This award will be used to support several research projects in the theory of automorphic forms especially in the context of the Langlands program. The projects chiefly concern the theory of both local and global theta correspondences. Among other things, the PI intends to solve the so-called non-vanishing problem of the theta correspondences by establishing the second term identity of the Siegel-Weil formula in full generality and obtaining certain expected analytic properties of local zeta integrals.
The theory of automorphic forms is one of the core themes of modern mathematics. This is not only because it has been found one of the deepest and most beautiful subjects, but also because it is closely connected to various areas of both pure and applied mathematics, in particular number theory and representation theory as well as harmonic analysis. Aside from those intellectual aspects, it should be mentioned that this award will be used to benefit a wide range of intellectual communities at various levels through publications and presentations in national and international professional meetings, through communications with researchers in related areas and other fields, and through formal and informal educational activities such as supervising students at both graduate and undergraduate levels.
I. Intellectual merits The PI finished the following seven papers: On the lattice model of the Weil representation and the Howe duality conjecture, J. Ramanujan Math Society, 29, (2014) 321-378 (with W. Gan and Y. Qiu) The regularized Siegel-Weil formula (the second term identity) and the Rallis inner product formula, Invent. Math.(to appear). Metaplectic tensor products of automorphic representations of $widetilde{GL}(r)$ (submitted) On a certain metaplectic Eisenstein series and the twisted symmetric square $L$-function (submitted) (with W. Gan) On the Howe duality conjecture in classical theta correspondence (submitted) A proof of the Howe duality conjecture (submitted) (A. Wood) Hecke algebra correspondences for the metaplectic group (preprint) In 1, the PI obtained a partial result toward a proof of the Howe duality conjecture via the method of lattice model, and then in 5 and 6, he with Gan proved the Howe duality conjecture in a different method. In 2, He with Gan and Qiu obtained the second term identity of the Siegel-Weil formula to obtain a fairy satisfactory result on the non-vanishing problem of global theta lifting. In 3 and 4, he completed his project on the holomorphy of the twisted symmetric square L-functions for GL(n), which was initiated by the PI in his previous work. Finally, in 7, the PI with Wood obtained Hecke algebra correspondences for the metaplectic group over any residual characteristic. II. Broader impacts The PI has been organizing a seminar at his home institute. All the talks given there are available on his website www.math.missouri.edu/~takedas/seminar.html The seminar has been attended usually by 5 to 12 people including graduate students and faculty, and has been successfully run. The PI together with others organized the following conferences The workshop ``the Future of Trace Formulas'' Jun. 2014, BIRS, Banff, Canada The AMS special session ``Recent Progress in the Langlands program'' Jan. 2014, Join meeting, Baltimore The special session ``Automorphic Forms and Representation Theory'' Oct. 2013, AMS sectional meeting, St. Louis and they were successfully run. The PI has been mentoring a postdoc (Aaron Wood) starting from Fall 2013 and has finished a joint paper. The PI gave talks at various meetings both nationally and internationally. To be more specific, he gave 5 seminar talks, 1 colloquium talk, 7 conference talks, and 2 mini courses. Also he attended 13 conferences in 5 different countries.
