18.318   M.I.T.   Spring 2012

# 18.318    topics in combinatorics   + + + + + Positive + + + + + GRASSMANNIAN

 Class meets: Wednesday, Friday   10:30 - 12:00 noon   room 2-102

Instructor: Alexander Postnikov   apost at math   room 2-389

Course webpage: http://math.mit.edu/~apost/courses/18.318/

Synopsis:

The course will focus on combinatorial structures that came up in the study of total positivity for the Grassmannian.

The Grassmannian is a classical geometrical object that leads to beautiful combinatorics. We will start with classical results on the Grassmannian and related combinatrial structures, such as Plucker coordinates, Schubert cells, Young diagrams and tableaux, Schur polynomials, Littlewood-Richardson rule, etc. We'll talk about Schubert calculus, which gives nice combinatorial answers to geometrical questions like ``How to find the number of lines in the 3-dimensional complex space that intersect with given four generic lines?''

We will talk about matroids, Gelfand-Goresky-Macpherson-Serganova theory, hypersimplices, matroid polytopes, moment map, etc. An example of combinatorial answer to a geometrical question: Degrees of torus orbits in the Grassmannian are the Eulerian numbers, or, more generally, volumes of matroid polytopes.

Then we'll move to the main player in the course --- the positive Grassmannian. Most of the results on the positive Grassmannian were found in the last decade, or even in the last few years. This is a very active area of current research in algebraic combinatorics, algebraic geometry, and physics.

We'll talk about various combinatorial objects that appear in this area, such as positroids, L-diagrams, plabic graphs, decorated permutations, alternating chord diagrams, etc.

Surpisingly, the same combinatorial objects appear in various areas of mathematics and physics, which are very far from each other, such as inverse boundary problems, study of quantum minors, super Yang-Mills theory, Teichmuller theory, statistical models, KP-equations and solitons, and many other areas. The positive Grassmannian is related to other areas of research in algebraic combinatorics, such as affine Stanley symmetric and k-Schur functions, Gromov-Witten invariants and quantum cohomology of the Grassmannian, and the theory of juggling. We will talk about some of these applications.

The course should be accessible to first year graduate students.

Lectures: Lecture Notes by Alejandro Morales.

These lecture notes will be updated periodically.

Problem Sets: Problem Set 1 (due March 23, 2012)

Texts and videos: The course material will be partially taken from (but not limited to) the following sources.

• Willam Fulton: Young Tableaux, Cambridge University Press, 1997.

• Alexander Postnikov: Total Positivity, Grassmannians, and Networks, arXiv:math/0609764

• Alexander Postnikov: Affine Approach to Quantum Schubert Calculus, arXiv:math/0205165

• Alexander Postnikov, David Speyer, Lauren Williams: Matching Polytopes, Toric Geometry, and the Non-Negative Part of the Grassmannian, arXiv:0706.2501

• Allen Knutson, Thomas Lam, David Speyer: Positroid Varieties: Juggling and Geometry, arXiv:1111.3660

• Nima Arkani-Hamed: Grassmannian Polytopes and Scattering Amplitudes, video of lecture at Perimeter Institute.

last updated: Feb 7, 2012