Annual Foundation School - Part III (2014)

Venue: Department of Mathematics, University of Pune
26th May to 21st June, 2014


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name S. A. Katre A. R. Shastri V. V. Joshi
Mailing Address Dept of Mathematics,
Univ. of Pune, Pune-7
Dept of Mathematics,
IIT Bombay,
Dept of Mathematics,
Univ. of Pune,


Speakers and Syllabus 

Each resource person has taken four lectures of 90 minutes each covering one module during one week. This was supported by two tutorial sessions of two hours each. The final timetable is enclosed herewith.

Name and affiliation of resource persons:

  • Diganta Bora IISER, Pune,
  • S. S. Bhoosnurmath Karnatak University, Dharwar
  • Ganesh Kadu Pune University, Pune
  • S. A. Katre Pune University, Pune
  • Dilip Patil IISc, Bangalore
  • Manoj Keshari IITB, Mumbai
  • Anant R. Shastri IITB, Mumbai
  • Parvati Shastri University of Mumbai
  • B. Subhash NIISER, Bhuvaneshwar
  • B. N. Waphare Pune University, Pune

Course Associates

  • Rajiv Garg IITB, Mumbai
  • Dipankar Ghosh IITB, Mumbai
  • Jai Laxmi IITB, Mumbai
  • Ojas Sahasrabudhe IITB, Mumbai
  • Prasant Singh IITB, Mumbai


  • Field Theory: Ist week by Prof. S. A. Katre
  1. Field Extensions and Examples. Algebraic and transcendental elements. Minimal polynomial. Degree of a field extension. Finite and infinite extensions. Simple extensions. Rationalising the denominator.
  2. Transitivity of finite/algebraic extensions. Compositum of two fields. Algebraically closed fields. Uncountably many algebraically closed subfields of the field of complex numbers
  3. Ruler and Compass constructions (in brief), Regular polygons. Squaring the circle etc.
  4. Existence of a splitting field of a polynomial.
  5. Symmetric polynomials. Newton’s theorem.
  6. Fundamental Theorem of Algebra using symmetric polynomials. Luroth’s theorem (Covered by Prof. Avinash Sathaye in a special lecture)
  • Field Theory: IInd week by Prof. B. N. Waphare
    In my lecture series we studied separable and inseparable extensions. An algebraic extension can be achieved in two stages: first, separable extension and then purely inseparable extension. The degree of a finite purely inseparable extension is a power of the characteristic of the field . The characterization of perfect fields and normal extensions are studied. The multiplicative group of a finite field is cyclic and determination of all subfields of a finite field are taken into consideration.


  • Field Theory: IIIrd week by Prof. Parvati Shastri
    Lecture 1. Galois extensions, equivalent conditions, examples, Galois group of quadratic, bi-quadratic extensions and extensions like Q(3√2, ω), where ω is a primitive cube root of unity, Galois group of finite a extension of a finite field.
    Lecture 2. Cyclotomic fields, cyclotomic polynomials, Galois group of cyclotomic fields; Artin’s theorem on the fixed field of a finite automorphism group of a field.
    Lecture 3. Fundamental theorem of Galois Theory (FTGT), illustrations, Galois group of separable cubic polynomials, roots of unity in any field, cyclicity of radical extensions K( n√a), a ∈ K,  when the ground  field K contains distinct nth roots of unity.
    Lecture 4. Applications of FTGT: Symmetric functions, symmetric functions are rational functions of elementary symmetric functions, discriminant of  polynomial, criterion for a Galois group to be subgroup of the alternating group, proof of Fundamental Theorem of Algebra via FTGT, constructibility of regular n-gons (Fermat primes), solution of inverse Galois problem for finite abelian groups.


  • Field Theory: IVth week by Prof. Dilip Patil
    Starting with equivalent definitions of Galois extensions, in some examples Galois groups were calculated using groups actions. A characterisation of Cyclic Galois extensions using Hilbert’s theorem 90 was done. Roots of unity, Galois groups of Cyclotomic extensions and Galois groups of finite field extensions were discussed. At the end solvability of radicals of polynomials over arbitrary fields and their relations to Galois groups of their splitting fields was done. The last part was sketchy and not all complete proofs were given as lack of time and prerequisites of the participants. Many of them were fresh even in AFS III without attending AFS I or AFS II. I hope these comments are taken in a right spirit and used to improve further AFSs.


  • Complex Analysis: Ist week by Dr. Diganta Borah


  1.  Definition of complex numbers, topology of the complex plane, the Riemann sphere
  2.  Complex differentiability, Cauchy-Riemann equations, complex versus Fr ́echet differentiability
  3.  Power series, Abel’s lemma, uniform convergence of power series, differentiation of power series
  4.  Exponential and trigonometric functions, branch of logarithm, branch of complex powers
  5.  Conformal mappings, M ̈obius transformations


  • Complex Analysis: II nd week by Dr. Ganesh Kadu
    Complex Integration (definition and some basic properties of contour integrals), Differentiation under integral sign (without proof), Existence of primitives, Primitive existence theorem, Cauchy-Goursat theorem, Cauchy’s theorem on convex region, Cauchy’s integral formula for derivatives, Morera’s theorem,Analyticity of complex differentiable functions, Taylor’s formula, Cauchy’s Estimate, Liouville’s theorem, Fundamental theorem of algebra, Gauss’ mean value theorem, maximum modulus theorem.


  • Complex Analysis: IIIrd week by Prof. A. R. Shastri
  1. Zeros of holomorphic functions, identity Theorem, open mapping theorem,
  2.  Isolated Singularities, Riemann’s removable singularity, Poles and essential singularities. Laurent series, residues.
  3.   Winding number, argument principle, Logarithmic residue theorem, Homotopy version of Cauchy’s theorem.
  4.  Homology version of Cauchy’s theorem, extension of residue theorems. Statement and sketch of proof of Riemann mapping theorem.
  • Complex Analysis: IVth Week by Prof. S. S. Bhoosnurmath
    Mean Value property, maximum principle, Schwarz’s reflection principle, Harmonic functions, Subhar-monic functions, Dirichlet’s problem, Green’s functions, Outline of a proof of Riemann mapping theorem. Comments to the Coordinator: From the students’ point of view and also from my own view
    point, material on harmonic functions from your book was very good and was extremely useful. Students’ response was very good. Some Participants felt that during 4th week some topics were much above their head. Can you help the students during the next AFS-III.


  • Topology: I st week by Prof. Manoj Keshri

                Four lectures of 90 minutes each.

Lecture 1. I introduced some examples asking them whether they are homeomorphic, and introduced homotopy of two maps, problem whether in a triagle, if two maps are given, can one find the third one so that the diagram commutes.

Lecture 2. I defined the cofibration, retraction, (strong) deformation retraction etc and relations between them ware done in tutorial.

Lecture 3. I introduced path homotopy, definition of π1(X) etc.

Lecture 4 was π1(S1) = Z (only statement), Van-Kampen theorem (statement) and some applications of V-K, e.g. π1 of torus, figure eight, R3 \ (x − axis ) etc.; Brouwer fixed point theorem was also done.


  • Topology: IInd week by Prof. A. R. Shastri


Lecture 1. Basic problem in Topology/algebraic topology: The workspace for algebraic topologists viz. CW-complexes. (By this time I realised that most of the students are very week in topology.) Many examples and only statements of basic topological properties of CW-complexes.

Lecture 2. Simplicial complexes, lots of examples, simplicial maps, barycentric subdivision.

Lecture 3. Simplicial approximation.

Lecture 4. Applications to Sperner lemma and Brouwer’s fixed point theorem. Brouwer’s invariance of domain (with proof of weaker version only, viz, Rn and Rm, m =! n are not homeomorphic).


  • Topology: IIIrd week by Dr. B. Subhash


My lectures started recalling some relevant material from the first week: The fundamental group of the circle Sl was computed highlighting the lifting properties of the exponential map exp : R → S1 In the second lecture, covering spaces were introduced and ample examples were discussed. Path and Homotopy lifting properties of the covering projection was discussed. The statement that covering projections are fibrations was proved.In the third lecture, the lifting problem was discussed with respect to the covering projection and necessary and sufficient conditions obtained for the same. The deck transformation groups were introduced, the relation of the deck transformation group with the fundamental group was discussed. Normal coverings were introduced and equivalent conditions for a covering to be normal were discussed.In the fourth lecture, the equivalence of coverings were introduced and the classification for coverings was taken up, leading to the one one correspondence between the equivalence classes of coverings and conjugacy classes of subgroups of the fundamental group for connected, locally path connected, semi locally simply connected spaces.


  • Topology: IVth week by Prof. A. R. Shastri


Lecture 1. Categories and functors, Chain complexes, homology and Snake lemma.

Lecture 2 Singular chain complex and statement of axioms: proof of the long exact sequence of a pair.

Lecture 3. Computation of homology using Mayer-Vietoris. Suspension theorem. Homology of spheres.

Lecture 4. Application to Brouwer’s fixed point theorem, invariance of domain and separation theorem etc.


Click here for tutorials and time table

Selected Applicants

Instructions to candidates:

  1. All further inquiries/communications, regarding the school should be addressed to Prof. S. A. Katre (
  2. Selected candidates should communicate the confirmation of  their participation on or before 2nd April 2014, along with a copy of their onward journey ticket where-ever applicable, failing which their selection will be cancelled automatically and some wait-listed candidate will be offered the chance.
  3. If you are a selected candidate  and  decide NOT to attend the school for any reason what-so-ever, at any stage even after after confirmation, please communicate this immediately to the organizers so that some wait-listed candidates may  benefit.
  4. Wat-listed candidates may therefore book their tickets so as not miss such a chance merely because they would not get a train-ticket.
  5. All the selected candidates should visit the following website immediately:
Sr. SID Full name Gender Affiliation State Position  University / Institute Year of passing Ph.d. degree date
1 5066 Mr. Santhakumar s M Periyar university Tamil nadu Research scholar T.d.m.n.s college, t. kallikulam 2012  
2 5071 Mr. Selvaganesh t M Periyar university Tamil nadu Phd, sivakasi. 2009  
3 5130 Mr. Vishal gupta M Swami vivekanand subharti university (svsu) Uttar pradesh Assistant profrssor IIT guwahati 2010  
4 5281 Ms.Sueet millon sahoo F National institute of science education and research Orissa Phd National institute of technology 2012  
5 5290 Ms. Moni kumari F Niser ,bbsr Orissa Phd Bhu 2012  
6 5377 Mr. Yogesh kumar M IIT Delhi Delhi Phd M.d.u. rohtak 2008  
7 5379 Mr. Chirag garg M IIT Delhi Delhi Student Punjabi university patiala / baba farid college, bathinda 2010  
8 5385 Mr. Rohit gupta M IIT Delhi Delhi Phd IIT Delhi 2011  
9 5386 Mr. Aashish kumar kesarwany M IIT Guwahati Assam Ph.d. IIT kanpur 2013  
10 5396 Mr. Vijay kumar yadav M Indian school of mines Jharkhand Ph.d Dr. ram manohar lohiya avadh university, faizabad 2003  
11 5401 Mr. Bharath kumar ethamakula M Indian institute of science, bangalore Karnataka Phd IIT, kanpur 2011  
12 5424 Ms. Manasa k j F National institute of technology karnataka Karnataka Phd Mangalore university, mangalore 2005  
13 5461 Mr. Debayudh das M Tifr mumbai Maharashtra Nbhm research scholar IIT, Madras 2013  
14 5488 Mr. Avijit nath M Institute of mathematical sciences Tamil nadu Nbhm ph.d. coursework student Ramakrishna mission vivekananda university 2013  
15 5497 Mr.Krishanu roy M Institute of mathematical sciences Tamil nadu Integrated ph.d student   appeared / awaiting result  
16 5503 Ms. Poonam kesarwani F Iit kanpur Uttar pradesh Phd IIT, Kanpur 2012  
17 5509 Ms. Meena pargaei F Indian institute of technology, kanpur Uttar pradesh Phd G.b.p.u.a.&t. pantnagar 2012  
18 5535 Ms. Sudha priya g F Periyar university Tamil nadu Research scholar Periyar university 2007  
19 5551 Mr.Nnirupam ghosh M Iit kharagpur West bengal Ph.d srudent IIT, Kanpur 2013  
20 5563 Mr. Jagannath bhanja M Sambalpur university Orissa First Utkal university 2012  
21 5568 Mr. Vaibhav pandey M National institute of science education and research Orissa Integrated student   appeared/ awaiting result  
22 5589 Dr. Sadiq basha M Anna university Tamil nadu Assistant professor Bharathidasan university 1990  
23 5618 Ms. Pooja punyani F Indian institute of technology, delhi Delhi Phd Universtiy of delhi/ st. stephens college/ mathematics 2012  
24 5629 Mr. Santosh kumar M Aligarh muslim university,aligarh Uttar pradesh Phd 2009  
25 5666 Ms. Srashti dwivedi F IIT delhi   Ph.d Delhi university 2012  
26 5680 Mr. Nabin kumar meher M HRI, allahabad Uttar pradesh Student IIT, Kanpur 2011  
27 5701 Mr. Koushik biswas M IIT ropar Punjab Phd Indian institute of technology, kanpur 2013  
28 5718 Ms. Kirandeep kaur F IIT ropar Punjab Phd G.n.d.u. amritsar 2011  
29 5721 Mr. Neeraj kumar tripathi M Indian institute of technology (bhu) varansi Uttar pradesh Research scholar Banaras hindu university 2009  
30* 5566 Mr. Nishad devendra mandlik M IISER pune Maharashtra Bs- ms fourth year student IISER pune appeared/ awaiting result  
* without accomodation
 Waiting list       
1 5050 Mr. Jalil sayyed yousuf sayyed M Swami ramanand tirth marathwada university,nanded Maharashtra Phd Swami ramanand tirth marathwada university,nanded. 2012  
2 5127 Mr. Pradeep kumar singh M Deen dayal upadhyaya gorakhpur university Uttar pradesh Ph.d. student Deen dayal upadhyaya gorakhpur university 2011  
3 5155 Mr. V. ramanathan M Manonmaniam sundarnar university Tamil nadu Phd Manonmaniam sundarnar university 2010  
4 5207 Ms. Preeti sharma F Sardar vallabhbhai national institute of technology, surat Gujarat Research scholar 2012  
5 5495 Mr. Jitendra kumar malik M Sambalpur university Orissa M.phil. Utkal university, vanivihar 2012  
6 5508 Mr. Anand pratap singh M Indian school of mines,dhanbad Jharkhand Doctoral student Mnnit allahabad 2012  
7 5522 Mr. Bikash chakraborty M University of kalyani West bengal Ph.d student University of kalyani 2013  
8 5526 Mr. Rakesh ramlal barai M Guru nanak khalsa college of arts science and commerce Maharashtra Assistant professor Mumbai university 2001  
9 5591 Mr. Sankarsan tarai M Sambalpur university Orissa M.phil 2012  
10 5621 Mr. Ashis kumar pati M IISER, Kolkata West bengal Int phd student IISER, Kolkata appeared/ awaiting result  
11 5697 Mr. Saurabh madan gosavi M IISER, Bhopal Madhya pradesh Msc student IISER,Bhopal appeared/ awaiting result  
12 5759 Mr. Dilip jain M S.v.n.i.t., surat Gujarat Teaching assistant IIT,Delhi 2012  



How To Reach

How to reach University of Pune

University of Pune is on the University Road, and it is about 5 km from the Main Railway Station and Main Bus stand.
The nearest Railway Station is Shivajinagar Railway Station (3km) and it is most convenient while coming from Mumbai side, but one should make sure that the train stops there. The  autorickshaw fare is about 10 times the meter reading.

From airport the University is about 16 km.

When one enters the university there are 2 roads, take the right road to go to Mathematics Dept. or to University Guest House/ Univ. SET Guest House. Dept. of Math. comes first on this road on the left side, and it is a building  with Green Colour Fence. To go to SET Guest House, one goes ahead and at the end turns to the left, to go near main Guest House. The main Guest House is very near the main building (with Flag) and is next to VC Bungalow and from the main Guest House, a road goes to IUCAA. The SET Guest House is located on this road about 150 meters from main Guest House on the right side.


More Info

WELCOME to AFS-III. This page will help you to take the best advantage from this programme. You should remember that in just four weeks duration you are going to be exposed some good mathematics at masters level which is normally covered in a semester. AFS-III is the third stage of such a programme. However, most of you have perhaps not attended the first two stages. Therefore, it becomes necessary for you to make at least a bit of preparation to attend this school. Even otherwise, it is good to come well-prepared to interact with several experts you will meet during this short duration of the school. The necessaary pre-requisites please click on: