Tahun Terbit

2010

We establish an essentially optimal estimate for the ninth moment of the exponential sum having argument αx3 + βx. The first substantial advance in this topic for over 60 years, this leads to improvements in Heath-Brown’s variant of Weyl’s inequality, and other applications of Diophantine type.

We present a motivated construction of large graphs not containing a given complete bipartite subgraph. The key insight is that the algebraic constructions yield very non-smooth probability distributions.

Frieze patterns of numbers, introduced in the early 1970s by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present survey aims to review the original work of Coxeter and the new developments around the notion of frieze, focusing on the representation theoretic, geometric and combinatorial approaches.

Misperceptions about the world’s grassy biomes contribute to their alarming rates of loss due to conversion for agriculture and tree plantations, as well as to forest encroachment. To illustrate the causes and consequences of these misperceptions, we show that the World Resources Institute and the International Union for Conservation of Nature misidentified 9 million square kilometers of anci…

The virtual dimensions of both framed and unframed SU(2) magnetic monopoles on asymptotically conic 3-manifolds are obtained by computing the index of a Fredholm extension of the associated deformation complex. The unframed dimension coincides with the one obtained by Braam for conformally compact 3-manifolds. The computation follows from the application of a Callias-type index theorem.

Given any ε > 0, we construct an orthonormal system of nk uniformly bounded polynomials of degree at most k on the unit sphere in Rm+1 where nk is bigger than 1 − ε times the dimension of the space of polynomials of degree at most k. Similarly, we construct an orthonormal system of sections of powers Ak of a positive holomorphic line bundle on a compact K¨ahler manifold with ca…

We prove an analog of Lagrange’s theorem for continued fractions on the Heisenberg group: points with an eventually periodic continued fraction expansion are those that satisfy a particular type of quadratic form, and vice-versa.

We show that the complement of the closure of the coamoeba of a variety of codimension k + 1 is k-convex, in the sense of Gromov and Henriques. This generalizes a result of Nisse for hypersurface coamoebas. We use this to show that the complement of the nonarchimedean coamoeba of a variety of codimension k + 1 is k-convex.

Let f : (Cn, 0) → (C, 0) be a semiquasihomogeneous function. We give a formula for the local Łojasiewicz exponent L0(f) of f, in terms of weights of f. In particular, in the case of a quasihomogeneous (QH) isolated singularity f, we generalize a formula for L0(f) of Krasiński, Oleksik and Płoski from 3 to n dimensions. This was previously announced in the paper [19] of Tan…