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# Hsiao, Chin-Yu

 My domain of research is Microlocal Analysis, Complex Geometry and CR(Cauchy-Riemann) Geometry. Mcrolocal Analysis is a term used to describe a technique developed from the 1950s by Kohn-Nirenberg, H\"{o}rmander, Sato, Boutet de Monvel, Sj\"{o}strand, Guillemin, Melrose, and others, based on Fourier transforms related to the study of variable coefficients linear and nonlinear partial differential operators. This includes pseudodifferential operators, wave front sets, Fourier integral operators and WKB constructions. The term Microlocal implies localisation not just at a point, but in terms of contangent space directions at a given point. This gains in importance on manifolds. Microlocal Analysis is a powerful analytic tool in Complex Geometry, CR Geometry, Spectral Theory and Theoretical Physics. Complex Geometry is the study of complex manifolds and functions of many complex variables. Complex Geometry is a highly attractive branch of modern mathematics that has witnessed many years of active and successful research. CR Geometry is the study of manifolds equipped with a system of CR-type. CR Geometry is a relatively young and nowadays intensively studied research area having interconnections with many other areas of mathematics and its applications. It deals with restrictions and boundary values of holomorphic functions (CR functions) and of holomorphic mappings (CR mappings) to real submanifolds. A phenomenon arising in dimension higher than one is the rich intrinsic structure that leads to the existence of real submanifolds of different non-equivalent types. The systems of tangential Cauchy-Riemann equations for functions and mappings present important examples of systems of partial differential equations. A celebrated example of a system of this kind due to Hans Lewy played a crucial role in the development of the solvability theory for more general classes of PDEs. A full understanding of modern CR Geometry requires knowledge of various topics such as real(complex) differential and symplectic geometry, foliation theory, the geometry of PDE's and microlocal analysis. Email : chsiao AT math.sinica.edu.tw Phone:+886 2 2368-5999 ext. 633 Fax: +886 2 2368-9771

### Publications : :

 1 (with Po-Lam Yung) Solution of the tangential Kohn Laplacian]{Solution of the tangential Kohn Laplacian on a class of non-compact CR manifolds, (2019-12)(50 pages, available at arXiv:1808.07212) 2 (with Hendrik Hermann, Xiaoshan Li) Torus equivariant Szeg\H{o} kernel asymptotics on strongly pseudoconvex CR manifolds, (2019-08)(available at arXiv:1808.01927) 3 (with Rung-Tzung Huang, Xiaoshan Li, Guokuan Shao) $S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary, (2017-12)(submitted, 34 pages, available at arXiv:1711.05537) 4 (with Hendrik Herrmann, Xiaoshan Li) Szeg\H{o} kernels and equivariant embedding theorems for CR manifolds, (2017-12)(submitted, 30 pages, Available at arXiv:1710.04910) 5 (with Guokuan Shao) Equidistribution theorems on strongly pseudoconvex domains, Transactions of the AMS. (2019-03)(to appear in Transactions of the AMS, 22 pages, Available at https://arxiv.org/pdf/1708.01094.pdf) 6 (with Hendrik Herrmann, Xiaoshan Li) An explicit formula for Szego kernels on the Heisenberg group, Acta Mathematica Sinica, English Series , 34(2018-07), 1225-1247 7 (with Rung-Tzung Huang) G -invariant Szegö kernel asymptotics and CR reduction, (2017-12)(submitted, 60 pages, Available at . arXiv:1702.05012) 8 (with Hendrik Herrmann, Xiaoshan Li) Szeg\H{o} kernel asymptotic expansion on strongly pseudoconvex CR manifolds with $S^1$ action, International Journal of Mathematics , 29(2018-08) 9 (with Xiaoshan Li, George Marinescu) On the stability of equivariant embedding of compact CR manifolds with circle action, Math. Z. , 289(2018-08), 201-222(Math. Z. 289 (2018), no. 1-2, 201–222.) 10 (with Rung-Tzung Huang) The asymptotics of the analytic torsion on CR manifolds with $S^1$ action, Communications in Contemporary Mathematics, (2018-11)(To appear in Communications in Contemporary Mathematics, 31 pages, arXiv:1605.07507) 11 (with Jeffrey S. Case, Paul Yang) Extremal metrics for the Q′-curvature in three dimensions., C. R. Math. Acad. Sci. Paris , 354(2016-04), 407-410 12 (with Xianshan Li, George Marinescu) Equivariant Kodaira embedding of CR manifolds with circle action, (2016-12)(arXiv:1603.08872 , 38 pages) 13 (with Hendrik Herrmann and Xiaoshan Li) Szegö kernel expansion and embedding of Sasakian manifolds, Ann. Global Anal. Geom. , 52(2017-10), 313-340 14 (with Jeffrey S. Case and Paul Yang) Extremal metrics for the $Q^\prime$-curvature in three dimensions, to appear in Journal of the European Mathematical Society (2018-12)(36 pages, arXiv:1511.05248,) 15 (with Jih-Hsin Cheng and I-Hsun Tsai) An index theorem for CR manifolds with $S^1$ action, (2016-12)(51 pages, arXiv:1511.00063) 16 (with Xiaoshan Li) Morse inequalities for Fourier components of Kohn–Rossi cohomology of CR manifolds with S1-action, Math. Z. (2016-07)(Math. Z. DOI 10.1007/s00209-016-1661-6) 17 (with Xiaoshan Li) Szeg\"o kernel asymptotics and Morse inequalities on CR manifolds with $S^1$ action, Asian Journal of Mathematics , 22(2018-07), 413-450(Asian Journal of Mathematics, Vol. 22, No. 3 (2018), pp. 413-450.) 18 (with George Marinescu) Szeg\H{o} kernel asymptotics and Kodaira embedding theorems of Levi-flat CR manifolds, Mathematical Research Letters , 24(2017-12), 1385-1451(Math. Res. Lett. 24 (2017), no. 5, 1385–1451.) 19 (with G. Marinescu) Berezin-Toeplitz quantization for lower energy forms, Comm. Partial Differential Equations , 42(2017-09), 895-942 20 (with G. Marinescu) On the singularities of the Szeg\"o projections on lower energy forms, Journal of Differential Geometry , 107(2017-09), 83-155 21 Szegö kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds, Memoirs of the American Mathematical Society , 254(2018-07), no. 1217, 1-146(Mem. Amer. Math. Soc. 254 (2018), no. 1217,) 22 The second coefficient of the asymptotic expansion of the weighted Bergman kernel on $\mathbb C^n$, Bulletin of the institute of Mathematics, Academia Sinica(New Series) , 11(2016-09), 521-570 23 On CR Paneitz operators and CR pluriharmonic functions, Math. Ann. , 362(2015-10), 903-926(27 pages, Math. Ann. DOI 10.1007/s00208-014-1151-2) 24 (with Po-Lam Yung) Solving Kohn Laplacian on asymptotically flat pseudohermitian 3-manifolds, Advances in Mathematics , 281(2015-08), 734-822(Advances in Mathematics, Volume 281, Pages 734–822, Available at arXiv:1303.6557) 25 Bergman kernel asymptotics and a pure analytic proof of Kodaira embedding theorem, (2015)(to appear in Complex Analysis and Geometry, Springer Proc. Math. Stat., Available at arXiv:1411.5441) 26 (with G. Marinescu) Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles, COMMUNICATIONS IN ANALYSIS AND GEOMETRY , 22(2014), 1-108(Available at arXiv:1112.5464) 27 (with Po-Lam Yung) The tangential Cauchy-Riemann complex on the Heisenberg group Via Conformal Invariance, Bulletin of the institute of Mathematics, Academia Sinica(New Series) , 8(2013), 359-375(Available at arXiv:1303.6547) 28 Existence of CR sections for high power of semi-positive generalized Sasakian CR line bundles over generalized Sasakian CR manifolds, (2015)(50 pages, , Ann. Glob. Anal. Geom., (2015), DOI 10.1007/s10455-014-9434-0., Available at arXiv:1204.4810) 29 On the coefficients of the asymptotic expansion of the kernel of Berezin-Toeplitz quantization, Annals of Global Analysis and Geometry, Annals of Global Analysis and Geometry , 2012(2012), no. 42, 207-245(Available at arXiv:1108.0498) 30 (with G. Marinescu) Szegö kernel asymptotics and Morse inequalities on CR manifolds, Math. Z. , 2012(2012), no. 271, 509-553(Available at arXiv:1005.5471) 31 Projections in several complex variables, Mémoires de la Société Mathématique de France , 2010(2010), no. 123, 1-136(Available at arXiv:0810.4083.) 32 Projecteurs en Plusieurs Variables Complexes, Phd Thesis at Ecole Polytechnique (2008), 1-254(Available at http://tel.archives-ouvertes.fr/tel-00332787)