NCMW - The Bloch-Kato conjecture (2019)

Speakers and Syllabus


The Bloch-Kato conjecture is a long-sought result relating Milnor K-theory 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 K-theory, 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 K-theory in the last few years. The aim of this workshop is to give a detailed account of Voevodsky’s proof of the Bloch-Kato 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 K-theory and Galois cohomology:

Lecture 1: Milnor K-theory and basic properties

Lecture 2: Norm map in Milnor K-theory

Lecture 3: Galois cohomology

Lecture 4: Statement of the Bloch-Kato 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 K-theory

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 Merkurjev-Suslin theorem:
K-theory of quadrics, a proof that the norm residue homomorphism is an isomorphism in degree 2 (Merkurjev-Suslin theorem), the norm residue homomorphism in degree 3.

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 Bloch-Kato conjecture:

Lecture 1: The skeleton of the proof and preliminary reductions

Lecture 2: The Beilinson-Lichtenbaum 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:

  1. V. Voevodsky, Motivic cohomology with Z/2-coefficients, Publ. Math. IHES, 98(2003), 59 - 104.
  2. V. Voevodsky, On motivic cohomology with Z/l-coefficients, Ann. of Math. 174(2011), 401 - 438.
  3. C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, 2006.
  4. 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
(4.00–4.30)

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

 

 

 

File Attachments: