No abstract uploaded!

Let$S$be a closed Shimura variety uniformized by the complex$n$-ball associated with a standard unitary group. The Hodge conjecture predicts that every Hodge class in $${H^{2k} (S, \mathbb{Q})}$$ , $${k=0,\dots, n}$$ , is algebraic. We show that this holds for all degrees$k$away from the neighborhood $${\bigl]\tfrac13n,\tfrac23n\bigr[}$$ of the middle degree. We also prove the Tate conjecture for the same degrees as the Hodge conjecture and the generalized form of the Hodge conjecture in degrees away from an interval (depending on the codimension$c$of the subvariety) centered at the middle dimension of S. These results are derived from a general theorem that applies to all Shimura varieties associated with standard unitary groups of any signature. The proofs make use of Arthur’s endoscopic classification of automorphic representations of classical groups. As such our results rely on the stabilization of the trace formula for the (disconnected) groups $${GL (N) \rtimes \langle \theta \rangle}$$ associated with base change.

We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that if$G$is a non-elementary, analytically finite Kleinian group, and its limit set Λ($G$) is connected, then Λ($G$) is either a circle or has dimension strictly bigger than 1.

In this paper, we study intermittent behaviors of coupled piecewise-expanding map lattices with two nodes and a weak coupling. We show that the successive phase transition between ordered and disordered phases occurs for almost every orbit. That is, we prove $\liminf_{n\rightarrow \infty}| x_1(n)-x_2(n)|=0$ and $\limsup_{n\rightarrow \infty}| x_1(n)-x_2(n)|\ge c_0>0$, where $x_1(n), x_2(n)$ correspond to the coordinates of two nodes at the iterative step $n$. We also prove the same conclusion for weakly coupled tent-map lattices with any multi-nodes.

We consider the Ising model at its critical temperature with external magnetic field ha15/8 on the square lattice with lattice spacing a. We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a ↓ 0. As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a = 1, the mass (inverse correlation length) is of order h8/15 as h ↓ 0; for the Euclidean field, it equals exactly Ch8/15 for some C. Although there has been much progress in the study of critical scaling limits, results on near-critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles.