NCMW  The BlochKato conjecture (2019)
Speakers and Syllabus
The BlochKato conjecture is a longsought result relating Milnor Ktheory and Galois cohomology, a proof of which was given by V. Voevodsky around 10 years ago. The conjecture, now known as the norm residue isomorphism theorem, is a more general statement than the Milnor conjecture relating seemingly unrelated areas of Milnor Ktheory, Galois cohomology and quadratic forms (for a proof of which Voevodsky received the Fields Medal in 2002). The norm residue isomorphism theorem has led to significant developments in the areas of motives, algebraic geometry and Ktheory in the last few years. The aim of this workshop is to give a detailed account of Voevodsky’s proof of the BlochKato conjecture building on many innovative ideas introduced by Voevodsky, Merkurjev, Suslin and Rost.
The workshop is targeted at advanced graduate students and researchers working in related areas and is aimed to equip them to access current developments in the subject. Basic knowledge of algebraic geometry will be assumed.
Name of the Speakers with their affiliation. 
No. of Lectures 
Detailed Syllabus 
Vivek Mohan Mallick (VM) 
4 
Milnor Ktheory and Galois cohomology: Lecture 1: Milnor Ktheory and basic properties Lecture 2: Norm map in Milnor Ktheory Lecture 3: Galois cohomology Lecture 4: Statement of the BlochKato conjecture 
Chetan Balwe (CB) 
6 
Motivic cohomology: Lecture 1: Finite correspondences and presheaves with transfers Lecture 2: Motivic complexes and motivic cohomology Lecture 3: Weight 1 motivic cohomology Lecture 4: Relation of motivic cohomology with Milnor Ktheory Lecture 5: The triangulated category of motives Lecture 6: Motivic cohomology in the critical range 
Vaibhav Vaish (VV) 
5 
Motivic homotopy theory: Lecture 1: Grothendieck topologies and simplicial sheaves Lecture 2: Introduction to motivic homotopy theory Lecture 3: The homotopy purity theorem Lecture 4: Symmetric powers of motives I Lecture 5: Symmetric powers of motives II 
V. Srinivas (VS) 
5 
The MerkurjevSuslin theorem: 
Amit Hogadi (AH) 
5 
Norm varieties: Lecture 1: Central simple algebras and the Brauer group of a field Lecture 2: Norm varieties and their properties Lecture 3: Rost’s chain lemma I Lecture 4: Rost’s chain lemma II Lecture 5: Summary of the formalism of Rost varieties 
Utsav Choudhury (UC) 
5 
Motivic Steenrod operations: Lecture 1: A review of Steenrod operations in topology Lecture 2: Cohomology operations I Lecture 3: Cohomology operations II Lecture 4: Motivic classifying spaces I Lecture 5: Motivic classifying spaces II 
Anand Sawant (AS) 
6 
Voevodsky’s proof of the BlochKato conjecture: Lecture 1: The skeleton of the proof and preliminary reductions Lecture 2: The BeilinsonLichtenbaum conjecture Lecture 3: The motivic Hilbert 90 theorem Lecture 4: Reduction to the existence of Rost motives Lecture 5: Existence of Rost motives Lecture 6: Conclusion of the proof of the main theorem. 
References:
 V. Voevodsky, Motivic cohomology with Z/2coefficients, Publ. Math. IHES, 98(2003), 59  104.
 V. Voevodsky, On motivic cohomology with Z/lcoefficients, Ann. of Math. 174(2011), 401  438.
 C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, 2006.
 C. Haesemeyer, C. Weibel, The norm residue theorem in motivic cohomology.
Time Table
Day 
Date 
Lecture 1 (9.30–11.00) 
Tea (11.–11.30) 
Lecture 2 (11.30–1.00) 
Lunch (1.00–2.30) 
Lecture 3 (2.30–4.00) 
Tea /Snacks 
Lecture 4 /Discussion (4.30–5.30) 
Mon 
23.12 
VM 1 

CB 1 

VM 2 

VM (Tutorial) 
Tue 
24.12 
VM 3 

CB 2 

AH 1 

VM (Tutorial) 
Wed 
25.12 
VM 4 

CB 3 

AH 2 

CB (Tutorial) 
Thu 
26.12 
CB 4 

VS 1 

VS 2 

VV 1 
Fri 
27.12 
VV 2 

VS 3 

VS 4 

Discussion 
Sat 
28.12 
VS 5 

CB 5 

VV 3 


Sun 
29.12 







Mon 
30.12 
AH 3 

CB 6 

UC 1 

VV (Tutorial) 
Tue 
31.12 
AS 1 

VV 4 

UC 2 

Discussion 
Wed 
01.01 
AS 2 

VV 5 

UC 3 

Discussion 
Thu 
02.01 
AS 3 

AH 4 

UC 4 

Discussion 
Fri 
03.01 
AS 4 

AH 5 

UC 5 

Discussion 
Sat 
04.01 
AS 5 

AS 6 




VM: Vivek Mohan Mallick
CB: Chetan Balwe
AH: Amit Hogadi
VS: V. Srinivas
VV: Vaibhav Vaish
UC: Utsav Choudhury
AS: Anand Sawant