Annual Foundation School - Part III (2013)

School Convener(s)

Speakers and Syllabus 

Field Theory:

1. S A. .Katre (Univ. of Pune)
2. Dilip Patil (IISc, Bangalore)
3. J. K. Verma (IIT, Mumbai)
4. Parvati Shastri (Mumbai Univ., Retd.)

Complex Analysis:

1. Diganta Borah (IISER, Pune)
2. Prachi Mahajan (IITB, Mumbai)
3. R. R. Simha (TIFR, Retd.)

1. A. R. Shastri (IITB, Mumbai)Mahuya Datta (ISI, Kolkata)
2. P. A. S. Sree Krishna (IITG, Guwahati)
3. Krishna Kaipa (IISER, Bhopal)
4. Mahuya Datta (ISI, Kolkata)

1. Module 1: 6 lectures: Algebraic and separable extensions, splitting field of polynomials, primitive element theorem, normal extensions.
2. Module 2: 6 lectures: Fundamental theorem of Galois theory (FTGT), applications to fundamental theorem of algebra, symmetric functions, solutions of cubic and quartic polynomaial equations via FTGT.
3. Module 3: 6 lectures: Galois group of X^n - a, solvable extensions, cyclotomic extensions, constructible regular polygons, inverse Galois problem for finite abelian extensions.
4. Module 4: 6 lectures: 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:

(Numbers in the bracket indicate section from [4])

1. Module 1: 6 lectures: Quick review of algebra and topology of complex plane, sequencesand series, uniform convergence, 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 Frechet Derivative, Geometric interpretation of holomorphy, formal differentiation (3,4,3,5), Mobius Transformation and the Riemann sphere. (3.6,3.7).
2. Module 2: 6 lectures: 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 theorem (4,2, 4.3), Cauchy's integral formula, Taylor's theorem, Liouville's theorem, 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. Module 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).
4. Module 4: 6 lectures: Simply 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, II edition GTM 11 Springer-Verlag (1973)
2. T. W. Gamelin, Complex Analysis, Springer-Verlag, UTM
3. R. Remmert, Theory of Complex Functions, GTM-122, Springer-Verlag(1980)
4. Anant R. Shastri, Basic Complex Analysis of One Variable, MacMillan Publishers India Ltd, Delhi (2011)

Important Note: The syllabus is likely to be modified in consultation with the speakers taking into consideration their previous experience in AFS-III in this topic and also the background of the participants.

Numbers in the bracket indicate the sections in [M]. (It is assumed that the participants are exposed to differential manifolds and Stokes' theorem earlier, which is likely if they have attended AFS-I.)

1. Module 1: 6 lectures: 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 of vectors, covariant derivatives, curvature (5.2).
2. Module 2: 6 lectures: 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) [Also see [B] section V.9, for a quick proof.] Applications of de Rham theorem: Hopf invariant, mapping degree, linking number, etc. (3.5) (2 lectures).
3. Module 3: 6 lectures: Riemannian metric on manifolds, Hodge *-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)
4. Module 4: 6 lectures: Connection 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 bunldes, Euler class (5.6). Applications: Gauss-Bonnet, Characteristic class of complex projective spaces, Characteristic numbers (5.7).

References:

1. [B] G. Bredon, Topology and Geometry, Springer Velag,GTM 139.
2. [M] S. Morita, Geometry of Differential Forms, AMS translations of Math. Monographs, 2011.

TIME/DAY
DATE

MONDAY
3/06/13

TUESDAY
4/06/13

WEDNESDAY
5/06/13

THURSDAY
6/06/13

FRIDAY
7/06/2013

SATURDAY
8/06/13

9.30am - 11.00 am

DB

MD

DB

MD

MD

MD

11.00am - 11.30am

TEA BREAK

11.30am - 1.00pm

SAK

SAK

SAK

SAK

DB

DB

1.00pm - 2.30 pm

LUNCH BREAK

2.30pm - 3.30pm

and

3.45pm - 4.45pm

SAK (T)

DB (T)

MD (T)

SAK (T)

MD(T)

DB(T)

TIME/DAY
DATE

MONDAY
10/06/13

TUESDAY
11/06/13

WEDNESDAY
12/06/13

THURSDAY
13/06/13

FRIDAY
14/06/13

SATURDAY
15/06/13

9.30am - 11.00 am

KV

HB

KV

HB

HB

HB

11.00am - 11.30am

TEA BREAK

11.30am - 1.00pm

DPP

DPP

DPP

DPP

KV

KV

1.00pm - 2.30 pm

LUNCH BREAK

2.30pm - 3.30pm

and

3.45pm - 4.45pm

DPP(T)

KV(T)

HB(T)

DPP(T)

KV(T)

HB(T)

TIME/DAY
DATE

MONDAY
17/06/13

TUESDAY
18/06/13

WEDNESDAY
19/06/13

THURSDAY
20/06/13

FRIDAY
21/06/13

SATURDAY
22/06/13

9.30am - 11.00 am

ARS

ARS

JKV

JKV

ARS

ARS

11.00am - 11.30am

TEA BREAK

11.30am - 1.00pm

JKV

JKV

RK

RK

RK

RK

1.00pm - 2.30 pm

LUNCH BREAK

2.30pm - 3.30pm

and

3.45pm - 4.45pm

JKV(T)

ARS(T)

RK(T)

JKV(T)

ARS(T)

RK(T)

TIME/DAY
DATE

MONDAY
24/06/13

TUESDAY
25/06/13

WEDNESDAY
26/06/13

THURSDAY
27/06/13

FRIDAY
28/06/13

SATURDAY
29/06/13

9.30am - 11.00 am

ARS

PS

ARS

ARS

ARS

RK

11.00am - 11.30am

TEA BREAK

11.30am - 1.00pm

PS

RK

PS

RK

RK

PS

1.00pm - 2.30 pm

LUNCH BREAK

2.30pm - 3.30pm

and

3.45pm - 4.45pm

PS(T)

ARS(T)

RK(T)

PS(T)

ARS(T)

RK(T)

Selected Applicants