On an asymptotically hyperbolic manifold $\left(X^{n+1},g\right)$, Mazzeo and Melrose [18] have constructed the meromorphic extension of the resolvent $R\left(\lambda \right):={\left({\Delta }_g-\lambda \left(n-\lambda \right)\right)}^{-1}$ for the Laplacian. However, there are special points on $\left(1/2\right)\left(n-\mathbb{N}\right)$ with which they did not deal. We show that the points of $\left(n/2\right)-\mathbb{N}$ are at most poles of finite multiplicity and that the same property holds for the points of $\left(\left(n+1\right)/2\right)-\mathbb{N}$ if and only if the metric is even. On the other hand, there exist some metrics for which $R\left(\lambda \right)$ has an essential singularity on $\left(\left(n+1\right)/2\right)-\mathbb{N}$, and these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of $R\left(\lambda \right)$ approaching an essential singularity.

Given a contact structure on a closed, oriented three-manifold $Y$, we describe an invariant that takes values in the three-manifold's Floer homology $\widehat{\mathrm{HF}}$. This invariant vanishes for overtwisted contact structures and is nonzero for Stein-fillable ones. The construction uses Giroux's interpretation of contact structures in terms of open-book decompositions.

Green's function for the Laplacian on surfaces is considered, and a mass-like quantity is derived from a regularization of Green's function. A heuristic argument, inspired by the role of the positive mass theorem in the solution to the Yamabe problem, gives rise to a geometrical mass that is a smooth function on a compact surface without boundary. The geometrical mass is shown to be independent of the point on the sphere, and it is also a spectral invariant. Moreover, a connection to a sharp Sobolev-type inequality reveals that it is actually minimized at the standard round metric. The behavior of the geometrical mass on the sphere is markedly different from that on other surfaces.

We generalize to Hilbert modular varieties of arbitrary dimension the work of W. Duke [16] on the equidistribution of Heegner points and of primitive positively oriented closed geodesics in the Poincaré upper half-plane, subject to certain subconvexity results. We also prove vanishing results for limits of cuspidal Weyl sums associated with analogous problems for the Siegel upper half-space of degree 2. In particular, these Weyl sums are associated with families of Humbert surfaces in Siegel 3-folds and of modular curves in these Humbert surfaces.

For two finite sets of real numbers $A$ and $B$, one says that $B$ is sum-free with respect to $A$ if the sum set $\left\{b+b\text{'}\mid b,b\text{'}\in B,b\ne b\text{'}\right\}$ is disjoint from $A$. Forty years ago, Erdőos and Moser posed the following question. Let $A$ be a set of $n$ real numbers. What is the size of the largest subset $B$ of $A$ which is sum-free with respect to $A$?

In this paper, we show that any set $A$ of $n$ real numbers contains a set $B$ of cardinality at least $g\left(n\right)\ln n$ which is sum-free with respect to $A$, where $g\left(n\right)$ tends to infinity with $n$. This improves earlier bounds of Klarner, Choi, and Ruzsa and is the first superlogarithmic bound for this problem.

Our proof combines tools from graph theory together with several fundamental results in additive number theory such as Freiman's inverse theorem, the Balog-Szemerédi theorem, and Szemerédi's result on long arithmetic progressions. In fact, in order to obtain an explicit bound on $g\left(n\right)$, we use the recent versions of these results, obtained by Chang and by Gowers, where significant quantitative improvements have been achieved.

Using a simple geometric argument, we obtain an infinite family of nontrivial relations in the tautological ring of ${ℳ}_g$ (coming, in fact, from relations in the Chow ring of ${\overline{ℳ}}_{g,2}$). One immediate consequence of these relations is that the classes ${\kappa }_1,\dots ,{\kappa }_{\left[g/3\right]}$ generate the tautological ring of ${ℳ}_g$, which was conjectured by Faber in [F] and recently proven at the level of cohomology by Morita in [M].

We give a generalization to higher genera of the famous formula $12\hspace{0.17em}\lambda =\delta$ for genus 1