1. Algebra II : Rings and Modules
• We will cover most of the material from Chapters 10, 11, and 12 of Michael Artin’s Algebra (Pearson, 1991). The detailed topics are given below.

• Chapter 10 (4 lectures): Definition of Rings, Examples, Ideals, Algebra of Ideals, Prime and Maximal ideals, Quotient Rings, Local Rings, Homomorphisms, Fundamental Theorems, Endomorphism Rings, Field of Fractions.
• Chapter 11 (8 lectures): Integral Domains, Factorisation, Euclidean Domains, Principal Ideals Domains, Unique Factorisation Domains, Gauss’s Lemma, Eisenstein Criteria;Primes in Z[i], Algebraic Integers, Factorisation in Imaginary quadratic Fields, Ideal Factorisation, Ideal Classes in Imaginary Quadratic Fields, Real Quadratic Fields, Applications to Diophantine Equations.
• Chapter 12 (4 lectures): Modules, Free Modules, Bases, Generators and Relations for  modules, Quotient Modules, Homomorphisms, Structure Theorem for Abelian groups, Modules over PIDs, Application to Linear Operators.

• Speakers: Prof. Nitin Nitsure NN (Retired Professor, TIFR Mumbai) for Module1, Dr. Pratyusha Chattopadhyay PC (Associate Professor, BITS Hyderabad) for Module 2 Dr. Pradipto Banerjee PB (Associate Professor, IIT Hyderabad) for Module 3, Prof. Narasimha Kumar NK (Professor, IIT Hyderabad) for Module4.
• Course Associates: Mr. Souman Manna (SOM)(IIT Hyderabad), Dr. Sohan Ghosh (SOG)(IISc Bengaluru), Dr. Sunil Kumar Pasupuleti (SUP)(CBS Mumbai), Ms. Dwipanjana Shit (DWS)(IIT Hyderabad).

References
1. Artin, Michael. Algebra, Pearson, 1991, Prentice-Hall of India, New Delhi, 2003.
2. Jacobson, N, Basic Algebra I and II, W. H. Freeman and Company, USA, 1974, 1980.–Indian Editions Published by Hindustan Publishing Corporation, Delhi, 1984.
3. Lang, Serge. Algebra. Revised third edition. Graduate Texts in Mathematics, 211. SpringerVerlag, New York, 2002.
4. Musili, C. Introduction to Rings and Modules. Revised third edition. Narosa Publishing House, New Delhi, 2003.

2. Analysis II: Measure and Integration
• We will cover most of the material in Chapters 1, 2 and 3 of Rudin’s Real and Complex Analysis. The detailed topics are given below.

• Chapter 1 (6 lectures) The concept of measurability, Simple functions, Elementary properties of measures, Arithmetic in [0, ∞], Integration of positive functions, Integration of complex functions, The role played by sets of measure zero.
• Chapter 2 (6 lectures) Vector spaces, Topological preliminaries, The Riesz representation theorem, Regularity properties of Borel measures, Lebesgue measure, Continuity properties of measurable functions
• Chapter 3 (4 lectures) Convex functions and inequalities, The Lp -spaces, Approximation by continuous functions.

• Speakers: Prof. EK Narayanan EK (Professor, IISc Bengaluru) for Module 1, Dr.S Aiyappan SA (Assistant Professor, IIT Hyderabad) for Module 2, Dr. Tanmoy Paul TP (Associate Professor, IIT Hyderabad) for Module 3, Prof. Venku Naidu VN (Professor, IIT Hyderabad) for Module 4.
• Course Associates: Mr. Ankush Sonu K (ASK)(IIT Hyderabad), Ms. Archana M P (AMP) (IISc Bengaluru), Mr. Ritesh Kumar (RIK)(IIT Hyderabad), Mr. Aakash R (AAR)(IIT Hyderabad), Dr. Shubham Bias (SHU)(IISc Bengaluru), Ms. Aditi Chattaraj (ADC) (IIT Hyderabad).

References
1. W. Rudin. Real and Complex Analysis. 3rd Edition, Tata McGraw-Hill Higher Education,New Delhi, 1987.
2. Stein, Elias; Shakarchi, Rami. Real Analysis. Princeton University Press, 2005.
3. Royden, R. L. Real Analysis. Macmillan Publishing Company, 3rd Ed., 1988.

3. Topology II: Introduction to Curves and Surfaces

We will cover most of the material in Chapters 1-11, (Chapter 9 on Minimal Surfaces may be left out), of A. Pressley’s Elementary Differential Geometry. The detailed topics are given below.

• Chapters 1-3 (3 lectures): Types of curves, Reparametrization, Curvature, Fundamental theorem of curves in plane and space, Isoperimetric Inequality, Four Vertex Theorem.
• Chapters 4-5 (3 lectures): Smooth and Quadric Surfaces, Orientability, Applications of Inverse Function Theorem, Lengths of curves on surfaces, Isometries and conformal mappings of surfaces, Equiareal maps.
• Chapters 6-7 (3 lectures): The Second Fundamental Form, Normal and principal curvatures with geometric interpretation, Gaussian and (constant) mean curvatures, Pseudo-
• sphere and flat surfaces, Gaussian of compact surfaces.
• Chapter 8-10 (5 lectures): Geodesics as shortest paths and their properties, Geodesic equations and co-ordinates. Plateu’s Problem, Gauss map on minimal surfaces, Gauss’s Remarkable Theorem, Isometries, Codazzi-Mainardi Equations, Compact surfaces of constant Gaussian curvature.
• Chapter 11 (2 lectures) Gauss-Bonnet Theorem for simple closed curves, curvilinear polygons and compact surfaces, Singularities of vector fields, Critical points.

• Speakers: Dr. Bhakti Bhusan Manna BM (Associate Professor, IIT Hyderabad) for Module 1, Prof. Mahuya Datta (MD)(Professor, ISI Kolkata) for Module 2, Dr. B. Subhash BS (Assistant Professor, IISER Tirupati) for Module 3, Dr. Archana Morye AM (Assistant Professor, University of Hyderabad) for Module 4.
• Course Associates: Mr. Sourav Nayak (SON)(IIT Hyderabad), Mr. Soumyadyuti Maiti (SMA) (IIT Hyderabad), Dr. Alok Kumar Sahoo (AKS) (Bhadrak College, Orissa), Dr. Arijit Mukherjee (ARM) (IIT Madras)

References
1. Pressley, Andrew. Elementary Differential Geometry. Springer, 2001.
2. Thorpe, J.A. Elementary Topics in Differential Geometry. Springer, 1979.