The purpose of this dissertation is to study several problems related to high-dimensional phenomena in analysis, geometry and probability. The first problem examines the behaviour of probability measures of dilations of sets possessing certain symmetries. We show that for the standard Gaussian measure on complex vector space, cylinders are optimal in the sense that, under dilations, the Gaussian measure grows no more rapidly for cylinders than for other domains possessing enough symmetries. We also prove an analogous result in the real case for Weibull and Gamma distributions. As a consequence, we derive optimal comparison of moments for these distributions. The second problem stems from the study of composite periodic quantum systems. It asks about the behaviour of certain random matrices when their size tends to infinity. We show that the spectrum of the tensor product of two large random unitary matrices is asymptotically Poissonian; what we would expect for diagonal matrices. The same conclusion is established for the tensor product of a large number of 2 2 random unitary matrices. The third problem concerns the invertibility of operators on L1. We construct an example of a locally invertible operator with kernel of arbitrarily large dimension. The construction is combinatorial, relying on expander graphs and recent results from computer science about the restricted isometry property on `1. We also establish some Sobolev-type inequalities and find a certain large class of convolution operators which are globally invertible on large subspaces.