In this paper, we study how the algebraic attack on the HFE multivariate public key cryptosystem works if we build an HFE cryptosystem on a finite field whose characteristic is not two. Using some very basic algebraic geometry we argue that when the characteristic is not two the algebraic attack should not be polynomial in the range of the parameters which are used in practical applications. We further support our claims with extensive experiments using the Magma implementation of F_4, which is currently the best publicly available implementation of the Gröbner basis algorithm. We present a new variant of the HFE cryptosystems, where we project the public key of HFE to a space of one dimension lower. This protects the system from the Kipnis-Shamir attack and makes the decryption process avoid multiple candidates for the plaintext. We propose an example for a practical application on GF(11) and suggest a test challenge on GF(7).

For two nearby disjoint coassociative submanifolds C and C′ in a G2-manifold, we construct thin instantons with boundaries lying on C and C′ from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.

We prove an approximation result showing how operators of the type (x ) in L^ 2 ({\bb R}^ 2), where is a graph, can be modelled in the strong resolvent sense by point-interaction Hamiltonians with an appropriate arrangement of the potentials. The result is illustrated on finding the spectral properties in cases when is a ring or a star. Furthermore, we use this method to indicate that scattering on an infinite curve which is locally close to a loop shape or has multiple bends may exhibit resonances due to quantum tunnelling or repeated reflections.

We describe the potential functions for N= 4B supersymmetric quantum mechanics with D (2, 1; ) symmetry.

We prove that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors.