Donaldson-Thomas theory and cluster algebras

We provide a transformation formula of non-commutative Donaldson-Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians. As an application, we give an alternative proof of Fomin-Zelevinsky's conjectures on $F$-polynomials and $g$-vectors.Comment: 39 pages, 8 figures, mostly rewritte

On continuous extension of grafting maps

The definition of the grafting operation for quasifuchsian groups is extended by Bromberg to all $b$-groups. Although the grafting maps are not necessarily continuous at boundary groups, in this paper, we show that the grafting maps take every "standard" convergent sequence to a convergent sequence. As a consequence of this result, we extend Goldman's grafting theorem for quasifuchsian groups to all boundary $b$-groups.Comment: 24 pages, 2 figure

Criteria for D_4 singularities of wave fronts

We give useful and simple criteria for determining D_4 singularities of wave fronts. As an application, we investigate behaviors of singular curvatures of cuspidal edges near D_4^+ singularities.Comment: 11 pages with 4 figure