NCMW - Sheaf Theory, Sheaf Cohomology and Spectral Sequences (2019)

Speakers and 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
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