Noam Lifshitz

Analysis of global sets

One of the main tools in analysis of Boolean functions is the hypercontractivity theorem of Bonami, Beckner and Gross. It says that a certain operator $T_\rho$ (called the 'noise operator') on the space of functions on the discrete cube $\{0,1\}^n$, satisfies $||T_{\rho}(f)||_4 \leq ||f||_2$ for all functions f. Generalising the analysis of Boolean functions to other domains is an active field of research with various applications in Extremal Combinatorics and Theoretical Computer Science. We consider settings such as the symmetric group and the n-dimensional m-grid. In these settings, the obvious generalisation of the Bonami-Beckner-Gross hypercontractivity theorem is not very useful. We give an effective generalisation of the hypercontractivity theorem to these settings, for a class of functions we call `global’. In this talk, we describe our hypercontractivity theorem for global functions, and show applications of it to intersection problems, isoperimetric inequalities, and Additive Combinatorics. Based on joint works with David Ellis (Bristol), Yuval Filmus (Technion), Peter Keevash (Oxford), Guy Kindler (HUJI), Eoin Long (Birmingham) and Dor Minzer (MIT).

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