College Graduate Courses
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 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.
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.
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.
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.
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.
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.
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 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.
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?
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.
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
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,
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
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
Apart from these also Softwares like MATLAB, Simulink, PSpice,
Cadence, Synopsis, Mathematica, PBasic, MS Office etc.
Go to the top | Contact us