Syllabus:

Algebra II: Rings and Modules. [Artin] Chapters 10,11,12.

• Chapter 10 [Rings]: AB (2 Lectures): The Definition of a Ring, Formal Construction of Integers and Polynomials, Homomorphisms and Ideals, Quotient Rings and Relations in a Ring, Adjunction of Elements, Integral Domains and Fraction Fields, Maximal Ideals, Algebraic Geometry.
• Chapter 11[Factorization]: AB (2 Lectures): Factorization of Integers and Polynomials, Unique Factorization Domains, Principal Ideal Domains and Euclidean Domains, Gauss's Lemma, Explicit Factorization of Polynomials. ISG (4 Lectures): Primes in the Ring of Gauss Integers, Algebraic Integers, Factorization in Imaginary Quadratic Fields, Ideal Factorization, The Relation Between Prime Ideals of R and Prime Integers.
• Chapter 12 [Modules]: AB (4 Lectures): The Definition of a Module, Matrices, Free Modules, and Bases, The Principle of Permanence of Identities, Diagonalization of Integer Matrices, JKV (4 Lectures): Generators and Relations for Modules, The Structure Theorem for Abelian Groups, Application to Linear Operators, Free Modules over Polynomial Rings.

Instructors:

1. Dr. Anirban Bose (AB), Assistant Professor, SRM University AP
2. Prof. Indranath Sengupta (ISG), Professor, IIT Gandhinagar
3. Prof. J. K. Verma (JKV), Professor, IIT Gandhinagar

Course Associates

1. Dr. Ranjana Mehta (RM), Assistant Professor, SRM University AP
2. Dr. Repaka Subha Sandeep (RSS), SRM University AP
3. Dr. Arpita Nayak (AN), Assistant Professor, SRM University AP
4. Dr. Krishanu Roy (KRoy), Assistant Professor, SRM University AP

Analysis II: Measure and Integration: [Rudin] Chapters 1,2,3.

• Chapter 1[ Abstract Integration]: CB (5 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 [Positive Borel Measures]: NA (5 Lectures): Vector spaces, Topological preliminaries, The Riesz representation theorem, Regularity properties of Borel measures. SR (2 Lectures): Lebesgue measure, Continuity properties of measurable functions.
• Chapter 3 [ L^P -Spaces]: SR (4 Lectures): Convex functions and inequalities, The L ^P -spaces, Approximation by continuous functions.

 Instructors:

1. Dr. Choiti Bandyopadhyay (CB), Assistant Professor, SRM University AP
2. Prof. Nikita Agarwal (NA), Professor, IISER Bhopal
3. Dr. Senthil Rani K S (SR), Assistant Professor, IISER Berhampur

Course Associates:

1. Dr. Sandeep Kumar Verma (SKV), Assistant Professor, SRM University AP
2. Dr. M Radhakrishnan (MR), Assistant Professor, SRM University AP
3. Dr. Manab Kundu (MK), Assistant Professor, SRM University AP
4. Dr. Chaitanya J Kulkarni (CJK), Postdoctoral Fellow, IISER Mohalii

Topology II: Introduction to Curves and Surfaces, [Pressley] Chapters 1-11, (Chapter 9 on Minimal Surfaces may be left out).

• Chapters 1-3 [Curves]: KR (2 Lectures): Different kinds of curves, Reparametrization, Isoperimetric Inequality, Four Vertex Theorem.
• Chapters 4-5 [Surfaces and The First Fundamental Form]: KR (3 Lectures): Smooth and Quadratic Surfaces, Orientability, Applications of Inverse Function Theorem, Lengths of curves on surfaces, Isometries and conformal mappings of surfaces, Equiareal maps.
• Chapters 6-7 [The Second Fundamental Form and Curvatures]: RK (3 Lectures): The Second Fundamental Form, Normal and principal curvatures with geometric interpretation, Gaussian and (constant) mean curvatures, Pseudosphere and flat surfaces, Gaussian of compact surfaces, Gauss map.
• Chapters 8-10 [Geodesics, Minimal Surfaces, Gauss’s Theorema Egregium]: RK (2 Lectures): Geodesics as shortest paths and their properties, Geodesic equations and co-ordinates. AM (3 Lectures): Plateu’s Problem, Gauss map on minimal surfaces, Gauss’s Remarkable Theorem, Isometries, Codazzi-Mainardi Equations, Compact surfaces of constant Gaussian curvature.
• Chapter 11 [The Gauss-Bonnet Theorem]: AM (3 Lectures) Gauss-Bonnet Theorem for simple closed curves, curvilinear polygons and compact surfaces, Singularities of vector fields, Critical points.

Instructors:

1. Prof. Kashyap Rajeevsarathy (KR), Professor, IISER Bhopal
2. Dr. Ramesh Kasilingam (RK), Assistant Professor, IIT Madras
3. Dr. Archana Morye (AM), Assistant Professor, University of Hyderabad

Course Associates:

1. Mr. Malegaonkar Swapnil Deepak (MS), JRF, SRM University AP
2. Dr. Animesh Bhandari (ABh), Assistant Professor, SRM University AP
3. Dr. Amit Kumar Singh (AK), Assistant Professor, SRM University AP
4. Dr. Namrata Arvind (NArvind), Post Doctoral Fellow, IMSC Chennai
5. Mr. Arkadeepta Roy (AR), SRF, HRI Prayagraj

References:

• [Artin] Michael Artin, Algebra, Pearson, 1991, Prentice-Hall of India, New-Delhi 2003.
• [Pressley] Andrew Pressley, Elementary Differential Geometry Springer, 2001.
• [Rudin] W. Rudin, Real and Complex Analysis, Mc-Graw-Hill 3rd ed.
• [Stein-Shakarchi] R. Shakarchi and E. M. Stein, Real Analysis, Princeton University Press, 2005.
• Please note that Chapters 6-11 and Chapters 13-14 in the I-edition of [Artin] correspond respectively to Chapters 7-12 and Chapters 15-16 of the II-edition of the book which is freely available online.