Cyclic cocycles on twisted convolution algebras

We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper \'etale groupoids, Tu and Xu provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to a construction of Mathai and Stevenson. When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras. The results in this article were originally published in the author's Ph.D. thesis.Comment: 39 page

Groupoids and an index theorem for conical pseudo-manifolds

We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold $M$. A main ingredient is a non-commutative algebra that plays in our setting the role of $C_0(T^*M)$. We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in $K$-theory. We then give a new proof of the Atiyah-Singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds

Overview of current trends in SOFC materials

Because of their potential to reduce the environmental impact and geopolitical consequences of the use of fossil fuels, fuel cells have emerged as sustainable power generation systems. Like a combustion engine, a fuel cell uses a chemical fuel as its energy source; but like a battery, the chemical energy is directly converted into electrical energy. In addition to high efficiency and low emissions, fuel cells are attractive for their modular and distributed nature, and zero noise pollution. SOFC single cells and stack systems are multilayer structures consisting of ceramic and metallic materials with different well defined electrical transport properties. All components have to show well-adjusted thermal expansion behaviour, chemical compatibility of material interfaces and chemical stability in the prevailing temperature and gas atmosphere. Performance and long-term stability improvements at affordable costs can be obtained by lowering the operating temperature. This goal requires drastic reduction of electrodes polarisation resistance as well as highly-conducting electrolyte materials. The use of new and nanostructured material and the application of appropriate technologies for the production of optimized microstructure becomes essential for highly efficient SOFCs operating in the medium and low temperature range. A review of state-of-art materials is given together with a perspective presentation of innovative materials and technologies for new types of SOFC