This Workshop is planned to introduce students to the some of the combinatorial aspects of commutative algebra. There will be three streams of lectures where three different types of combinatorial commutative algebraic topics will be discussed.
Speakers:
(a) Huy Tài Hà (Tulane University, USA)
(b) K. N. Raghavan (Institute of Mathematical Sciences, Chennai)
(c) Indranath Sengupta (IIT Gandhinagar)
(d) A. V. Jayanthan (IIT Madras)
(e) Madhusudan Manjunath (IIT Bombay)
Syllabus:
| Title: | Matrix Schubert Varieties |
| Speaker: | K. N. Raghavan |
| Topics | We will cover the material in two papers of Knutson and Miller (Advances 2004 & Annals 2005) on "Gröbner geometry of Schubert polynomials". Recall that Schubert polynomials are certain special representatives of Schubert classes in the integral cohomology ring of the full flag manifold. One of the main theorems of Knutson and Miller gives a geometric interpretation of Schubert polynomials as multi-degrees associated to matrix Schubert varieties. The multi-degree, which is a combinatorial commutative algebraic notion, works as a more versatile proxy for equivariant cohomology. (Double Schubert polynomials, Grothendieck polynomials, and double Grothendieck polynomials can also be analogously treated.) Combining this interpretation with another result about a Gröbner basis for matrix Schubert varieties, one gets elegant and satisfactory explanations for previously discovered "positive" formulas for Schubert polynomials. |
| Title: | Six Lectures on Toric Algebra |
| Speaker: | Indranath Sengupta |
| Topics: | 1. Numerical Semigroups: Monoids and monoid homomorphisms. Multiplicity and embedding dimension. Frobenius number and Genus. 2. Semigroup rings: Semigroups and lattice ideals. Affine semigroups and polyhedral cones. Hilbert bases. Initial ideals of Lattice ideals. 3. Toric Ideals |
| Title: | Graphs, Hypergraphs and Associated Ideals |
| Speakers: | A. V. Jayanthan and Huy Tài Hà |
| Topics: | 1) Basic commutative algebra such as resolution, regularity, depth etc. 2) Edge ideals — the algebra-combinatorics framework 3) Associated primes of powers of squarefree monomial ideals 4) Regularity of powers of edge ideals 5) Combinatorial structures through algebraic lenses. |
| Title: | Commutative Algebra of Chip Firing and Generalised Frobenius Numbers |
| Speaker: | Madhusudan Manjunath |
| Topics: | 1. The G-parking function ideal and the toppling ideal of a graph, their Hilbert series, minimal free resolutions, connection to Tutte polynomials. 2. Generalised Frobenius numbers: the case k = 2, commutative algebraic aspects of general k, applications to sequences of generalised Frobenius numbers |
Pre-requisites: Familiarity with a graduate course in commutative algebra is assumed.
Time-table:
| 9:00-10:30 | 10:30-11:00 | 11;00-12:30 | 12:30-2:00 | 2:00-3:30 | 3:30-4:00 | 4:00-5:30 | |
| 18.06 | AVJ |
T E A |
KNR | L U N C H | AVJ |
T E A |
MM |
| 19.06 | KNR | MM | IS | IS | |||
| 20.06 | TH | KNR | IS | MM | |||
| 21.06 | TH | IS | MM | KNR | |||
| 22.06 | KNR | IS | TH | MM | |||
| 23.06 | KNR | TH | MM | IS |
AVJ – A V Jayanthan
KNR – K N Raghavan
MM – Madhusudan Manjunath
IS – Indranath Sengupta
TH – Huy Tai Ha