FLAGS Seminar - Fall 2013
University of California, Berkeley

Fridays 12-1pm, 939 Evans Hall

Organisers: George Melvin and Damien Mondragon



Seminar Outline

This seminar will be an introduction to some of the geometry, combinatorics and representation theory related to (partial) flag varieties. There will be an initial focus on Schubert calculus of Grassmannians and full flag varieties; this is the study of the ring structure of the cohomology ring of these varieties. There is then a possibility of extending this study to the equivariant/quantum Schubert calculus, or moving in a different direction and investigating Springer theory, smoothness of Schubert varieties, Soergel bimodules... The interests of participants will motivate these latter investigations.


(Tentative) Schedule of Talks + Speakers

  1. 9/20 - Introduction to Schubert calculus on Grassmannians. Brief Summary of Poincare Duality. (Fulton, 9.4, App. B; Block-Bray; Kresch) Speaker: Maria Monks Abstract. Some handwritten notes (All errors, inaccuracies and inconsistencies are due to me.)
  2. 9/27 - Schubert calculus on flag varieties; the Borel/Bruhat description and introduction to Schubert polynomials (Fulton, 10.2-10.3; Block-Bray; Kresch; Knutson (SchP)) Speaker: Steven Karp Abstract. Some handwritten notes (All errors, inaccuracies and inconsistencies are due to me.)
  3. !!891 Evans!! 10/4 - Schubert polynomials. Bruhat Order. Examples. Applications to Grassmannian. (Fulton, 10.4-10.6; Knutson (SchP)) Speaker: Damien Mondragon Cancelled!
  4. 10/11 - Schubert Calculus and Puzzles (Purbhoo) Speaker: Bo Lin Abstract. Some notes from the talk. (All errors, inaccuracies and inconsistencies are due to me.)
  5. 10/18 - A Geometric Littlewood-Richardson Rule (Vakil; Knutson (PuzN)) Speaker: Pablo Solis Abstract.
  6. 10/25 - Equivariant Schubert Calculus and Puzzles (Knutson-Tao; Knutson (PuzN)) Speaker: Harrison Chen Abstract. Some notes on equivariant cohomology by Harrison; some notes from the talk (by Harrison).
  7. 11/1 - Proof of Equivariant Puzzle Rule via Yang-Baxter (Zinn-Justin) Speaker: Gus Schrader Abstract. Some notes from the talk. (All errors, inaccuracies and inconsistencies are due to me.)
  8. 11/8 - (Guest lecture) Restricting Equivariant Classes to Fixed Points Speaker: Allen Knutson Abstract. Some notes from the talk. (All errors, inaccuracies and inconsistencies are due to me.)
  9. 11/15 - (Guest lecture) Restricting Equivariant Classes to Fixed Points - Part 2 Speaker: Allen Knutson Some notes from the talk. (All errors, inaccuracies and inconsistencies are due to me.)

References

Old-School Schubert Calculus and History
'Schubert Calculus' - S. Kleiman, D. Laksov
'Schubert Calculus according to Schubert' - F. Ronga

'Current' Developments
'Young Tableaux: with Applications to Representation Theory and Geometry' - W. Fulton
'Two Languages of Schubert Calculus: Grassmannians and Flag Manifolds' - F. Block, N. Bray
'Flag Varieties and Schubert Calculus' - A. Kresch
'Littlewood-Richardson Coefficients and Integrable Tilings' - P. Zinn-Justin
'Puzzles, Tableaux and Mosaics' - K. Purbhoo
'Schubert Polynomials and Symmetric Functions" - A. Knutson (SchP)
'Schubert Calculus and Puzzles' - A. Knutson (PuzN)
'Puzzles and (equivariant) cohomology' - A. Knutson, T. Tao
'A Geometric Littlewood-Richardson Rule' - R. Vakil
'Combinatorial Construction of Schubert Polynomials' - N. Bergeron

General Notes
'Lecture Notes on Flag Varieties' - M. Brion


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