# Results on the Parameter Space of Certain Lattices in Mixed Characteristic

The work on an arithmetic analogue (i.e., in mixed characteristic) of a certain geometric object, called an affine Grassmannian, that are common over equal-characteristic local fields. A finite-dimensional Grassmannian variety is the parameter space of subspaces of a given vector space; the points of this space (corresponding to subspaces) form a continuous family. Affine Grassmannian is a generalization of this when the given vector space is defined over a local field (think of the field of quotients of formal power series), and that the subspaces are replaced by certain lattices, that is, maximal-rank submodules over the ring of integers of the local filed.

When the local field is in mixed characteristic, such parameter space was constructed and studied by W. Haboush. It is a union of finite dimensional projective varieties over the residue field, each of which has singularities. The results that are obtained is that this singular variety is still normal (i.e. singularities are "not bad") and locally complete intersection, making it the next best thing to have, after a nonsingular variety.

Terminologies appearing above, in terms of examples will be mentioned. Some ideas will be going into constructing this parameter space, and if time permits, relevance to other branches of mathematics such as number theory, representation theory or physics will be mentioned.