On Uniqueness And Existence of Conformally Compact Einstein Metrics with Homogeneous Conformal Infinity

In this paper we show that for a generalized Berger metric $\hat{g}$ on $S^3$ close to the round metric, the conformally compact Einstein (CCE) manifold $(M, g)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if $\hat{g}$ is an $\text{SU}(k+1)$-invariant metric on $S^{2k+1}$ for $k\geq1$, the non-positively curved CCE metric on the $(2k+1)$-ball $B_1(0)$ with $(S^{2k+1}, [\hat{g}])$ as its conformal infinity is unique up to isometries. In particular, since in \cite{LiQingShi}, we proved that if the Yamabe constant of the conformal infinity $Y(S^{2k+1}, [\hat{g}])$ is close to that of the round sphere then any CCE manifold filled in must be negatively curved and simply connected, therefore if $\hat{g}$ is an $\text{SU}(k+1)$-invariant metric on $S^{2k+1}$ which is close to the round metric, the CCE metric filled in is unique up to isometries. Using the continuity method, we prove an existence result of the non-positively curved CCE metric with prescribed conformal infinity $(S^{2k+1}, [\hat{g}])$ when the metric $\hat{g}$ is $\text{SU}(k+1)$-invariant.Comment: Comments are welcome

QBDT, a new boosting decision tree method with systematic uncertainties into training for High Energy Physics

A new boosting decision tree (BDT) method, QBDT, is proposed for the classification problem in the field of high energy physics (HEP). In many HEP researches, great efforts are made to increase the signal significance with the presence of huge background and various systematical uncertainties. Why not develop a BDT method targeting the significance directly? Indeed, the significance plays a central role in this new method. It is used to split a node in building a tree and to be also the weight contributing to the BDT score. As the systematical uncertainties can be easily included in the significance calculation, this method is able to learn about reducing the effect of the systematical uncertainties via training. Taking the search of the rare radiative Higgs decay in proton-proton collisions $pp \to h + X \to \gamma\tau^+\tau^-+X$ as example, QBDT and the popular Gradient BDT (GradBDT) method are compared. QBDT is found to reduce the correlation between the signal strength and systematical uncertainty sources and thus to give a better significance. The contribution to the signal strength uncertainty from the systematical uncertainty sources using the new method is 50-85~\% of that using the GradBDT method.Comment: 29 pages, accepted for publication in NIMA, algorithm available at https://github.com/xialigang/QBD

Measuring the ratio of HWWHWW and HZZHZZ couplings through W+W−HW^+W^-H production

For a generic Higgs boson, measuring the relative sign and magnitude of its couplings with the $W$ and $Z$ bosons is essential in determining its origin. Such a test is also indispensable for the 125-GeV Higgs boson. We propose that the ratio of the $HWW$ and $HZZ$ couplings $\lambda_{WZ}$ can be directly determined through the $W^+W^-H$ production, where $H$ denotes a generic Higgs boson, owing to the tree-level interference effect. While this is impractical at the LHC due to the limited sensitivity, it can be done at future $e^+e^-$ colliders, such as a 500-GeV ILC with the beam polarization $P(e^-,e^+)=(-0.8,+0.3)$ in the $jj\ell^{\pm}bb$ and $\ell^{\pm}\ell^{\pm}\ell^{\mp}jj$ channels. The discovery potential of a general ratio and the power to discriminate it from the SM value are studied in detail. Combining the cross section of $e^+e^-\to W^+ W^- H$ with the measurements of $HZZ$ coupling at the HL-LHC, one can further improve the sensitivity of $\lambda_{WZ}$.Comment: 24 pages, 10 figures, 2 table

Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

We introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex. We apply this method to constructing the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to any smooth projective variety. As an application, we give an algebraic definition of GW-invariants for any smooth projective variety.Comment: 68 pages, Amstex. Revised version. It appeared in J. of Amer. Math. Soc. Jan 9