Venue:	CEMS, Department of Mathematics, SSJ Campus, Kumaun University, Almora.
Dates:	1st - 28th May, 2014

 

Convener(s)

Speakers, Syllabus and Time table

Applicants/Participants

 

School Convener(s)
Name	K.N. Raghavan	H.S. Dhami	Sanjay K Pant
Mailing Address	Professor, Institute of Mathematical Sciences (IMSc), CIT Campus, Taramani, Chennai - 600113	Vice-Chancellor, Kumaun University, Nainital-263001	Associate Professor, Deen Dayal Upadhyaya College, (University of Delhi),Shivaji Marg, Karampura, New Delhi-110015

 

Speakers and Syllabus

1. Ring Theory.

1. Modules over Principal Ideal Domains Modules, direct sums, free modules, finitely geneover a PID, structure of finitely generated abelian groups, rational and Jordan canonical form.
2. Basics Commutative rings, nil radical, Jacobson radical, localization of rings and modulrings, primary decomposition of ideals and modules.
3. Integral extensions of rings, Going up and going down theorems, finiteness of intdiscrete valuation rings, Krull's normality criterion, Noether normalization lemma, Hilbert's Nullstellensatz
4. Semisimple rings Wedderburn's Theorem, Rings with chain conditions and ArWedderburn’s main theorem.

Speakers: Parvati Shastri (Mumbai University),    Ronnie Sebastian (IISER,Pune),    K.N.Raghavan (IMSc, Chennai)

Course Associates: TBA (To Be Announced)

References:

S. Lang, Algebra, 3rd edition, Addison-Wesley.

1. D. S. Dummit and R. M. Foote, Abstract Algebra, 2nd edition John-Wiley.
2. N. Jacobson, Basic Algebra, Vol. 1 and 2, Dover, 2011.
3. A. W. Knapp, Advanced Algebra, Birkhauser, 2011.

2. Functional Analysis.

1. Normed linear spaces, Continuous linear transformations, application to differential equBanach theorems-analytic and geometric versions, vector valued integration.
2. Bounded Linear maps on Banach Spaces Baire's theorem and applications: Uniformprinciple and application to Fourier series, Open mapping and closed graph theoremscomplemented subspaces, unbounded operators and adjoints.
3. Bounded linear functionals Weak and weak* topologies, Applications to reflexive sepauniformly convex spaces, Application to calculus of variations.
4. Hilbert spaces, Riesz representation theorem, Lax-Milgram lemma and applicationinequalities, Orthonormal bases, Applications to Fourier series and examples of special funcLegendre and Hermite polynomials.

Speakers: Anant R Shastri (IIT,Mumbai),   R. Srinivasan (CMI,Chennai), Sanjay K Pant (CEMS, Almora),  Abhay Bhatt (ISI,Delhi)**

Course Associates: TBA
** to be confirmed.

References

1. J. B. Conway, A Course in Functional Analysis, II edition, Springer, Berlin 1990.
2. C. Goffman, G. Pedrick, First Course in Functional Analysis, Prentice-Hall, 1974.
3. S. Kesavan, Functional Analysis Volume 52 of Texts and readings in mathematics, HAgency (India), 2009.
4. B. B. Limaye, Functional Analysis, II edition New Age International, 1996.
5. A. Tayor and D. Lay, Introduction to Functional Analysis, Wiley, New York, 1980.

3. Differential Topology.

1. Review of differential calculus of several variables: Inverse and implicit function theoremsmooth functions; smooth partition of unity, Submanifolds of Euclidean spaces (withboundary) Tangent space, embeddings, immersions and submersions, Regular valutheorem, Transversality and Stability. [The above material should be supported amply by exand examples from matrix groups.]
2. Abstract topological and smooth manifolds, partition of unity, Fundamental gluing lemmfor Hausdorffness of the quotient, classification of 1-manifolds. Definition of a vector bundbundle as an example. Sard's theorem. Easy Whitney embedding theorems.
3. Vector fields and isotopies Normal bundle and Tubular neighborhood theorem. Omanifolds and on normal bundles. Vector fields. Isotopy extension theorem. Disc Theorem. neighborhood theorem.
4. Intersection Theory: Transverse homotopy theorem and oriented intersection numbmaps both oriented and non oriented cases, winding number, Jordan Brouwer separation theorem, Borsuk- Ulam theorem.

Speakers: Amiya Mukherjee (ISI, Kolkata), Gautam Mukherjee (ISI, Kolkata), Samik Basu (RKMVU, Kolkata)

Course Associates: TBA

References:

1. V. Gullemin and A. Pollack, Differential Topology, Englewood Cliff, N.J. Prentice Hall (19(2) W. Hirsch, Differential Topology, Springer-Verlag.
2.  Morris W. Hirsch: Differential Topology,Springer-Verlag, New York Inc., 1976.
3. J. W. Milnor, Topology from the Differential Viewpoint, Univ. Press, Verginia.
4. Anant R. Shastri, Elements of Differential Topology, CRC Press, 2011.

 

Time Table (tentative)

Time Table

Week - 1
Date	09:30 to 11:00	11:00-11:30	11:30-13:00	13:00 -14:30	14:30 -15:30	15:30 - 16:00	16:00 -17:00
May 01, 2014	RT1	T E A	FA1	L U N C H	TRT1	T E A	TFA1
May 02, 2014	FA2		DT1		TFA2		TDT1
May 03, 2014	DT2		RT2		TDT2		TRT2
May 05, 2014	RT3		FA3		TRT3		TFA3
May 06, 2014	FA4		DT3		TFA4		TDT3
May 07, 2014	DT4		RT4		TDT4		TRT4

 

Week - 2
Date	09:30 - 11:00	11:00 - 11:30	11:30 - 13:00	13:00 - 14:30	14:30 - 15:30	15:30 - 16:00	16:00 - 17:00
May 08, 2014	RT5	T E A	FA5	L U N C H	TRT5	T E A	TFA5
May 09, 2014	FA6		DT5		TFA6		TDT5
May 10, 2014	DT6		RT6		TDT6		TRT6
May 12, 2014	RT7		FA7		TRT7		TFA7
May 13, 2014	FA8		DT7		TFA8		TDT7
May 14, 2014	DT8		RT8		TDT8		TRT8

 

Week-3
Date	09:30 - 11:00	11:00 - 11:30	11:30 - 13:00	13:00 - 14:30	14:30 - 15:30	15:30 - 16:00	16:00 - 17:00
May 15, 2014	RT9	T E A	FA9	L U N C H	TRT9	T E A	TFA9
May 16, 2014	FA10		DT9		TFA10		TDT9
May 17, 2014	DT10		RT10		TDT10		TRT10
May 19, 2014	RT11		FA11		TRT11		TFA11
May 20, 2014	FA12		DT11		TFA12		TDT11
May 21, 2014	DT12		RT12		TDT12		TRT12

 

Week - 4
Date	09:30 - 11:00	11:00 - 11:30	11:30 - 13:00	13:00 - 14:30	14:30 - 15:30	15:30 - 16:00	16:00 - 17:00
May 22, 2014	RT13	T E A	FA13	L U N C H	TRT13	T E A	TFA13
May 23, 2014	FA14		DT13		TFA14		TDT13
May 24, 2014	DT14		RT14		TDT14		TRT14
May 26, 2014	RT15		FA15		TRT15		TFA15
May 27, 2014	FA16		DT15		TFA16		TDT15
May 28, 2014	DT16		RT16		TDT16		TRT16

 

RTn -  nth lecture in ring theory	TFAn - nth tutorial in functional analysis
TRTn -  nth tutorial in ring theory	DTn -  nth lecture in differential topology
FAn -  nth lecture in functional analysis	TDTn -  nth tutorial in differential topology

 

Selected Applicants

Please read the following instructions carefully.

1.  All further inquiries/communications, regarding the school should be addressed to the respective local organizers. 
2. Selected candidates should communicate the confirmation of their participation on or before 05th March 2014, along with a copy of their onward journey ticket where-ever applicable, failing which their selection will be cancelled automatically.
3. 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.
4. Organizers are requested to fill-up the vacancies as and when they occur by informing the candidates from the wait-list. 
5. Wait-listed candidates may therefore book their tickets so as not miss such a chance merely because they would not get a train-ticket.

Serial	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	5095	Ms. Namita Sharma	Female	Department of mathematics, delhi university	Delhi	M.Phil	St. Stephen's college	2012
2	5100	Mr. Uma Shankar Verma	Male	University of allahabad	Uttar Pradesh	Research Scholar	University of Lucknow, Lucknow	2006
3	5129	Mr. Vijay Kumar Yadav	Male	Ndian school of mines	Jharkhand	Ph.D	Dr. RAM MANOHAR LOHIYA AVADH UNIVERSITY, FAIZABAD	2003
4	5187	Mr Sumit Kumar	Male	Imsc, chennai	Tamil Nadu	JRF	IISC	2013
5	5203	Mr Anindya Ghatak	Male	National institute and science education and research	Orissa	PhD	Calcutta University	2012
6	5226	Mrs Priyanka Ashok Harjule	Female	Mnit jaipur	Rajasthan	Ph.D	Bangalore University	2012
7	5244	Mr. Ashish Kumar Kesarwany	Male	Iit guwahati	Assam	Ph.D.	IIT Kanpur	2012
8	5279	Ms Sueet Millon Sahoo	Female	National institute of science education and research,bhubaneswar	Orissa	PhD	NATIONAL INSTITUTE OF TECHNOLOGY	2012
9	5289	Ms Moni Kumari	Female	Niser bhubanswar	Orissa	Phd	BHU	2012
10	5292	Mr. Suraj Krishna M S	Male	Chennai mathematical institute		PhD	University of Hyderabad	2013
11	5305	Ms Moni Kumari	Female	Niser	Orissa	Ph.D	BHU	2012
12	5309	Mr Sapan Kumar Das	Male	Nit jamshedpur research scholar	Jharkhand	RESEARCH SCHOLAR	fakir mohan (auto) college	2012
13	5335	Ms. Anupam Gumber	Female	Indian institute of technology indore	Madhya Pradesh	PhD Student	G.N.D.U Amritsar/Lyallpur Khalsa College Jalandhar	2007
14	5372	Mr Harish D	Male	Department of mathematics, bangalore university.	Karnataka	PhD	Bangalore University	2012
15	5384	Ms. Kavita Khandelwal	Female	Malaviya national institute of technology jaipur	Rajasthan	PhD	University of Rajasthan	2008
16	5403	Ms. Sheela Verma	Female	Indian institute of technology kanpur	Uttar Pradesh	PhD	IIT Kanpur	2012
17	5406	Ms Poonam Rani	Female	Department of mathematics	Delhi	phD	Jamia Millia Islamia	2012
18	5437	Ms. Sweta Kumari	Female	University of allahabad, allahabad	Uttar Pradesh	D. Phil. Student	University of Allahabad	2009
19	5444	Ms. Shelly Verma	Female	University of delhi	Delhi	M.Phil student	University of Delhi	2010
20	5448	Mr. Ganga Prasad Yadav	Male	University of allahabad, allahabad	Uttar Pradesh	D.Phil. Student	University of Allahabad, Allahabad	2011
21	5457	Mr. Sanjeev Kumar Maurya	Male	Indian institute of technology (bhu) varanasi	Uttar Pradesh	Ph.D	VBS PURVANCHAL UNIVERSITY JAUNPUR	2010
22	5459	Mr. Alok Kumar Yadav	Male	Jawaharlal neharu university (jnu)	Delhi	Research Scholar	D.D.U. Gorakhpur University (U.P.)	2011
23	5466	Mr. Nitin Bisht	Male	Indian institute of technology indore	Madhya Pradesh	PhD	H.N.B. GARHWAL UNIVERSITY	2011
24	5470	Ms. Debaroti Das	Female	Nit silchar	Assam	PhD Student	Tripura University	2011
25	5471	Mr. Krishanu Roy	Male	Institute of mathematical sciences	Tamil Nadu	Integrated Ph.d student		Appeared / Awaiting Result
26	5486	Mr. Avijit Nath	Male	Institute of mathematical sciences	Tamil Nadu	NBHM Ph.D. coursework student	Ramakrishna Mission Vivekananda University	2013
27	5502	Ms Poonam Kesarwani	Female	IIT Kanpur	Uttar Pradesh	PhD	IIT Kanpur	2012
28	5507	Ms. Meena Pargaei	Female	IIT Kanpur	Uttar Pradesh	PhD	G.B.P.U.A.&T. Pantnagar	2012
29	5510	Ms. Meena Pargaei	Female	IIT Kanpur	Uttar Pradesh	PhD	G.B.P.U.A.&T. Pantnagar	2012
30	5512	Mr Dillip Kumar Pradhan	Male	National institute of technology, rourkela	Orissa	PhD	Sambalpur University	2011
Waiting List
Serial	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	5288	Mr. Bikramaditya Sahu	Male	Niser , bhubaneswar	Orissa	PhD	Indian Institute of Technology Bombay	2013
2	5388	Mr. Ranadip Gangopadhyay	Male	Banaras hindu university	Uttar Pradesh	PhD	NETAJI SUBHAS OPEN UNIVERSITY	2013
3	5400	Ms. Meena Kumari Gurjar	Female	Malaviya national institute of technology, jaipur	Rajasthan	Research Scholar	University of Rajasthan Jaipur	2008
4	5511	Mr. Ratan Kumar Giri	Male	Nit rourkela	Orissa	PhD	IIT Guwahati	2012
5	5517	Mr Gireesh D.S	Male	Bangalore university	Karnataka	PhD	Bangalore University	2011
6	5521	Mr. Arun Kumar	Male	Indian institute of science education and research, mohali	Punjab	Int.Phd	MOHALI	Awaiting Result
7	5534	Mr. Abhishake Rastogi	Male	IIT Delhi	Delhi	Ph.D.	IIT Delhi	2013
8	5538	Mr. Anuj Kumar	Male	IIT Delhi	Delhi	PhD	IIT Delhi	2011
9	5543	Ms. Saroj Rani	Female	IIT Delhi	Delhi	Ph.D. Student	KURUKSHETRA UNIVERSITY	2007
10	5544	Mr Aranya Lahiri	Male	IISER Bhopal	Madhya Pradesh	BS-MS 5th year student	IISER Bhopal	Appeared / Awaiting Result

 

More Information

Advice to participants:

• Keep up with updates to this pageand follow instructions posted here.
• You must prepare yourselves carefully in order to gain effectively from the school.
• Solving problems: This is extremely important---it is perhaps the most important component of learning mathematics---but there may not be enough time during the school itself to work out in detail sufficiently many problems. Therefore, go through, well in advance, the material on the syllabus (as in any of the standard sources) and give the exercises listed below a try.

Syllabus: The syllabi  (pdf file available at the end of page) for all levels of the AFS: I, II, and III. Particularly relevant for us are sections 2.1 (ring theory or RT), 2.2 (functional analysis or FA), and 2.3 (differential topology or DT), for these are the ones to be covered in AFS-II. Each section has four subsections or "modules".

Notation: For subsections or modules, this is fixed as follows: DT-III refers to the third module of differential topology, etc. On the timetable, FA10 stands for the tenth lecture of section FA, etc. Assuming that the modules are covered one after another in succession (which may not be the case!) and that there are four lectures per module, FA10 would be the second lecture of the FA-III module.

Resource persons (listed, as is customary in mathematics literature, by alphabetical order on last name):

    Ring theory: K. N. Raghavan (IMSc), Ronnie Sebastian (IISER, Pune), Parvati Shastri (University of Mumbai)
    Functional Analysis: Sanjay Pant (Delhi University), Anant Shastri (IITB), R. Srinivasan (CMI)
    Differential topology: Samik Basu (RKMVU), Amiya Mukherjee (ISI, Kolkata), Gautam Mukherjee (ISI, Kolkata)

General advice to participants about RT-I: Revise linear algebra and basic facts about PIDs. The theorems will be proved for modules over a general PID although in the applications the PID will invariably be either the ring of integers or the ring of polynomials in one variable over a field. Specifically come prepared with knowledge of the following material:

•     PIDs (as, for example, in Artin's algebra text)
•     definitions and very basics of commutative rings (again as, for example, in Artin's algebra text)
•     matrices over commutative rings: determinants, cofactors, invertibility
•     linear algebra (including eigenvalues and canonical forms) from any standard book on linear algebra, e.g., Hoffman and Kunze.

 Note on references for RT-II and RT-III: The following text is suggested as an additional important reference:
M. F. Atiyah and I. G. Macdonald: Introduction to Commutative Algebra, 1969.

Exercises for RT-II and RT-III: from the book of Atiyah and Macdonald mentioned above:
Chapter 1 : 1-12, 14
Chapter 2 : 1-13
Chapter 3 : 1, 2, 5, 6, 7, 9, 12, 13, 14, 17
Chapter 4 : 2, 4, 5, 7, 8, 14, 15, 16
Chapter 5 : 2-7, 11-14
Chapter 7 : 2, 5, 6, 8, 10, 11, 15

 

Notes (All notes are available as single download zip file at the end of the page):

    Samik Basu: Additional problems for DT.
    Amiya Mukherjee: Part 1, Part 2, Part 3, Part 4 of DT; Tutorial problems
    K. N. Raghavan: Integral extensions, Simple and semisimple finite dimensional algebras
    Anant Shastri: Normed linear spaces, Hahn-Banach theorems, Hilbert spaces
    Parvati Shastri: Lectures on modules over PIDs
    Ronnie Sebastian: Chapters 1 and 2 (of Atiyah-Macdonald), Chapter 3
    R. Srinivasan: tutorial sheet 1, sheet 2

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How to reach CEMS Almora

By Air:
The nearest airport is New Delhi. From Delhi one can reach Almora by train or by road as given below.

By Train:

The nearest train station is Kathgodam. Several important rail stations in India have direct connection to Kathgodam. From Delhi, there three main trains run almost daily from Delhi to Kathgodam: 12040 ANVT-KGM Shatabdi (06:15 AM), 15013 Ranikhet Express (22:40) and 15035 UTR-SAMPARK Express (16:00). We recommend the first two connections. One may also get down at Haldwani (the station just before Kathgodam). Please note that train number 12040 starts from ANAND VIHAR railway station at Delhi and the other two trains depart from OLD DELHI railway station. So, if you plan to reach Kathgodam by train, after arriving at the airport or New Delhi railway station, you should direct yourself towards the Anand Vihar railway terminus or the Old Delhi rail station in Delhi. The metro stops most nearest to these rail stations are Anand Vihar and Chandni Chowk respectively.

Summary of trains:

1. *Delhi - Kathgodam by Ranikhet Express. (Train No. 15013)
2. *Anand Vihar (ANVT)-Kathgodam (KGM) Shatabdi (Train o. 12040)
3. Delhi - Kathgodam by Uttarakhand Smprk Kranti Express (Train No.15035)
4. Delhi - Lal Kuan by Pooja Express. (Train No. 0517)

Trains with asterix are recommended.

By Road:
This option is little expensive, but direct and in many cases, would be flexible. This option is to travel to Almora from New Delhi Airport by road. You can book a (private) taxi at the airport or in advance. By car it generally takes 8-10 hours to Almora (from Delhi airport) assuming that you will take one or two breaks in between the journey.

AC Volvo buses are also available from ISBT Delhi to Kathgodam.

Car Direction:
Delhi - Hapur bypass - Moradabad bypass - Rampur - Bilaspur - Rudrapur - Haldwani - Kathgodam [approximately 270 kms]- Ranibagh - Bhimtal - Bhowali - Khairna Bridge - Kwarah Bridge - Almora. From Kathgodam/Haldwani to Almora: One has to travel this route by car or bus. It takes sround 3 hrs to reach Almora from Kathgodam. We recommend you to hire a car or shared taxi at Kathgodam/Haldwani. If you intend to travel by a shared taxi, it is better to get down at Haldwani since the availability of shared taxis are better there.

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