Approximate representations of groups

In this thesis, we consider various notions of approximate representations of groups. Loosely speaking, an approximate representation is a map from a group into the unitary operators on a Hilbert space that satisfies the homomorphism equation up to a small error. Maps that are close to actual representations are trivial examples of approximate representations, and a natural question to ask is whether all approximate representations of a given group arise in this way. A group with this property is called stable. In joint work with Lev Glebsky, Alexander Lubotzky and Andreas Thom, we approach the stability question in the setting of local asymptotic representations. We provide sufficient condition in terms of cohomology vanishing for a finitely presented group to be stable. We use this result to provide new examples of groups that are stable with respect to the Frobenius norm, including the first examples of groups that are not Frobenius approximable. In joint work with Narutaka Ozawa and Andreas Thom, we generalize a theorem by Gowers and Hatami about maps with non-vanishing uniformity norm. We use this to prove a very general stability result for uniform epsilon-representations of amenable groups which subsumes results by both Gowers-Hatami and Kazhdan

A correlational study of areal surface texture parameters on some typical machined surfaces

A number of areal surface texture parameters have been adopted by standards bodies, namely ISO 25178-2, in which forty-one parameters within six groups are defined. The selection of the suitable areal parameters becomes an issue for a designer. The study of correlation among parameters is one of the ways to find the most suitable parameters for a specification. This paper presents a Spearman’s correlation study of areal surface texture parameters on some typical machined surfaces. Sixty surfaces, produced by nineteen machining methods, have been assessed by the use of an optical instrument; the operators adhered to ISO 25178-3; and parameters defined by ISO 25178-2. The correlation results are classified by using five correlation levels. It details the correlations between different groups of parameters, together with the correlation of parameters within the same group. The results are presented in Spearman’s rank correlation coefficient matrix and charts. A three-layer parameters tree is then proposed to help engineer in the selection of parameters