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We introduce and physically motivate the following problem in geometric combinatorics, originally inspired by analysing Bell inequalities. A grasshopper lands at a random point on a planar lawn of area 1. It then jumps once, a fixed distance d, in a random direction.What shape should the lawn be to maximize the chance that the grasshopper remains on the lawn after jumping? We show that, perhaps surprisingly, a disc-shaped lawn is not optimal for any d>0. We investigate further by introducing a spin model whose ground state corresponds to the solution of a discrete version of the grasshopper problem. Simulated annealing and parallel tempering searches are consistent with the hypothesis that, for d<π-1/2, the optimal lawn resembles a cogwheel with n cogs, where the integer n is close to π(arcsin(√πd/2))-1. We find transitions to other shapes for d ≥ π-1/2. [ABSTRACT FROM AUTHOR]

We show that for any dimension d ≥ 3, one can obtain Wolff's L(d+2)/2 bound on Kakeya-Nikodym maximal function in Rd for d ≥ 3 without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional L² estimate with an auxiliary maximal function. We also prove that the same L(d+2)/2 bound holds for Nikodym maximal function for any manifold (Md, g) with constant curvature, which generalizes Sogge's results for d = 3 to any d ≥ 3. As in the 3-dimensional case, we can handle manifolds of constant curvature due to the fact that, in this case, two intersecting geodesics uniquely determine a 2-dimensional totally geodesic submanifold, which allows the use of the auxiliary maximal function. [ABSTRACT FROM AUTHOR]

We reprove Wolff's L5/2-bound for the R³-Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on Nikodym-type sets in curved spaces. Although the equivalence between these two type maximal functions is well known, our proof may shed light on some new geometric observations which is interesting in its own right. [ABSTRACT FROM AUTHOR]

The Erdős–Falconer distance problem in Z q d asks one to show that if E ⊂ Z q d is of sufficiently large cardinality, then the set of distances determined by E satisfies Δ ( E ) = Z q . Previous results were known only in the case q = p ℓ , where p is an odd prime, and as such only showed that all units were obtained in the distance set. We give the first such result over rings Z q where q is no longer confined to be a prime power, and despite this, we show that the distance set of E contains all of Z q whenever E is of sufficiently large cardinality. [ABSTRACT FROM AUTHOR]

We show that an arrangement $\mathcal{A}$ of n lines in general position in the plane has an inducing polygon of size O(n). Additionally, we present a simple algorithm for finding an inducing n-path for $\mathcal{A}$ in O(n log n) time and an algorithm that constructs an inducing n-gon for a special class of line arrangements within the same time bound. [ABSTRACT FROM AUTHOR]

Abstract: We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to -maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, -categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph-associahedra. [Copyright &y& Elsevier]

Abstract: In this paper, optimal estimates are obtained for operators which average functions over polynomial submanifolds, generalizing the k-plane transform. An important advance over previous work (e.g., [P. Gressman, -improving properties of X-ray like transforms, Math. Res. Lett. 13 (5) (2006) 787–803]) is that full estimates are obtained by methods which have traditionally yielded only restricted weak-type estimates. In the process, one is led to make coercivity estimates for certain functionals on for . [Copyright &y& Elsevier]