NCMW  Hilbert Functions and Local Cohomology (2023)
Speakers and 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 blowup algebras.
 Tony Puthenpurakal: Definition of local cohomology, Cech cohomology, vanishing theorems, nonvanishing theorems, Grothendieck duality for modules over local and graded rings, GrothendieckSerre formula for HilbertSamuel 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. RiemannRoch theorem for curves and its applications to HilbertSamuel 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, ValabregaValla, HuckabaMarley and RossiValla.
 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 GrothendieckSerre formula and BlancafortJohnstonVerma formula for HilbertSamuel functions of filtrations of ideals to postulation and reduction number of a good filtration of ideals, Sally’s formulas for Hilbertcoefficients.
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 RiemannRoch Theorem 
H. Ananthnarayan 
HA 
IIT Bombay 
Hilbert functions and local cohomology 
Jugal Verma 
JKV 
IIT Bombay 
The Chern number of a good filtration and CoehnMacaulay 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 