Annual Foundation School  Part III (2012)
Venue:  HRI, Allahabad 
Dates:  6^{th} Jul to 31^{st} Jul, 2012 
Convener(s)  Speakers, Syllabus and Time table  Applicants/Participants 
Name  N. Raghavendra 
Mailing Address 
HarishChandra Research Institute 
1. Field theory
Speakers
Name  Aﬃliation 
D. P. Patil  Indian Institute of Science, Bangalore 
K. N. Raghavan  Institute of Mathematical Sciences, Chennai 
J. K. Verma  Indian Institute of Technology Bombay, Mumbai 
Manoj Keshari  IIT Bombay 
1.2 Syllabus
1. Algebraic and separable extensions, splitting ﬁeld of polynomials, primitive element theorem, normal extensions;
2. Fundamental theorem of Galois theory (FTGT), applications to fundamental theorem of algebra, symmetric functions, solutions of cubic and quartic polynomial equations via FTGT;
3. Galois group of X n − a, solvable extensions, cyclotomic extensions, constructible regular polygons, inverse Galois problem for ﬁnite abelian extensions; and
4. Hilbert’s theorem 90 and structure of cyclic extensions, Dedekind’s reduction mod p technique, and construction of polynomials with Galois group Sn and An .
2. Complex analysis
2.1 Speakers
Name  Aﬃliation 
R. S. Kulkarni  Indian Institute of Technology Bombay, Mumbai 
R. R. Simha  
K. Verma  Indian Institute of Science, Bangalore 
2.2 Syllabus
Numbers in parentheses indicate sections from reference [4] below.
1. Quick review of algebra and topology of complex plane, sequences and series, uniform convergenc Weierstrass M test (1.1–1.7), complex diﬀerentiability, basic properties, analytic functions, power series, Abel’s theorem, examples (2.1–2.4), CauchyRiemann equations,Cauchy derivative versus Fr´chet derivative, geometric interpretation of holomorphy, formale diﬀerentiation (3.4, 3.5), Mobius Transformation and the Riemann sphere. (3.6, 3.7);
2. Line integrals, basic properties, diﬀerentiation under integral sign (4.1), primitive existence theorem, CauchyGoursat theorem (statements and sketch of the proof only), Cauchy’s theorem on a convex domain (4.2, 4.3), Cauchy’s integral formula, Taylor’s theorem, Liouville,maximum modulus principle, (4.5–4.7), zeros of holomorphic functions, identity theorem, open mapping theorem and isolated singularities (5.1–5.2), Laurent series and residues (5.3–
5.4), winding number and argument principle (5.5, 5.6);
3. Schwartz lemma, inverse function theorem, Rouche’s theorem (7.1, 7.2), convergence of sequences of holomorphic and meromorphic functions, theorems of Weierstrass and Hurwitz(8.1, 8.2), Runge’s approximation theorem (8.6, 7.4), homology form of Cauchy’s theorem,MittagLeﬄer theorem (8.6, 7.4), inﬁnite products Weierstrass’s theorems on products (8.6);and
4. Simple connectivity, homotopy version of Cauchy’s theorem (7.3), harmonic functions, mean value property, maximum principle, etc., Schwartz reﬂection principle (4.8), Harnack’s principle, subharmonic functions (9.2, 9.3), Dirichlet’s problem, Perron’s solution (9.4), Green’s function and a proof of Riemann mapping theorem (9.6, 8.11), multiply connected domains
(9.7).
References
1. J. B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics,11, SpringerVerlag, 1973.
2. T. W. Gamelin, Complex analysis, Undergraduate Texts in Mathematics, SpringerVerlag,2001.
3. R. Remmert, Theory of complex functions, Graduate Texts in Mathematics, 122, SpringerVerlag, 1980.
4. A. R. Shastri, Basic complex analysis of one variable, MacMillan Publishers India Ltd., 2011.
3. Diﬀerential geometry
3.1 Speakers
Name  Aﬃliation 
R. S. Kulkarni  Indian Institute of Technology Bombay, Mumbai 
N. Raghavendra  HarishChandra Research Institute, Allahabad 
3.2 Syllabus
Numbers in parentheses indicate sections from reference [2]. It is assumed that the participants have been exposed to diﬀerential manifolds and Stokes’ theorem, which is likely if they have attended AFS I.
1. Quick review of diﬀerential forms on manifolds, exterior diﬀerentiation, Lie derivatives and Cartan’s formula (2.2), Frobenius theorem (2.3), the MaurerCartan form of a Lie group (2.4),vector bundles on Manifolds, various operations (5.1), geodesics and parallel translation ofvectors, covariant derivatives, curvature (5.2); and
2. Smooth triangulation of smooth manifolds, smooth singular chain complex (3.1), integration and Stokes theorem (3.2). de Rham cohomology, Poincare lemma (3.3). Cech cohomology; proof of de Rham theorem (3.4) (see also [1, Section V.9] for a quick proof), applications of de Rham theorem, Hopf invariant, mapping degree, linking number, etc. (3.5);
3. Riemannian metric on manifolds, Hodge star operator (4.1), harmonic forms (4.2), Hodge theorem, outline of a proof of Hodge decomposition (4.3), applications of Hodge theorem, Poincare duality, Euler characteristic, intersection number, etc. (4.4); and
4. Connections on vector bundles, curvature (5.3), invariant polynomials, Pontrjagin classes, LeviCivita connection (5.4), Chern classes, relation with Pontrjagin classes (5.5), Orientation on vector bundles, Euler class (5.6), applications: GaussBonnet, characteristic classes of complex projective spaces, characteristic numbers (5.7).
References
1. G. Bredon, Topology and geometry, Graduate Texts in Mathematics, 139, SpringerVerlag,1993.
2. S. Morita, Geometry of diﬀerential forms, Translations of Mathematical Monographs, 201, Iwanami Series in Modern Mathematics, American Mathematical Society, 2001.
Selected Applicants 
 Following applicants are shortlisted to participate in the school.
 The selected participants are requested to confirm their participation by 14th May 2012 by sending an email to N. Raghavendra at raghu at hri.res.in
 If a selected participant does not confirm his/her participation by 14th May, 2012, it will be assumed that he/she is not interested in attending the school, and his/her name will be removed from the list of selected applicants.
 Participants are eligible for train travel by Third AC class.
 Participation and payment of TA/DA will be conditional upon receipt of application forms in hard copy duly signed by the head of the institution.
Sr  SID  Full Name  Gender  Affiliation  State  Position  Univ./Institute M.Sc./M.A.  Year of Passing M.Sc./M.A  Ph.D. Degree Date 
1  1307  Mr. Hitesh Ramesh Raundal  M  IISER Pune  Maharashtra  2nd year Ph.D. Student  Univ. of Hyderabad  2010   
2  1315  Mr. Manoj Kumar Verma  M  DSTCIMS, BHU  Uttar Pradesh  PhD  Dr.R.M.L.Awadh.Univ. Faizabad U.P.  2008   
3  1318  Mr. Laxmi Kant Mishra  M  Univ. of Allahabad  Uttar Pradesh  D.Phil.  Univ. of Allahabad  2009   
4  1333  Mr. Abhash Kumar Jha  M  NISER, Bhubaneswar  Orissa  Ph.D.  B.H.U, Varanasi  2010   
5  1341  Ms. Rana Noor  F  B.H.U, Varanasi  Uttar Pradesh  Research Scholar  B.H.U, Varanasi  2010   
6  1363  Mrs. Vasundhara Gadiyaram  F  None (Housewife)  Karnataka  None  S.K. Univ.  2007   
7  1376  Mr. Nitin Shridhar Darkunde  M  Dept. of Mathematics, S.R.T.M. Univ.. 
Maharashtra  Assistant Professor  Univ. of Pune  2008   
8  1379  Mr. Sachin Pandurang Basude  M  S R T M Univ., Nanded.  Maharashtra  PhD student  S R T M Univ., Nanded.  2011   
9  1387  Mr. Sheo Kumar Singh  M  B.H.U, Varanasi  Uttar Pradesh  PhD Student  B.H.U, Varanasi  2006   
10  1388  Mr. Ashok Kumar Sah  M  Univ. of Delhi  Delhi  M.Phil  Univ. of Delhi  2010   
11  1404  Mr. Pritam Rooj  M  Univ. of Calcutta, Kolkata  West Bengal  JRF  Ramakrishna Mission Vidyamandira, Belurmath  2010   
12  1417  Ms. Manidipa Pal  F  IISER Pune  Maharashtra  Research Fellow  IIT Bombay  2011   
13  1418  Mr Sushil Bhunia  M  IISER Pune  Maharashtra  Research Fellow  Jadavpur Univ.  2010   
14  1446  Ms. Saudamini Nayak  F  National Institute of Technology, Rourkela  Orissa  PhD  Sambalpur Univ.  2008   
15  1448  Mr. Prabhat Kumar Kushwaha  M  IISER Pune  Maharashtra  Research Fellow  B.H.U, Varanasi  2011   
16  1464  Mr. Chandra Prakash  M  Univ. of Delhi  Delhi  M.PHIL.  B.H.U, Varanasi  2008   
17  1482  Ms. Swagatika Sahoo  F  Dept. of Mathematics, Sambalpur Univ.  Orissa  M.Phil. Student  P.G. Department of Mathematics, Utkal Univ.,  2010   
18  1484  Mr. Pralhad Mohan Shinde  M  IISER Pune  Maharashtra  Ph.D student  Univ. of Pune  2011   
19  1503  Ms. Shittal Sharma  F  Univ. of Jammu  Jammu and Kashmir  PhD  Univ. of Jammu  2010   
20  1504  Mr. Gaurav Kumar  M  Univ. of Jammu  Jammu and Kashmir  PhD  Univ. of Jammu  2011   
21  1525  Mr. Ramu Geddavalasa  M  National Institute of Technology  Karnataka  Research Scholar  Andhra Univ.  2007  22 Dec 2011 
22  1528  Ms. Mitu Gupta  F  Univ. of Jammu  Jammu and Kashmir  PhD  Univ. of Jammu  2010   
23  1530  Mr. Heera Saini  M  Univ. of Jammu  Jammu and Kashmir  PhD  Univ. of Jammu  2010   
24  1531  Mr. Banarsi Lal  M  Univ. of Jammu  Jammu and Kashmir  PhD  Univ. of Jammu  2009   
25  1538  Mr. Anish Mallick  M  IMSc, Chennai  Tamil Nadu  PhD  IISER  2011   
26  1539  Mr. Uday Bhaskar Sharma  M  IMSc, Chennai  Tamil Nadu  PhD  Univ. of Hyderabad  2011   
27  1541  Ms. Rekha Biswal  F  IMSc, Chennai  Tamil Nadu  JRF  IIT Bombay  2011   
28  1549  Mr. Devendra Tiwari  M  None    None  B.H.U, Varanasi  2010   
29  1550  Mr. Gugan Chandrashekhar Thoppe  M  TIFR, Mumbai  Maharashtra  Research Scholar  TIFR  Appeared/ Awaiting Result   
30  1556  Mr. Shailendra Kumar Bharati  M  B.H.U, Varanasi  Uttar Pradesh  Pd.D, Research Scholar  M.Sc  2010   
Wait Listed Candidates


1  1457  Ms. Sangita Vinod Choudhary  
2  1533  Ms. Priya Verma  
3  1544  Ms Sueet Millon Sahoo  
4  1437  Mr. Suresh M  
5  1327  Mr Amit Ranjan  
6  1299  Mr. Piyush Goyal  
7  1542  Mr Sahil Gupta 
How to reach 
1.By train
Allahabad is well connected by train from various parts of the country. The most convenient trains from New Delhi to Allahabad are 12418 Prayagraj Express, and 12276 Duronto Express. Trains passing through Allahabad include 12311 Howrah–Kalka Mail, 12322 Mumbai–Howrah Mail, and 12669 Chennai–Chhapra Ganga Kaveri Express. For detailed information on these and other trains to Allahabad, please see the Indian Railways Web sites.
2.By air
There is a single daily ﬂight, AI 9811 of Air India, from Delhi to Allahabad on all days of the week except Sunday. For details, please see the Air India Web site.
3.By road
Allahabad is 238 km. from Lucknow, and 125 km. from Varanasi. There are frequent bus services between these cities and Allahabad, typically once every half hour. Private taxis also operate on these routes.
The road journey from Lucknow to Allahabad takes about 5 hours. Buses leave Lucknow from the Charbagh Bus Depot, and terminate at the Civil Lines Bus Depot in Allahabad.
The road journey from Varanasi to Allahabad takes about 3 hours. Most vehicles enter Allahabad along the Shastri Bridge over the river Ganga. In that case, one reaches Chak, a place near the Institute (see below), before Allahabad city.
4.From Allahabad city to HRI
The HRI campus is at Jhusi, about 13 km. from Allahabad city towards Varanasi. The Institute operates a regular bus service from the HRI City Guest House, to the campus. Please see http://www.hri.res.in/contact.html for the address and phone number of the City Guest House, and http://www.hri.res.in/∼yashpal/busschedule.pdf for the timetable of bus services between the City Guest House and the campus.
If you are coming to Allahabad by train, then come out through the Civil Lines Exit of the station, and take a rickshaw or a tempo from the station to the HRI City Guest House. The distance from the station to the City Guest House is about 4 km.
If you are coming from the city to HRI by a vehicle other than an HRI bus, the following information may be useful. To reach the campus from the city, one goes along the Shastri bridge over the river Ganga. Three km. beyond the end of the bridge, along the Grand Trunk Road (G. T. Road), there is an HRI sign, where one should turn right, oﬀ the highway and into Chhatnag Road. This point on the G. T. Road is called Chak. The HRI campus is a further 3 km. from Chak, along Chhatnag Road. The entrance to the campus is about 100 m. before the Birla Guest House, an establishment owned by a private industrial company.