We compute the open Gromov-Witten invariants for every compact semi-Fano toric surface, i.e. a toric surface X with nef anticanonical bundle. Unlike the Fano case, this involves non-trivial obstructions in the corresponding moduli problem. As an application, an explicit expression of the superpotential W for the mirror of X is obtained, which in turn gives an explicit ring presentation of the small quantum cohomology of X. We also give a computational verification of the natural ring isomorphism between the small quantum cohomology of X and the Jacobian ring of W.

In this paper, we construct an adaptive multiscale method for solving H(curl)-elliptic problems in highly heterogeneous media. Our method is based on the generalized multiscale finite element method. We will first construct a suitable snapshot space, and a dimensional reduction procedure to identify important modes of the solution. We next develop and analyze an a posteriori error indicator, and the corresponding adaptive algorithm. In addition, we will construct a coupled offlineonline adaptive algorithm, which provides an adaptive strategy to the selection of offline and online basis functions. Our theory shows that the convergence is robust with respect to the heterogeneities and contrast of the media. We present several numerical results to illustrate the performance of our method.

An irreducible integrable connection (E,∇) on a smooth projective complex variety X is called rigid if it gives rise to an isolated point of the corresponding moduli space M_{dR}(X). According to Simpson’s motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gauss-Manin connection of a family of smooth projective varieties defined on an open dense subvariety of X. In this article we study mod-p reductions of irreducible rigid connections and establish results which confirm Simpson’s prediction. In particular, for large p, we prove that p-curvatures of mod-p reductions of irreducible rigid flat connections are nilpotent, and building on this result, we construct an F-isocrystalline realization for irreducible rigid flat connections. More precisely, we prove that there exist smooth models X_R and (E_R,∇_R) of X and (E,∇), over a finite-type ring R, such that for every Witt ring W(k) of a finite field k and every homomorphism R→W(k), the p-adic completion of the base change (\hat E_{W(k)}, \hat ∇_{W(k)}) on \hat X_{W(k)} represents an F-isocrystal. Subsequently, we show that irreducible rigid flat connections with vanishing p-curvatures are unitary. This allows us to prove new cases of the Grothendieck–Katz p-curvature conjecture. We also prove the existence of a complete companion correspondence for F-isocrystals stemming from irreducible cohomologically rigid connections.

This research, initiated by a problem excerpted from Baidu, one of the most famous internet search engines in China, explores two types of movements, “converging curve” and “tracing curve” named by this research group, which really act in a regular pattern. In the research, by applying scores of mathematical methods, we found that the key to converging curves is to spot the invariant elements from the changeful motions. For tracing curves, we adopt estimating equations to get the solution. Due to the limit of time and the relevant knowledge of the research group, many unexpected problems constantly appeared which are quite far beyond our ability. Yet we still tried to apply some promising mathematical thoughts, and have achieved some exciting results. In the process of discovering and rediscovering, we did not succeed in finding a perfect solution to the problem. After all, as a group of middle school students fascinated by mathematics, creating and discovering more are the greatest delight of us all.

Making use of the extended flux homomorphism defined in [13] on the group Symp§g of symplectomorphisms of a closed ori- ented surface §g of genus g ≥ 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to g and we show that they are highly non- trivial. We also prove that the second homology of the group Ham§g of Hamiltonian symplectomorphisms of §g, equipped with the discrete topology, is very large for all g ≥ 2.