» MTH100: Mathematics I  Linear Algebra

Description: This first level math course covers basics of linear algebra including vector spaces, matrix algebra, linear transformations, eigenvalues and eigenvectors, orthogonality, properties of symmetric matrices, positive definite matrices, and SVD. The course is developed with an aim to provide a strong foundation in linear algebra which will be used in the subsequent curriculum by both CS and ECE students. Time permitting, some applications of linear algebra in engineering disciplines will be introduced. The course also attempts to increase the mathematical maturity of students by introducing proofs and mathematical rigour.
Course Content: Systems of linear equations, row reduction and echelon forms, matrix equation of the form Ax = b, invertibility of matrices, Vector spaces and subspaces, linear dependence/independence, dimension, span, applications. Fundamental subspaces, Linear transformation, rank. Matrix of linear transformation, effect of change of basis, similarity transformation. Algebra of linear transformations. Determinants, properties of determinants, Cramers rule, volume, Eigenvalues and eigenvectors, diagonalization of a matrix, eigenvectors and linear transformations, complex eigenvalues, Orthogonality and least squares, inner product, length, orthogonal projections, GramSchmidt orthogonalization, QR decomposition. Symmetric matrices and Quadratic forms, diagonalization of symmetric matrices, positive definite matrices, SVD, application to image processing.
» MTH200: Introduction to Mathematical Logic

Description: This is an introductory course in mathematical logic. The subject is of interest to students in both Mathematics and Computer Science. Topics covered include propositional logic, first order logic, consistency, satisfiability, soundness, completeness, and compactness. We will also discuss some basic set theory and axiomatic number theory (Peano's arithmetic). If time permits we will discuss the famous incompleteness theorems of Gödel.
Course Content: Introduction, language of propositional logic, wellformed formulas, induction. Truthassignments, truthtables, parsing, alternative notations, the deduction theorem. Soundness and completeness of propositional logic. Compactness theorem for propositional logic. Language of firstorder logic, wellformed formulas, structures. Satisfiability and validity of formulas, models, logical implication, definability in a structure, equality.Consistency, soundness for firstorder logic. Completeness for firstorder logic. Compactness theorem for firstorder logic, applications of compactness theorem, proof systems. Theories and models, formal number theory. Formal number theory, naïve set theory, cardinality, axiom of choice. Set theory  naïve to axiomatic. Additional topics to be chosen from computability, the halting problem, and Gödel's incompleteness theorems.
» MTH201: Mathematics II  Probability & Statistics

Description: The course introduces students to probability theory and how it can be applied to statistical problems. They learn about probabilistic models that occur in common applications. They are introduced to probability mass and density functions and statistics like expectation, correlation, and covariance. This is followed by an introduction to sampling statistics like the sample mean and variance, statistical hypothesis testing, and parameter estimation.
Course Content: Set Theory, Experiments, Observations, Axioms of Probability, classical and frequentist methods, describing data sets, conditional probability, Bayes' rule, law of total probability, tree diagrams, Counting, Discrete RV, PMF, Common Discrete RV models, CDF, Expectation, Functions of Discrete RV, Second order statistics, Conditional PMF, Discrete to Continuous RV, CDF, PDF, Expectation, Commonly used models, Functions of continuous RV, RV(s) as functions of Gaussian RV, Conditioning, Sample mean and variance from data, Moment generating functions, Pairs of RV(s), Joint and marginal PMF and PDF. Independence, Correlation, Covariance, Scatter Plots, Calculating from data sets, Expected value of sum of two RV(s), Extension of pairs to vectors of RV(s), Vector representation, Independence, Expected value and covariance of sums of RV(s), Chebyshev Inequality, Sample mean and variance, Central Limit Theorem (Statement, intuition and application), Weak Law of Large Numbers (Statement, intuition and application), Sampling from a normal population, Sampling from a finite population, Maximum Likelihood Estimation, Interval estimates, Point estimator properties, Bayes estimator, Hypothesis Testing, Tests concerning Normal, Bernoulli and Poisson
» MTH203: Mathematics III  Multivariate Calculus

Description: This course covers topics in multivariable calculus, vector calculus and complex analysis. The course starts with an extension of concepts like limits, continuity and differentiation to functions of several variables. Partial differentiation is defined and applied. The idea of Taylor series is extended to functions of two variables. This is followed by a definition and application of integration in more than one dimensions: double and triple integrals. We then work with vector functions and develop the ideas of vector fields and differentiation and integration as applicable to vector calculus (gradient, divergence, curl, line and surface integrals, etc. The ideas of circulation and flux are developed and Green's, Stoke's and divergence theorems are covered. The final module of the course deals with complex numbers and functions and an extension of concepts of calculus to complex variables.
Course Content:Functions of several variables, limits and continuity in higher dimensions, partial derivatives, chain rule, Directional derivatives and gradient vectors, tangent planes and differentials, extreme values and saddle points, Lagrange multipliers, partial derivatives with constrained variables, Taylor's formula for two variables, Double integrals, Areas, moments and center of mass, Double integrals in polar form, triple integrals in rectangular coordinates, Masses and moments in three dimensions, triple integrals in cylindrical and spherical coordinates, substitutions in multiple integrals, Vectors in space, dot product, cross product, lines and planes in space; vector functions, Arc length and the unit tangent vector, curvature and the unit normal vector, torsion and the unit binormal vector, Line integrals, vector fields, gradient of a scalar field, work, circulation, flux, path independence, potential functions, conservative fields, divergence and curl of a vector field, Green's theorem, Surface area and surface integrals, parametrized surfaces, Stoke's theorem, Divergence theorem, Complex numbers, polar form, derivatives, analytic functions, CauchyRiemann equations, Exponential functions, trigonometric and hyperbolic functions, logarithms, general power, Complex integration, line integrals, Cauchy's theorem, Cauchy's integral formula, derivatives of analytic fucntions, Sequences, series, convergence tests, power series, functions given by power series, Taylor series.
» MTH210: Discrete Structures

Description: This is a basic course in discrete mathematics, tailored for students in the computer science and applied math program. The course is intended to give students an exposure to formal mathematical language, the notion of proofs, and basic discrete structures such as graphs and permutations, as well as discrete probability and the probabilistic method.
Course Content: Propositional and Predicate Calculus. Functions, Relations, Order. Combinatorial Counting, Estimates and asymptotic notation. Recurrences and solving recurrences,Graphs, Drawing graphs in the plane, Double counting, number of spanning trees. Finite projective planes, Ramsey Theory, Generating functions, Probabilistic method.
» MTH211: Number Theory

Description: This course is an elementary introduction to number theory with no algebraic prerequisites. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, Arithmetic functions, Lagrange's foursquares theorem and partitions.
Course Content: Divisibility: basic definition, properties, prime numbers, some results on distribution of primes, Congruences, Complete and reduced residue systems, theorems of Fermat, Euler & Wilson, application to RSA cryptosystem. Linear congruences and Chinese Remainder Theorem, Primitive roots and indices, Quadratic congruences, and Quadratic Reciprocity law, Arithmetical functions: examples, with some properties and their rate of growth, Numbers of special form: Perfect numbers; Mersenne Primes and Amicable numbers; Fermat's numbers; Fibonacci numbers, Diophantine equations: linear and quadratic, some general equations, Representation of integers as sums of squares, Partition: basic properties and results.
» MTH212: Abstract Algebra I

Description: Algebraic structures are of fundamental importance in mathematics and play an increasingly significant role in many application areas, including computer science and communication. This course introduces the main algebraic structures: groups, rings and fields. Fundamental concepts and properties of these structures are studied, both in the abstract and in relation to concrete families of structures. Furthermore, algebra is an excellent vehicle for training students in mathematical rigour, proofs, and problem solving
Course Content: Equivalence relation and partition. Groups and subgroups, Cyclic groups, Abelian groups, Matrix groups, Quaternionic group, Lagrange's theorem, normal subgroup, quotient groups, isomorphism theorems, Group actions, Permutation groups, Simple groups, Simplicity of the alternating group, Centralizers and Normalizers, the class equation and its applications, Dihedral groups, Sylow's Theorems and their applications, Direct product of groups, Fundamental Theorem of Finite Abelian Groups, Rings and Fields, Matrix rings and Group rings, units and zero divisors, ideals, quotient rings, principal ideal, prime ideal, maximal ideal, nil and nilpootent ideals, integral domain, Isomorphism of rings, Direct product of rings, Polynomial rings, The Chinese Remainder Theorem, Principal ideal domain, Euclidean domain, Unique factorization domain (basic algebraic properties only), Fields extenions, Finite fields. Some applications to coding theory.
» MTH240: Real Analysis I

Description: This course covers topics in real analysis, singlevariable and multivariable calculus, and vector calculus. The course starts with the real number system, then discusses sequences and series in detail. Singlevariable calculus is covered in detail with emphasis on rigour and proofs. In the final section of the course, concepts and methods of multivariable calculus are covered but with less emphais on proofs.
Course Content: Real Numbers: Countable and uncountable sets, Supremum, Infimum, Completeness Axiom, Archimedean Property, Density Property , Sequences and Series of real numbers: Definition of a sequence, bounded sequence, convergence & divergence of a sequence, Operations on sequences, Cauchy sequence, Cauchy’s criterion for convergence, Monotone sequence, BolzanoWeierstrass theorem, limit superior and inferior, examples of some special sequences. Definition of a series, convergence and divergence of series, absolute and conditional convergence, Alternating series. Comparison, Ratio, Root, Cauchy’s condensation and Leibnitz tests for convergence of series. Rearrangements, Riemann’s Theorem (statement only), Singlevariable Differential & Integral Calculus: Limits of realvalued functions, continuity, Boundary value theorem, Intermediate Value Theorem, uniform continuity, differentiability, Mean Value Theorems and applications, Taylor's theorem, L’ Hospital’s Rule, Taylor series and applications. Definition of Riemann integral, Properties of the integral, fundamental theorems, Applications to area, arc length, volume and surface area, Improper integrals of first and second kind, Beta and Gamma functions, Differentiation under the integral sign and applications, Multivariable Differential and Integral Calculus in several variables: Functions of several variables, limits, continuity, Partial derivatives, Mixed derivative theorem (statement only), chain rule, directional derivatives, Maxima & minima and method of Lagrange multiplies. Double & triple integrals, Fubini’s Theorem for regular domains (statement only), the Jacobian & change of variables, Applications to area, volume, mass and moments.
» MTH310: Graph Theory

Description: This course is aimed at giving students an introduction to the theory of graphs. The course will introduce concepts that are widely used such as matchings, colorings, etc and study relations between various graph parameters such as matching number, chromatic number, clique number, etc. The emphasis will be on common proof techniques, and applying them to prove properties of graphs in general and for specific families of graphs.
Course Content: Elementary notions: degree, directed, undirected, bipartite, complete, trees, cycles, paths, graphic sequence, isomorphism, spanning trees etc. Connectivity: Menger's theorem, notions such as strong connectivity, cut edges, cut vertices, bridges, finding blocks, etc. Matching (and covering): Hall's theorem, Konig's theorem, vertex cover, matching in bipartite graphs, matching in general graphs, factors, Cuts and Flows: maxflow/mincut: FordFulkerson Theorem
Coloring: vertex coloring chromatic number, clique number, independence number, clique cover, Brook's theorem, ddegeracy, edgecoloring, Vizing's theorem, existence of trianglefree graphs with arbitrarily large chromatic number, Paths and cycles: Euler tours, Hamiltonian cycles and paths, Chinese Postman Problem, TSP, Planarity: plane graphs, Kuratowski's theorem, dualilty, testing planarity, Euler's formula, 5coloring planar graphs, Ramsey theory: Ramsey numbers and generalizations (optional if time permits) » MTH311: Combinatorics and its Applications

Description: The aim of this course is to familiarize students with fundamental concepts in combinatorics, especially those used in enumeration. Topics covered include permutation groups, linear codes, Stirling and Bell numbers. Generating functions are introduced and their applications are discussed. Applications of group theory to enumeration: Burnside’s Lemma and Polya’s theory of counting are covered. Students are introduced to error correcting codes and linear codes over finite fields.
Course Content: Introductory Ideas: Solving Linear Recurrences, generating functions, Fibonacci numbers, Stirling numbers of the second kind, Formal Power Series, Bell Numbers, Catalan Numbers, Stirling numbers of the first kind, Applications of generating functions to unlabeled counting: Integer Partitions, Money changing problem etc. Permutation groups, symmetry groups, Burnside’s lemma with applications to enumeration, The cycle index and Polya’s theory of counting, Error correcting codes, Linear Codes, Graph enumeration/Systems of distinct Representatives/de Bruijn sequences (optional)
» MTH340: Real Analysis II

Description: Much of mathematics relies on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of socalled "analysis", which in many respects is just Calculus in very general settings. Real AnalysisII is a course that develops this basic material in a systematic and rigorous manner in the context of realvalued functions of a real variable. Topics covered are: Real numbers and their basic properties, Sequences: convergence, subsequences, Cauchy sequences, Open, closed, and compact sets of real numbers, Continuous functions and uniform continuity. Lebesgue out measure, Lebesgue integral, sigma algebra of Lebesgue measurable sets.
Course Content: Real number system, ordered set, LUB, GLB property, Metric space, open, closed, complete, connected, perfect sets, Sequence, cauchy sequence, complete metric space, Sequence, metric space (Contd.), Contruction of real numbers, Limits, continuity, Contraction map, fixed points on a complete metric space, Differentiability, Taylor's theorem, Sequence and series of functions, Uniform convergence and continuity, integration, differentiation, real, continuous function which is nowhere differentiable, Lebesgue Measure, Lebesgue measurable functions, Lebesgue integral.
» MTH341/MTH541: Complex Analysis

Description: This course gives students an introduction to the theory of a function of a complex variable. A function of a complex variable has some remarkable properties which do not hold for a function of a real variable. In this course, we discuss many such properties. Topics include complex numbers and their properties; analytic functions and their derivatives, integrals and power series expansions; singularities and zeros of a complex function. The course also covers applications to physics and engineering
Course Content: Review of Complex variables and Analytic functions, Elementary functions: Exponential, logarithmic, trigonometric and hyperbolic, Contour Integrals, Branch cuts, Antiderivatives, CauchyGoursat theorem, Cauchy Integral formula, Liouville’s theorem, Maximum modulus principle, Convergence of sequences and series, Taylor’s theorem, Laurent’s theorem, Absolute and uniform convergence, integration and differentiation of power series. Residues, Singularities, Cauchy’s Residue theorem, zeros and poles. Applications of residues, Jordan’s Lemma, Argument principle, Rouchés theorem, Conformal Mapping
» MTH343/MTH543: Introduction to Dynamical Systems

Description: Dynamical systems is the mathematical formalism that describes the time evolution of a point in an appropriate space. This formalism has lead to the development of a rich theory which is of interest to both pure and applied mathematicians as it has found application in physics, biology, chemistry and several branches of engineering. This course starts with a systematic study of features and mechanisms dictating behaviour of 1D dynamical systems (bifurcation, transitivity, periodic points, Devaney chaos) and their application to well known examples (logistic family, tent map, shift map, etc). The second half of the course is dedicated to the study of phenomena from dynamical systems of dimension greater than one. It discusses applications of the PerronFrobenius Theorem, concept of hyperbolicity, LotkaVolterra equations, and time permitting symbolic dynamical systems.
Course Content: Overview of dynamical systems; First examples, Newton's method; Attractors and Repellers; Cobweb diagrams, Bifurcation analysis; Examples: Logistic family of maps, Sharkovsky's Theorem, Singers theorem; Examples: Logistic family of maps, Equivalence, Transitivity, Devaney Chaos, Sensitivity to initial conditions; Examples: Logistic family of maps, tent map, sawtooth transformation, shift map., The PerronFrobenius Theorem (statement only) and its applications; Examples: Leslie model of population growth, Markov chains, Review of analysis (no proofs): Contraction mapping principle, Inverse function theorem, Implicit function theorem, Existence theorem for solutions of differential equations, Hyperbolicity; Examples: Arnold's cat map, LotkaVolterra equations; Rössler attractor; Lorenz system, Symbolic dynamics, Topological Entropy, connections to Information theory.
» MTH371: Introduction to Stochastic Processes

Description: The course provides an introduction to different types of stochastic processes and its application in the field of engineering. A short review of probability and distributions will be done. The concepts of random process, random walk and Markov chain will be introduced. Various examples will be discussed and simulations will be taught to illustrate random processes.
Course Content: Review of Probability and Statistics: Probability, Random variables, Conditional distributions, Characteristic functions, Distributions (discrete and continous), Asymptotic distributions, Random Process: Introduction, Random variables (Joint), Conditional distributions, Conditional expectations, Functions of random variables, Random Process:Sequence of random variables, Mean square estimation, Weak law of large numbers, Strong law of large numbers, Convergence in Probability, Central Limit Theorem, Stocahstic convergence, Random number generation, Stochastic Process: Introduction, Bernoulli process, Poisson process, Random walk, Covariance and Correlation, Stationarity, Discrete Time Markov Chain: Introduction, Transition probabilities, Classification of states, Steady state behavior, Transient states and Absorption probabilities, Branching Processes, Continous Markov Chain, Queing Theory (Time permitting)
» MTH373/MTH573: Scientific Computing

Description: This is an overview course to be offered to 3rd and 4th year undergraduate, and postgraduate students. The course is structured to systematically build on and provide an overview of several ideas and topics that comprise the basics of discretizations of continuous mathematics. In this setup, we will concern ourselves with computational as well as stability analyses of both methods and algorithms.We will begin with an introduction to scientific computing. Then we will analyze and study methods in numerical linear algebra: matrix factorizations, direct solution of linear systems, solutions of linear least square problems, and solutions to eigenvalue problems.This will be followed by solutions of nonlinear equations in 1d and then more generally. We will apply this learning to unconstrained optimization in 1d and again more generally. We will also delve into some constrained optimization and nonlinear least squares problems. The next part of the course will discuss polynomial interpolation (including using splines) of discrete data in 1d. This will be utilized in methods for numerical differentiation of sampled data, and for numerically carrying out integration in 1d (also known as quadrature).The last part of the course will deal with numerical solutions of initial and boundary value ordinary differential equations in 1d. This will be followed by a foray into numerical solution of model partial differential equations. A student registering for the MTH573 version of the course will be required to work on an additional Pass/No Pass project for getting the course credits. Graduate students, in particular, will not be allowed to register for the 373 listing of this course.
Course Content: Introduction: computer arithmetic, roundoff, error propagation, stability of algorithms; Linear systems: existence and uniqueness, sensitivity and conditioning, direct methods: Gaussian elimination and LU factorization, Cholesky for symmetric positive definite,Overview of iterative methods for linear systems, Linear least squares, existence and uniqueness, sensitivity and conditioning, normal equations, QR factorization (Householder, Givens and GramSchmidt), Singular value decomposition, eigenvalue problems, existence and uniqueness, sensitivity and conditioning, similarity transformation, power iteration and variants, deflation, QR iteration, Krylov subspace methods; generalized eigenvalue problems, nonlinear equations, existence and uniqueness (contraction mapping theorem), sensitivity and conditioning, bisection, fixed point iteration, Newton’s method and variations ,Methods for system of nonlinear equations (Newton’s, Broyden’s); Optimization problems, existence, uniqueness, sensitivity, and conditioning; unconstrained optimization (steepest descent, Newton’s, BFGS, conjugategradient), nonlinear least squares (GaussNewton, LevenbergMarquardt), Constrained optimization (sequential quadratic and linear programming); Interpolation: existence, uniqueness and conditioning; polynomial and piecewise polynomial interpolation, Integration: existence, uniqueness and conditioning; numerical quadrature (NewtonCotes, Gaussian, progressive and composite quadrature); numerical differentiation (finite difference and Richardson extrapolation), Initial value problems for ordinary differential equations (ODEs): existence, uniqueness and conditioning, Euler’s methods, accuracy and stability, stiffness, higherorder singlestep methods, RungeKutta methods (Heun’s, RK4), multistep methods (AdamsBashforth, AdamsMoulton); methods for systems of ODEs ,Boundary value problems for ODEs: existence, uniqueness, and conditioning; shooting method, finite differences, collocation, Galerkin’s method; overview of sparse linear solvers, Partial differential equations: introduction and classification; timedependent problems, finitedifference schemes for onedimensional parabolic and hyperbolic model problems (heat, advection and wave equations), Consistency and stability (Lax equivalence theorem), CFL condition, vonNeumann analysis and stability condition,Timeindependent problems, boundary conditions, finite difference methods for one and twodimensional Poisson’s, Linear finite element method for one and twodimensional Poisson’s
» MTH374/MTH574: Linear Optimization

Description: This course aims at introducing students to the application of optimization techniques to various areas of CSE and ECE. We will primarily focus on linear optimization (linear programming) and learn about the structural and algorithmic aspects of optimization problems. The theoretical assignments will aim at developing the necessary skills for analysing algorithms and formulation of LPs. Computational assignments will complement the theory by modeling realworld problems as linear programs and solve them using publicly available solvers. Towards the end of the course, we will briefly discuss convex programs and semidefinite programs (SDPs) with realworld applications and point to some of the existing solvers for this class of problems.
Course Content: Linear Algebra Review, Linear Optimisation Problems  Modeling through examples, Geometry of Linear Programming, Simplex Method, Duality Theory, Theorem of the Alternative, Ellipsoid Method/Interior Point Methods, Network Flow Problems, SDP , Integer Programming Formulations, Overflow.
» MTH510: Advanced Linear Algebra

Description: This course is designed to enhance the understanding of the principles underlying the subject and to prepare students to take more advanced courses in Mathematics and engineering (e.g. Machine learning, Advanced Matrix Theory, Algebraic Coding Theory).
Course Content:Review of Fields, Finite Fields, vector spaces, Direct sum of subspaces, linear transformations (vector space homomorphisms), Definition of linear algebra, Matrix of a linear transformation and change of basis. Isomorphisms, Quotient spaces, Fundamental theorem of vector space homomorphism, Linear functionals, Dual space, double dual, transpose of a linear transformation, Eigenvalues and eigenvectors, Diagonalization, Simultaneous triangulation and simultaneous diagonalization, The primary decomposition theorem, Generalized CayleyHamilton Theorem, Cyclic decomposition and the rational and Jordan canonical forms. Computation of invariant factors, Inner product spaces, unitary operators, spectral theorem for normal operators, polar decomposition, Bilinear and quadratic forms, Symmetric and skewsymmetric bilinear forms, Generalized inverse of a matrix and its applications.
» MTH512: Algebraic Number Theory

Description: An algebraic number field is a field obtained by adjoining to the rational numbers the roots of an irreducible rational polynomial. Algebraic number theory is the study of properies of such fields. This course will cover the following topics: number fields, rings of integers, factorization in Dedekind domains, class numbers and class groups, units in rings of integers , valuations and local fields, and if time permits decomposition of primes, and zeta functions of number fields. These tools provide solutions to several problems which are elementary to state but surprisingly difficult to resolve, including Pell’s equation, quadratic reciprocity, the two squares theorem and the four squares theorem (every positive integer is a sum of four square integers).
Course Content: Review of required tools from the theory of fields and rings, Field extensions, ideals, maximal ideals, prime ideals, Number Fields  Definitions and basic examples, Embeddings into the real and complex numbers, Field norms and trace, Rings of IntegersDefinitions and basic properties,Integral closures,discriminants, quadratic field extensions, cyclotomic fields, Unique factorisation of ideals Prime ideals in rings of integers of number fields, Unique factorisation into prime ideals, Geometry of numbers ( Lattices, The Minkowski bound), Failure of unique factorisation Examples, Definition and finiteness of the class group, examples of computing class numbers using Minkowski bound. Applications Applications to nonlinear Diophantine equations, Some cases of Fermat’s last theorem.
» MTH513: Abstract Algebra II

Course Description: This course discusses factorization theory in integral domains, and basic properties of solvable groups, nilpotent groups and rings with chain conditions. This course also introduces Galois Theory, which was initiated by Galois in the 19th century. As applications of Galois Theory, solvability of polynomials by radicals, and classical straightedge (ruler) and compass constructions will be discussed.
Course Content: Solvable groups, Nilpotent groups, JordanHolder Theorem, Factorization Theory in Integral domains: Prime and irreducible elements, Principal ideal domains, Euclidean domains, Unique factorization domains, Eisenstein's Irreducibility Criterion., Rings with Chain conditions: Noetherian rings, Artinian rings, Fields: Prime fields, Extension of fields, Algebraic extensions, Separable and Inseparable extensions, Cyclotomic polynomials and extensions., Galois Theory: Galois fields and their structure, The Galois group of a polynomial, Normal extensions and Fundamental Theorem of Galois Theory, Radical extensions and solvability of polynomials by radicals, applications to the classical straightedge and compass constructions.
» MTH542: Introduction to Functional Analysis

Description: Functional analysis is the next step after linear algebra. In linear algebra, you deal with vectors which lie in a finitedimensional space. What if these vectors come from an infinitedimensional space? Can we still apply the results we have learned in linear algebra or they breakdown? This course offers answers to some of these questions. When you go beyond linear algebra, you require a more rigorous analytical framework and that's why this course is called "Functional analysis". It is a vast subject and this course is designed to introduce students to this area. Functional analysis is a valulable tool in theoretical mathematics as well as engineering and it is at the very core of numerical simulation. The course should be accessible to any advanced BTech, MTech or PhD student of any department who has some training in linear algebra and real analysis.
Course Content: Review of vector spaces, Normed spaces, Linear operators, Dual spaces, HahnBanach theorem, Cantor intersection theorem, Baire Category theorem, Open mapping theorem, Closed graph theorem, Uniform boundedness principle, Weak, weak* and strong topologies, Alaoglu's theorem, Inner product spaces, Orthonormal bases, The adjoint operator, Riesz Representation theorem, L^p spaces, Sobolev spaces, Distributions, Distributional solutions of PDEs, Compact operators, Spectral theory of compact operators.
» MTH598: Numerical Partial Differential Equations

Description: This course will be provide an overview of two standard numerical methods for partial differential equations (PDEs). The focus will be on essential theoretical analysis as well parabolic and hyperbolic partial differential equations. This will be followed by a short foray into linear system solvers and finite difference scheme for twodimensional Poisson’s (elliptic) problem. The second part of the course will deal with finite element methods exclusively for elliptic problems. The core ideas in functional analysis, variational formulation, error analysis, and computer implementation will be presented for the onedimensional problem. This will be followed be a more practical treatment of twodimensional problems. The last part will consist of an overview of the specialized topics of mixed and adaptive finite element methods.as computer implementation. The first part will be on finite difference methods. Key numerical schemes and underlying theory will be provided for onedimensional.
Course Content: Introduction to partial differential equations (PDEs) including classification, initial and boundaryvalue problems, boundary conditions and common PDEs; Python tutorial, Overview of onedimensional parabolic PDEs (heat and convectiondiffusion equations); introduction to finite differences; explicit and implicit schemes for onedimensional parabolic equations; Consistency, stability and Fourier analysis; maximum principle in parabolic PDEs; Overview of onedimensional hyperbolic PDEs (advection equation); finite difference schemes (method of lines discretizations and Lax Wendroff schemes) for onedimensional hyperbolic PDEs; CourantFriedrichsLewy (CFL) condition, Lax equivalence theorem; vonNeumann analysis and stability condition, Order of accuracy of solution; dissipation and dispersion in finite difference schemes for advection equation, Overview of twodimensional elliptic PDEs (Laplacian); Maximum principle for Laplacians; reentrant corner singularities, Interregnum: Direct and iterative methods for linear system solution; finite differences for twodimensional Poisson’s; Sobolev norms and spaces; weak derivatives; variational formulation; finite element method in onedimensions and error estimates, Hilbert spaces; Riesz representation theorem; LaxMilgram theorem, Meshing; quadrature; twodimensional finite element spaces, Implementation of twodimensional linear finite element for Poisson’s; Use of FEniCS package for other elements, Adaptive and mixed finite elements in onedimension; FEniCS implementation
» MTH571: Integral Transforms and their Applications

Description: The course is designed as an introduction to the theory and applications of integral transforms to problems in linear differential equations, and to boundary and initial value problems in partial differential equations. The course assumes very limited knowledge of vector calculus, ordinary differential equations, complex variables contour integration, partial differential equations and continuum mechanics. Many new applications in applied mathematics, physics, chemistry, biology and engineering are included. This course will serve as a reference for advanced study and research in this subject as well as for its applications in the fields of neuroscience, signal processing, informatics and communications. The course is open to advanced undergraduates, honors, MTech and PhD students through instructor consent.
Course Content: Basic concepts of integral transforms. Fourier transforms: Introduction, basic properties, applications to solutions of ODEs, PDEs. Fourier sine and cosine transforms, application to solutions of PDEs, evaluation of definite integrals. Laplace transforms: Introduction, existence criteria. Laplace transforms: Convolution, differentiation, integration, inverse transform, Tauberian Theorems, Watson’s Lemma, solutions to ODEs, PDEs (IVP, BVP). Applications of joint FourierLaplace transform, definite integrals, summation of infinite series, transfer functions, impulse response function of linear systems. Fractional Calculus and its applications: Intro, fractional derivatives, integrals, applications. Integral transforms in fractional equations: Intro, laplace transform of fractional integrals & derivatives, fractional ODEs, integral equations . IVPs for fractional DEs, fractional PDEs, green’s function for fractional DEs. Mellin transforms: Intro, properties, applications; Generalized Mellin transforms.Hankel Transforms: Intro, properties and applications to PDEs. Hilbert Transforms: Intro, definition, basic properties, Hilbert transforms in complex plane, applications; asymptotic expansions of 1sided Hilbert transforms. Stieltjes Transform: definition, properties, application, inversion theorems, properties of generalized Stieltjes Transform. Z Transforms: Intro, definition, properties; dynamic linear system and impulse response, inverse Z transforms, summation of infinite series, applications to finite diff. eqns. Legendre transforms: Intro, definition, properties, applications. Hermite transforms: Intro, definition, properties. Radon transforms: Intro, properties, derivatives, convulsion theorem, applications, inverse radon transform.
» MTH599: Variational Calculus and their Applications

Description: Variational Calculus is the simplest and the most direct means of unifying all branches under the discipline of Applied Mathematics. This course assumes very limited knowledge of vector calculus, ordinary differential equations and basic mechanics. Many new applications in applied mathematics, physics, chemistry, biology and engineering are included. This course will serve as a reference for advanced study and research in this subject as well as for its applications in the fields of neuroscience, signal processing, informatics and communications. The course is open to advanced MTech and PhD students through instructor consent only.
Course Content: First variation: Intro., weak variations, Eulerian equations, Legendre test; Applications: Catenary, optics, geodesics on a sphere. Applications: Brachistochrone, minimal surfaces, fluid motion, Newton’s solid of minimum resistance; Principle of least action, discontinuous solutions. Second Variation: Intro., Jacobi’s accessory equation, conjugate points (kinetic foci), property of conjugate points. Analytical methods for conjugate points, conjugate points on catenary, parabolic trajectory, spherical geodesics, orbits under inverse square law. Generalization of 1st and 2nd variation: Intro., Maxima/minima of functionals of multiple variables, lemma on double integration, Application to other physical problems, theory of minimal surfaces. Relative max/min for isoperimetric problems: Several examples of relative max/min, subsidiary equations of nonintegral type, nonholonomic dynamical constraints, Isoperimetric problems using second variations. Principle of least action: Intro., degrees of freedom, holonomic & nonholonomic systems, conservative/nonconservative systems, Hamilton’s principle and proof. Lagrange’s equation of motion, energy equation for conservative force fields, special variation in externals, geodesics on hypersurfaces, Path of minimum time in streamflow. Hamilton’s principle in Relativity theory: Physical basis, Michelson & Morley experiment, spacetime continuum: Newtonian vs. relativistic concept, Hamilton’s principle in relativity mechanics, mass & energy in relativity mechanics. Applications to problems in elasticity: Illustration using Euler’s equation, RayleighRitz method, StrumLiouville functions, strepstrain relations, SaintVenant torsion problem, Applications via Trefftz maethod, Galerkin method, torsion in beams. Hilbert Integrals: Problems with variable endpoints, determination of focal points via geometric, analytic methods, fields of extremals, method of Caratheodory, Bliss condition. Strong variations: Weierstrassian function in simplified form, Weierstrassian theory for parametric form, Applications to geodesics on surfaces, special cases.