A Gr ̈obner basis is a set of multivariate polynomials that has several algorithmic properties. Any set of polynomials can be transformed into a Gr ̈obner basis. Gr ̈obner bases have been used in several applications such as solving equations, graph theory, optimization, cryptography, coding theory, robotics,control theory, statistics and automatic geometric theorem proving, to name a few. This workshop aims to introduce some of these applications after covering basics of this beautiful theory. The prerequisites are linear algebra and a basic course in abstract algebra.
Eligibility: The school will admit 30 outstation and 10 local participants. Research scholars and Net qualified teachers pursuing Ph. D. in algebra are eligible.
Resource person  Affiliation  Topics 
H. Ananthnarayan (HA)  IIT Bombay  Introduction to Gr ̈obner bases 
Indranath Sengupta (IS)  IIT Gandhinagar  First applications of Gr ̈obner bases 
Dilip Patil (DP)  IISc, Bangalore  Algebraic varieties and and dimension theory 
Manoj Kummini (MK)  CMI, Chennai  Gr ̈obner bases and multivariate splines 
S. R. Ghorpade (SRG)  IIT Bombay  Gr ̈obner bases and coding theory 
A. V. Jayanthan (AVJ)  IIT Madras  Solving polynomial equations using Gr ̈ obner bases 
R. Balasubramanian (RB)  IIT Bombay  Introduction to Cryptography 
M. Prem Laxman Das (PLD)  SETS, Chennai  Gr ̈obner bases and cryptography 
Anurag Singh (AS)  University of Utah  Computation of integral closure of an ideal 
Vivek Mukundan (VM)  Univ. of Virginia  Lab/Tutorial sessions 
Clare D’Cruz (SM)  CMI, Chennai  Lab/Tutorial sessions 
Shreedevi Masuti (SM)  Univ. of Genoa  Lab/Tutorial sessions 
Jugal Verma (JKV)  IIT Bombay  Tutorial sessions 
Shameek Paul  Ramakrishna Mission Vivekananda Univerisity  Labs/Tutorial 
Syllabi of Courses of Lectures and Lab Sessions
Name  Lecture  Reference 
Dilip Patil  Basic Algebraic Geometry : Affine and projective varieties, Hilbert’s Nullstellensatz, Noether Normalisation, dimension theory of graded rings and projective and affine varieties and their computation using Gr ̈obner bases. 
Chapter 1, 8 and 9 of CoxLittleO’Shea: Ideals, Varieties and Algorithms 
H. Ananthnarayan  Basics of Gr ̈obner bases: monomial orders and initial ideals inthe polynomial ring S = k[x1 , . . . , xn], where k is a field, important examples of monomial orderings and their basic properties, Dickson’s Lemma and Hilbert basis Theorem, using initial ideals to find kbasis for S/I, where I is an ideal in S, division algorithm in S, and Buch berger’s algorithm, reduced Gr ̈obner bases.  Chapter 2 of CoxLittleO’Shea: Ideals, varieties and Algorithms 
Indranath Sengupta  First applications of Gr ̈obner Bases: Elimination, computing kernel and image of a polynomial map of affine algebras, resultants, computation of radicals,intersection, colon and saturation of ideals, geometric meaning of elimination, minimal polynomials of algebraic elements over a field.  Chapter 2 and 3 of CoxLittleO’Shea: Ideals, Varieties and Algorithms 
Manoj Kummini  Multivariate polynomial splines:Recent applications of Gr ́obner bases to the problem of construction and anaysis of piecewise polynomial splines with specified degree of smoothness. Gr ̈obner bases of modules over polynomial rings will be used in these applications. In particular Schreyer’s algorithm will be discussed for computing syzygies. A conjecture of Strang about dimension of the vector space of C r functions on polyhedral complexes will be discussed.  D. Cox, J. Little and D. O’Shea: Using Algebraic Geometry. 
Sudhir R. Ghorpade  Gr ̈obner bases and coding theory:Basics of errorcorrecting codes, cyclic codes, ReedSolomon code, ReedMuller codes, Applications of Gr ̈obner bases to determining the minimum distance of ReedMuller type codes, Footprint bound, Some recent results.  (a) C. Carvalho, Applications of results from commutative algebra to the study of certain evaluation codes, Lecture notes of CIMPA Research School on Algebraic Methods in Coding Theory, Sao Paulo, Brazil, July 2017. https : //www.ime.usp.br/ cimpars/notes/sc401.pdf (b) D. Cox, J. Little, D. O’Shea, Ideals, Varieties and Algorithms, 3rd ed.,Springer,2007. (c) D. Cox, J. Little, D. O’Shea,Using Algebraic Geometry, 2nd ed., Springer,2006. (d) M. Sala et al (Eds), Gr ̈obner Bases, Coding, and Cryptography, Springer 2009. 
A. V. Jayanthan  Solving polynomial equations : Checking consistency of a system of polynomial equations using constructive Nullstellensatz, bounding number of solutions, multi variate Lagrange interpolation, eigenvector theorems, Stickelberger’s Theorem, counting real solutions using quadratic forms.  Chapter 2 of CoxLittleO’Shea: Using Algebraic Geometry 
R. Balasubramanian  Introduction to cryptography: Introduction to cryptography, symmetric key: block and stream ciphers, public key: elliptic curve based systems, multivariate cryptosystems, hard problems  
M. Prem Laxman Da  Gr ̈obner bases in cryptography: Algebraic attacks on block and stream ciphers, attacks on HFE and multivariate cryptosystems, index calculus attacks on elliptic curves  
Anurag Singh  Computation of integral closure of ideals: An algorithm for computation of integral closure of ideals will be explained.  Reference: Anurag K. Singh and Irena Swanson, An algorithm for computing the integral closure. Algebra and Number Theory 3 (2009), no. 5, 587–595. 
First Week:  Laboratory sessions: Computing Gr ̈obner basis, elimination, saturation, com puting kernel and image of algebra homomorphisms, division algorithm, computing basis of a zerodimensional affine algebra, computation of radicals, colons and intersection of ideals, calcula tion of Hilbert series and Hilbert polynomial of a graded algebra, minimal polynomial of algebraic elements, resultants and discriminants via elimination. 

Second Week:  Laboratory sessions: Solving polynomial equations, computing integral closure of ideals, computing splines, computing sygygies and free resolutions, cryptosystems, construction of codes using Gr ̈obner bases. 
Schedule of Lectures and Tutorials
Day  Dec  9.00  10.15  11.30  11.45  1.00  2.30 (Tutorial )  3.30  4.00 Comp. Lab 
Mon  11  DP  HA  Tea  IS  Lunch  1 (DP, JKV, SM)  2 (HA, IS, SM)  
Tue  12  DP  HA  IS  3 (HA, JKV, SM)  4 (IS, HA, SM)  
Wed  13  DP  HA  IS  5 (IS, JKV, SM)  6 (HA, IS, SM)  
Thu  14  DP  HA  IS  7 (DP, JKV, SM)  8 (IS, HA, SM)  
Fri  15  DP  HA  IS  9 (HA, JKV, SM)  10 (HA, IS, SM)  
Sat  16  DP  MK  MK  11 (IS, JKV, SM)  12 (IS, HA, SM)  
Sun  17  Delhi Darshan  
Mon  18  AVJ  MK  RB  13 (MK, HA, AVJ)  14 (MK, CD, AVJ)  
Tue  19  AVJ  MK  RB  15 (AVJ, HA, MK)  16 (AVJ, CD, MK)  
Wed  20  AVJ  SRG  RB, PLD, SP)  17 (MK, HA, VM)  18 (MK, CD, VM)  
Thu  21  AVJ  SRG  PLD  19 (AVJ, HA, VM)  20 (AVJ, CD, VM)  
Fri  22  AS  SRG  (PLD, CD, SP)  21 (SRG, CD, SP)  22 (SRG, CD, SP)  
Sat  23  AS  SRG  PLD  23 (AS, JKV, VM)  Val function. 