This paper studies a notion of enumerative invariants for stable A-branes, and discusses its relation to invariants defined by spectral and exponential networks. A natural definition of stable A-branes and their counts is provided by the string theoretic origin of the topological A-model. This is the Witten index of the supersymmetric quantum mechanics of a single D3 brane supported on a special Lagrangian in a Calabi-Yau threefold. Geometrically, this is closely related to the Euler characteristic of the A-brane moduli space. Using the natural torus action on this moduli space, we reduce the computation of its Euler characteristic to a count of fixed points via equivariant localization. Studying the A-branes that correspond to fixed points, we make contact with definitions of spectral and exponential networks. We find agreement between the counts defined via the Witten index, and the BPS invariants defined by networks. By extension, our definition also matches with Donaldson-Thomas invariants of B-branes related by homological mirror symmetry.

We prove the endpoint case of the multilinear Kakeya conjecture of Bennett, Carbery and Tao. The proof uses the polynomial method introduced by Dvir.

Authenticated Key Exchange (AKE) is a cryptographic scheme with the aim to establish a high-entropy and secret session key over a insecure communications network. \emph{Password}-Authenticated Key Exchange (PAKE) assumes that the parties in play share a simple password, which is cheap and human-memorable and is used to achieve the authentication. PAKEs are practically relevant as these features are extremely appealing in an age where most people access sensitive personal data remotely from more-and-more pervasive hand-held devices. Theoretically, PAKEs allow the secure computation and authentication of a high-entropy piece of data using a low-entropy string as a starting point. In this paper, we apply the recently proposed technique introduced in~\cite{DXX2012} to construct two lattice-based PAKE protocols enjoying a very simple and elegant design that is an parallel extension of the class of Random Oracle Model (ROM)-based protocols \msf{PAK} and \msf{PPK}~\cite{BMP2000,M2002}, but in the lattice-based setting. The new protocol resembling \msf{PAK} is three-pass, and provides \emph{mutual explicit authentication}, while the protocol following the structure of \msf{PPK} is two-pass, and provides \emph{implicit authentication}. Our protocols rely on the Ring-Learning-with-Errors (RLWE) assumption, and exploit the additive structure of the underlying ring. They have a comparable level of efficiency to \msf{PAK} and \msf{PPK}, which makes them highly attractive. We present a preliminary implementation of our protocols to demonstrate that they are both efficient and practical. We believe they are suitable quantum safe replacements for \msf{PAK} and \msf{PPK}.

We propose a fast local level set method for the inverse problem of gravimetry. The theoretical foundation for our approach is based on the following uniqueness result: if an open set D is star-shaped or x₃-convex with respect to its center of gravity, then its exterior potential uniquely determines the open set D. To achieve this purpose constructively, the first challenge is how to parametrize this open set D as its boundary may have a variety of possible shapes. To describe those different shapes we propose to use a level-set function to parametrize the unknown boundary of this open set. The second challenge is how to deal with the issue of partial data as gravimetric measurements are only made on a part of a given reference domain . To overcome this difficulty we propose a linear numerical continuation approach based on the single layer representation to find potentials on the boundary of some artificial domain

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