The Bloch-Kato conjecture (2019)

Convener(s)

 
Name: Anand Sawant Amit Hogadi
Mailing Address:

Reader,
Tata Institute of Fundamental Research,
Dr. Homi Bhabha Road, Colaba,
Mumbai-400005.

Associate Professor
IISER Pune,
Dr Homi Bhabha Road, Pashan,
Pune-411008.
Email:

asawant at math.tifr.res.in

 amit at iiserpune.ac.in

Please Note:  Participants have to arrange for their own travel.

 

Dates: 

Monday, December 23, 2019 - 09:30 to Saturday, January 4, 2020 - 18:45

Venue: 

Venue Address: 

Dr Homi Bhabha Road, Pashan, Pune-411008.

Venue State: 

Venue City: 

PIN: 

411008

Chrono Order: 

341

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

 

 

 

Selected Applicants: 

Instructions for participants:

  1. Selected participants are required to confirm their participation by replying to the selection email on or before October 4, 2019; failing to do so may result in .
  2. Participants are required make their own travel arrangements; no travel support will be provided.
Sr.n SID Full Name Gender Affiliation Position in College/University University/Institute M.Sc./M.A. Year of Passing M.Sc./M.A Ph.D. Degree Date
1 28923 Ms Aparna Pradeep V K Female Cochin University of Science and Technology PhD Student Cochin University of Science and Technology 2016  
2 28987 Prof. Manoj Keshari Male IIT Bombay Professor IIT Kanpur 1996 10/10/2004
3 29063 Mr. Shravan Ganesh Patankar Male University of Illinois at Chicago PhD University of Illinois at Chicago    
4 29117 Dr. Sarang Sane Male IIT Madras Assistant Professor M.Stat., ISI 2004 15/10/2010
5 29126 Mr. Yong-Gyu Choi Male Korea Advanced Institute of Science and Technology Integrated MSc-PhD student Korea Advanced Institute of Science and Technology    
6 29167 Mr Jitendra Rathore Male TIFR Phd IISER Mohali 2017  
7 29214 Mr. Sourjya Banerjee Male ISI, Kolkata PhD University of Calcutta 2016  
8 29250 Dr. Rakesh Pawar Male IISER Mohali Postdoc IIT Mumbai 2012 18/08/2017
9 29269 Mrs. Bandna Rani Female IISER Mohali PhD Punjabi University 2016  
10 29366 Mr. Avijit Nath Male The Institute of Mathematical Sciences Ph.D. Ramakrishna Mission Vivekananda University, Belur Math, Howrah 2013  
11 29380 Mr. Subhajit Chanda Male IIT MADRAS PhD IIT MADRAS 2015  
12 29429 Mr. Onkar Kamlakar Kale Male IISER Tirupati PhD Student M.Sc. 2016  
13 29448 Mr. Wataru Kai Male Tohoku University Assistant Professor The University of Tokyo 2013 24/03/2016
14 29487 Mr Sourav Sen Male Ramakrishna Mission Vidyamandira PhD Ramakrishna Mission Vivekananda University 2013  
15 29497 Dr. Girja Shanker Tripathi Male IISER Tirupati Asst Professor University of Allahabad 2002  
16 29545 Dr Vivek Sadhu Male IISER Bhopal Assistant Professor Jadavpur University 2007  
17 29573 Mr. Ankit Rai Male TIFR Research Scholar Chennai Mathematical Institute 2014  
18 29587 Dr. Sampat Kumar Sharma Male ISI Kolkokata Visiting Scientist IISER Mohali   26/06/2019
19 29669 Mr. Ritankar Nath Male TIFR, Mumbai Integrated PhD TIFR, Mumbai    
20 29796 Mr. Sourayan Banerjee Male IISER Integrated PhD. IISER 2019  
21 29956 Dr. Umesh V Dubey Male Harish-Chandra Research Institute Asst. Prof. Mumbai University 2005 01/10/2012
22 30063 Ms Maria Ann Mathew Female IIT Bombay Ph.D. IIT delhi 2013  
23 30151 Dr. Shameek Paul Male RKMVERI Assistant Professor University of Mumbai 2005 19/01/2012
  Local participants      
1 29000 Mr. Abhishek Gupta Male IISER Project student IISER 2019  
2 29486 Mr. Suraj Prakash Yadav Male IISER Pune PhD IIT Kanpur 2016  
3 29829 Mr Kuntal Chakraborty Male Phd   Indian Statistical Institute Kolkata 2014  
4 29830 Mr. Dhruva Rasesh Kelkar Male University of Duisburg-Essen ALGANT Master Student      
5  - Girish Kulkarni   IISER Pune PhD student      
6 29731 Mr. Neeraj Deshmukh   IISER Pune  PhD  IISer Mohali  2015  
  Waiting List      
1 29218 Mr Subham Sarkar Male TIFR, Mumbai. Postdoctoral Fellow RKMVU 2010 01/04/2019
2 29520 Mr. Pradeep Das Male Harish-Chandra Research Institute PhD Student IIT Kanpur 2011  
3 29536 Mr. Anoop Singh Male Harish-Chandra Research Institute PhD University of Allahabad 2013  

 

 

How to Reach: 

TBA

School Short Name: 

TBKC

Last Date Application: 

Monday, November 19, 2018

List of Conveners: 

  • Anand Sawant
  • Amit Hogadi

School Type: 

NCMW