Annual Foundation School - Part III (2012)

 1. Field theory

  Speakers

 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 

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), Cauchy-Riemann 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, Cauchy-Goursat 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,Mittag-Leffler 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, Springer-Verlag, 1973.

 2. T. W. Gamelin, Complex analysis, Undergraduate Texts in Mathematics, Springer-Verlag,2001.

 3. R. Remmert, Theory of complex functions, Graduate Texts in Mathematics, 122, Springer-Verlag, 1980.

 4. A. R. Shastri, Basic complex analysis of one variable, MacMillan Publishers India Ltd., 2011.

3. Differential geometry

 3.1 Speakers

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 Maurer-Cartan 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, Levi-Civita connection (5.4), Chern classes, relation with Pontrjagin classes (5.5), Orientation on vector bundles, Euler class (5.6), applications: Gauss-Bonnet, characteristic classes of complex projective spaces, characteristic numbers (5.7).

References

 1. G. Bredon, Topology and geometry, Graduate Texts in Mathematics, 139, Springer-Verlag,1993.

2. S. Morita, Geometry of differential forms, Translations of Mathematical Monographs, 201, Iwanami Series in Modern Mathematics, American Mathematical Society, 2001.

• 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 e-mail 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.

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.