Abstract. We consider the isoperimetric problem in planar sectors with density r p , and with density a > 1 inside the unit disk and 1 outside. We characterize solutions as a function of sector angle. We also solve the isoperimetric problem in R n with density r p , p < 0.

To every object X of a symmetric tensor category over a field of characteristic p > 0 we attach p-adic integers Dim + (X) and Dim − (X) whose reduction modulo p is the categorical dimension dim(X) of X, coinciding with the usual dimension when X is a vector space. We study properties of Dim ± (X), and in particular show that they don't always coincide with each other, and can take any value in Z p . We also discuss the connection of p-adic dimensions with the theory of λ-rings and Brauer characters.Etingof:

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The partition algebra P k (n) and the symmetric group S n are in Schur-Weyl duality on the k-fold tensor power M ⊗k n of the permutation module M n of S n , so there is a surjection, which is an isomorphism when n ≥ 2k. We prove a dimension formula for the irreducible modules of the centralizer algebra Z k (n) in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities of the irreducible S n -modules in M ⊗k n . Our dimension expressions hold for any n ≥ 1 and k ≥ 0. Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on M ⊗k n and the quasi-partition algebra corresponding to tensor powers of the reflection representation of S n .

We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.