This paper attempts to relate some ideas of Grothendieck in his Esquisse d'un programme and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss the Teichmller theory, the mapping class groups, $ SL (2,\bold C) $ representation variety of surface groups, and Thurston's theory of measured laminations.

As a discrete-time quantum walk model on the one-dimensional integer lattice Z, the quantum walk recently constructed byWang and Ye [CaishiWang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2, by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z^d, which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

We determine a basis of the (twisted) cocenter of graded affine Hecke algebras with arbitrary parameters. In this setting, we prove that the kernel of the (twisted) trace map is the commutator subspace (the Density theorem) and that the image is the space of good forms (the trace PaleyWiener theorem).

We prove that a random map drawn from any class \mathfrak{C} of polynomial maps from (F_q)^n to (F_q)^{n+r} that is (i) totally random in the affine terms, and (ii) has a negligible chance of being not strongly one-way, provides a secure PRNG (hence a secure stream cipher) for any q. Plausible choices for are semi-sparse (i.e., the affine terms are truly random) systems and other systems that are easy to evaluate from a small (compared to a generic map) number of parameters. To our knowledge, there are no other positive results for provable security of specialized polynomial systems, in particular sparse ones (which are natural candidates to investigate for speed). We can build a family of provably secure stream ciphers a rough implementation of which at the same security level can be more than twice faster than an optimized QUAD (and any other provably secure stream ciphers proposed so far), and uses much less storage. This may also help build faster provably secure hashes. We also examine the effects of recent results on specialization on security, e.g., Aumasson-Meier (ICISC 2007), which precludes Merkle-Damgaard compression using polynomials systems {uniformly very sparse in every degree} from being universally collision-free. We conclude that our ideas are consistent with and complements these new results. We think that we can build secure primitives based on specialized (versus generic) polynomial maps which are more efficient.

Given a closed strictly convex hypersurface M in the Euclidean space I?"+*, the Gauss map of M defines a homeomorphism between M and the unit sphere S". Therefore the Gauss-Kronecker curvature of M can be transplanted via the Gauss map to a function defined on S". If this function is denoted by K, then Minkowski observed that K must satisfy the equation where xi are the coordinate functions on S". Minkowski then asked the converse of the problem. Namely, given a positive function K defined on S" satisfying the above integral conditions, can we find a closed strictly convex hypersurface whose curvature function is given by K? Minkowski solved the problem in the category of polyhedrons. Then AD Alexandrov and others solved the problem in general. However, this last solution does not provide any information about the regularity of the (unique) convex hypersurface even if we assume K is smooth. In the two