Annual Foundation School - Part III (2013)

Venue:  University of Pune
Dates: 3rd  to 29th  Jun, 2013

 

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

 

Speakers and Syllabus 

 

Speakers:

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.)
Differential Geometry:
  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)
Unity of Mathematics: Dinesh Thakur (Univ. of Arizona, USA)

 

Topics:

3.1 Field Theory:
  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:
3.2 Complex Analysis:

(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)
3.3 Differential Geometry:

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 TABLE  (To Be Changed)

WEEK 1

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)

 

WEEK 2

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)

 

WEEK 3

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)

 

WEEK 4

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)

 The timing of Unity of Mathematics Lectures to be decided later.

 

Selected Applicants

       

  • The selected participants should confirm their participation or otherwise by email to:
    sakatre at gmail.com and to vinayakjoshi111 at gmail.com.
  • The confirmation should be done latest by 15th February 2013 failing which wait-listed candidates may be considered in their place.
  • If the candidate(s) are unable to come for some reason, they should inform so at the earliest. This will help wait-listed students.
  • For some students the signed form is not received and it is essential that it should reach latest by 15th February.
  • Wait-listed students may book travel, so that if they are selected, there is no last minute inconvenience.
Sr SID Full Name Gender Affiliation State Position in College/ Univ. Univ./ Institute M.Sc./ M.A. Year of Passing M.Sc./ M.A Ph.D. Degree Date
1 2864 Mr. Ajeet  Kumar Male Banaras Hindu Univ. Uttar Pradesh PhD Banaras Hindu Univ., Varanasi 2010  
2 2880 Mr. Goutam  Anand Male IISER, Thiruvanantha- puram Kerala Integrated PhD Student IISER, Thiruvanantha- puram Appeared/ Awaiting Result  
3 2886 Ms Priya  Das Female IISER, Kolkata West Bengal JRF 1ST YEAR Univ. of Hyderabad 2010  
4 2931 Mr. Biswajyoti  Saha Male The Institute of Mathematical Sciences Tamil Nadu PhD Student Univ. of Hyderabad 2012  
5 2932 Ms. Ekata  Saha Female The Institute of Mathematical Sciences Tamil Nadu PhD Student Univ. of Hyderabad 2012  
6 2953 Mr Prashanth  Raman Male BITS, Pilani Hyderabad Campus Andhra Pradesh student BITS, Pilani Hyderabad Campus 2011  
7 2980 Mrs. Yogalakshmi  T. Female Sri Sarada College for Women (autonomous) Tamil Nadu Research Scholar M.Sc. 2008  
8 2995 Mr. Mushtaq Ahmad Shah Male Kashmir Univ. Jammu and Kashmir Ph.D Scholar Kashmir Univ. 2010  
9 3031 Ms. Snehal Suryakant Mitragotri Female Shivaji Univ., Kolhapur Maharashtra Asst. Prof. IIT, Roorkee 2009  
10 3040 Mrs Visalakshi  V Female Sri Sarada College for Women (Autonomous) Tamil Nadu Ph.D student Sri Sarada College for Women, Salem, Tamilnadu 2008  
11 3050 Mr Azeef muhammed  P ajmal Male Univ. Of Kerala Kerala PhD Student Univ. Of Hyderabad 2011  
12 3065 Ms. Vidhya  Dhanaraj Female Sri Sarada College for Women Tamil Nadu Ph.  D(Student) Sri Sarada College for Women 2008  
13 3112 Mr. Akhilesh  Yadav Male Univ. of Allahabad, Allahabad. Uttar Pradesh Ph.D. Student. Univ. of Allahabad, Allahabad. 2008  
14 3131 Mr. Amit Kumar Laha Male The Univ. Of Burdwan West Bengal Phd IIT Guwahati 2011  
15 3133 Mr. Sasi kumar  Palanisamy Male Gobi Arts & Science College, Gobichettipalayam Tamil Nadu Ph.D., M.Sc. Mathematics 2004  
16 3153 Mr Irshad Ahmad Sheikh Male NIT Srinagar Jammu and Kashmir SCHOLAR Kashmir Univ. 2008  
17 3170 Mr. Tarun Kumar Chakra Male IIT Bhubaneswar Orissa Ph.D PG Dept. of mathematics, Utkal Univ.. vanivihar 2007  
18 3242 Mr. Bimolchand Singh Leiphrakpam Male Assam Univ., Silchar Assam PhD Manipur Univ., Canchipur, Imphal 2009  
19 3259 Ms. Dakjum  Eshi Female North Eastern Regional Institute of Science and technology Arunachal Pradesh Ph.D. student North Eastern Regional Institute of Science and Technology 2011  
20 3275 Mr. Gopal  Datt Male Department of Mathematics, Univ. of Delhi Delhi PhD Delhi Univ. 2009  
21 3342 Mr. Rameez  Raja Male Univ. Of Kashmir Jammu and Kashmir PhD Student Univ. Of Kashmir 2008  
22 3343 Mr. Neeraj  Sangwan Male IISER Mohali Punjab Integrated MS student IISER Mohali Appeared/ Awaiting Result  
23 3365 Mr. M.  Nagarajan Male Bharathiar Univ. Tamil Nadu PhD Bharathiar Univ. 2007  
24 3480 Ms. Sneha Sanjay Kshemkalyani Female B.P.H.E. Society's Ahmednagar college Maharashtra Asst.Prof. Univ. of Pune 2011  
25 3489 Ms. Jyoti  Thorwe Female Shivaji Univ., Kolhapur. Maharashtra Assistant Professor Pune Univ., Pune 2008  
26 3496 Ms. Shikha  Singh Female IIT Kharagpur West Bengal Integrated MSc Student IIT Kharagpur Appeared/ Awaiting Result  
27 3498 Mr. Bulusu Naga venkata Satish Male ADIKAVI NANNAYA Univ. Andhra Pradesh Ph.D Alagappa Univ. 2010  
28 3513 Ms Varsha Hambirrao Shinde Female Tuljaram Chaturchand College Maharashtra Assistant Professor Univ. Of Pune, Pune 2011  
29 3560 Mr Paltu  Sarkar Male Jadavpur Univ. West Bengal PhD Student IIT Kharagpur 2008  
30 3210 Mrs Neeta Vikas Shinde Female College of Engineering Pune Maharashtra Asst. Prof. M. Sc in Pune Univ 2000  
31 3278 Mr. Sagar Nandlal Vispute Male Department of mathematics Univ. of pune. Maharashtra Student Univ. of pune Appeared/ Awaiting Result  
32 3504 Ms. Ishrat jahan Md. Haroon Ansari Female M.C.E.Society's Abeda Inamdar Sr.College Maharashtra Asst. Prof. Univ. of Pune/ Department of Mathematics 2007  
33 3508 Ms. Nasreen bano Muslim Shaikh Female M.C.E Society's Abeda Inamdar Sr. College Maharashtra Asst. Prof. Univ. of Pune/ Abeda Inamdar Sr. College 2010  
   
Waiting list    
1 2921 Mr. Suhail Gulzar Mattoo Male Kashmir Univ. Jammu and Kashmir Research Scholar Kashmir Univ./ M. Sc. 2007  
2 2927 Mr C  Gangadhar Male Univ. Of Hyderabad Andhra Pradesh PhD Univ. Of Hyderabad 2009  
3 2957 Mr. Arvind  Kumar Male Harish-Chandra Research Institute Uttar Pradesh PhD Banaras Hindu Univ. 2012  
4 3083 Ms. Revathi  G.k Female Sri sarada college for women Tamil Nadu Ph. D., (Student) Sri Sarada College for Women 2006  
5 3236 Ms Syed Zarqua Mohammadi Female SGGS IE&T Maharashtra LECTURER S.R.T.M.Univ., Nanded-431605 2011  
6 3329 Mr. Soumya  Dey Male Ramakrishna Mission Vivekananda Univ. West Bengal MSc Student Ramakrishna Mission Vivekananda Univ. Appeared/ Awaiting Result  
7 3466 Mr Ramu  Geddavalasa Male National Institute Of Technology Karnataka Karnataka Research Scholar Andhra Univ. 2007  
8 3509 Mr Suresh Kumar M Male National Institute Of Technology Karnataka Karnataka Ph. D., (Student) Manonmaniam Sundaranar Univ. 2010