We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. These provide a solution to the long-standing problem of finding a good notion of Ricci flow through singularities, in the 3-dimensional case. We prove that Ricci flow with surgery, starting from a fixed initial condition, subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.

We study pretty good quantum state transfer (ie, state transfer that becomes arbitrarily close to perfect) between vertices of graphs with an involution in the presence of an energy potential. In particular, we show that if a graph has an involution that fixes at least one vertex or at least one edge, then there exists a choice of potential on the vertex set of the graph for which we get pretty good state transfer between symmetric vertices of the graph. We show further that in many cases, the potential can be chosen so that it is only non-zero at the vertices between which we want pretty good state transfer. As a special case of this, we show that such a potential can be chosen on the endpoints of a path to induce pretty good state transfer in paths of any length. This is in contrast to the result of [6], in which the authors show that there cannot be perfect state transfer in paths of length 4 or more, no matter what potential is chosen.

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In this paper, we give a complete classification of cotorsion pairs in a cluster category C of type A^\infty_\infty via certain configurations of arcs, called τ-compact Ptolemy diagrams, in an infinite strip with marked points. As applications, we classify t-structures and functorially finite rigid subcategories in C, respectively. We also deduce Liu-Paquette's classification of cluster tilting subcategories of C and Ng's classification of torsion pairs in the cluster category of type A^\infty_\infty.

In G2 manifolds, 3-dimensional associative submanifolds (instantons) play a role similar to J-holomorphic curves in symplectic geometry. In [21], instantons in G2 manifolds were constructed from regular J-holomorphic curves in coassociative submanifolds. In this exposition paper, after reviewing the background of G2 geometry, we explain the main ingredients in the proofs in [21]. We also construct new examples of instantons.