# 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 withtheir affiliation. No. of Lectures Details Vivek Mohan Mallick (VM) 5 Milnor K-theory and Galois cohomology:Introduction to Milnor K-theory, construction of the residue and norm homomoprhisms, recollection of Galois cohomology, the norm residue homomorphism. Chetan Balwe (CB) 5 Motivic cohomology:Finite correspondences, presheaves with transfers, motivic cohomology, the triangulated category of motives, relation with Milnor K-theory. Vaibhav Vaish (VV) 5 Motivic homotopy theory:Introduction to motivic homotopy theory, motivic Eilenberg-MacLane spaces, symmetric powers. 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) 6 Norm varieties: Norm varieties and their properties, Rost’s chain lemma and degree formulas. Utsav Choudhury (UC) 5 Motivic Steenrod operations:Construction of the motivic Steenrod algebra and its dual, Milnor operations and reduced power operations. Anand Sawant (AS) 5 Voevodsky’s proof of the Bloch-Kato conjecture:Rost’s degree formula, existence of splitting varieties and the Rost motive, motivic Hilbert 90 theorem and its relationship with the Bloch-Kato and Beilinson-Lichtenbaum conjectures, 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.

(Lecture notes from the speakers, if available)

Names of the tutors with their affiliation : (To be decided)

## Time Table

 Day Date Lecture 1(9.30–11.00) Tea(11.–11.30) Lecture 2(11.30–1.00) Lunch(1.00–2.00) Lecture 3(2.00–3.30) Tea /Snacks(3.30-4.00) Lecture 4/ Discussion(4.00-5.00) (name of the speaker) (name of the speaker) (name of the speaker) (Name of the speaker) Mon V M C B V V Discussion Tues V M C B V V Discussion Wed V M C B V V Discussion Thu V M C B V V Discussion Fri V M C B V V Discussion Sat A H U C V S Discussion Sun Mon A H U C V S Discussion Tues A H V S V S Discussion Wed A S A H V S Discussion Thu A S U C U C Discussion Fri A S U C A H Discussion Sat A S A S
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