On G/N-Hilb of N-Hilb

In this paper we consider the iterated G-equivariant Hilbert scheme G/N-Hilb(N-Hilb) and prove that G/N-Hilb(N-Hilb(C^3)) is a crepant resolution of C^3/G isomorphic to the moduli space of \theta-stable representations of the McKay quiver for certain stability condition \theta. We provide several explicit examples to illustrate this construction. We also consider the problem of when G/N-Hilb(N-Hilb) is isomorphic to G-Hilb showing the fact that these spaces are most of the times different.Comment: Final version. Explanations improved throughout the paper and mistakes in some statements have been corrected; (old) sections 2.2 and 3.1 have been expanded into new sections; (old) section 5 have been reorganised and several results in it have been extended. To appear in Kyoto Journal of Mathematic

Dihedral groups and G-Hilbert schemes

Let G ⊂ GL(2,C) be a finite subgroup acting on the complex plane C2, and consider the following diagram C2 -> X <- π:Y where π is the minimal resolution of singularities. Since Du Val in the 1930s the explicit calculation of Y was made from X by blowing up the singularity at the origin, where we lose any information about the group G in the process. But, is there a direct relation between the resolution Y and the group G? McKay [McK80] in the late 1970s was the first to realise the link between the group action and the resolution Y , thus giving birth to the so called McKay correspondence. This beautiful correspondence establishes an equivalence between the geometry of the minimal resolution Y of the quotient singularity C2/G, and the G-equivariant geometry of C2

Dimer models and group actions

We construct a consistent dimer model having the same symmetry as its characteristic polygon. This produces examples of non-commutative crepant resolutions of non-toric non-quotient Gorenstein singularities in dimension 3.Comment: 23 pages, 5 figure

The balance between GMD and OFUT1 regulates notch signaling pathway activity by modulating notch stability

The Notch signaling pathway plays an important role in development and physiology. In Drosophila, Notch is activated by its Delta or Serrate ligands, depending in part on the sugar modifications present in its extracellular domain. O-fucosyltransferase-1 (OFUT1) performs the first glycosylation step in this process, O-fucosylating various EGF repeats at the Notch extracellular domain. Besides its O-fucosyltransferase activity, OFUT1 also behaves as a chaperone during Notch synthesis and is able to down regulate Notch by enhancing its endocytosis and degradation. We have reevaluated the roles that O-fucosylation and the synthesis of GDP-fucose play in the regulation of Notch protein stability. Using mutants and the UAS/Gal4 system, we modified in developing tissues the amount of GDP-mannosedeshydratase (GMD), the first enzyme in the synthesis of GDP-fucose. Our results show that GMD activity, and likely the levels of GDPfucose and O-fucosylation, are essential to stabilize the Notch protein. Notch degradation observed under low GMD expression is absolutely dependent on OFUT1 and this is also observed in Notch Abruptex mutants, which have mutations in some potential O-fucosylated EGF domains. We propose that the GDP-fucose/OFUT1 balance determines the ability of OFUT1 to endocytose and degrade Notch in a manner that is independent of the residues affected by Abruptex mutations in Notch EGF domains.This work was funded by ICM P06-039F grant to A.G. and by a BFU2009-09403 grant of the M.E.C. to J.F.dC. An institutional grant from the Ramón Areces Foundation to the CBMSO is also acknowledged.Peer Reviewe

Dihedral groups and G-Hilbert schemes

Let G ⊂ GL(2,C) be a finite subgroup acting on the complex plane C2, and consider the following diagram C2 -> X <- π:Y where π is the minimal resolution of singularities. Since Du Val in the 1930s the explicit calculation of Y was made from X by blowing up the singularity at the origin, where we lose any information about the group G in the process. But, is there a direct relation between the resolution Y and the group G? McKay [McK80] in the late 1970s was the first to realise the link between the group action and the resolution Y , thus giving birth to the so called McKay correspondence. This beautiful correspondence establishes an equivalence between the geometry of the minimal resolution Y of the quotient singularity C2/G, and the G-equivariant geometry of C2.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

The City as a Tool for STEAM Education: Problem-Posing in the Context of Math Trails

This study presents an experience that combines problem-posing and Math Trails in the context of future teachers’ instruction. Pre-service teachers in the third year of their studies were faced with the design of tasks to be included in Math Trails for primary school students. The study analyzes, from a quantitative approach, 117 tasks contained in 11 Math Trails. The analysis was performed on the basis of classification variables (grade, mathematical content and object or real element involved in every task) and research variables which provide information about the nature of the tasks (procedural vs. problem-solving, level of cognitive demand, degree of contextualization, openness and creativity). Additionally, relationships between the different categories of analysis have been studied. The results reveal certain biases in the tasks in relation to the contents addressed (an abundance of tasks with a geometric component and a scarcity of tasks involving algebra or probability concepts). Most of the tasks are presented in a real context. However, a higher percentage of procedural tasks than problem-solving tasks is observed, with a predominance of low openness, creativity and cognitive demand. These results provide useful lines of work to address difficulties faced by future teachers in the STEAM field

A novel method to construct NSSD molecular graphs

A graph is said to be NSSD (=non-singular with a singular deck) if it has no eigenvalue equal to zero, whereas all its vertex-deleted subgraphs have eigenvalues equal to zero. NSSD graphs are of importance in the theory of conductance of organic compounds. In this paper, a novel method is described for constructing NSSD molecular graphs from the commuting graphs of the Hv-group. An algorithm is presented to construct the NSSD graphs from these commuting graphsThis research is partially funded through Quaid-i-Azam University grant URF-201