Abstract: We will show how localic groupoids model a generalized notion of 'topological space' or topos. This talk will introduce toposes, monadic descent, and give an overview of extended Grothendieck-Galois theory developed by A. Joyal and M. Tierney.

Abstract:

Abstract: Let $F$ be any field of characteristic $p$ and let $C_p$ denote the cyclic group of order $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,\dots,V_p$ of $C_p$ defined over $F$. It is also well-known that there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $\textrm{SL}_2(\mathbb{C})$ given by the action on the space $R_d$ of $d$ forms. In this talk I will describe my recent result which reduces the computation of the ring of $C_p$-invariants of a $C_p$-representation $V=\oplus_{i=1}^k V_{n_i+1}$ to the computation of the classical ring of invariants (or covariants) $C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\textrm{SL}_2(\mathbb{C})}$.

This allows us to compute for the first time the ring of invariants for many representations of $C_p$.