This is an expositiry article on collapsing theory written for the Modern Encyclopedia of Mathematical Physics (MEMPhys). We focus on describing the geometric and topological structure of collapsed/non-collapsed regions in Riemannian manifold under various curvature assumptions. Numerous applications of collapsing theory to Riemannian geometry are not discussed in this survey, due to page limit...

The abelian-nonabelian correspondence outlined in [BCFK08] gives a broad conjectural relationship between (twisted) Gromov-Witten invariants of related GIT quotients. This paper proves a case of the correspondence explicitly relating genus zero mpointed Gromov-Witten invariants of Grassmannians Gr(2, n) and products of projective space P ×P. Computation of the twisted Gromov-Witten invariants o...

In this paper, usng the gluing formula of Gromov-Witten invariants under symplectic cutting, due to Li and Ruan, we studied the Gromov-Witten invariants of blow-ups at a smooth point or along a smooth curve. We established some relations between Gromov-Witten invariants of M and its blow-ups at a smooth point or along a smooth curve.

The Virasoro conjecture predicts that the generating function of Gromov-Witten invariants is annihilated by infinitely many differential operators which form a half branch of the Virasoro algebra. This conjecture was proposed by Eguchi, Hori and Xiong [EHX2] and also by S. Katz [Ka] (see also [EJX]). It provides a powerful tool in the computation of Gromov-Witten invariants. In [LT], the author...

We represent stationary descendant Gromov-Witten invariants of projective space, up to explicit combinatorial factors, by polynomials. One application gives the asymptotic behaviour of large degree behaviour of stationary descendant Gromov-Witten invariants in terms of intersection numbers over the moduli space of curves. We also show that primary Gromov-Witten invariants are ”virtual” stationa...

We extend several notions and results from the classical Patterson-Sullivan theory to setting of Anosov subgroups higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove equality between Hausdorff dimensions flag limit sets, computed respect a suitable Gromov (pre-)metric manifold, critical exponents subgroups.