In this article, we apply spectral invariants constructed in [Oh5] and [6] to the study of Hamiltonian diffeomorphisms of closed symplectic manifolds $(M,\omega )$. Using spectral invariants, we first construct an invariant norm called the spectral norm on the Hamiltonian diffeomorphism group and obtain several lower bounds for the spectral norm in terms of the $\epsilon$-regularity theorem and the symplectic area of certain pseudoholomorphic curves. We then apply spectral invariants to the study of length minimizing properties of certain Hamiltonian paths among all paths. In addition to the construction of spectral invariants, these applications rely heavily on the chain-level Floer theory and on some existence theorems with energy bounds of pseudoholomorphic sections of certain Hamiltonian fibrations with prescribed monodromy. The existence scheme that we develop in this article in turn relies on some careful geometric analysis involving adiabatic degeneration and thick-thin decomposition of the Floer moduli spaces which has an independent interest of its own. We assume that $(M,\omega )$ is strongly semipositive throughout this article

In an earlier article with Gurevich and Jiang [GGJ], we constructed the cubic unipotent Arthur packets of the exceptional group $G_2$ and showed that the discrete multiplicities of the representations in these packets are at least those predicted by Arthur's multiplicity formula. In this article, we establish the multiplicity formula by showing the reverse inequality. We also show that the parts of the discrete spectrum described by some of these packets are full near-equivalence classes

Basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered. $L^p$ norm inequalities are established for multilinear integral operators of Calderón-Zygmund type which incorporate oscillatory factors $e^{iP}$, where $P$ is a real-valued polynomial

In the first half of the article, we introduce the notion of the universal unitary completion of a continuous representation of a $p$-adic reductive group on a locally convex $p$-adic vector space, and we prove that such a completion exists under appropriate hypotheses. The problem of studying unitary completions has been raised by Breuil [4] in connection with his work on a possible $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2$, and we relate our construction to certain conjectures of Breuil in [4, Sec. 1.3] for the group ${\mathrm{GL}}_2({\mathbb{Q}}_p)$. In particular, we show that the universal unitary completion of the locally analytic parabolic induction of a locally algebraic character coincides with the universal unitary completion of the corresponding locally algebraic induction, provided that the character being induced satisfies a noncritical slope condition (see Prop. 2.5). In the second half of the article, we consider a certain unitary Banach space representation of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$ obtained by $p$-adically completing the cohomology of classical modular curves. The mere existence of this representation implies that those locally algebraic, parabolically induced representations of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$ which arise from classical finite-slope newforms have a nontrivial universal unitary completion (verifying a conjecture of Breuil in [4, Sec. 1.3] for these representations), while applying Proposition 2.5 in this context enables us to give a new construction of $p$-adic $L$-functions attached to $p$-stabilized newforms of noncritical slope. Combining our construction with the representation-theoretic definition of $\mathcal{L}$-invariants implicit in [5, Cor. 1.1.7], we are able to give a simple proof of the Mazur-Tate-Teitelbaum exceptional zero conjecture in [18, p. 46] (in terms of Breuil's definition of the $\mathcal{L}$-invariant)

We consider Schrödinger operators with ergodic potential $V_{\omega }(n)=f(T^n(\omega ))$, $n\in \mathbb{Z}$, $\omega \in \Omega$, where $T:\Omega \to \Omega$ is a nonperiodic homeomorphism. We show that for generic $f\in C(\Omega )$, the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani theory