ABSTRACT SEMI-INFINITE FLAGS AND ZASTAVA SPACES SEPTEMBER 2023 ANDREAS HAYASH, B.A., HAMPSHIRE COLLEGE M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D, UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Ivan Mirković We give an interpretation of Dennis Gaitsgory’s semi-infinite intersection cohomol- ogy sheaf associated to a semisimple simply-connected algebraic group in terms of finite-dimensional geometry. Specifically, we construct machinery to build factoriza- tion spaces over the Ran space from factorization spaces over the configuration space, and show that under this procedure the compactified Zastava space is sent to the support of the semi-infinite intersection cohomology sheaf in the Beilinson-Drinfeld Grassmannian. We also construct a partial resolution of singularities of the compact- ified Zastava space and show that the Zastava version of the semi-infinite intersection cohomology sheaf is pulled back to the ordinary (perverse) intersection cohomology sheaf of the partial resolution. Lastly, we show that there is a monad acting on sheaves over the resolution whose category of modules embeds fully faithfully in sheaves on the affine Grassmannian.