We introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero. The theory is based on Kirillov's method of orbits for nilpotent groups over local fields. When the semisimple group is a classical group, we prove that the new theory is essentially equivalent to Howe's theory of $N$-rank (see [Ho4], [L2], [Sc]). Therefore our results provide a systematic generalization of the notion of a small representation (in the sense of Howe) to exceptional groups. However, unlike previous works that used ad hoc methods to study different types of classical groups (and some exceptional ones; see [We], [LS]), our definition is simultaneously applicable to both classical and exceptional groups. The most important result of this article is a general “purity” result for unitary representations which demonstrates how similar partial results in these authors' works should be formulated and proved for an arbitrary semisimple group in the language of Kirillov's theory. The purity result is a crucial step toward studying small representations of exceptional groups. New results concerning small unitary representations of exceptional groups will be published in a forthcoming paper [S]

Consider the Schrödinger operators $H_{\pm }=-d^2/{\mathrm{dx}}^2\pm V(x)$. We present a method for estimating the potential in terms of the negative eigenvalues of these operators. Among the applications are inverse Lieb-Thirring inequalities and several sharp results concerning the spectral properties of $H_{\pm }$

We construct a virtual fundamental class on the Quot scheme parametrizing quotients of a trivial bundle on a smooth projective curve. We use the virtual localization formula to calculate virtual intersection numbers on Quot. As a consequence, we re-prove the Vafa-Intriligator formula; our answer is valid even when the Quot scheme is badly behaved. More intersections of Vafa-Intriligator type are computed by the same method. Finally, we present an application to the nonvanishing of the Pontryagin ring of the moduli space of bundles

We investigate the moments of a smooth counting function of the zeros near the central point of $L$-functions of weight $k$ cuspidal newforms of prime level $N$. We split by the sign of the functional equations and show that for test functions whose Fourier transform is supported in $(-1/n,1/n)$, as $N\to \infty$ the first $n$ centered moments are Gaussian. By extending the support to $(-1/(n-1),1/(n-1))$, we see non-Gaussian behavior; in particular, the odd-centered moments are nonzero for such test functions. If we do not split by sign, we obtain Gaussian behavior for support in $(-2/n,2/n)$ if $2k\ge n$. The $n$th-centered moments agree with random matrix theory in this extended range, providing additional support for the Katz-Sarnak conjectures. The proof requires calculating multidimensional integrals of the nondiagonal terms in the Bessel-Kloosterman expansion of the Petersson formula. We convert these multidimensional integrals to one-dimensional integrals already considered in the work of Iwaniec, Luo, and Sarnak [ILS] and derive a new and more tractable expression for the $n$th-centered moments for such test functions. This new formula facilitates comparisons between number theory and random matrix theory for test functions supported in $(-1/(n-1),1/(n-1))$ by simplifying the combinatorial arguments. As an application we obtain bounds for the percentage of such cusp forms with a given order of vanishing at the central point

We prove that the ascending chain condition (ACC) for log canonical (lc) thresholds in dimension $d$, existence of log flips in dimension $d$, and the log minimal model program (LMMP) in dimension $d-1$ imply termination of any sequence of log flips starting with a $d$-dimensional effective lc pair and also imply termination of flops in dimension $d$. In particular, the latter terminations in dimension $4$ follow from the Alexeev-Borisov conjecture in dimension $3$

Let $O$ be a compact orientable $3$-orbifold with nonempty singular locus and a finite volume hyperbolic structure. (Equivalently, the interior of $O$ is the quotient of hyperbolic $3$-space by a lattice in $\mathrm{PSL}(2,\mathbb{C})$ with torsion.) Then we prove that $O$ has a tower of finite-sheeted covers $\{O_i\}$ with linear growth of mod $p$ homology for some prime $p$. This means that the dimension of the first homology, with mod $p$ coefficients, of the fundamental group of $O_i$ grows linearly in the covering degree. The proof combines techniques from $3$-manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality and results about $p$-adic analytic pro-$p$ groups. This has several consequences. First, the fundamental group of $O$ has at least exponential subgroup growth. Second, the covers $\{O_i\}$ have positive Heegaard gradient. Third, we use the existence of this tower of covers to show that a group-theoretic conjecture of Lubotzky and Zelmanov implies that $O$ has a large fundamental group. This implication uses a new theorem of the author, which will appear in a forthcoming paper. These results all provide strong evidence for the conjecture that any closed orientable hyperbolic $3$-orbifold with nonempty singular locus has large fundamental group. Many of these results also apply to $3$-manifolds commensurable with an orientable finite-volume hyperbolic $3$-orbifold with nonempty singular locus. This includes all closed orientable hyperbolic $3$-manifolds with rank-two fundamental group and all arithmetic $3$-manifolds