I will describe an explicit geometric Littlewood-Richardson rule,

interpreted as deforming the intersection of two Schubert varieties so

that they break into Schubert varieties. There are no restrictions on

the base field, and all multiplicities arising are 1; this is

important for applications. This rule should be seen as a

generalization of Pieri's rule to arbitrary Schubert classes, by way

of explicit homotopies. It has a straightforward bijection to other

Littlewood-Richardson rules, such as tableaux and Knutson and Tao's

puzzles.

This gives the first geometric proof and interpretation of the

Littlewood-Richardson rule. It has a host of geometric consequences,

which I may describe, time permitting. The rule also has an

interpretation in K-theory, suggested by Buch, which gives an

extension of puzzles to K-theory, and in fact a Littlewood-Richardson

rule in equivariant K-theory (ongoing work with Knutson). The rule

suggests a natural approach to the open question of finding a

Littlewood-Richardson rule for the flag variety, leading to a

conjecture, shown to be true up to dimension 5. Finally, the rule

suggests approaches to similar open problems, such as

Littlewood-Richardson rules for the symplectic Grassmannian and

two-flag varieties.

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