AFS-II - Annual Foundation School - II (2026) Hyderabad

Speakers and Syllabus

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 Module 1, Dr. Pradipto Banerjee PB (Associate Professor, IIT Hyderabad) for Module 2, Prof. Narasimha Kumar NK (Professor, IIT Hyderabad) for Module 3, Dr. Pratyusha Chattopadhyay PC (Assistant Professor, BITS Hyderabad) for Module 4
• Course Associates: K. Ganapathy (KGA)(IITM), Tiasa Datta (TID) (IITH), Dr.Sohan Ghosh (SOG)(IISER Mohali), Dwipanjana Shit (DWS) (IITH), Dr. Sunil Pasupuleti (SUP)(IMSc Chennai), Srijan Das (SRD)(IISER Pune), Sushant Kala (SUK)(IMSc Chennai), Dhrubajyoti Das (DHD)(IISER Pune).

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 Bangalore) for Module 1,Prof. Venku Naidu VN (Professor, IIT Hyderabad) for Module 2, Dr. Tanmoy Paul TP (Associate Professor, IIT Hyderabad) for Module 3, Dr. S Aiyappan SA (Assistant Professor, IIT Hyderabad) for Module 4.
• Course Associates: Dr T. Renjith (TRE) (ISI Bangalore), Ankush Sonu K (ASK)(IITH),Archana M P (AMP) (IISc Bangalore), Aditi Chattaraj (ADC) (IITH), Ritesh Kumar (RIK)(IITH), Sumit Kumar (SMK)(JNU), Sainik Karak (SAK) (IITH), ,Dipika Robi Das (DRD)(IITH)

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: Prof. Mahuya Datta (MD)(Professor, ISI Kolkata) for Module 1, Dr. Bhakti Bhusan Manna BM (Associate Professor, IIT Hyderabad) for Module 2,Dr. Soumen Sarkar SS (Associate Professor, IIT Madras) for Module 3, Dr. Archana Morye AM (Assistant Professor, University of Hyderabad) for Module 4.
• Course Associates: Sourav Nayak (SON)(IITH), Arijit Mukherjee (ARM) (IITM), Atanu Manna (ATM) (IITH), Dr. Alok Kumar Sahoo (AKS) (Bhadrak College,Orissa), Souptik Chakraborty (SOC) (TIFR-CAM), Rahul Roy(RAR) (IITH),Kartik Mahata (KAM)(IITH), Anusree (ASR) (IITH).

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

Time Table

   Week -One   
Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
1 Jun RM1 NN -T- MI1 EK L TRM1 -T- TMI1
2 Jun CS1 MD -T- RM2 NN L TCS1 -T- TRM2
3 Jun MI2 EK -T- CS2 MD L TMI2 -T- TCS2
4 Jun RM3 NN -T- MI3 EK L TRM3 -T- TMI3
5 Jun CS3 MD -T- RM4 NN L TCS3 -T- TRM4
6 Jun MI4 EK -T- CS4 MD L TMI4 -T- TCS4
   Week -Two   
Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
8 Jun RM5 PB T MI5 SA L TRM5 T TMI5
9 Jun CS5 BM T RM6 PB L TCS5 T TRM6
10 Jun MI6 SA T CS6 BM L TMI6 T TCS6
11 Jun RM7 PB T MI7 SA L TRM7 T TMI7
12 Jun CS7 BM T RM8 PB L TCS7 T TRM8
13 Jun MI8 SA T CS8 BM L TMI8 T TCS8
   Week -Three   
Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
15 Jun RM9 NK T MI9 TP L TRM9 T TMI9
16 Jun CS9 SS T RM10 NK L TCS9 T TRM10
17 Jun MI10 TP T CS10 SS L TMI10 T TCS10
18 Jun RM11 NK T MI11 TP L TRM11 T TMI11
19 Jun CS11 SS T RM12 NK L TCS11 T TRM12
20 Jun MI12 TP T CS12 SS L TMI12 T TCS12
   Week -Four   
Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
22 Jun RM13 PC T MI13 VN L TRM13 T TMI13
23 Jun CS13 AM T RM14 PC L TCS13 T TRM14
24 Jun MI14 VN T CS14 AM L TMI14 T TCS14
25 Jun RM15 PC T MI15 VN L TRM15 T TMI15
26 Jun CS15 AM T RM16 PC L TCS15 T TRM16
27 Jun MI16 VN T CS16 AM L TMI16 T TCS16

• RMn : nth lecture in Rings and Modules
• TRMn: nth tutorial in Rings and Modules
• MIn : nth lecture in Measure and Integration
• TMIn: nth tutorial in Measure and Integration
• CSn : nth lecture in Curves and Surfaces
• TCSn: nth tutorial in Curves and Surfaces
• T : Tea
• L : Lunch
• CA: Research Scholars from many academic institutions, like IITS, IISERs, IMSc, IIT Hyderabad, UoH, etc..
• TRM: TRM1 - TRM 4: NN, KGA, TID. TRM5 - TRM 8: PB, SOG, DWS. TRM9 - TRM 12: NK, SUP, SRD. TRM13 - TRM 16: PC, SUK, DHD.
• TMI: TMI1 - TMI 4: EK, TRE, ASK. TMI5 - TMI 8: VN, AMP, ADC. TMI9-12: TP, RIK,SMK TMI13-16: SA, SAK, DRD.
• TCS: TCS1 - TCS4 - MD, SON, ARM. TCS5- TCS8 - BM, ATM, AKS. TCS9 - TCS13 - SS,SOC, RAR. TCS14- TCS18 - AM, KAM, ASR.

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