The purpose of this paper is to solve a problem posed by Strominger in constructing smooth models of superstring theory with flux. These are given by non-K¨ahler manifolds with torsion

Let (M, !) be a compact symplectic manifold, and L a compact embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono [8] construct Lagrangian Floer cohomology, yielding groups HF(L, b;nov) for one or HF􀀀 (L1, b1), (L2, b2);nov
for two Lagrangians, where b, b1, b2 are choices of bounding cochains, and exist if and only if L,L1,L2 have unobstructed Floer cohomology. These are independent of choices up to isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M, !), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich. The goal of this paper is to extend [8] to immersed Lagrangians  : L ! M, with transverse self-intersections. In the embedded case, Floer cohomology HF(L, b;nov) is a modified, ‘quantized’ version of singular homology Hn−(L;nov) over the Novikov ring nov. In our immersed case, HF(L, b;nov) turns out to be a quantized version of Hn−(L;nov)  L (p−,p+)2R nov · (p−, p+), where R =  (p−, p+) : p−, p+ 2 L, p− 6= p+, (p−) = (p+) is a set of two extra generators for each self-intersection point of L, and (p−, p+) has degree (p−,p+) 2 Z, an index depending on how L intersects itself at (p−) = (p+). The theory becomes simpler and more powerful for graded Lagrangians in Calabi–Yau manifolds, when we can work over a smaller Novikov ring CY. The proofs involve associating a gapped filtered A1 algebra over 0 nov or 0 CY to  : L ! M, which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called AN,0 algebras for N = 0, 1, 2, . . ..

Let ($M$,$ω$) be a connected, symplectic 4-manifold. A$semitoric integrable system$on ($M$,$ω$) essentially consists of a pair of independent, real-valued, smooth functions$J$and$H$on$M$, for which$J$generates a Hamiltonian circle action under which$H$is invariant. In this paper we give a general method to construct, starting from a collection of five ingredients, a symplectic 4-manifold equipped a semitoric integrable system. Then we show that every semitoric integrable system on a symplectic 4-manifold is obtained in this fashion. In conjunction with the uniqueness theorem proved recently by the authors, this gives a classification of semitoric integrable systems on 4-manifolds, in terms of five invariants.

A Riemannian or Finsler metric on a compact manifoldM gives rise to a length function on the free loop space M, whose critical points are the closed geodesics in the given metric. If X is a homology class on M, the “minimax” critical level cr(X) is a critical value. Let M be a sphere of dimension > 2, and fix a metric g and a coefficient field G. We prove that the limit as deg(X) goes to infinity of cr(X)/ deg(X) exists. We call this limit = (M, g,G) the global mean frequency of M. As a consequence we derive resonance statements for closed geodesics on spheres; in particular either all homology on  of sufficiently high degreee lies hanging on closed geodesics of mean frequency (length/average index) , or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to . The proof uses the ChasSullivan product and results of Goresky-Hingston [7].

We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future asymptotically flat null cone, and show that the solution exists uniformly around the null cone for general such initial data. Therefore, the solution contains a piece of the future null infinity. The initial data are not required to be small and the decaying condition is consistent with those in the works of [8] and [11].