The length classification of threefold flops via noncommutative algebras

Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that can occur as generic hyperplane sections, and the simultaneous resolutions associated to such a partial resolution produce the universal flop of length l. In this paper we translate these ideas into noncommutative algebra. We introduce the universal flopping algebra of length l from which the universal flop of length l can be recovered by a moduli construction, and we present each of these algebras as the path algebra of a quiver with relations. This explicit realisation can then be used to construct examples of NCCRs associated threefold flops of any length as quiver with relations defined by superpotentials, to recover the matrix factorisation description of the universal flop conjectured by Curto and Morrison, and to realise examples of contraction algebras

Deformations of algebras defined by tilting bundles

In this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles. We do this in two settings, first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformation of that scheme and then expanding this result to C⁎ -equivariant deformations over schemes with a good C⁎ -action. In both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra, and this is a graded deformation in the C⁎ -equivariant case. We apply our results to rational surface singularities, generalising the deformed preprojective algebras of Crawley-Boevey and Holland, and also to symplectic situations where the deformations produced are related to symplectic reflection algebras

Computational Investigation of Biological Membranes

Lipids are the building blocks of biological membranes, and the types of lipids that compose these cellular envelopes influence the physicochemical properties of the chemicals that can enter or exit the cell across the membrane. This work focuses on the lipid membrane compositions of eukaryotic (red blood cells) and prokaryotic (Pseudomonas aeruginosa) membranes. By analyzing the lipid-lipid and lipid-protein interactions results of the computational simulations, insights into lipid aggregation, bilayer leaflet behavior, membrane asymmetry, and small molecule transport through protein channels were obtained. The differences between prokaryotic and eukaryotic cell membranes are qualitative known; however, this work provides these concepts through quantitative evidence using multiscale molecular dynamics simulations. The results show that membrane leaflet asymmetry affects the membrane properties and protein-lipid interactions. The CLASP algorithm was employed to efficiently simulate the transport of small molecules through a bacterial membrane porin and analyze the resulting contacts of that molecule with the pore-lining residues

Ringel duality for certain strongly quasi-hereditary algebras

We study quasi-hereditary endomorphism algebras defined over a new class of finite dimensional monomial algebras with a special ideal structure. The main result is a uniform formula describing the Ringel duals of these quasi-hereditary algebras. As special cases, we obtain a Ringel duality formula for a family of strongly quasi-hereditary algebras arising from a type A configuration of projective lines in a rational, projective surface as recently introduced by Hille and Ploog, for certain Auslander–Dlab–Ringel algebras, and for Eiriksson and Sauter’s nilpotent quiver algebras when the quiver has no sinks and no sources. We also recover Tan’s result that the Auslander algebras of self-injective Nakayama algebras are Ringel self-dual