topics such as the ones listed here, though the topics that follow are not entirely exhaustive.

Linear Algebra

Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, Dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps, Determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization. Spectral Theory for General Maps Finite Dimensions: The Eigenvalue Problem, Characteristic and Minimal Polynomials, Cayley-Hamilton Theorem, Spectral Mapping Theorem, Generalized Eigenvectors, Similarity Transformations, Similar Matrices. The Adjoint, Euclidean Structure on Linear Spaces. Vector norms, Orthogonal Projections & Complements, Orthonormal Basis, Matrix Norm, Isometry, Complex Euclidean Space. Spectral Theory for Selfadjoint Mappings, Quadratic Forms, Spectral Resolution, Orthogonal, Unitary, Symmetric, Hermitian, Skew-Symmetric, Skew-Hermitian and Positive Definite Matrices and Operators. Normal Maps, Commuting Maps and Simultaneous Diagonalization of Matrices. Rayleigh Quotient, The Minmax Principle.

Algebra

Basic concepts of groups, rings and fields. Symmetry groups, linear groups, Sylow theorems; quotient rings, polynomial rings, ideals, unique factorization, Nullstellensatz; field extensions, finite fields.

Representations of finite groups. Characters, orthogonality of the characters of irreducible representations, a ring of representations. Induced representations, Artin’s theorem, Brauer’s theorem. Representations of compact groups and the Peter-Weyl theorem. Lie groups, examples of Lie groups, representations and characters of Lie group. Lie algebras associated with Lie groups. Applications of the group representations in algebra and physics. Elements of algebraic geometry.

Number Theory

Introduction to elementary methods of number theory. Topics: arithmetic functions, congruences, the prime number theorem, primes in arithmetic progression, quadratic reciprocity, the arithmetic of number fields, approximations and transcendence theory, p-adic numbers, diophantine equations of degree 2 and 3.

Cryptography

The primary focus of this course is on definitions and constructions of various cryptographic objects, such as pseudorandom generators, encryption schemes, digital signature schemes, message authentication codes, block ciphers, and others time permitting. The class tries to understand what security properties are desirable in such objects, how to properly define these properties, and how to design objects that satisfy them. Once a good definition is established for a particular object, the emphasis will be on constructing examples that provably satisfy the definition. Thus, a main prerequisite of this course is mathematical maturity and a certain comfort level with proofs. Secondary topics, covered only briefly, are current cryptographic practice and the history of cryptography and cryptanalysis.

Topology

After introducing metric and general topological spaces, the emphasis will be on the algebraic topology of manifolds and cell complexes. Elements of algebraic topology to be covered include fundamental groups and covering spaces, homotopy and the degree of maps and its applications. Some differential topology will be introduced including transversality and intersection theory. Some examples will be taken from knot theory. Homology and cohomology from simplicial, singular, cellular, axiomatic and differential form viewpoints. Axiomatic characterizations and applications to geometrical problems of embedding and fixed points. Manifolds and Poincare duality. Products and ring structures. Vector bundles, tangent bundles, De Rham cohomology and differential forms.

Differential Geometry

Differentiable manifolds, tangent bundle, embedding theorems, vector fields and differential forms. Riemannian metrics and connections, geodesics, exponential map, and Jacobi fields. Generalizations of differential geometric concepts and applications.

Differential forms. Integration on manifolds. Sard's Theorem. DeRham cohomology. Morse theory. Submanifolds and second fundamental form. Applications to geometric problems.

Advanced Topics in Geometry

Asymptotic geometry is concerned with properties of metric spaces which are insensitive to small-scale structure. It is a well-known theme in many areas of mathematics, such as the geometry of Riemannian manifolds or singular spaces, geometric group theory, the theory of discrete subgroups of Lie groups, geometric topology (especially 3-manifolds), graph theory, and recently in theoretical computer science. The course will begin with asymptotic invariants such as growth rates, isoperimetric inequalities, coarse topology, and boundaries, followed by a discussion of Mostow rigidity and variants. Subsequent topics will chosen according to the interests of the audience.

Analysis

Functions of one variable: rigorous treatment of limits and continuity. Derivatives. Riemann integral. Taylor series. Convergence of infinite series and integrals. Absolute and uniform convergence. Infinite series of functions. Fourier series. Functions of several variables and their derivatives. Topology of Euclidean spaces. The implicit function theorem, optimization and Lagrange multipliers. Line integrals, multiple integrals, theorems of Gauss, Stokes, and Green.

Complex Variables

Complex numbers; analytic functions, Cauchy-Riemann equations; linear fractional transformations; construction and geometry of the elementary functions; Green's theorem, Cauchy's theorem; Jordan curve theorem, Cauchy's formula; Taylor's theorem, Laurent expansion; analytic continuation; isolated singularities, Liouville's theorem; Abel's convergence theorem and the Poisson integral formula. The fundamental theorem of algebra, the argument principle; calculus of residues, Fourier transform; the Gamma and Zeta functions, product expansions; Schwarz principle of reflection and Schwarz-Christoffel transformation; elliptic functions, Riemann surfaces; conformal mapping and univalent functions; maximum principle and Schwarz's lemma; the Riemann mapping theorem.

Ordinary Differential Equations

Existence theorem: finite differences; power series. Uniqueness. Linear systems: stability, resonance. Linearized systems: behavior in the neighborhood of fixed points. Linear systems with periodic coefficients. Linear analytic equations in the complex domain: Bessel and hypergeometric equations.

Functional Analysis

The course will concentrate on concrete aspects of the subject and on the spaces most commonly used in practice such as Lp(1<= p <= ?), C, C?, and their duals. Working knowledge of Lebesgue measure and integral is expected. Special attention to Hilbert space (L2, Hardy spaces, Sobolev spaces, etc.), to the general spectral theorem there, and to its application to ordinary and partial differential equations. Fourier series and integrals in that setting. Compact operators and Fredholm determinants with an application or two. Introduction to measure/volume in infinite-dimensional spaces (Brownian motion). Some indications about non-linear analysis in an infinite-dimensional setting. General theme: How does ordinary linear algebra and calculus extend to d=? dimensions?

Numerical Analysis

Floating point arithmetic; conditioning and stability; numerical linear algebra, including systems of linear equations, least squares, and eigenvalue problems; LU, Cholesky, QR and SVD factorizations; conjugate gradient and Lanczos methods; interpolation by polynomials and cubic splines; Gaussian quadrature. Computer programming assignments form an essential part of the course.

This course will cover fundamental methods that are essential for numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments form an essential part of the course. The course will introduce students to numerical methods for (1) nonlinear equations, Newton’s method; (2) ordinary differential equations, Runge-Kutta and multistep methods, convergence and stability; (3) finite difference and ;finite element methods; (4) fast solvers, multigrid method; (5) parabolic and hyperbolic partial differential equations.

Engineering Course Descriptions

Sensor Based Robotics

Robot Mechanisms, Robot arm Kinematics (direct and inverse kinematics), Robot Arm Dynamics (Euler-Lagrange, Newton-Euler, and Hamiltonian Formulations), Six DOF rigid body kinematics and dynamics, Quaternion, Nonholonomic systems, Trajectory planning, various sensors and actuators for robotic applications, End-Effector mechanisms, Force and Moment analysis, Introduction to Control of Robotic Manipulators.

Applied Matrix Theory

In-depth introduction to theory and application of linear operators and matrices in finite-dimensional vector space; Determinants, Eigen values and eigenvectors; Theory of Linear Equations; Canonical forms and Jordan Canonical form; Matrix analysis of Differential and Difference equations; Singular value decomposition; Variational Principles and Perturbation Theory; Numerical methods.

Linear Systems

Basic System concepts. Equations describing Continuous and Discrete-time Linear Systems; Time domain analysis, State Variables, Transition Matrix and Impulse Response; Transform Methods; Time-variable systems; Controllability, Observability and stability; SISO pole placement, observer design. Sampled data systems.

System Optimization Methods

Formulations of System Optimization problems; Elements of Functional Analysis Applied to System Optimization; Local and Global system optimization with and without constraints; Variational methods, calculus of variations, and linear, nonlinear and dynamic programming iterative methods; Examples and applications; Newton and Lagrange multiplier algorithms, convergence analysis.

System Theory and Feedback Control

Design of Single-Input-Output and Multivariable Systems in Frequency domain; Stability of interconnected systems from component transfer functions; Parameterization of stabilizing controllers; Introduction to optimization (Wiener-Hopf design).

State Space Design for Linear Control Systems

Topics to be covered include canonical forms; control system design objectives; feedback system design by MIMO pole placement; MIMO linear observers; the separation principle; linear quadratic optimum control; random processes; Kalman filters as optimum observers; the separation theorem; LQG; Sampled-data systems; microprocessor-based digital control; robust control. and the servo-compensator problem.

Applied Non-Linear Control Theory

Stability and stabilization for Nonlinear systems; Lyapunov stability and functions, input-output stability, and control Lyapunov functions. Differential geometric approaches for analysis and control of nonlinear systems: controllability, Observability, feedback linearization, normal form, inverse dynamics, stabilization, tracking, and disturbance attenuation. Analytical approaches: recursive Backstepping, input-to-state stability, nonlinear small-gain methods, and passivity. Output feedback designs. Various application examples for nonlinear systems including robotic and communication systems.

Introduction to Electrical Power Systems

Basic concepts: Single and Three-Phase circuits, Power triangle; Transmission lines parameters: Resistance, Inductance, Capacitance, Transformers, and Generators; Lumped-component pi-equivalent circuit representation; Per-Unit Normalization; symmetrical phase components; load-flow program.

Digital Signal Processing

Properties and applications of the discrete Fourier transform and FFT; Frequency measurement; Properties and design of linear--phase FIR digital filters by windowing, least-squares, and Minimax criterion; Spectral factorization and design of minimum--phase FIR filters; Design of recursive digital filters; Short--time Fourier transform; Finite precision effects; Multi-rate systems; Basic Spectral Estimation; Basic adaptive filtering (LMS algorithm); Computer-based exercises will be given regularly.

Mechatronics

Introduction to Theoretical and Applied Mechatronics, design and operation of Mechatronics systems; Mechanical, Electrical, Electronic, and Opto-electronic components; Sensors and Actuators including signal conditioning and Power Electronics; Microcontrollers--fundamentals, Programming, and Interfacing; and Feedback control. Includes structured and term projects in the design and development of proto-type integrated Mechatronic systems.

Apart from these also Softwares like MATLAB, Simulink, PSpice, Cadence, Synopsis, Mathematica, PBasic, MS Office etc.

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