Geometric positivity in the  Schubert calculus -- Ravi Vakil

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The last decade has seen vigorous advances in our understanding of the
Schubert calculus, which in a geometric setting means the cohomology of
Grassmannians and other partial flag varieties.  I will discuss one theme,
that of geometric positivity, which encompasses work of Graham, Knutson,
Tao, Woodward, Buch, Brion, Griffeth, Ram, Miller, Shimozono,
Coskun, and many
others.  I will end with a proposed new geometric positivity in
equivariant K-theory (joint with Allen Knutson), unifying the geometric
Littlewood-Richardson rule and puzzles.