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Equivariant cohomology attaches to a G-equivariant sheaf \mathcal{F} on a variety X with a G action a set of invariants H^_G(X,\mathcal{F}) which have been useful in representation theory and enumerative geometry. My talk will be in two parts: The first part will be technical-focusing on equivariant sheaves, defining H^_G(X,\mathcal{F}) and sketching the ideas behind localization theorems. The second part will be computational using the GKM localization theorems to compute equivariant cohomologies of Schubert cells on Grassmanians.

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