> Graduate
> Undergraduate
> High School
> College Preparation
> Entrance Exam Prep
> Career Counseling
> Undergraduate
> High School
> College Preparation
> Entrance Exam Prep
> Career Counseling
College Graduate Courses
Math | Computer Science | 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.
Computer Science Course Descriptions
Foundations of Computer Science
Logic, Sets, functions, relations, asymptotic notation, proof techniques, induction, combinatorics, discrete probability, recurrences, graphs, trees, mathematical models of computation, undecidability.Introduction to Programming
An introduction to computer programming and problem solving. General topics covered include the fundamentals of programming, good software development practices and solving problems using computer programming. Specific topics include compiling, running and debugging a program, program testing, documentation, variables and data types, assignments, arithmetic expressions, input and output, top-down design and procedures, the random number generator, conditionals and loops functions, arrays, and an introduction to classes and object oriented programming.Object-Oriented Programming
An intermediate-level programming course teaching object-oriented programming in C++. Pointers, dynamic memory allocation, and recursion. Classes and objects including constructors, destructors, methods (member functions) and data members. Access and the interface to relationships of classes including composition, association, and inheritance. Polymorphism through function overloading operators. Inheritance and templates. The standard template library will be used to introduce elementary data structures and their use.Data Structures and Algorithms
Abstract data types and the implementation and use of standard data structures. Fundamental algorithms and the basics of algorithm analysis.Digital Logic and State Machine Design
Combinational and sequential digital circuits. An introduction to digital systems. Number systems and binary arithmetic. Switching algebra and logic design. Error detection and correction. Combinational integrated circuits, including adders. Timing hazards. Sequential circuits, flip-flops, state diagrams and synchronous machine synthesis. Programmable Logic Devices, PLA, PAL and FPGA. Finite-state machine design. Memory elements.Computer Architecture and Organization
A top-down approach to computer design. Computer architecture: introduction to assembly language programming and machine language set design. Computer organization: logical modules; CPU, memory and I/O units. Instruction cycles, the data-path and control unit. Hardwiring and microprogramming. The memory subsystem and timing. I/O interface, interrupts, programmed I/O and DMA. Introduction to pipelining and memory hierarchies. Fundamentals of computer networks.Operating Systems
Fundamental concepts and principles of operating systems. Batch, spooling, and multiprogramming systems are introduced. The parts of an operating system are described in terms of their functions, structure and implementation. Basic policies for allocating resources are also discussed.Introduction to Parallel and Distributed Systems
Basic issues and techniques of parallel and distributed computing. The material we cover will cover the spectrum from theoretical models of parallel and distributed systems to actual programming assignments.Design and Implementation of Programming Languages
The design of high-level programming languages, along with elements of the compiler technology used to translate those languages into executable code. Formal description of language syntax, parsing, memory management attributes of variables and their binding times, control and data abstraction mechanisms and object-oriented language features. Imperative and object-oriented languages, with brief introduction to functional and logic programming paradigms.Design and Analysis of Algorithms
Fundamental principles of the design and analysis of algorithms. Asymptotic notation, recurrences, randomized algorithms, sorting and selection, balanced binary search trees, augmented data structures, advanced data structures, algorithms on strings, graph algorithms, geometric algorithms, greedy algorithms, dynamic programming, and NP completeness.Software Engineering
Software engineering techniques to specify, design, test and document medium and large software systems. Design techniques include information engineering, object-oriented, and complexity measures. Testing methods such as path testing, exhaustive test models, and construction of test data. An introduction to software tools and project management techniques is presented.Scientific and Engineering Computing
Using programming skills to exploit the broad power of modern computing related to science and engineering disciplines. Computational techniques are taught in parallel with programming and problem solving methodologies. Students learn how to recognize a good or bad formulation of a problem, select the proper algorithm to solve a given computational problem and interpret the results; thus, learning to become intelligent users, rather than creators, of computational software. Computational developments that have the greatest influence on the development and practice of science and engineering in the last century. Course draws upon a variety of computational problems from the breadth of science and engineering to interest students and establishes the relevance of the computational problem solving approach.Secure Information Systems Engineering
An approach to secure information systems engineering is developed consistent with today’s vulnerabilities, threats and risks. Grounding is established in the basic security technologies and strategies in use today. A concept of security engineering is constructed for whole elements of the critical infrastructure (e.g., utilities, government services, financial services, etc.) including legacy environments, the Internet, wireless and the coming evolution of “ubiquitous computing.” Specifically.UNIX System Programming
Programming and system administration of UNIX systems. Covers Shell programming, special purpose languages, UNIX utilities, UNIX programming tools, systems programming and system administration.Assembly Language and Systems Programming
Internal representation of numeric and character data. Machine organization and machine language programming. Assembly language, assemblers. Assembly language programming: branching, arrays, lists, arithmetic and bit manipulation, macros, stacks, subroutines, parameter passing, recursion. Linking and loading, position-independent and reentrant code. Traps and interrupts.Introduction to Databases
Introduction to database systems and database approach as a mechanism for modeling the real world. The course will cover data models (relational, object-oriented), physical database design, query languages, query processing and optimization, as well as transaction management techniques. Implementation issues, object-oriented and distributed databases will also be introduced.Algorithms for Parallel and Distributed Systems
Covers the design, implementation and evaluation of algorithms for parallel and distributed systems. Scheduling and load balancing, parallel and distributed information retrieval and database operations, parallel scientific algorithms. Concurrency control. Security in distributed systems.Java and Web Design
Programmers familiar with C or C++ will learn how to develop Java applications and applets. Syntax of the Java language, object-oriented programming in Java, creating graphical user interfaces (GIU) using the Java 2 Platform technology event model, Java exceptions, file input/output (I/O) using Java Foundation Class threads and networking.Design and Analysis of Algorithms
Review of basic data structures and mathematical tools. Data structures: priority queues, binary search trees, balanced search trees. B-trees. Algorithm design and analysis techniques illustrated in searching and sorting: heap-sort, quick-sort, sorting in linear time, medians and order statistics. Design and analysis techniques: dynamic programming, greedy algorithms. Graph algorithms: elementary graph algorithms (breadth-first search, depth-first search, topological sort, connected components, strongly connected components), minimum spanning tree, shortest path. String algorithms. Geometric algorithms. Linear programming. Brief introduction to NP-completeness. Advanced design and analysis techniques: amortized analysis of algorithms. Advanced data-structures: binomial heaps, Fibonacci heaps, data structures for disjoint sets, analysis of union by rank with path compression. Graph algorithms: elementary graph algorithms, maximum flow, matching algorithms. Randomized algorithms. Theory of NP completeness and approach to finding (approximate) solutions to NP-complete problems.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.
Go to the top | Contact us
