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Abstract: Fix the coordinate ring R of a complex non-singular affine algebraic variety V, and a radical ideal I in R. The set of polynomial functions on V vanishing to order at least N at all points of the zero locus of I in V has the structure of an ideal -- the N-th symbolic power I^{(N)} of I. In particular, while I^N always lies in I^{(N)}, the latter may be bigger, and the latter encodes geometric information at the expense of being grossly incomputable algebraically. My thesis is guided by sleuthing for containments of type I^{(N)} ⊆ I^c, both in this classical algebro-geometric context and the more broad context of Noetherian commutative rings. In the latter setting, I am in fact interested in finding classes of rings R such that there's a positive integer D such that the Dc-th symbolic power of P is contained in the c-th ordinary power of P for all prime ideals P in R and all positive integers c. I first survey some state-of-the art results in this vein, starting with the celebrated Ein-Lazarsfeld-Smith Theorem. I'll then motivate and record a theorem from my thesis that covers a large class of algebro-geometric surface singularities R (e.g., toric, du Val (ADE)), drawing ties with 1960s work of Joseph Lipman.

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