Jonathan Hickman Title: Fourier restriction over rings of integers modulo N Abstract: Recently substantial progress has been made on several major open problems in harmonic analysis by considering discrete models formulated over finite fields. In particular, Wolff's finite field Kakeya conjecture was famously resolved by Dvir; the techniques introduced in this proof were then developed by Guth et al to study the original Euclidean Kakeya and restriction conjectures. Finite fields do not encapsulate all aspects of analysis over R, however, and the lack of any non-trivial notion of scale in the former leads to divergence between the two theories. One approach to modelling multiple scales in a discrete setting is to work over rings of integers Z/p^kZ (or more generally, Z/NZ for composite N). In this case, each `scale' corresponds to a possible number of divisors and the resulting theory closely parallels that of the Euclidean case. This in turn naturally leads one to consider analysis over non-archimedean local fields (such as the p-adics or p-series fields), a topic which already has a long history dating back to the 1960s and efforts to generalise classical Calderon-Zygmund theory. In this talk I will discuss some joint work with J. Wright, presenting some formulations of the Fourier restriction problem over Z/NZ. I will draw particular attention to how the algebraic structure of the rings can be used to effectively model familiar localisation and discretisation phenomena from the Euclidean setting.