Research. My research is on:
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Langlands functoriality and reciprocity principles, non-abelian class field theory.
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The theory of Eisenstein series, Arthur trace formula and Jacquet-Zagier trace formula.
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Higher-dimensional (K-theoretic) class field theory, theory of motives,
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dg-categories, NC-schemes, NC-motives.
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Conformal field theory and topological quantum field theory.
I am especially interested in
1-dimensional case: The local-global Langlands functoriality and reciprocity principles including “geometrization” (local case Fargues-Scholze FF-curve method, global case analytic stacks of Clausen-Scholze using condensed mathematics formalism under construction), Scholze's “perfectoid tecniques” (applied to the work of V. Lafforgue on char. p>0 global Langlands reciprocity principle) and the relative Langlands duality of Ben-Zvi, Sakellaridis and Venkatesh (BZSV-duality), char. p>0 case of geometric theory of Beilinson-Drinfel'd, in the local-global non-abelian class field theory à la Koch (local case: Fesenko-Laubie theory, global case: WA-idèles), and other approaches to local-global non-abelian class field theory (Kim's diophantine geometric iterative global approach, Nikolaev's operator theoretic global approach, Ono's generalized global Artin reciprocity applied to absolute arithmetic and F1-geometry, Weng's Tannakian geometric and local approach); their relationship with each other via higher ramification theory, local ϵ-factors, and Howe theory in the local case, and WA-parameters in the global case to form a “1-dimensional grand unified theory”.
Higher-dimensional case: The higher-dimensional generalizations of above theories to the K-theoretic (higher-dimensional) setting (that is, non-abelianization à la Kapranov of the local-global class field theory of Kato-Parshin: the 1-dimensional theory is the local-global Langlands functoriality and reciprocity principles; Scholl's higher-dimensional field of norms and perfectoid techniques to construct the local-global higher-dimensional non-abelian class field theory à la Koch: 1-dimensional theory is for the local case Fesenko-Laubie theory and for the global case WA-idèles; higher diophantine theoretic, operator theoretic, algebraic, and Tannakian methods to construct the global higher-dimensional non-abelian class field theories of Kim, of Nikolaev, of Ono, and geometric and local higher-dimensional non-abelian class field theories of Weng respectively), i.e., to n-dimensional local fields, to semi-global fields and to schemes (that is, local-global Kapranov functoriality and reciprocity principles, the K-theoretic Fesenko-Laubie theory in the local case and Lichtenbaum-Weil-Arthur idèles in the global case, K-theoretic global non-abelian class field theories of Kim, of Nikolaev, of Ono, and K-theoretic geometric and local non-abelian class field theories of Weng), complex curve case of geometric theory of Beilinson-Drinfel'd; their relationship with each other via Zhukov's and Abbes-Saito’s higher ramification theories, local ϵ-factors, and higher-dimensional Howe theory in the local case, and Lichtenbaum-Weil-Arthur parameters in the global case to form an “n-dimensional grand unified theory”. Completing the Kapranov functoriality and reciprocity formalism with the content of Ginzburg-Kapranov-Vasserot (that is, char. p>0 surface setting) and the content of analytic Langlands correspondence of Etingof-Frenkel-Kazhdan (that is, in the semi-global and archimedean setting the higher-dimensional analytic Beilinson-Drinfel'd theory). The place of Arthur trace formula, Eisenstein series and spectral decomposition, and Langlands L-functions (that is, the analytic theory of automorphic higher-representations and “automorphic L-functions like” analytic objects attached to automorphic higher-representations) in these general higher-dimensional settings following Garland and Kapranov.
I am also interested in the dg-(A∞-, stable (∞,1)-)versions (that is, possible extensions of the above theories to the non-commutative spaces of Kontsevich-Rosenberg-Soibelman and the derived algebraic geometry of Töen, namely to dg-categories, that is to the NC schemes and to the NC motives of Pippi, Robalo and Tabuada), and in the applications and possible analogues of these very general theories in algebraic topology (theory of topological modular forms and topological automorphic forms), arithmetic topology of Kapranov, Mazur, Morishita, and mathematical physics and in string theory (quantum and quantum analytic Langlands program).