Relative Orbifold Donaldson-Thomas Theory and the Degeneration Formula

We generalize the notion of expanded degenerations and pairs for a simple degeneration or smooth pair to the case of smooth Deligne-Mumford stacks. We then define stable quotients on the classifying stacks of expanded degenerations and pairs and prove the properness of their moduli's. On 3-dimensional smooth projective DM stacks this leads to a definition of relative Donaldson-Thomas invariants and the associated degeneration formula.Comment: 59 pages. Final versio

Virtual Coulomb branch and vertex functions

We introduce a variant of the $K$-theoretic quantized Coulomb branch constructed by Braverman--Finkelberg--Nakajima, by application of a new virtual intersection theory. In the abelian case, we define Verma modules for such virtual Coulomb branches, and relate them to the moduli spaces of quasimaps into the corresponding Higgs branches. The descendent vertex functions, defined by $K$-theoretic quasimap invariants of the Higgs branch, can be realized as the associated Whittaker functions. The quantum $q$-difference modules and Bethe algebras (analogue of quantum $K$-theory rings) can then be described in terms of the virtual Coulomb branch. As an application, we prove the wall-crossing result for quantum $q$-difference modules under the variation of GIT. Nonabelian cases are also treated via abelianization.Comment: 53 pages, 2 figures; to appear in Duke Mathematical Journa

Draft Genome of the Leopard Gecko, \u3cem\u3eEublepharis Macularius\u3c/em\u3e

Background Geckos are among the most species-rich reptile groups and the sister clade to all other lizards and snakes. Geckos possess a suite of distinctive characteristics, including adhesive digits, nocturnal activity, hard, calcareous eggshells, and a lack of eyelids. However, one gecko clade, the Eublepharidae, appears to be the exception to most of these ‘rules’ and lacks adhesive toe pads, has eyelids, and lays eggs with soft, leathery eggshells. These differences make eublepharids an important component of any investigation into the underlying genomic innovations contributing to the distinctive phenotypes in ‘typical’ geckos. Findings We report high-depth genome sequencing, assembly, and annotation for a male leopard gecko, Eublepharis macularius (Eublepharidae). Illumina sequence data were generated from seven insert libraries (ranging from 170 to 20 kb), representing a raw sequencing depth of 136X from 303 Gb of data, reduced to 84X and 187 Gb after filtering. The assembled genome of 2.02 Gb was close to the 2.23 Gb estimated by k-mer analysis. Scaffold and contig N50 sizes of 664 and 20 kb, respectively, were compble to the previously published Gekko japonicus genome. Repetitive elements accounted for 42 % of the genome. Gene annotation yielded 24,755 protein-coding genes, of which 93 % were functionally annotated. CEGMA and BUSCO assessment showed that our assembly captured 91 % (225 of 248) of the core eukaryotic genes, and 76 % of vertebrate universal single-copy orthologs. Conclusions Assembly of the leopard gecko genome provides a valuable resource for future comptive genomic studies of geckos and other squamate reptiles