Let A , A be two regular commutative unital Banach algebras such that A is integral over A . In 2003, Dawson and Feinstein showed that the topological stable rank A whenever A . In this note, we investigate whether we will have A in general. For instance, when A is a commutative unital A -algebra, we show that A , and the equality holds at least when the integral extension is separable. In general, A and A have the same Bass stable ranks A .

In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form Y_1\cup_D Y_2 that arises from a degeneration Y_1\cup_D Y_2 and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair Y_1\cup_D Y_2 . As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber Y_1\cup_D Y_2 of Y_1\cup_D Y_2 to the Gromov-Witten invariants of the singular fiber Y_1\cup_D Y_2 via gluing the relative pairs Y_1\cup_D Y_2 and Y_1\cup_D Y_2 . The finite sum mentioned above depends on a relative ample line bundle Y_1\cup_D Y_2 on Y_1\cup_D Y_2 . His theory has already applications to string theory and mathematics alike. For other new applications of Jun Li's theory, one needs a refined degeneration formula that depends on a curve class Y_1\cup_D Y_2 in Y_1\cup_D Y_2 or Y_1\cup_D Y_2 , rather than on the line bundle Y_1\cup_D Y_2 . Some monodromy effect has to be taken care of to deal with this. For the simple but useful case of a degeneration Y_1\cup_D Y_2 that arises from blowing up a trivial family Y_1\cup_D Y_2 , we explain how the details of Jun Li's work can be employed to reach such a desired degeneration formula. The related set Y_1\cup_D Y_2 of admissible triples adapted to Y_1\cup_D Y_2 that appears in the formula can be obtained via an analysis on the intersection numbers of relevant cycles and a study of Mori cones that appear in the problem. This set is intrinsically determined by Y_1\cup_D Y_2 and the normal bundle Y_1\cup_D Y_2 of the smooth subscheme Y_1\cup_D Y_2 in Y_1\cup_D Y_2 to be blown up.

The classification and construction of symmetry protected topological (SPT) phases have been intensively studied in interacting systems recently. To our surprise, in interacting fermion systems, there exists a new class of the so-called anomalous SPT (ASPT) states which are only well defined on the boundary of a trivial fermionic bulk system. We first demonstrate the essential idea by considering an anomalous topological superconductor with time-reversal symmetry T^2 = 1 in 2D. The physical reason for this is that the fermion parity might be changed locally by certain symmetry action, but it is conserved if we introduce a bulk. Then we discuss the layer structure and systematical construction of ASPT states in interacting fermion systems in 2D with a total symmetry G_f = G_b × \mathbb{Z}^f_2. Finally, potential experimental realizations of ASPT states are also addressed.

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.

For two nearby disjoint coassociative submanifolds C and C′ in a G2-manifold, we construct thin instantons with boundaries lying on C and C′ from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.