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.

No abstract uploaded!

Many computer graphics problems require computing geometric shapes subject to certain constraints. This often results in non-linear and non-convex optimization problems with globally coupled variables, which pose great challenge for interactive applications. Local-global solvers developed in recent years can quickly compute an approximate solution to such problems, making them an attractive choice for applications that prioritize efficiency over accuracy. However, these solvers suffer from lower convergence rate, and may take a long time to compute an accurate result. In this paper, we propose a simple and effective technique to accelerate the convergence of such solvers. By treating each local-global step as a fixed-point iteration, we apply Anderson acceleration, a well-established technique for fixed-point solvers, to speed up the convergence of a local-global solver. To address the stability issue of classical Anderson acceleration, we propose a simple strategy to guarantee the decrease of target energy and ensure its global onvergence. In addition, we analyze the connection between Anderson acceleration and quasi-Newton methods, and show that the canonical choice of its mixing parameter is suitable for accelerating local-global solvers. Moreover, our technique is effective beyond classical local-global solvers, and can be applied to iterative methods with a common structure. We evaluate the performance of our technique on a variety of geometry optimization and physics simulation problems. Our approach significantly reduces the number of iterations required to compute an accurate result, with only a slight increase of computational cost per iteration. Its simplicity and effectiveness makes it a promising tool for accelerating existing algorithms as well as designing efficient new algorithms.

Let F be a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from GLn+1(F) to GLn(F). A main result shows that each Bernstein component of an irreducible smooth representation of GLn+1(F) restricted to GLn(F) is indecomposable. We also classify all irreducible representations which are projective when restricting from GLn+1(F) to GLn(F). A main tool of our study is a notion of left and right derivatives, extending some previous work joint with Gordan Savin. As a by-product, we also determine the branching law in the opposite direction.

No abstract uploaded!