Niziol intends to continue her research into properties of p-adic Galois representations arising from geometry as well as families of such. For isolated representations her main goal is to understand those coming from ´etale and motivic cohomologies as well as the related p-adic regulator maps (into the de Rham and syntomic cohomologies) and the induced p-adic period maps. As a corollary she expect to obtain a uniqueness criterium for p-adic period maps for semistable varieties with a compactification by a normal crossing divisor, an equality of the p-adic period maps defined by Tsuji, Faltings, and Niziol, a compatibility of these maps with the weight spectral sequences, as well as a factorization of p-adic regulators through certain Selmer groups.
For families, she plans to study the category of p-adic local systems on varieties over local fields, its subcategories of crystalline and semistable local systems, and the induced ´etale and syntomic cohomologies. In particular she expect to be able to extend the category of Breuil modules to the relative setting, to prove p-adic comparison theorems for them, and to apply them to the study of some automorphic forms.
The study of Galois representations coming from geometry in various ways is of fundamental importance in modern number theory, representation theory, and algebraic geometry. The related p-adic methods are also becoming increasingly important in computational number theory and its applications. In the past Niziol has mentored both graduate and undergraduate students as well as postdoctoral fellows and she expects to do the same in directions suggested by this project.
