Sheaf Theory, Sheaf Cohomology and Spectral Sequences (2019)

Convener(s)

 
Name: Prof Satya Deo Dr. Hemant Kumar Singh Dr Satya Prakash Tripathi
Mailing Address:  Harish Chandra Research Institute,
Chhatnaag Road, Jhusi, Allahabad- 211019		 Department of Mathematics,
University of Delhi, Delhi-110007		 Department of Mathematics,
Kirori Mal College, University of Delhi, Delhi-110007
Email: sdeo at hri.res.in
sdeo94 at gmail.com		 hksinghdu at gmail.com
hemantksingh at maths.du.ac.in		 drsptripathi at yahoo.com
satyapt93 at yahoo.com

Please Note:  Participants have to arrange for their own travel.

Sheaf Theory is a very powerful tool for studying problems of algebraic Topology, algebraic Geometry, several complex variables and many other areas.of mathematics. The fundamental contributions made by H.Cartan, J.Leray and J-P Serre etc using the concepts of sheaf theory and sheaf cohomology are indispensable in the study of above subjects. Sheaf cohomology presents a unified approach to all kinds of cohomology theories.

Spectral sequences are generalizations of exact sequences. It was introduced by Jean Leray in 1946, and now have become important computrational tools, particularly in algebraic topology, algebraic geometry and homological algebra. In algebraic topology, for a pair of topological spaces (X, A), the relationship between certain homotopy, homology, or cohomology groups is expressed perfectly by an exact sequence. But for any sequence of spaces this relationship is very commplicated and a more powerful algebraic tool is needed, and the spectral sequences provides such a tool. Also, the Mayer-Vietoris sequence for a decomposition of a space X by two sets generalizes to a spectral sequence associated to a cover of X by any number of sets.

Discussion classes of the workshop will supplement and enhance the material covered in lectures along with problem solving.

Dates: 

Monday, November 18, 2019 - 09:00 to Saturday, November 30, 2019 - 18:00

Venue: 

Venue Address: 

Department of Mathematics, University of Delhi

Venue State: 

Venue City: 

PIN: 

110007

Chrono Order: 

329

Syllabus: 

Tentative Syllabus:

  1. Classical Cohomology Theories: Cohomology groups, Cup products and cohomology algebra of topological spaces Singular Coholomogy, Alexander-Spanier Cohomology, de Rham Cohomology and Cech Cohomology (without using Sheaf Theory).
  2.  Basic Sheaf Theory: Definitions of Sheaves and Presheaves, Homomorphisms, subsheaves and quotient sheaves, Direct and inverse images, cohomomorphisms, Classical Cohomology theories, Differential Sheaves and resolution, The canonical resolution and Sheaf Cohomology.
  3. Sheaf Cohomolgy: Axioms for cohomology and the cup product, The Vietoris mapping theorem and homotopy invariance, Mayer-Vietoris theorems, The Kunneth and Universal Coefficient theorems, Dimensions, The transfer homomorphism and the smith sequences, Steenrod cyclic reduced powers, The steenrod operations.
  4. Spectral Sequences-I: Informal Introduction- (Filtration, gradded algebra, derivation, Poincare series and Euler Characteristic), What is spectral sequence ?- (Basic properties and how does it arise), Convergence of spectral sequences-(limit, colimit and Zeeman’s comparison theorem), Filtration, Hopf Algebras and Steenrod Algebras.
  5. Spectral Sequences-II: Construction of Leray-Serre spectral sequence and its applications, transgression, classifying spaces and characteristic classes, Other construction of spectral sequences.
  6. Applications of Spectral Sequences: Spectral sequence of a differential sheaf, The fundamental theorem of sheaves, The Leray Sheaf, The Leray spectral sequence of a map, Fibre bundles, The spectral sequences of Borel and Cartan, Sphere bundles with singularities, the Oliver transfer and the Conner conjecture.

List of Confirmed Speakers:

  • SD: Prof Satya Deo, HRI, Allahabad
  • SSK: Prof S S Khare, NEHU, Shillong
  • AM: Prof Amiya Mukherjee, ISI, Kolkata
  • HKM: Prof Himadri Mukherji, NEHU, Shillog
  • BS: Dr B Subhash, IISER, Tirupati
  • HKS: DrHemant Kumar Singh, University of Delhi, Delhi

Topics to be covered:

Speakers No. of Lectures Topics
Topic-1 (SSK) 6 (90 Minutes) Classical Cohomology Theories: Cohomology groups, Cup products and cohomology algebra of cohomology theories: Singular Coholomogy, Alexander-Spanier Cohomology, de Rham Cohomology and Cech Cohomology (without using Sheaf Theory).
Topic-2 (HKS) 6 (90 Minutes) Basic Sheaf Theory: Sheaves, Presheaves, Differential Sheaves and resolution, Sheaf Cohomology, Axioms for cohomology and the cup product, Mayer-Vietoris theorems, The Kunneth and Universal Coefficient theorems, The steenrod operations
Topic-3 (SD) 6 (90 Minutes) Sheaf Cohomolgy: Axioms for cohomology and the cup product, The Vietoris mapping theorem and homotopy invariance, Mayer-Vietoris theorems, The Kunneth and Universal Coefficient theorems, Dimensions, The transfer homomorphism and the smith sequences, Steenrod cyclic reduced powers, The steenrod operations.
Topic-4(HKM) 6 (90 Minutes) Spectral Sequences-I: Informal Introduction- (Filtration, gradded algebra, derivation, Poincare series and Euler Characteristic), What is spectral sequence ?- (Basic properties and how does it arise), Convergence of spectral sequences-(limit, colimit and Zeeman’s comparison theorem), Filtration, Hopf Algebras and Steenrod Algebras
Topic-5 (BS) 6 (90 Minutes) Spectral Sequences-II: Construction of Leray-Serre spectral sequence and its applica- tions, transgression, classifying spaces and characteristic classes, Other construction of spectral sequences.
Topic-6 (AM) 6 (90 Minutes) Applications of Spectral Sequences: Spectral sequence of a differential sheaf, The fun- damental theorem of sheaves, The Leray Sheaf, The Leray spectral sequence of a map, Fibre bundles, The spectral sequences of Borel and Cartan, Sphere bundles with singular- ities, the Oliver transfer and the Conner conjecture.

 

Special Lectures:

  • SD:Prof Satya Deo, HRI, Allahabad

List of Other Confirmed Speakers:

  • ATS: Dr. Angom Tiken Singh, North-Eastern Hill University, Shillong
  • SPT: Dr. Satya Prakash Tripathi, Kirori Mal College, University of Delhi, Delhi

Recommended Texts:

  1. G. E. Bredon, Sheaf Theory, Springer-Verlag, 2nd Edition, 1997.
  2. J. McCleary, A User’s Guide to Spectral Sequence, Cambridge University Press, 2001.
  3. C.H. Dowker, Lectures on Sheaf Theory, TIFR, Bombay, 1956.
  4. R. Godement, Topology Algebrique et Theory des Faisceaux, Heman, Paris 1958.

Time Table: 

    Lecture-1 Tea Lecture-2 Lunch Lecture-3 Tea Discussion Class Tea and Snacks
Day Date 9:30 -11:00 11-11:30 11:30 -1:00 1:00-2:30 2:30-4:00 4:00-4:30 4:30-5:30 5:30-6:00
Monday 18/11/19 Topic-2 (HKS)   Topic-1 (SSK) L Topic-4 (HKM)   ATS/SPT  
Tuesday 19/11/19 Topic-2 (HKS) T Topic-1 (SSK) U Topic-4 (HKM) T ATS/SPT T
Wednesd ay 20/11/19 Topic-2 (HKS) E Topic-1 (SSK) N Topic-4 (HKM) E ATS/SPT E
Thursday 21/11/19 Topic-2 (HKS) A Topic-1 (SSK) C Topic-4 (HKM) A Special Lecture A
Friday 22/11/19 Topic-2 (HKS)   Topic-1 (SSK) H Topic-4 (HKM)   ATS/SPT  
Saturday 23/11/19 Topic-2 (HKS)   Topic-1 (SSK)   Topic-4 (HKM)   ATS/SPT  
Sunday 24/11/19 S U N D A Y                                                B R E A K
Monday 25/11/19 Topic-3 (SD)   Topic-6 (AM)   Topic-5 (BS)   ATS/SPT  
Tuesday 26/11/19 Topic-3 (SD) B Topic-6 (AM) B Topic-5 (BS) B ATS/SPT B
Wednesd ay 27/11/19 Topic-3 (SD) R Topic-6 (AM) R Topic-5 (BS) R ATS/SPT R
Thursday 28/11/19 Topic-3 (SD) E Topic-6 (AM) E Topic-5 (BS) E Special Lecture E
Friday 29/11/19 Topic-3 (SD) A Topic-6 (AM) A Topic-5 (BS) A ATS/SPT A
Saturday 30/11/19 Topic-3 (SD) K Topic-6 (AM) K Topic-5 (BS) K ATS/SPT K

Selected Applicants: 

 General Instruction for the candidates:

  1. All selected candidates are required to confirm their participation and travel plan on or before July 21, 2019, by sending the scanned copy of their forward train/air journey tickets wherever applicable, to the following emails: hksinghdu@gmail.com; ncmtopologydu@gmail.com

  1. After the deadline for confirming the participation by the selected participants, the wait listed candidates will be considered if the seats are vacant. The candidates who will selected from the waiting list will be informed separately through e-mail by the organizers. Such candidates will then be required to confirm their participation (as indicated in point 1 above) within a given specific timeline.

  1. Lodging and boarding will be free of cost.

  1. No travel allowance is permissible.

  1. It is assumed that all the participants in the workshop are familiar with the basic concepts of Homotopy Theory (Fundamental Groups), Simplicial Homology Theory and Homological Algebra. All the participants are, therefore, advised to revise these topics before coming to attend the workshop.

  Outstation Participants      
Sr.n SID Full Name Gender Affiliation Position in College/ University University/Institute M.Sc./ M.A. Year of Passing M.Sc./ M.A Ph.D. Degree Date
1 25320 Mr. Arindam Das Male BITS Pilani PhD Student Nit Rourkela 2016  
2 25566 Mr Gimnitz Simon S Male The American College MSc Student The American College    
3 25904 Mr. Umar Mohammad Khan Male Aligarh Muslim University PhD Student Aligarh Muslim University 2015  
4 26023 Mr Abhishek Shrivastava Male Aligarh Muslim university PhD University of Delhi 2016  
5 26274 Dr. Jitender Singh Male Guru Nanak Dev University Amritsar Assistant Professor Panjab University Chandigarh 2002 07/04/2008
6 26281 Mr Priyabrata Mandal Male IIT BOMBAY PhD IIT KANPUR, M.Sc 2016  
7 26322 Ms. Pratibha Pandey Female University of Allahabad Ph. D. Student University of Allahabad 2015  
8 26498 Mr. Sachchidanand Prasad Male IISER Kolkata PhD National Institute of Technology Rourkela 2017  
9 27041 Mr Mohan R Male Indian Statistical Institute PhD student Indian Institute of Technology Madras 2012  
10 27137 Ms. Arifa Naz Female Aligarh Muslim University PhD Aligarh Muslim University 2016  
11 27470 Mr. Rampao Koksi Sangma Male North Eastern Hill University Ph.D. North Eastern Hill University, Shillong 2013  
12 27472 Mr Shanborlang Bynnud Male North Eastern Hill University Ph.D. North Eastern Hill University, Shillong 2016  
13 28345 Mr Sumanta Das Male Indian Statistical Institute, Bangalore Centre MSc Student. Indian Statistical Institute, Bangalore Centre. Appeared  
14 28394 Mr. Soham Mondal Male IIT MADRAS, PH.D VISVA-BHARATI 2017  
15 28624 Mr Dilawar Juneed Mir Male Central University of Kashmir Integrated MPhil-PhD University of Kashmir Srinagar 2014  
16 28626 Mr. Mohd Rafiq Parray Male Central University of Kashmir Integrated M.Phil- PhD Central University of Kashmir 2014  
17 28663 Mr. Kushal Bhowmick Male Harish Chandra Research Institute. PhD Indian Statistical Institute. 2016  
18 28670 Mr. Chandramouli Chakraborty Male Indian Institute of Technology Madras Ph.d Indian Institute of Technology Madras 2018  
19 28734 Ms. Kriti Goel Female IIT Bombay PhD student IIT Bombay 2015  
20 28814 Mr. Saikat Panja Male IISER ,Pune PhD Indian Institute of Science Education and Research Kolkata 2017  
21 28824 Dr. Puneet Kumar Gupt Male University of Lucknow, Lucknow PhD University of Lucknow, Lucknow 2012 03/04/2019
22 28831 Dr. Snigdha Bharati Choudhury Female Harish-Chandra Research Institute Post-doctoral Fellow Utkal University, Bhubaneswar 2011 20/01/2018
23 28973 Dr. Ananya Chaturvedi Female TIFR, Mumbai Postdoctoral Fellow University of Houston 2012 15/12/2016
24 28976 Dr Shine C Mathew Male St Berchmans College Assistant Professor CMS College Kottayam 2004 16/02/2012
25 28985 Ms. Parul Keshari Female Harish Chandra Research Institute PhD Student Allahabad University 2015  
26 28995 Mr Anantadulal Paul Male NISER, Bhubaneswar Ph.D. STUDENT University Of Pune, Pune 2015  
27 29008 Ms Gunja Sachdeva Female TIFR, Mumbai Postdoc Dayalbagh Educational Institute 2011 04/08/2017
28 29028 Mr. Cyril J Jacob Male St. Berchmans College MSc Student St. Berchmans College Appeared  
29 29059 Mr. Somrik Bhattacharya Male Techno India university MSc student Techno India university    
30 29083 Mr. Sumit Roy Male TIFR, Mumbai PhD Ramakrishna Mission Vivekananda University, Belur 2016  
  Local Participants      
1 25425 Ms. Shivani Sharma Female Ambedkar Unioversity Delhi Mphil Student Hansraj College, University Of Delhi 2016  
2 25426 Ms. Charu Gupta Female Dr B.R Ambedkar University Ph.D Student Delhi University 2017  
3 26520 Ms. Arushi Arushi Female Ambedkar University Delhi PhD STUDENT Hindu College , University Of Delhi 2017  
4 27076 Mr Harshit Mathur Male Department of Mathematics, University of Delhi PhD Scholar Hindu College, University of Delhi 2013  
5 27351 Ms. Anju Kumari Female Department of Mathematics, University of Delhi PhD student Hindu College, University of Delhi 2014  
6 28252 Mr. Gaurav Kumar Male University of Delhi M.Phil Student Hindu College, University of Delhi 2017  
7 28260 Ms. Dimpi . Female University of Delhi M.Phil Student I. P. College, University of Delhi 2016  
8 28969 Mr. Nitin Sharma Male University of Delhi PhD Scholar University of Delhi 2016  
9 28993 Mr. Ananya Manas Male University of Delhi PhD Student IIT Kanpur 2016  
10 29001 Mr Konthoujam Somorjit Singh Male Department of Mathematics, University of Delhi Ph.D. Student University of Delhi 2010  
11 29061 Mrs. Jaspreet Kaur Female St. Stephen's College Assistant Professor Delhi 2004 08/11/2007
12 29084 Mr Sumit Mittal Male University of Delhi, Delhi PhD Student University of Delhi, Delhi 2016  
  Waiting List Outstation      
1 28346 Mr Pritthijit Biswas Male Chennai Mathematical Institute. PhD Ramakrishna Mission Vivekananda University 2017  
2 28937 Mr. Manoj Mahadev Patil Male G.M.Vedak Institute Of Technology,Tala Assistant Professor Willingdon College Sangli 2017  
3 28726 Ms. Sakeena Bano Female Central University of Kashmir Int. MPhil/Phd Jamia Millia Islamia University New Delhi 2014  
4 28724 Mr. Muneer Nabi Male Central University of Kashmir Int. MPhil/Phd Jamia Millia Islamia University New Delhi 2015  
5 28361 Ms. Rashmirekha Patra Female Sambalpur University Institute Of Information Technology Asst. Prof. Utkal University 2010  
6 27719 Mr Shankhadeep Mondal Male IISERThiruvananthapuram Integrated PhD IISER-TVM 2018  
7 27048 Mr. Mahesh Krishna K Male National Institute Of Technology Karnataka (NITK) Surathkal
Mangalore		 Research Scholar Mangalore University 2014  
8 26368 Ms. Archita Mondal Female IIT BOMBAY PhD IIT MADRAS 2015  
9 26183 Mrs. Bandna Rani Female Iiser mohali PhD Punjabi University 2016  
10 28916 Ms. Shreya Goyal Female Roorkee Institute of Technology Lecturer Department of Mathematics, Panjab University 2018  
11 29080 Dr. Makarand Sanjay Sarnobat Male ISSER, Bhopal Post doctoral fellow Pune University 2012 27/02/2019
12 27363 Mr. Subham Bhakta Male Georg-August-Universität Göttingen PhD Chennai Mathematical Institute 2018  
13 28130 Mr. Gaurav Tyagi Male M. M. Degree College , Khekra , Baghpat Assistant professor C C S University , Meerut 2009  
14 28986 Ms Radhika M M Female TIFR Mumbai Visiting fellow NIT Calicut    
15 25466 Mr. Khairatkar Anant Avinash Male D.B.F Dayanand college of Arts and Science MSc Student D.B.F Dayanand college of Arts and Science 2018  
16 26096 Mr. Jeyarambharathi A.M. Male Hindustan Institute Of Technology And Science, PHD Student MAHER 2016  
17 28652 Mr Vijay Saini Male Chandigarh university Pass Out M.SC Student M.SC 2018  
18 28754 Mr. Sauvik Mondal Male Indian Statistical Institute Msc Indian Statistical Institute    
19 28870 Dr. Gopal Datt Male Indian Institute of Science Postdoctoral fellow University of Delhi 2009 02/06/2017
Waiting list Local     
1 29087 Ms. Bharti Garg Female Hindu college ,university of delhi MSCstudent Hindu college ,university of delhi Appeared  
2 27904 Mr. Aakash Gupta Male University of Delhi, Department of Mathematics MSc student University of Delhi Appeared  

 

 

How to Reach: 

TBA

School Short Name: 

stscss

Last Date Application: 

Saturday, February 23, 2019

School Type: 

NCMW