Speaker
Description
We survey classical results in additive combinatorics and develop linear analogues over field extensions, with an emphasis on Kneser-type phenomena. In addition to recalling Kneser's theorem and stabilizer methods (including Cauchy--Davenport and DeVos's refinement), we present a rigidity theorem showing that if $|A+B|=|A|+|B|-1$ with $A+B\neq G$, then $A+B$ is a subgroup and $A$ is a coset; an $n$-fold Kneser bound expressed in terms of stabilizers; a coset-sparsity condition that yields near Cauchy--Davenport bounds; and a density estimate guaranteeing large sumsets.
We then establish linear analogues for finite-dimensional $K$-subspaces $A,B\subseteq L$: the $K$-span $\langle AB\rangle$ admits stabilizers that are intermediate fields, leading to a linear rigidity theorem and sharpened Hou--Leung--Xiang bounds under separability.
We conclude with open problems on weakening separability assumptions and on bridging the group and linear settings via stabilizer geometry and coset projections.