Luke,
Yes, I think this is what it would look like:
ModuleName.Application_Institution/ModuleName.Institution/ModuleName.User_Institution = '[%CurrentUser%]'
Will display those Applications for the current user. If you want to show those Applications for a different user, embed the datagrid inside of a dataview which displays a user and the XPath would be:
ModuleName.Application_Institution/ModuleName.Institution/ModuleName.User_Institution = '[%CurrentObject%]'
Hope that helps,
Mike