18.318 M.I.T. Spring 2003
Topics in Combinatorics:
``Schubert Calculus and Combinatorics''
"Problem 15: To establish a rigorous foundation of Schubert's enumerative calculus."

Mathematische Probleme by David Hilbert

Class meets: Tuesday, Thursday 1112:30
room 2102 
Instructor: Alexander Postnikov
apost at math
room 2389
Course webpage:
http://wwwmath.mit.edu/~apost/courses/18.318/
Synopsis:
The course is devoted to combinatorial aspects of Schubert calculus
of the Grassmannian and the flag manifold.
It is a classical area of enumerative geometry whose purpose is
to calculate various intersection numbers and solve problems like the following:
Find the number of lines in the 3dimensional complex space that intersect
with given four generic lines.

Schubert calculus has links with combinatorics
of symmetric functions and representation theory of the general linear
group.
The course will include the following topics:
GrassmannPlucker relations,
Schubert cells, Pieri's formula, Young tableaux, Schur symmetric polynomials,
JacobiTrudy and Giambelli's formulas,
LittlewoodRichardson rule, GelfandSerganova cells and matroids,
Bruhat order, ChevalleyMonk's formula,
Schubert polynomials, BernsteinGelfandGelfandDemazure theorem,
Cauchy formula, RCgraphs, etc.
We will also discuss some recent results related to quantum cohomology and
GromovWitten invariants,
total positivity and links with inverse boundary problem for planar networks.
Preference will be given to explicit combinatorial constructions and proofs.
The course will be selfcontained. All required notions and definitions will
be given.
There are no any special prerequisites for the course.
Course Level: Graduate
Texts:
Recommended (but not required) textbooks are:
[F] 
W. Fulton: Young Tableaux, Cambridge University Press, 1997.

[M] 
L. Manivel: Symmetric Functions,
Schubert Polynomials and Degeneracy Loci,
AMS, 2001.

[EC2] 
R. P. Stanley: Enumerative Combinatorics,
Vol 2, Cambridge University Press, 1999.

Lectures:
 T 02/04/03. Course overview. Grassmannian: main definitions. [M, 3.1.1]
 R 02/06/03. Application: qbinomial coefficients. Plucker relations.
[F, 9.1], [M, 3.1.2].
 T 02/11/03. Schubert cells in the Grassmannian: 4 definitions.
Schubert varieties.
[F 9.4], [M 3.2].
 R 02/13/03. Matroids. Matroid stratification of the Grassmannian.
[Gelfand, Goresky, MacPherson, Serganova,
Combinatorial geometries, convex
polyhedra, and Schubert cells, Adv. Math. 63 (1987) 301316.]
T 02/18/03. [no class, Monday schedule]
 R 02/20/03. Cohomology ring the Grassmannian.
Duality theorem. [F, 9.4, Appendix B], [M, 3.2, Appendix].
 T 02/25/03. Cohomology ring the Grassmannian (cont'd).
Pieri's formula.
[F, 9.4], [M, 3.2].
 R 02/27/03. Symmetric polynomials. Schur polynomials.
[EC2, 7.17.10, 7.15], [F, 6], [M, 1.11.2]
 T 03/04/03. JacobiTrudy identity and Giambelli formula.
[EC2, 7.9, 7.10, 7.16], [F, 6], [M, 1.2]
 R 03/06/03. LittlewoodRichardson rule: Classical rule and
Zelevinsky's pictures.
[EC2, A1], [F, 5], [M 1.5].
 T 03/11/03. LR rule (cont'd): BerensteinZelevinsky
triangles, KnutsonTao honeycombs.
[EC2, A1], [Knutson, Tao, The honeycomb model of GL(n)
tensor products I: proof of the saturation conjecture,
math.RT/9807160].
Problem Set 1 is due.
 R 03/13/03.
LR rule (cont'd): Klyachko cone, BerensteinZelevinsky polytope,
GelfandTsetlin polytope.
 T 03/18/03. Symmetric group, reduced decompositions, wiring diagrams,
weak Bruhat order. [M 2.1]
 R 03/20/03. Strong Bruhat order. [M 2.1]
T 03/25/03. [no class, Spring break]
R 03/27/03. [no class, Spring break]
 T 04/01/03. Schubert polynomials, divided differences, RCgraphs.
[M 2.3]
 R 04/03/03.
Schubert polynomials (cont'd): nilHecke algebra, YangBaxter equation,
Cauchy formula. [M 2.3, 2.4]
 T 04/08/03. Schubert polynomials (con'd): Grassmannian permutations,
relations to Schur polynomials, RCgraphs and families nonintersecting paths,
GesselViennot method.
 R 04/10/03. Schubert polynomials (cont'd): ChevalleyMonk formula,
Pieri formula, FominKirillov quadratic algebra.
 T 04/15/03. Geometry of the flag manifold: Schubert cells and
varieties, cohomology ring, ChevalleyMonk formula (geometric variant).
 R 04/17/03. Borel theorem, coinvariant algebra.
Grobner bases.
T 04/22/03. [no class, Patriots day]
 R 04/24/03. Grobner bases. Calculating the generalized
LRcoefficients for the flag manifold.
 T 04/29/03. Guest lecture by Rom Pinchasi.
 R 05/01/03. Verma's theorem.
 T 05/06/03.
 R 05/08/03.
 T 05/13/03.
 R 05/15/03.