We study the pointwise properties of $k$-subharmonic functions, that is, the viscosity subsolutions to the fully nonlinear elliptic equations $F_k[u]=0$, where $F_k[u]$ is the elementary symmetric function of order $k,1\leq k\leq n$, of the eigenvalues of $[D\sp 2u]$, $F_1[u]=\Delta u,F_n[u]=\det D^2u$. Thus $1$-subharmonic functions are subharmonic in the classical sense; $n$-subharmonic functions are convex. We use a special capacity to investigate the typical questions of potential theory: local behaviour, removability of singularities, and polar, negligible, and thin sets, and we obtain estimates for the capacity in terms of the Hausdorff measure. We also prove the Wiener test for the regularity of a boundary point for the Dirichlet problem for the fully nonlinear equation $F_k[u]=0$. The crucial tool in the proofs of these results is the Radon measure $F_k[u]$ introduced recently by N. Trudinger and X.-J. Wang for any $k$-subharmonic $u$. We use ideas from the potential theories both for the complex Monge-Ampère and for the $p$-Laplace equations.

We establish a twisted analog of our recent work on vertex representations and the McKay correspondence. For each finite group $\Gamma$ and a virtual character of $\Gamma$, we construct twisted vertex operators on the Fock space spanned by the super spin characters of the spin wreath products $\Gamma\wr \tilde {S}_n$ of $\Gamma$ and a double cover of the symmetric group $S_n$ for all $n$. When $\Gamma$ is a subgroup of ${\rm SL}_2(\mathbb {C})$ with the McKay virtual character, our construction gives a group-theoretic realization of the basic representations of the twisted affine and twisted toroidal Lie algebras. When $\Gamma$ is an arbitrary finite group and the virtual character is trivial, our vertex operator construction yields the spin character tables for $\Gamma\wr \tilde {S}_n$.

The standard eigenfunctions $\phi_\lambda=e^{i\langle\lambda,x\rangle}$ on flat tori $\mathbb {R}^n/L$ have $L^\infty$-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that $L^2$-normalized eigenfunctions have uniformly bounded $^\infty$-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with quantum completely integrable Laplacians.

Estimates of the form $L^{(j)}(s,\mathscr{A})\ll_{\epsilon,j,\mathscr {D_A}}\mathscr {R}^\epsilon_{\mathscr {A}}$ in the range $|s-1|\ll 1/\log \mathscr {R_A}$ for general $L$-functions, where $\mathscr {R_A}$ is a parameter related to the functional equation of $L(s,\mathscr {A})$, can be quite easily obtained if the Ramanujan hypothesis is assumed. We prove the same estimates when the $L$-functions have Euler product of polynomial type and the Ramanujan hypothesis is replaced by a much weaker assumption about the growth of certain elementary symmetrical functions. As a consequence, we obtain an upper bound of this type for every $L(s, \pi)$, where $\pi$ is an automorphic cusp form on ${\rm GL}(\mathbf {d},\mathbb {A}_K)$. We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the third symmetric power of a Maass form.

We study algebraic fiber spaces $f : X \longrightarrow Y$ where $Y$ is of maximal Albanese dimension. In particular, we give an effective version of a theorem of Y. Kawamata: If $P_m(X)=1$ for some $m\geq 2$, then the Albanese map of $X$ is surjective. Combining this with [1], it follows that $X$ is birational to an abelian variety if and only if $P_2(X)=1$ and $q(X)=\dim(X)$.

In this paper we establish the higher-dimensional energy bubbling results for harmonic maps to spheres. We have shown in particular that the energy density of concentrations has to be the sum of energies of harmonic maps from the standard 2-dimensional spheres. The result also applies to the structure of tangent maps of stationary harmonic maps at either a singularity or infinity.