@article{AGeneralFrameworkforRepresentationStability:1807,
      recid = {1807},
      author = {Gadish, Nir},
      title = {A General Framework for Representation Stability, with  Applications to Arrangements and Arithmetic},
      publisher = {The University of Chicago},
      school = {Ph.D.},
      address = {2019-06},
      pages = {153},
      abstract = {This thesis sets out to explore and implement the paradigm  of representation stability, specifically in the study of  sequences of linear subspace arrangements, their stable  combinatorics, topology and arithmetic.

In a traditional  sense, the sequences of arrangements in question do not  exhibit any form of stability, e.g. the Betti numbers of  their complements grow to infinity. But when one considers  the symmetries at play and the various maps between the  arrangements, a new notion of stability presents itself:  representation stability, where a finite collection of  patterns is merely translated around by increasingly large  groups. In this sense, stability is understood as a notion  of finite generation.

One way to encode the structure of  various symmetry groups and intertwining maps between them  is using the language of diagrams. These are functors from  a fixed category into the category of arrangements, vector  spaces or any other target category. Thus a system of  intertwined group actions can be treated as a single  mathematical object, and finite generation gets a precise  meaning. Representation stability therefore consists of two  main aspects: identify finitely generated diagrams and  operations that preserve finite generation (e.g.  Noetherianity theorems), then extract stable invariants  from a finitely generated diagram (e.g. polynomial  characters).

This work addresses both of the above aspects  in a general axiomatic framework. We define and study  finitely generated diagrams of linear subspace arrangements  - these occur in many natural examples from algebraic  geometry and combinatorics, such as colored configuration  spaces, $k$-equals arrangements and covers of moduli spaces  of rational maps. Such collections of arrangements exhibit  finite generation in their intersection posets, and this in  turn leads to finite generation in the cohomology of their  complements. We then study the reprecautions of this finite  generation using an adaptation of character theory to the  study of diagrams. Lastly, we adapt the  Grothendieck-Lefschetz fixed point formula to provide a  bridge between cohomological representation stability  results and asymptotic factorization statistics of orbits  over finite fields.},
      url = {http://knowledge.uchicago.edu/record/1807},
      doi = {https://doi.org/10.6082/uchicago.1807},
}