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

Syllabus:

• Chapter 10 [Rings]:
  GKB (4): 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]:
  DS (2): Factorization of Integers and Polynomials, Unique Factorization Domains, Principal Ideal Domains and Euclidean Domains, Gauss's Lemma, Explicit Factorization of Polynomials.
  CK (4): 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, Ideal Classes in Imaginary Quadratic Fields, Real Quadratic Fields.
• Chapter 12 [Modules]:
  OP (6): The Definition of a Module, Matrices, Free Modules, and Bases, The Principle of Permanence of Identities, Diagonalization of Integer Matrices, Generators and Relations for Modules, The Structure Theorem for Abelian Groups, Application to Linear Operators, Free Modules over Polynomial Rings.

Instructors:

1. Dr. Gurmit Kaur Bakshi (GKB), Professor, Punjab University Chandigarh.
2. Dr. Chanchal Kumar (CK), Professor, IISER Mohali
3. Dr. Om Prakash (OP), Professor, IIT Patna
4. Dr. Deep Singh (DS), Associate Professor, Dr. B. R. Ambedkar University, Delhi

Course Associates/Tutors:

1. Dr. Shashank Vikram Singh (SVS) (PDF), IISER Mohali
2. Mr. Shiva Barman (SB) (4th year Ph. D. student), IISER Mohali
3. 3.Mr. Nihal Khetan (NK), (3rd year Ph. D. student), CUP Bathinda
4. 4.Ms. Adeeba Zaidi (AZ), (5th year Ph.D. student), CUP Bathinda

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

Syllabus:

• Chapter 1 [Abstract Integration]:
  SKP (4): The concept of measurability, Simple functions, Elementary properties of measures, Arithmetic in [0, ∞], Integration of positive functions.
  SK (2): Integration of complex functions, The role played by sets of measure zero.
• Chapter 2 [Positive Borel Measures]:
  SPSK (6): Vector spaces, Topological preliminaries, The Riesz representation theorem, Regularity properties of Borel measures, Lebesgue measure, Continuity properties of measurable functions
•  Chapter 3 [L ^P -Spaces]:
  MMM (4): Convex functions and inequalities, The L P -spaces, Approximation by continuous functions.

Instructors:

1. Dr. Surinder Pal Singh Kainth (SPSK), Associate Professor, PU Chandigarh
2. Dr. Santosh Kumar Pamula (SKP), Assistant Professor, IISER Mohali
3. Dr. M. M. Mishra (MMM), Associate Professor, Hansraj College, University of Delhi
4. Dr. Sachin Kumar (SK), Associate Professor, CUP Bathinda

Course Associates/Tutors:

1. Dr. Rachna Agrawal (RA), Assistant Professor, Amity University, Mohali
2. Dr. Ankit Yadav (AY), Assistant Professor, Indira Gandhi University, Haryana
3. Mr. Aakash Gupta (AG), Research scholar, IISER Pune
4. Mr. Shivank Pal (SP), (2nd year Ph. D. student), CUP Bathinda

Topology II: Introduction to Curves and Surfaces, [Pressley] Chapters 1-11.

• Chapters 1-3 [Curves]:
  KG (4): Different kinds of curves, Reparametrization, Curvature and Torsion, Fundamental theorem of curves in plane and space, Isoperimetric Inequality, Four Vertex Theorem.
• Chapters 4-5 [Surfaces and The First Fundamental Form]:
  GS (4): 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 [The Second Fundamental Form and Curvatures]:
  PS (4): 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, 10-11 [Geodesics, Gauss’s Theorema Egregium and The Gauss-Bonnet Theorem]:
  BT (4): Geodesics as shortest paths and their properties, Geodesic equations and co- ordinates, Gauss’s Remarkable Theorem, Isometries, Codazzi-Mainardi Equations, Compact surfaces of constant Gaussian curvature, Gauss-Bonnet Theorem for simple closed curves, curvilinear polygons and compact surfaces, Singularities of vector fields, Critical points.

Instructors:

1. Dr. Krishnendu Gongopadhyay (KG), Professor, IISER Mohali
2. Dr. Pranab Sardar (PS), Associate Professor, IISER Mohali
3. Dr. Bankteshwar Tiwari (BT), Professor, BHU Varanasi
4. Dr. Gauree Shanker (GS), Professor, CUP Bathinda

Course Associates/Tutors:

1. Dr. Ashok Kumar (AK), (PDF), HRI, Prayagraj
2. Mr. Debattam Das (DD), (4th year Ph. D. Student), IISER Mohali
3. Mr. Rahul Mondal (RM), (4th year Ph. D. Student), IISER Mohali
4. Ms. Harmandeep Kaur (HK), (4th year Ph.D. student), CUP Bathinda

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 first edition of [Artin] correspond respectively to Chapters 7-12 and Chapters 15-16 of the second edition of the book which is freely available online.