Expository Writings

1. Grothendieck-Serre Formula for Hilbert Functions: These notes written jointly with Parangama Sarkar and Shreedevi Masuti, are published in 2016 by Springer Proceedings of a conference at AMU. 

2. Solving Polynomial Equations: These are notes of lectures first delivered in an instructional school at IISER, Pune. Since then I have presented several talks on this very interesting topic. The tools developed here are very useful in many applications of Groebner bases. See the file polynomialequations.pdf below.

3. Normal Hilbert Polynomials: Jointly written with Mousumi Mandal and Shreedevi Masuti. See the file MandalMasutiVerma.pdf below.

4. Hilbert functions of multigrades algebras, mixed multiplicities of ideals and their applications, jointly written with N. V. Trung. See the file mixed_survey.pdf below.

5. Negativity of the Chern numbers of parameter ideals: Jointly written with Shiro Goto and Mousumi Mandal. See the file GotoMandalVerma.pdf below.

6. Hilbert coefficients and depth of the associated graded ring of an ideal: This paper gives a simple proof of an interesting theorem of Huckaba and Marley about the first Hilbert coefficient of the Hilbert-Samuel polynomial of an ideal. Several basic techniques required in the study of Hilbert-Samuel polynomial are discussed with complete proofs. See the file 2007MathStudent.pdf below.

7. Zariski's Theory of complete ideals: These are notes of my lectures in Hanoi in 2012. These are based on an expository paper of Craig Huneke, thesis of Vijay Kodiyalam  and the notes of a course offered by Craig Huneke at Purdue that I attended in 1986. See the file Hanoi_Zariski_complete.pdf below.

8. Stanley's solution of the Anand-Dumir-Gupta conjectures: This expository paper is based on my lecture at the Indian Mathematical Society in 2013. See the file 2013Stanleys.pdf below.

9. Rings of invariants and Gorenstein rings: Notes of lectures delivered at KSOM leading to K.-I. Watanabe's Theorem about Gorenstein rings of invariants.   See the file gorenstein.pdf

10. Basics of  multigraded Hilbert-Samuel functions: Notes of lectures at IIT Bombay. 

See MultigradedHilbertFunctions.pdf

11. Basic notions of algebraic geometry: Notes of four lectures delivered in the Annual Foundation School-II at Bhaskaracharya Pratishthana, Pune during 22-25 May, 2007. See the file AFSII_Pune2017.pdf below.

12. Groups of Symmetries: Notes of six lectures delivered in the Instructional school on algebra held at Himachal Pradesh University, Shimla during 5-9 June, 2017. See the file 

IST@Shimla2017.pdf

13. IIT Kharagpur Lectures.  I delivered 10 lectures on Groebner bases and integer programming during 27-31 August 2018. The file IITKGP2018.pdf has the notes of these lectures.

14. Lectures on Hilbert functions and local cohomology. These are notes of 16 lectures delivered at the University of Genoa during May 2018. See the file Genoa_May2018.pdf.