Teaching

DISCIPLINE SPECIFIC CORE(DSC) – 1: FIELD THEORY 

Semester: First

Total credits: 4 (3 Lectures + 1 Tutorial)
Pre-requisite of the course: Basics of Groups and Rings  

Course Objectives: The primary objective of this course is to: 

•  Study the ancient problem of solvability of polynomials over the rational field. 

•  Understand a necessary and sufficient condition for the solvability of polynomials in terms of radical expressions.

•  Apply Galois theory to classical problems, such as the insolvability of the general quintic. 

Course Learning Outcomes: This course will enable the students to: 

•  Identify and construct examples of fields, distinguish between algebraic and transcendental extensions, characterize normal extensions in terms of splitting fields and prove the existence of algebraic closure of a field.

•  Characterize perfect fields using separable extensions, construct examples of automorphism group of a field and Galois extensions as well as prove Artin’s theorem and the fundamental theorem of Galois theory.

•  Classify finite fields using roots of unity and Galois theory and prove that every finite separable extension is simple.

•  Use Galois theory of equations to prove that a polynomial equation over a field of characteristic is solvable by radicals if and only if its group (Galois) is a solvable group and hence deduce that a general quintic equation is not solvable by radicals. 

Contents: Unit I: Fields and their extensions, Splitting fields, Normal extensions, Algebraic closure of a field.
Unit II: Separability, Perfect fields, Automorphisms of field extensions, Artin’s theorem, Galois extensions, Fundamental theorem of Galois theory.
Unit III: Roots of unity, Cyclotomic polynomials and extensions, Finite fields, Theorem of primitive element, and Steinitz’s theorem.
Unit IV: Galois theory of equations, Theorem on natural irrationalities, Radical extension and solvability by radicals. 

Suggested Readings:
[1] P. M. Cohn, Basic Algebra, Springer International Edition, 2003
[2] D. S. Dummit & R. S. Foote, Abstract Algebra, Wiley Student Edition, 2011
[3] T. W. Hungerford, Algebra, Springer-Verlag, 1981.
[4] N. Jacobson, Basic Algebra, Volume I, Dover Publications Inc., 2009. 
[5] I. Stwart, Galois Theory, CRC Press, 2015.

DISCIPLINE SPECIFIC CORE(DSC) – 2: INTRODUCTION TO TOPOLOGY 

Semester: First

Total credits: 4 (3 Lectures + 1 Tutorial)
Pre-requisite of the course: Knowledge of metric spaces.

Course Objectives: The primary objective of this course is to introduce: 

•  Basic principles of point-set topology, including bases and subbases for a topology. 

• Continuity, homeomorphisms, and different types of topologies, such as product and box topologies. 

•  Key notions of connectedness and local connectedness. 

• Compactness and its significance in topological spaces. 

Course Learning Outcomes: This course will enable the students to: 

• Analyze subsets of topological spaces by determining their interior, closure, boundary, and limit points, as well as identifying bases and subbases. 

•  Identify continuous functions between topological spaces, analyze mappings into product spaces, and compare topological properties of different spaces .

•  Evaluate the connectedness and path connectedness of the product of an arbitrary family of spaces. 

•  Understand key classifications of topological spaces, including Hausdorff spaces, first and second countable spaces, and separable spaces. 

• Explore advanced concepts such as limit point compactness and Tychonoff’s theorem.

Contents: Unit I: Topological spaces, Basis, Order topology, Subspace topology, Metric topology, Closed set and limit points, Hausdorff spaces.  
Unit II: Continuous functions, Homeomorphism, The box and product topologies, Metrizability of products of metric spaces, Connected and path connected spaces.  
Unit III: Locally connected and locally path connected spaces, Connectedness of product of spaces, First and second countable spaces, Separable spaces.  
Unit IV: Compact spaces, The Tychonoff theorem, Limit point compactness, Sequential compactness. 

Suggested Readings:
[1] J. R. Munkres, Topology, Updated Second Edition, Pearson, 2021.  
[2] T. B. Singh, Introduction to Topology, Springer Nature, 2019.  
[3] G. E. Bredon, Topology and Geometry, Springer, 2014.  
[4]  J Dugundji, Topology, Allyn and Bacon Inc., Boston, 1978.  
[5]  J L. Kelley, General Topology, Dover Publications, 2017.
 

DISCIPLINE SPECIFIC ELECTIVE(DSE) – 5 (i): ALGEBRAIC TOPOLOGY  

Semester: Third

Total credits: 4 (3 Lectures + 1 Tutorial)
Pre-requisite of the course: Basics of Topology.

Course Objectives: The primary objective of this course is to:

• Understand the concepts of homotopic maps, homotopy type, retracts, and deformation retracts, and their role in algebraic topology.

• Compute fundamental groups of basic topological spaces. 

• Acquire knowledge of covering projections, lifting properties, the Borsuk–Ulam theorem, and classification techniques of covering spaces.

• Learn free groups and free products to understand and apply the Seifert–Van Kampen theorem 

Course Learning Outcomes: This course will enable the students to:

•  Distinguish between spaces with the same homotopy type.

•  Compute fundamental groups of standard spaces such as n-sphere and punctured planes.  

• Apply the concept of fundamental groups to prove Brouwer’s fixed-point theorem and the Fundamental theorem of Algebra.

• Explain the lifting theorems and their implications in topology.  

• Classify covering spaces for given base spaces.

• Apply Seifert–Van Kampen theorem to compute fundamental groups of glued spaces. 
  
   

Contents: Unit I: Homotopic maps, Homotopy type, Retract and Deformation retract.  
Unit II: Fundamental group, Calculation of fundamental groups of n-sphere, cylinder, torus, and punctured plane, Brouwer’s fixed-point theorem, Fundamental theorem of Algebra.
Unit III: Covering projections, Lifting theorems, Borsuk–Ulam theorem, Classification of covering spaces.
Unit IV: Free products, Free groups, Seifert–Van Kampen theorem and its applications. 

Essential Readings :
[1] J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley Publishing Company, 1984.
[2] T. B. Singh, Introduction to Topology, Springer Nature Singapore, 2019.

Suggested Readings:
[1] G. E. Bredon, Topology and Geometry, Springer, 2014.
[2] W. S. Massey, A Basic Course in Algebraic Topology, World Publishing Corporation, 2009.
[3] J. J. Rotman, An Introduction to Algebraic Topology, Springer, 2011.
[4] T. B. Singh, Elements of Topology, CRC Press, Taylor & Francis, 2013.
[5] E. H. Spanier, Algebraic Topology, Springer-Verlag, 1989.  

MMATH18-401(i): Advanced Group Theory 

Semester IV 

Total Marks: 100 (70 End Semester Examination + 30 Internal Assessment)  

Course Objectives: 

•  The course aims to introduce the learner to the concepts of normal series, composition series and Zessenhaus lemma. 

•  A study of solvable groups, nilpotent group and fitting and Frattini subgroup will be conducted and the students will be introduced to free group, presentation of a group and properties of a free group. 

Course Learning Outcomes: After doing this course student will be able to 

• Prove Schreier’s refinement theorem and Jordan-Hölder theorem, and also to prove fundamental theorem of arithmetic using Jordan-Hölder theorem. 

•  Prove Hall’s theorem, Schur’s theorem and Burnside basis theorem.

•  Identify indecomposable spaces and prove Krull-Schmidt theorem.

•  Determine distinct presentations of a group. 

Contents: Unit I: Normal series, composition series, Zessenhaus lemma, Schreier’s refinement theorem, Jordan-Hölder theorem. 
Unit II: Solvable groups, derived series, supersolvable groups, minimal normal subgroup, Hall’s theorem, Hall subgroup, p-complements, central series, Schur’s theorem. 
Unit III: Nilpotent groups, Fitting subgroup, Jacobi identity, Three subgroup lemma, Frattini subgroup, Burnside basis theorem. Indecomposable groups, Fitting’s lemma, KrullSchmidt theorem, Semidirect product. 
Unit IV: Free group, Generators and relations of a group, Rank of free group, Projective and injective property of a free group, Free semigroup and representation of a quotient semigroup. 

Suggested Readings:
[1] T.W. Hungerford, Algebra, Springer-Verlag, 1981. 
[2] D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, 1996.
[3] J.S. Rose, A Course on Group Theory, Dover Publications, 2012. 
[4] J.J. Rotman, An Introduction to the Theory of Groups, Springer, 1995. 
[5] M. Suzuki, Group Theory-I, Springer, 2014. 

Previously taught courses:
M.Sc. Mathematics: 

• MMATH18-101: Field Theory

• MMATH18-201: Module Theory 

• MMATH18-202: Introduction to Topology 

• MMATH18-301(i):  Algebraic Topology

• MMATH18-401(i): Advanced Group Theory 

• MMATH18-401(iii): Simplicial Homology Theory

• MMATH18-403(ii): Differential Geometry 
  
  

B.Sc. Mathematics: 

List of courses taught at Satyavati College, Kirori Mal College and Keshav Mahavidyalaya at B.Sc. level:
Calculus, Algebra, Analysis, Differential Equations, Mechanics, Probability theory, Statistics, Multivariate calculus, etc.