We study categories associated to a decorated marked surface S_△, which is obtained from an unpunctured marked surface S by adding a set of decorating points. For any triangulation T of S_△, let Γ_T be the associated Ginzburg dg algebra. We show that there is a bijection between reachable open arcs in S_△ and the reachable rigid indecomposables in the perfect derived category perΓ_T. This is the dual of the bijection, between simple closed arcs in S_△ and reachable spherical objects in the 3-Calabi-Yau category D_{fd}(ΓT), constructed in the prequel (Qiu in Math Ann 365:595–633, 2016). Moreover, we show that Amiot’s quotient perΓ_T/D_{fd}(ΓT) that defines the generalized cluster categories corresponds to the forgetful map S_△→S (forgetting the decorating points) in a suitable sense.

We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

Let Q be an acyclic quiver and s be a sequence with elements in the vertex set Q_0. We describe an induced sequence of simple (backward) tilting in the bounded derived category D(Q), starting from the standard heart H_Q = modkQ and ending at another heart H_s in D(Q). Then we show that s is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to HQ[−1]; it is maximal if and only if H_s = H_Q[−1]. This provides a categorical way to understand green mutations. Further, fix a Coxeter element c in the Coxeter group W_Q of Q, which is admissible with respect to the orientation of Q. We prove that the sequence \tilde{w} induced by a c-sortable word w is a green mutation sequence. As a consequence, we obtain a bijection between c-sortable words and finite torsion classes in H_Q. As byproducts, the interpretations of inversions, descents and cover reflections of a c-sortable word w are given in terms of the combinatorics of green mutations.

We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface S with marked points and non-empty boundary, which generalizes Brüstle–Zhang’s result for the puncture free case. As an application, we show that the intersection of the shifts in the 3-Calabi–Yau derived category D(Γ_S) associated to the surface and the corresponding Seidel–Thomas braid group of D(Γ_S) is empty, unless S is a polygon with at most one puncture (i.e. of type A or D).

We study the homogenization of a stationary conductivity problem in a random heterogeneous medium with highly oscillating conductivity coefficients and an ensemble of simply closed conductivity resistant membranes. This medium is randomly deformed and then rescaled from a periodic one with periodic membranes, in a manner similar to the random medium proposed by Blanc, Le Bris, and Lions (2006). Across the membranes, the flux is continuous but the potential field itself undergoes a jump of Robin-type. We prove that, for almost all realizations of the random deformation, as the small scale of variations of the medium goes to zero, the random conductivity problem is well approximated by that of an effective medium which has deterministic and constant coefficients and contains no membrane. The effective coefficients are explicitly represented. One of our main contributions is to provide a solution to the associated auxiliary problem that is posed on the whole space with infinitely many interfaces, a setting that is out of the standard stationary ergodic framework.