Hilbert Functions and Local Cohomology (2023)

Convener(s)

 
Name: Prof. J.K. Verma Prof. Tony Puthenpurakal
Mailing Address: IIT Bombay,
Department of Mathematics,
Powai, Mumbai - 400076		 IIT Bombay,
Department of Mathematics,
Powai, Mumbai - 400076
Email:  jkv.iitb at gmail.com  tputhen at math.iitb.ac.in

Hilbert functions were introduced for graded modules by David Hilbert in his study of invariant subrings.  Pierre Samuel introduced the Hilbert-Samuel coefficients of any m-primary ideal of a Noetherian local ring. The Hilbert coefficients have been found to be very useful in the study of singularities of algebraic varieties. Grothendieck and Serre found a deep connection between local cohomology and Hilbert functions. This helped understanding the computation of the postulation number in terms of largest non-vanishing degree of graded local cohomology modules of blowup algebras. A related notion is that of Castelnuovo-Mumford regularity of graded modules which helps in the study of Hilbert functions. Wolmer Vasconcelos introduced the Sally modules of ideals which have been effectively used to find properties of Hilbert polynomials and and blowup algebras. This workshop will introduce the fundamental techniques of local cohomology modules, Sally modules and Castelnuovo-Mumford regularity with applications to Hilbert functions of blowup algebras, bounds on Hilbert coefficients, behaviour of Hilbert functions under perturbations and connections with Sheaf cohomology and local cohomology.

Dates: 

Monday, November 27, 2023 - 19:00 to Saturday, December 2, 2023 - 19:00

Venue: 

Venue Address: 

bombay

Venue State: 

Venue City: 

PIN: 

400076

Chrono Order: 

461

Syllabus: 

Syllabi for the Courses of lectures

  • Shreedevi Masuti: The Sally module was introduced by W. V. Vasconcelos.  The Sally module is an important tool to interplay between Hilbert coefficients and the properties of the associated graded ring. In these series of lectures we will discuss the properties of the Sally module and its relation to the normal Hilbert coefficients and blow-up algebras.
  • Tony Puthenpurakal: Definition of local cohomology, Cech cohomology, vanishing theorems, non-vanishing theorems, Grothendieck duality for modules over local and graded rings, Grothendieck-Serre formula for Hilbert-Samuel functions
  • Vivek Mukundan: Itoh’s work on the normal Hilbert polynomial and his solution of Itoh’s conjecture for the normal Hilbert polynomial of maximal ideal in a Gorenstein local ring. Vanishing of the second local cohomology module of the extended Rees algebra and its applications to normal Hilbert polynomials.
  • Suprajo Das:Sheaf cohomology of projective varieties and its connections with local cohomology of graded modules. Riemann-Roch theorem for curves and its applications to Hilbert-Samuel coefficients and epsilon multiplicity.
  • Jugal Verma: Superficial sequences for good filtrations of ideals and their applications to the study of Hilbert coefficients, Theorems of Nagata, Huneke -Ooishi, Valabrega-Valla, Huckaba-Marley and Rossi-Valla.
  • V. Srinivas and V. Trivedi: Bounds on Hilbert coefficients, Hilbert functions under small perturbations.
  • N. V. Trung: Extended degree and its application to bounds on  Hilbert functions and regularity.
  • H. Ananthnarayan: Applications of Grothendieck-Serre formula  and Blancafort-Johnston-Verma formula for Hilbert-Samuel functions of filtrations of ideals to postulation and reduction number of a good filtration of ideals, Sally’s formulas for Hilbert-coefficients.

Applicants: Must know the contents of "Homological Methods in commutative algebra available for free at https://mathweb.tifr.res.in/sites/default/files/publications/pamphlets/homological.pdf
The application must have the following compulsory question: List the chapters that you know of the book above.

 

 

 

Time Table: 

 

 

Date

9.30

11.00

11.00

11.15

11.15

12.45

1.00

2.00

2.15

3.45

3.45

4.00

4.00

5.30

5.30

6.00

27 Nov

JKV

Tea

TJP

Lunch

SKM

Tea

SKM

Snacks

28 Nov

JKV

 

TJP

 

SD

 

SKM

 

29 Nov

JKV

 

TJP

 

SD

 

VT

 

30 Nov

VM

 

TJP

 

SD

 

VT

 

1 Dec

VM

 

NVT

 

HA

 

VS

 

2 Dec

VM

 

NVT

 

HA

 

VS

 

Titles of Lecture courses

Speaker

Abbv.

Affiliation

Title

Tony Puthenpurakal

TJP

IIT Bombay

Local cohomology modules

Vivek Mukundan

VM

IIT Delhi

Normal Hilbert polynomials

Shreedevi Masuti

SKM

IIT Dharwad

Hilbert functions and Sally modules

Suprajo Das

SD

IIT Bombay

Sheaf cohomology and

Riemann-Roch Theorem

H. Ananthnarayan

HA

IIT Bombay

Hilbert functions and local cohomology

Jugal Verma

JKV

IIT Bombay

The Chern number of a good filtration and Coehn-Macaulay associated graded rings

Ngo Viet Trung

NVT

IM, Hanoi

Extended degree,Hilbert functions and regularity

Vasudevan Srinivas

VS

IIT Bombay

Bounds on Hilbert coefficients

Vijaylaxmi Trivedi

VT

TIFR, Mumbai

Hilbert functions under small perturbations

 

 

Selected Applicants: 

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 52856 Mr Omkar Deepak Javadekar Male IIT Bombay MSc & PhD Dual Degree Student Indian Institute of Technology, Bombay 2020  
2 51994 Ms Shruti Priya Female IIT Kharagpur PhD NIT Rourkela 2020  
3 52043 Mr. Siddhartha Pramanik Male Indian Institute of Technology Kharagpur PhD Student Indian Institute of Technology Madras 2022  
4 52061 Ms Nidhi Gupta Female Central University of Jammu Ph.D Scholar Central University of jammu 2019  
5 52062 Ms. Mouma Samanta Female Indian Institute of Technology Kharagpur Ph.D. Indian Institute of Technology Kharagpur 2022  
6 52110 Dr. Aarti Patle Female Swami Vivekanand Government College Assistant Professor Visvesvaraya National Institute of Technology (VNIT Nagpur) 2014 07/08/2023
7 52127 Mr. Anoot Kumar Yadav Male Indian Institute of Technology Patna PhD Student Ewing Christian College (University of Allahabad) 2018  
8 52143 Mr Nikhil Purushottam Zade Male Visvesvaraya National Institute of Technology PhD Visvesvaraya National Institute of Technology 2020  
9 52167 Mr. Sayed Sadiqul Islam Male IIT Bombay PhD Jadavpur University 2021  
10 52173 Mr. Aniruddha Saha Male INDIAN INSTITUTE OF TECHNOLOGY, HYDERABAD. Phd UNIVERSITY OF HYDERABAD 2018  
11 52222 Mr. Adarsh Subhash Narendulwar Male Visvesvaraya National Institute of technology, Nagpur PhD Indian Institute of Technology, Mandi 2022  
12 52235 Ms Meghana Bhat Female Indian Institute of Technology Dharwad PhD Student Pondicherry University 2021  
13 52256 Mr. Mohamed Harith Male IIT Madras PhD IIT Bombay 2020  
14 52278 Mr. Vivek Bhabani Lama Male Indian Institute of Technology Kharagpur PhD St. Joseph's College (Autonomous), Bangalore City University 2021  
15 52304 Mr Sirajul Haque Male RAMAKRISHNA MISSION VIVEKANANDA EDUCATIONAL AND RESEARCH INSTITUTE PhD student University of Calcutta 2019  
16 52315 Mr. Ganapathy K Male Indian Institute of Technology Madras PhD student IIT Madras    
17 52327 Dr. Dibyendu Mondal Male ISI Kolkata Postdoctoral fellow IIT Bombay 2015 10/08/2021
18 52359 Mr. Saikat Manna Male Indian Institute Of Technology, Palakkad PhD Indian Institute Of Technology, Gandhinagar 2023 09/08/2023
19 52363 Mr. Manohar Kumar Male Indian Institute of Technology Kharagpur Indian Institute of Technology Kharagpur Indian Institute of Technology, Kharagpur 2020  
20 52522 Dr. Ankit Mishra Male UM-DAE CEBS Research Associate University of Allahabad 2015 20/08/2022
21 52573 Mr. Shrikant Shekhar Male IISER BHOPAL PhD IIT GANDHINAGAR 2019  
22 52575 Ms. Saipriya Dubey Female IIT Dharwad Research Associate   2016 25/02/2023
23 52588 Ms. Prativa Biswas Female IIT Kharagpur Ph.D IIT Kharagpur 2021  
24 52616 Mr. Nirmal Kotal Male Chennai Mathematical Institute PhD IIT Kharagpur 2017  
25 52628 Mr. Kanoy Kumar Das Male Ramakrishna Mission Vivekananda Educational and Research Institute PhD Ramakrishna Mission Vidyamandira 2018  
26 52723 Dr. Om Prakash Male Indian Institute of Technology Bombay, Powai Institute Post Doctoral Fellow Maharshi Dayanand university Rohtak 2016 29/07/2023
27 52734 Mr. Abhiram Subramanian Male Chennai Mathematical Institute PhD Student Chennai Mathematical Institute 2022  
28 52764 Mr. Ritam Halder Male IIT Kharagpur PhD Ramakrishna Mission Vivekananda Educational and Research Institute(RKM VERI), Belur 2022  
29 52871 Ms. Deblina Dey Female IIT MADRAS Ph.D Ramakrishna Mission and Vivekananda Educational and Research Institute 2020  
30 52950 Ms. Sutapa Dey Female IIT Hyderabad PhD BIT,Mesra 2017  
31 52967 Dr. Buddhadev Hajra Male TIFR Mumbai Postdoctoral Fellow IIT Bombay, Mumbai 2016 20/08/2022
32 53041 Mr. Sai Krishna P M S Male IIT Bombay Ph.D. ISI kolkata 2020  
33 53057 Ms. Anupama O S Female INDIAN INSTITUTE OF TECHNOLOGY JUNIOR RESEARCH FELLOW COCHIN UNIVERSITY OF SCIENCE AND TECHNOLOGY 2023  
34 53155 Mr. Pritam Roy Male Indian Institute of Technology Kharagpur PhD The University of Burdwan 2021  
35 53278 Dr. Sudeshna Roy Female Tata Institute of Fundamental Research, Mumbai Visiting Fellow Jadavpur University 2013 04/06/2020
36 53336 Dr Arusha C Female IIT Bombay Postdoctoral Fellow M.S. University of Baroda 2015 31/03/2021
37 53464 Mr. Samarendra Sahoo Male IIT BOMBAY PhD Ravenshaw University, Cuttack 2017  
Waiting list
1 52020 Mr. Paramhans Kushwaha Male IIT JAMMU PhD IIT DELHI 2020  
2 52364 Mr Jebasingh R Male IISER Kolkata PhD IIT Madras 2019  
3 52653 Mr. Shashi Ranjan Sinha Male Indian Institute of Technology Hyderabad PhD Patna University 2020  
4 52934 Mr. Amit Roy Male National Institute of Science Education and Research (NISER) Bhubaneswar Postdoctoral fellow Chennai Mathematical Institute 2015 28/06/2021
5 52754 Mr. Bijender Dharan Male Indian Institute of Technology, Jammu PhD Kurukshetra University, Kurukshetra 2015  

 

 

How to Reach: 

Click here

School Short Name: 

hflc

Last Date Application: 

Saturday, November 18, 2023

School Type: 

NCMW

Separate faculty form: 

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