Next-generation sequencing technology enables the routine detection of bacterial pathogens for clinical diagnostics and genetic research. Whole-genome sequencing has been of importance in the epidemiologic analysis of bacterial pathogens. However, few whole-genome sequencing-based genotyping pipelines are available for practical applications. Here, we present the whole-genome sequencing-based single nucleotide polymorphism(SNP) genotyping method and apply to the evolutionary analysis of methicillin-resistant Staphylococcus aureus. The SNP genotyping method calls genome variants using next-generation sequencing reads of whole genomes and calculates the pair-wise Jaccard distances of the genome variants. The method may reveal the high-resolution whole-genome SNP profiles and the structural variants of different isolates of methicillin-resistant S. aureus(MRSA) and methicillin-susceptible S. aureus(MSSA) strains. The phylogenetic analysis of whole genomes and particular regions may monitor and track the evolution and the transmission dynamic of bacterial pathogens. The computer pro- grams of the whole genome sequencing-based SNP genotyping methods are available to the public at https://github. com/ cyinbox/NGS.

In this paper, we propose a conjectural multiplicity formula for general spherical varieties. For all the cases where a multiplicity formula has been proved, including Whittaker model, Gan-Gross-Prasad model, Ginzburg-Rallis model, Galois model and Shalika model, we show that the multiplicity formula in our conjecture matches the multiplicity formula that has been proved. We also give a proof of this multiplicity formula in two new cases.

By applying the residue method for period integrals and Langlands-Shahidi’s theory for residues of Eisenstein series, we study the period integrals for six spherical varieties. For each spherical variety, we prove a relation between the period integrals and certain automorphic L-functions. In some cases, we also study the local multiplicity of the spherical varieties.

We calculate a G_2-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group E_6, and relate this period to the Ginzburg-Rallis period of cusp forms on GL_6. This gives us a relation between the Ginzburg-Rallis period and the central value of the exterior cube L-function of GL_6.

Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, we are able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, we prove a multiplicity formula of the GinzburgRallis model for the supercuspidal representations. Using that multiplicity formula, we prove the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.