Abstract:
Background Pleurocybella porrigens is a mushroom-forming fungus, which has been consumed as a traditional food in Japan. In 2004, 55 people were poisoned by eating the mushroom and 17 people among them died of acute encephalopathy. Since then, the Japanese government has been alerting Japanese people to take precautions against eating the P. porrigens mushroom. Unfortunately, despite efforts, the molecular mechanism of the encephalopathy remains elusive. The genome and transcriptome sequence data of P. porrigens and the related species, however, are not stored in the public database. To gain the omics data in P. porrigens, we sequenced genome and transcriptome of its fruiting bodies and mycelia by next generation sequencing. Methodology/Principal Findings Short read sequences of genomic DNAs and mRNAs in P. porrigens were generated by Illumina Genome Analyzer. Genome short reads were de novo assembled into scaffolds using Velvet. Comparisons of genome signatures among Agaricales showed that P. porrigens has a unique genome signature. Transcriptome sequences were assembled into contigs (unigenes). Biological functions of unigenes were predicted by Gene Ontology and KEGG pathway analyses. The majority of unigenes would be novel genes without significant counterparts in the public omics databases. Conclusions Functional analyses of unigenes present the existence of numerous novel genes in the basidiomycetes division. The results mean that the omics information such as genome, transcriptome and metabolome in basidiomycetes is short in the current databases. The large-scale omics information on P. porrigens, provided from this research, will give a new data resource for gene discovery in basidiomycetes.

Abstract:
We construct geometrically a gerbe assigned to a connection on a principal SU(2)-bundle over an oriented closed 1-dimensional manifold. If the connection is given by the restriction of a connection on a bundle over a compact 2-manifold bounding the 1-manifold, then we have a natural object in the gerbe. The gerbes and the objects satisfy certain fundamental properties, e.g. gluing law.

Abstract:
From a certain strongly equivariant bundle gerbe with connection and curving over a smooth manifold on which a Lie group acts, we construct under some conditions a bundle gerbe with connection and curving over the quotient space. In general, the construction requires a choice, and we can consequently obtain distinct stable isomorphism classes of bundle gerbes with connection and curving over the quotient space. A bundle gerbe naturally arising in Chern-Simons theory provides an example of the reduction.

Abstract:
We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations is finite, and is determined by the cohomology of the manifold.

Abstract:
Based on projective representations of smooth Deligne cohomology groups, we introduce an analogue of the space of conformal blocks to compact oriented (4k+2)-dimensional Riemannian manifolds with boundary. For the standard (4k+2)-dimensional disk, we compute the space concretely to prove that its dimension is finite.

Abstract:
For twisted K-theory whose twist is classified by a degree three integral cohomology of infinite order, universal even degree characteristic classes are in one to one correspondence with invariant polynomials of Atiyah and Segal. The present paper describes the ring of these invariant polynomials by a basis and structure constants.

Abstract:
Mickelsson's invariant is an invariant of certain odd twisted K-classes of compact oriented three dimensional manifolds. We reformulate the invariant as a natural homomorphism taking values in a quotient of the third cohomology, and provide a generalization taking values in a quotient of the fifth cohomology. These homomorphisms are related to the Atiyah-Hirzebruch spectral sequence. We also construct some characteristic classes for odd twisted K-theory in a similar vein.

Abstract:
A twist is a datum playing a role of a local system for topological K-theory. In equivariant setting, twists are classified into four types according to how they are realized geometrically. This paper lists the possible types of twists for the torus with the actions of the point groups of all the 2-dimensional space groups (crystallographic groups), or equivalently, the torus with the actions of all the possible finite subgroups in its mapping class group. This is carried out by computing Borel's equivariant cohomology and the Leray-Serre spectral sequence. As a byproduct, the equivariant cohomology up to degree three is determined in all cases.

Abstract:
We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures.

Abstract:
We formulate the Chern-Simons action for any compact Lie group using Deligne cohomology. This action is defined as a certain function on the space of smooth maps from the underlying 3-manifold to the classifying space for principal bundles. If the 3-manifold is closed, the action is a function with values in complex numbers. If the 3-manifold is not closed, then the action is a section of a Hermitian line bundle associated with the Riemann surface which appears as the boundary.