(click on each title to show descriptions of the research)
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Beilinson's Conjectures
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Beilinson's conjecture is a deep conjecture relating higher regulators of pure motives and non-critical values of motivic
L-functions. Instead of working on the general conjecture, I mainly work on examples of Beilinson's conjecture for automorphic motives using
the tools from motives, Hodge theory and automorphic representations
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1. In my forthcoming Ph.D. thesis, I proved an identity relating the motivic class constructed in [LSZ] and a non-critical value of the motivic L-function associated to some
cuspidal automorphic representation of GU(2,1).
2. In a joint work in progress with Wenxuan Qi (Peking University), Liang Xiao (Peking University) and Yichao Zhang (Hunan Normal University)，borrowing ideas from
the Kudla Program (See next part), we have been constructing new motivic classes in higher Chow groups of general orthogonal Shimura varieties and proving new classes
of Beilinson's conjecture for automorphic representations of higher rank orthogonal group.
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Kudla Program
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Inspired by the ideas from theta correspondence, Kudla and his collaborators built up the theory of geometric theta series and arithmetic theta series and used those theta series to construct
cohomology classes (Kudla-Millson Theory) and algebraic cycles (arithmetic theta lifting).
In joint work in progress with Wenxuan Qi (Peking University), Liang Xiao (Peking University) and Yichao Zhang (Hunan Normal University)
as mentioned above, we have built up the theory of ``Deligne theta series'' with values in Deligne cohomology and have been building up the theory of ``motivic theta series'' with values in higher Chow group. Then using the ``motivic theta correspondence'', we hope to construct new classes in higher Chow groups of
general orthogonal Shimura varieties which will be used to prove new cases of Beilinson's conjecture.
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Relative Langlands Program
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The relative Langlands program aims to explore the relations between period integrals, special values of automorphic L-functions and Langlands functoriality. I mainly
work on regularized periods of Eisenstein series using Truncation operator and Residue method and its applications.
1. I prove a formula relating regularized periods of cuspidal Eisenstein series to cuspidal intertwining periods for
linear models. Using this formula, I prove
a formula relating regularized periods of discrete Eisenstein series to discrete intertwining periods for GL2n/GLn ✕ GLn.
2. In a work in progress, I have been trying to prove a global spectral expansion for the spherical variety GL2n/GLn ✕ GLn using the formula I proved (stated in 1.) and for GL2n/Sp2n using the formula proved by Yamana.
3. In a joint work in progress with Yiyang Wang (Kyoto University), we have been working to factorize some regularized periods of Eisenstein series for GLn + m/GLn ✕ GLm into local linear forms and proving local Plancherel formula for the spherical variety using the local linear forms.
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