Ezra Miller's title and abstract for the 2005 Summer
  Insitute in Algebraic Geometry Warmup Workshop


  Title:
  ------
  
  Combinatorial positivity in algebraic geometry
  
  
  Abstract:
  ---------
  
  Combinatorics can be a great excuse to talk about
  really cool properties of familiar spaces from
  algebraic geometry.  These spaces run the gamut from
  flag varieties and degeneracy loci for vector bundle
  morphisms to toric varieties and Hilbert schemes.  The
  combinatorics can often be phrased as the question,
  "You've just handed me a certain polynomial; why do
  its coefficients seem to be positive integers?"  The
  algebraic geometry can involve calculations in various
  kinds of cohomology and K-theory accompanied by
  various kinds of flat degenerations.


  Non-exhaustive list of examples, some of which we
  could study during the workshop:

  1. Haiman's positivity proof for the coefficients of
     Macdonald polynomials using the K-theory of Hilbert
     schemes of points in the plane.

  2. Knutson-Miller-Shimozono degenerations of matrix
     Schubert varieties and quiver loci to explain the
     positivity of Schubert polynomials and prove
     positivity for the Buch-Fulton quiver polynomials
     in Chow cohomology.

  3. Speyer's degeneration of the apiary ("triple flag")
     variety to the hive toric variety to explain
     geometrically the Knutson-Tao honeycomb description
     of Littlewood-Richardson numbers.

  4. Related to number 3: the Gonciulea-Lakshmibai
     degeneration of the ordinary flag variety to the
     Gelfand-Tsetlin toric variety.

  5. Vakil's geometric Littlewood-Richardon rule by
     successively degenerating intersections of Schubert
     varieties.


  References and suggested preparatory reading:
  ---------------------------------------------

  For topic 1:
  ------------

  Mark Haiman,
  Hilbert schemes, polygraphs and the Macdonald positivity conjecture.
  J. Amer. Math. Soc. \textbf{14} (2001), no. 4, 941-1006 (electronic).
  
  Mark Haiman,
  Macdonald polynomials and geometry.
  New perspectives in algebraic combinatorics (Berkeley, CA, 1996-97), 207-254,
  Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.
  
  Chapter 18 of
  Ezra Miller and Bernd Sturmfels,
  Combinatorial commutative algebra.
  Graduate Texts in Mathematics, 227.
  Springer-Verlag, New York, 2005. xiv+417 pp.
  
  
  For topic 2:
  ------------
  
  Allen Knutson and Ezra Miller,
  Gr\"obner geometry of Schubert polynomials
  Annals of Mathematics \textbf{161} (2005) no. 3.
  Available online at 
      http://www.math.princeton.edu/%7Eannals/issues/2005/161_3.html
  or more precisely,
  http://www.math.princeton.edu/%7Eannals/issues/2005/May2005/KnutsonMiller.pdf
  
  Anders Buch and William Fulton,
  Chern class formulas for quiver varieties.
  Invent. Math. \textbf{135} (1999), no. 3, 665-687.
  
  Allen Knutson, Ezra Miller, and Mark Shimozono,
  Four positive formulae for type A quiver polynomials.
  Invent. Math., to appear.  (The arXiv version is out 
  of date; please get the most recent version online at 
  http://www.math.umn.edu/~ezra/papers.html )
  
  
  For topic 4:
  ------------

  Chapter 14 of
  Ezra Miller and Bernd Sturmfels,
  Combinatorial commutative algebra.
  Graduate Texts in Mathematics, 227.
  Springer-Verlag, New York, 2005. xiv+417 pp.