A Note on Direct Products, Subreducts and Subvarieties of PBZ*--lattices

PBZ*--lattices are bounded lattice--ordered structures arising in the study of quantum logics, which include orthomodular lattices, as well as antiortholattices. Antiortholattices turn out not only to be directly irreducible, but also to have directly irreducible lattice reducts. Their presence in varieties of PBZ*--lattices determines the lengths of the subposets of dense elements of the members of those varieties. The variety they generate includes two disjoint infinite ascending chains of subvarieties, and the lattice of subvarieties of the variety of pseudo--Kleene algebras can be embedded as a poset in the lattice of subvarieties of its subvariety formed of its members that satisfy the Strong De Morgan condition. We obtain axiomatizations for all members of a complete sublattice of the lattice of subvarieties of this latter variety axiomatized by the Strong De Morgan identity with respect to the variety generated by antiortholattices.Comment: 18 page

Demazure resolutions as varieties of lattices with infinitesimal structure

Let k be a field of positive characteristic. We construct, for each dominant coweight \lambda of the standard maximal torus in the special linear group, a closed subvariety D(\lambda) of the multigraded Hilbert scheme of an affine space over k, such that the k-valued points of D(\lambda) can be interpreted as lattices in k((z))^n endowed with infinitesimal structure. Moreover, for any \lambda we construct a universal homeomorphism from D(\lambda) to a Demazure resolution of the Schubert variety associated with \lambda in the affine Grassmannian. Lattices in D(\lambda) have non-trivial infinitesimal structure if and only if they lie over the boundary of the big cell.Comment: 24 pages, added the missing bibliograph

Spaces of Lattices in Equal and Mixed Characteristics

We investigate the affine Grassmannian for the special linear group, and its Schubert varieties. In (positive) equal characteristic, we give an alternative way to construct Demazure resolutions of these Schubert varieties as subvarieties of a multigraded Hilbert scheme. In mixed characteristics, we investigate the approach by Haboush to construct in a similar way Schubert varieties for the p-adic affine Grassmannian. Moreover, we investigate which results on the affine Grassmannian can be carried over from the function field case to the p-adic case

Versality in Mirror Symmetry

One of the attractions of homological mirror symmetry is that it not only implies the previous predictions of mirror symmetry (e.g., curve counts on the quintic), but it should in some sense be `less of a coincidence' than they are and therefore easier to prove. In this survey we explain how Seidel's approach to mirror symmetry via versality at the large volume/large complex structure limit makes this idea precise.Comment: 43 pages, 4 figures. Survey for the proceedings of the conference Current Developments in Mathematics 201