# Annual Foundation School - Part III (2014)

Venue: |
Department of Mathematics, University of Pune |

Dates: |
26th May to 21st June, 2014 |

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

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, Mumbai-76 |
Dept of Mathematics, Univ. of Pune, Pune-7 |

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**

- Field Extensions and Examples. Algebraic and transcendental elements. Minimal polynomial. Degree of a field extension. Finite and infinite extensions. Simple extensions. Rationalising the denominator.
- Transitivity of finite/algebraic extensions. Compositum of two fields. Algebraically closed fields. Uncountably many algebraically closed subfields of the field of complex numbers
- Ruler and Compass constructions (in brief), Regular polygons. Squaring the circle etc.
- Existence of a splitting field of a polynomial.
- Symmetric polynomials. Newton’s theorem.
- 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 n^{th}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**

- Definition of complex numbers, topology of the complex plane, the Riemann sphere
- Complex differentiability, Cauchy-Riemann equations, complex versus Fr ́echet differentiability
- Power series, Abel’s lemma, uniform convergence of power series, differentiation of power series
- Exponential and trigonometric functions, branch of logarithm, branch of complex powers
- 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**

- Zeros of holomorphic functions, identity Theorem, open mapping theorem,
- Isolated Singularities, Riemann’s removable singularity, Poles and essential singularities. Laurent series, residues.
- Winding number, argument principle, Logarithmic residue theorem, Homotopy version of Cauchy’s theorem.
- 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 π_{1}(S^{1}) = Z (only statement), Van-Kampen theorem (statement) and some applications of V-K, e.g. π_{1} of torus, figure eight, R^{3} \ (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, R^{n} and R^{m}, 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 S^{l} was computed highlighting the lifting properties of the exponential map exp : R → S^{1 }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: **

- All further inquiries/communications, regarding the school should be addressed to Prof. S. A. Katre (sakatre@gmail.com).
- 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.
- 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.
- Wat-listed candidates may therefore book their tickets so as not miss such a chance merely because they would not get a train-ticket.
- All the selected candidates should visit the following website immediately:

www.math.iitb.ac.in/~ars/afsIIIPune2014.html

Sr. |
SID |
Full name |
Gender |
Affiliation |
State |
Position |
University / Institute M.sc./M.a. |
Year of passing M.sc./M.a |
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 | A.n.j.a.college, 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 m.sc 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/ m.sc. mathematics | 2012 | |

24 | 5629 | Mr. Santosh kumar | M | Aligarh muslim university,aligarh | Uttar pradesh | Phd | M.sc. | 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 | M.sc. | 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 | M.sc | 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: