# 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

VS: V. Srinivas

VV: Vaibhav Vaish

UC: Utsav Choudhury

AS: Anand Sawant

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