This is an informal seminar on algebraic geometry. In Fall 2020 we plan to look at various key concepts in
algebraic geometry applied to ** flag varieties ** and their relatives.

The seminar meets at 3-4 on Fridays. Please email Erik Carlsson ,
Roger Casals or Eugene Gorsky for details.

** Warm-up problem: ** How many lines intersect four given lines in general position in
three-dimensional space?

10/9 Roger: intro to cohomology and intersection theory notes

Exercise set 1

10/16 Erik: intro to Schubert calculus notes , problems , solutions

10/23 Ashleigh: homology of Grassmannians [C,EH]

10/30 Ashleigh continues notes

11/6 Yuze: Pieri and Giambelli rules [C,EH] notes

11/11 Eugene: recap on homology and cohomology notes

Exercise set 2

11/13 Raymond: equivariant cohomology and puzzles [Knu,T] notes

11/20 Raymond continues notes

11/27 Thanksgiving break

12/4 Raymond continues notes

- Key objects: Grassmannians, flag varieties, partial flags. Schubert cells, Schubert varieties, Plucker coordinates, incidence varieties.
- Tautological bundles. Cohomology, relation to symmetric functions. Schubert polynomials. Applications to enumerative problems. [B, BV, T, Knu]
- Bott-Samelson varieties and resolution of singularities. [Brion]
- Cluster structure and positivity. [FWZ, SSW]
- Derived category of coherent sheaves. Exceptional collections. Braid group action. [Kap, Kuz, KT]

- [B] S. Billey. Tutorial on Schubert varieties and Schubert calculus. slides
- [BV] S. Billey, R. Vakil. Intersections of Schubert varieties and other permutation array schemes. arXiv:0502468
- [Brion] M. Brion. Lectures on the geometry of flag varieties. arXiv:math/0410240
- [C] I. Coskun. Lectures on homogeneous varieties. lectures
- [EH] D. Eisenbud, J. Harris. 3264 & All That. Intersection Theory in Algebraic Geometry. book
- [FWZ] S. Fomin, L. Williams, A. Zelevinsky. Introduction to Cluster Algebras. arXiv:1608.05735
- [Kap] M. Kapranov. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), no. 3, 479-508. file .
- [KT] M. Khovanov, R. Thomas. Braid cobordisms, triangulated categories, and flag varieties. arXiv:0609335 [Nicolle]
- [Knu] A. Knuntson. Schubert calculus and puzzles. notes
- [Kuz]A. Kuznetsov. Semiorthogonal decompositions in algebraic geometry. arXiv:1404.3143
- [SSW] K. Serhiyenko, M. Sherman-Bennett, L. Williams. Cluster structures in Schubert varieties in the Grassmannian. arXiv:1902.00807
- [T] J. Tymoczko. Permutation actions on equivariant cohomology. arXiv:0706.0460 [Raymond]