This an expository article on Givental's axiomatic Gromov--Witten theory and some of its applications.

Sparse coding and dictionary learning have seen their applications in many vision tasks, which usually is formulated as a non-convex optimization problem. Many iterative methods have been proposed to tackle such an optimization problem. However, it remains an open problem to have a method that is not only practically fast but also is globally convergent. In this paper, we proposed a fast proximal method for solving \ell_0 norm based dictionary learning problems, and we proved that the whole sequence generated by the proposed method converges to a stationary point with sub-linear convergence rate. The benefit of having a fast and convergent dictionary learning method is demonstrated in the applications of image recovery and face recognition.

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The chapter discusses the equivariant loop theorem for three-dimensional manifolds that is needed in settling the Smith conjecture and reviews the existence theorems for minimal surfaces. The equivariant version of the loop theorem says that there are a finite number of properly embedded disks in <i>M</i> that satisfy the required properties. The loop theorem respects the action of the group <i>G</i> in a suitable manner. The chapter puts a metric on <i>M</i> so that the group <i>G</i> acts isometrically and so that <i>M</i> is convex with respect to the outward normal. Then with respect to this metric, the existence of an immersed disk <i>D</i>₁ is demonstrated, in <i>M</i> whose boundary <i> D₁</i>, represents a nontrivial element in ₁(<i>S</i>) and whose area is minimal among all such disks. The chapter describes Morrey's solution for the plateau problem in a general Riemannian manifold. The existence theorem for manifolds with boundary

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