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 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

Titles of Lecture courses