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."
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Mathematische Probleme by David Hilbert
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Class meets: Tuesday, Thursday 11-12:30
room 2-102 |
Instructor: Alexander Postnikov
apost at math
room 2-389
Course webpage:
http://www-math.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 3-dimensional complex space that intersect
with given four generic lines.
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Schubert calculus has links with combinatorics
of symmetric functions and representation theory of the general linear
group.
The course will include the following topics:
Grassmann-Plucker relations,
Schubert cells, Pieri's formula, Young tableaux, Schur symmetric polynomials,
Jacobi-Trudy and Giambelli's formulas,
Littlewood-Richardson rule, Gelfand-Serganova cells and matroids,
Bruhat order, Chevalley-Monk's formula,
Schubert polynomials, Bernstein-Gelfand-Gelfand-Demazure theorem,
Cauchy formula, RC-graphs, etc.
We will also discuss some recent results related to quantum cohomology and
Gromov-Witten 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 self-contained. 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.
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[M] |
L. Manivel: Symmetric Functions,
Schubert Polynomials and Degeneracy Loci,
AMS, 2001.
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[EC2] |
R. P. Stanley: Enumerative Combinatorics,
Vol 2, Cambridge University Press, 1999.
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Lectures:
- T 02/04/03. Course overview. Grassmannian: main definitions. [M, 3.1.1]
- R 02/06/03. Application: q-binomial 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) 301-316.]
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, 7.1-7.10, 7.15], [F, 6], [M, 1.1-1.2]
- T 03/04/03. Jacobi-Trudy identity and Giambelli formula.
[EC2, 7.9, 7.10, 7.16], [F, 6], [M, 1.2]
- R 03/06/03. Littlewood-Richardson rule: Classical rule and
Zelevinsky's pictures.
[EC2, A1], [F, 5], [M 1.5].
- T 03/11/03. LR rule (cont'd): Berenstein-Zelevinsky
triangles, Knutson-Tao 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, Berenstein-Zelevinsky polytope,
Gelfand-Tsetlin 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, RC-graphs.
[M 2.3]
- R 04/03/03.
Schubert polynomials (cont'd): nilHecke algebra, Yang-Baxter equation,
Cauchy formula. [M 2.3, 2.4]
- T 04/08/03. Schubert polynomials (con'd): Grassmannian permutations,
relations to Schur polynomials, RC-graphs and families non-intersecting paths,
Gessel-Viennot method.
- R 04/10/03. Schubert polynomials (cont'd): Chevalley-Monk formula,
Pieri formula, Fomin-Kirillov quadratic algebra.
- T 04/15/03. Geometry of the flag manifold: Schubert cells and
varieties, cohomology ring, Chevalley-Monk 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
LR-coefficients 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.