II Sem

Galois Theory-M201

Paper - I 

Course Outcomes

1.Explain the fundamental concepts of advanced algebra and their role in modern mathematics and applied contexts 

2.Demonstrate accurate and efficient use of advanced algebraic techniques 

3.Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from advanced algebra 

4.Apply problem-solving using advanced algebraic techniques applied to diverse situations in physics, engineering and other mathematical sciemces 

Unit-I

•  Algebraic extensions of fields: Irreducible polynomials and Eisenstein criterion.

• Adjunction of roots.

• Algebraic extensions.

• Algebraically closed fields.

 Unit-II

• Splitting fields. 

• Normal extensions.

• Multiple roots.

• Finite fields.

• Separable extensions. 

 Unit-III

•  Galois theory: Automorphism groups and fixed fields. 

• Fundamental theorem of Galois theory.

• Fundamental theorem of Algebra. 

Unit-IV 

• Applications of Galois theory to classical problems.

• Roots of unity and cyclotomic polynomials. 

• Cyclic extensions.

• Polynomials solvable by radicals.

• Ruler and Compass constructions. 

Text Book: 

Basic Abstract Algebra by S.K. Jain, P.B. Bhattacharya, S.R. Nagpaul.

 References: 

[1] Topics in Algebra by I.N. Herstein. 

[2] Elements of Modern Algebra by Gibert& Gilbert. 

[3] Abstract Algebra by Jeffrey Bergen. 

[4] Basic Abstract Algebra by Robert B Ash. 

 Lebesgue Measure & Integration-M202

Paper-II

Course Outcomes

1.Define concepts within measurement theory 

2.Formulate and prove theorems in measurement theory 

3.Define basic concepts within Banach and operator theory 

4.Use above-mentioned theory within applications 

Unit-I

• Algebra of sets-Borelsets.

• Outermeasure.

• Measurable sets and Lebesgue  measure.

• Anon-measurable set.

• Measurable functions.

• Littlewood three principles.

Unit-II

• The Riemannintegral.

• TheLebesgue integral of a bounded function over a set of finitemeasure.

• The integral of a non-negative function.

• The general Lebesgue integral.

Unit-III

• Convergence in measure.

• Di erentiation of a monotone functions.

• Functions of bounded variation.

Unit-IV

• Differentiation of an integral-Absolutecontinuity.

• TheLp-spaces.

• The Minkowski and Holder's inequalities.

• Convergence and completeness.

TextBook:

Real Analysis (3rd Edition)(Chapters3,4,5)byH.L.RoydenPearsonEducation(LowPriceEdition).

References:

[1] LebesguemeasureandIntegration byG.deBarra.

[2] Measure andIntegral byRichardL.Wheeden,AnotoniZygmund.

 Complex Analysis-M203

Paper-III

Course Outcomes

1.Define a function of complex variable and carry out basic mathematical operations with complex numbers. 

2.Know the condition(s) for a complex variable function to be analytic and/or harmonic 

3.State and prove the Cauchy Riemann Equation and use it to show that a function is analytic 

4.Define singularities of a function, know the different types of singularities, and be able to determine the points of singularities of a function 

Unit-I

• Regions in the Complex Plane.

• Functions of a Complex Variable-Mappings-Mappings by the Exponential Function.

• Limits-Limits Involving thePoint at In nity.

• Continuity-Derivatives-Cauchy{RiemannEquations -Su cientConditionsforDi erentiability.

• Analytic Functions.

• Harmonic Functions.

• Uniquely Determined Analytic Functions.

• Reection Principle.

• The Exponential Function.

• The Logarithmic Function -Some Identities Involving Logarithms.

• ComplexExponents-Trigonometric Functions-Hyperbolic Functions.

Unit-II

• Derivatives of Functions w(t) -De niteIntegrals of Functions w(t).

• Contours-Contour Integrals-Some Examples -Examples with BranchCuts Upper Bounds for Moduli of Contour Integrals{Antiderivatives.

• Cauchy{GoursatTheorem-Simply Connected Domains-Multiply Connected Domains-Cauchy Integral Formula- An Extension of the Cauchy Integral Formula.

• Liouville'sTheorem.

• The Fundamental Theorem of Algebra-Maximum Modulus Principle.

Unit-III

• Convergence of Sequences.

• Convergence of Series.

• Taylor Series-Laurent Series.

• Absolute and Uniform Convergence of Power Series.

• Continuity of Sums of Power Series.

• Integration and Di erentiation of Power Series.

• Uniqueness of Series Representations.

• Isolated Singular Points.

• Residues-Cauchy's ResidueThe-orem -Residue at In nity.

• TheThreeTypes of Isolated Singular Points-Residues at Poles-Examples-

• Zeros of Analytic Functions-Zeros and Poles.

• Behavior of Functions Near Isolated Singular Points.

Unit-IV

• Evaluation of Improper Integrals.

• Improper Integrals from Fourier Analysis.

• Jordan'sLemma-Indented Paths

• De niteIntegrals nvolving Sines and Cosines.

• Argument Principle.

• Rouche's Theorem.

• Linear Transformations-TheTransformation w = 1=z - Mappingsby1=z . 

• Linear Fractional Transformations.

• AnImplicit Form-Mappings of the Upper Half Plane.

TextBook:

Complex Variableswithapplications byJamesWardBrown,RuelVChurchill.

References:

[1] ComplexAnalysisbyDennisG.Zill.

[2] ComplexVariablesbyStevanG.Krantz.

[3] ComplexVariableswithApplicationsbyS.Ponnusamy,HerbSilverman.

[4] ComplexAnalysisbyJosephBak,DonaldJ.Newman.

Topology-M204 

Paper-IV

Course Outcomes

1.The definitions of standard terms in topology

2.Reading and writing proofs in topology and a variety of examples and counterexamples in topology.

3.The fundamental group and covering spaces.

4.Students will understand the machinery needed to define homology and cohomology.

Unit-I

• TopologicalSpaces: The De nition and examples.

• Elementary concepts.

• Open bases and open sub bases.

• Weak topologies.

Unit-II

• Compactness: Compact spaces.

• Products of spaces.

• Tychono 's theorem.

• locally compact spaces-Compactness for metrics paces.

• Ascoli's theorem.

Unit-III

• Separation: T1 - spaces and Hausdor spaces.

• Completely regular spaces andn ormalspaces.

• Urysohn'slemma.

• The Tietze extension theorem.

• The Urysohn imbedding theorem.

Unit-IV

• Connectedness: Connected spaces.

• The components of a spaces.

• Totally disconnected spaces.

• Locally connected spaces.

TextBook:

IntroductiontoTopologyandModernAnalysis (Chapters 3,4,5,6)ByG.F.Simmon'sTataMcGraw

Hill Edition.

References:

[1] IntroductoryTopologybyMohammedH.Mortad.

[2] ExplorationsinTopologybyDavidGay.

[3] EncyclopediaofGeneralTopologybyHart,Nagata,Vanghan.

[4] ElementaryTopologybyMichaelC.Gemignani.

Theory of Ordinary Differential Equations-M205

Paper-V

Course Outcomes

1.Analyze real world scenarios to recognize when ordinary differential equations (ODEs) or systems of ODEs are appropriate, formulate problems about the scenarios, creatively model these scenarios (using technology, if appropriate) in order to solve the problems using multiple approaches, judge if the results are reasonable, and then interpret and clearly communicate the results.

2.Recognize ODEs and system of ODEs concepts that are encountered in the real world. 

3.Work with ODEs and systems of ODEs in various situations and use correct mathematical terminology, notation, and symbolic processes in order to engage in work, study, and conversation on topics involving ODEs and systems of ODEs with colleagues in the field of mathematics, science or engineering

4.Understand and be able to communicate the underlying mathematics involved to help another person gain insight into the situation.

Unit-I (Linear differential equations of higher order)

•  Introduction-Higher order equations.

• A Modelling problem, Linear Independence.

• Equations with constant coe cients

• Equations with variable coe cients.

• Wronskian.

• Variation of parameters.

• Some Standard methods.

Unit-II (Existence and uniqueness of solutions)

• Introduction-Preliminaries.

• Successive approximations,Pi-card's theorem.

• Continuation and dependence on intial conditions.

• Existence of solutions in the large.

• Existence and uniqueness of solutions of systems.

•  Fixed point method.

Unit-III (Analysis and methods ofnon-linear differential equations)

•  Introduction-Existence theorem.

•  Extremal solutions-Upper and Lower solutions.

•  Monotone iterative method.

•  method of quasi linearization.

•  Bihari'sinequality,Application of Bihari's inequality.

Unit-IV (Oscillation theory for linearDifferential Equation of Second order)

• The adjoint equation-Self adjoin.

• linear di erential equation of second order.

• Abel'sformula.

• The number of zeros in a niteinterval.

• The sturm separation theorem.

• The strum comparison theorem.

• The sturm picone theorem.

• Bocher Osgood theorem.

• A special pair of solution-Oscillation on half axis.

TextBook:

[2] TextbookofOrdinaryDi erentialEquation, ByS.G.Deo,V.LakshmiKantham,V.Raghavendra,

TataMc.GrawHillPub.CompanyLtd.

References:

[1] TextBookofOrdinaryDi erentialEquations byEarlACoddington.

[2] Di erentialEquations byEdward,Penny,Calvis.

[3] Di erentialEquation byHarryHochstardt.