Research Fellow |   Hsiao, Chin-Yu
Contact Information
• chsiao$\color{red}{@}$math.sinica.edu.tw
• +886 2 2368-5999 ext. 633
• +886 2 2368-9771
Research Interests
• Microlocal Analysis
• Complex Geometry and Cauchy-Riemann Geometry
Education
• Ph.D. in Mathematics CMLS,  Ecole polytechnique, Palaiseau, France (2005/10 - 2008/07)
• M.S. in Mathematics National Taiwan University (2001/9 - 2004/6)
• B.S. in Mathematics National Taiwan University (1996/9 - 2001/6)

Work Experience
• Research Fellow Institute of Mathematics, Academia Sinica, Taiwan 2017/5-
• Associate Research Fellow Institute of Mathematics, Academia Sinica, Taiwan 2015/5-2017/5
• Assistant Professor Institute of Mathematics, Academia Sinica, Taiwan 2013/8-2015/5
• Postdoctoral Universität zu Köln, Mathematisches Institut 2010/12 - 2013/7
• Postdoctoral Chalmers University of Technology 2009/7 - 2010/11
• Postdoctoral Chalmers University of Technology 2008/11 - 2009/6

Awards
• Post-doctoral fellowship of the Royal Swedish research council, 2008
• École polytechnique Excellence Post-doctoral fellowship, 2008
• École polytechnique Excellence Scholarship, 2005
• Dean of the Faculty of Science Award, National Taiwan University, 2000

Research Descriptions

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.

Selected Publications
1. (with Weixia Zhu) "Heat kernel asymptotics for Kohn Laplacians on CR manifolds" , 2021-01. (arXiv:2106.09268)
2. (with Chin-Yu Hsiao and Wei-Chuan Shen) "On the second coefficient of the asymptotic expansion of Boutet de Monvel--Sj\"ostrand" , 2020-12. (arXiv:2012.15456, to appear in Bulletin of the institute of Mathematics, Academia Sinica)
3. (with Chin-Yu Hsiao, Huan Wang and George Marinescu) "Szeg\H{o} kernel asymptotics on some non-compact complete CR manifolds" , 2020-12. (arXiv:2012.11457)
4. (with Rung-Tzung Huang and Guokuan Shao) "On the coefficients of the equivariant Szeg\H{o} kernel asymptotic expansions" , 2020-12. (arXiv:2009.10529)
5. (with Nikhil Savale) "Bergman-Szeg\H{o} kernel asymptotics in weakly pseudoconvex finite type cases" , 2020-12. (44 pages, arXiv:2009.07159)
6. (with George Marinescu) "On the singularities of the Bergman projections for lower energy forms on complex manifolds with boundary" , 2020-12. (arXiv:1911.10928, 54 pages)
7. (with Hendrik Herrmann, Xiaoshan Li) "Szego kernels and equivariant embedding theorems for CR manifolds" , 2020-12. (60 pages, to appear in Mathematical Research letters)
8. (with Xiaonan Ma, George Marinescu) "Geometric quantization on CR manifolds" , 2020-12. (arXiv:1906.05627, 47 pages)
9. (with Kevin Fritsch, Hendrik Herrmann) "$G$-equivariant embedding theorems for CR manifolds of high codimension" , 2020-12. (36 pages, to appear in Michigan Mathematical Journal, available at arXiv:1810.09629)
10. (with Rung-Tzung Huang) "G -invariant Szegö kernel asymptotics and CR reduction" , 2020-12. (to appear in Calculus of Variations and Partial Differential Equations, 50 pages, Available at . arXiv:1702.05012)
11. (with Xianshan Li, George Marinescu) "Equivariant Kodaira embedding of CR manifolds with circle action" , Michigan Mathematical Journal. , 2020-12. (arXiv:1603.08872 , 38 pages, to appear in Michigan Mathematical Journal.)
12. (with Rung-Tzung Huang, Xiaoshan Li, Guokuan Shao) "$S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary" , Journal of Functional analysis , 2020-09. (DOI: 10.1016/j.jfa.2020.108558)
13. (with Hendrik Hermann, Xiaoshan Li) "Torus equivariant Szeg\H{o} kernel asymptotics on strongly pseudoconvex CR manifolds" , Acta Math. Vietnam. , 45 (1), 113--135-, 2020-04.
14. (with Po-Lam Yung) "Solution of the tangential Kohn Laplacian on a class of non-compact CR manifolds" , Calculus of Variations and PDE , 58 (2), 1-62, 2019-5.
15. (with Guokuan Shao) "Equidistribution theorems on strongly pseudoconvex domains" , Transactions of the AMS. , 37 (2), 1113-1137, 2019-06.
16. (with Rung-Tzung Huang) "The asymptotics of the analytic torsion on CR manifolds with $S^1$ action" , Communications in Contemporary Mathematics, , 21 (4), 1-35, 2019-06.
17. (with Jih-Hsin Cheng and I-Hsun Tsai) "Heat kernel asymptotics, local index theorem and trace integrals for CR manifolds with $S^1$ action" , Mém. Soc. Math. Fr. , 162, 1-139, 2019-05.
18. (with Jeffrey S. Case and Paul Yang) "Extremal metrics for the $Q^\prime$-curvature in three dimensions" , Journal of the European Mathematical Society , 21 (2), 582-626, 2019-04.
19. (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 (9), 2018-08.
20. (with Xiaoshan Li, George Marinescu) "On the stability of equivariant embedding of compact CR manifolds with circle action" , Math. Z. , 289, 201-222, 2018-08. (Math. Z. 289 (2018), no. 1-2, 201–222.)
21. (with Hendrik Herrmann, Xiaoshan Li) "An explicit formula for Szego kernels on the Heisenberg group" , Acta Mathematica Sinica, English Series , 34 (8), 1225-1247, 2018-07.
22. (with Xiaoshan Li) "Szeg\"o kernel asymptotics and Morse inequalities on CR manifolds with $S^1$ action" , Asian Journal of Mathematics , 22 (3), 413-450, 2018-07. (Asian Journal of Mathematics, Vol. 22, No. 3 (2018), pp. 413-450.)
23. "Szegö kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds" , Memoirs of the American Mathematical Society , 254 (1217), 1-146, 2018-07. (Mem. Amer. Math. Soc. 254 (2018), no. 1217,)
24. (with George Marinescu) "Szeg\H{o} kernel asymptotics and Kodaira embedding theorems of Levi-flat CR manifolds" , Mathematical Research Letters , 24 (5), 1385-1451, 2017-12. (Math. Res. Lett. 24 (2017), no. 5, 1385–1451.)
25. (with Hendrik Herrmann and Xiaoshan Li) "Szegö kernel expansion and embedding of Sasakian manifolds" , Ann. Global Anal. Geom. , 52 (3), 313-340, 2017-10.
26. (with G. Marinescu) "Berezin-Toeplitz quantization for lower energy forms" , Comm. Partial Differential Equations , 42 (6), 895-942, 2017-09.
27. (with G. Marinescu) "On the singularities of the Szeg\"o projections on lower energy forms" , Journal of Differential Geometry , 107 (1), 83-155, 2017-09.
28. "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 (3), 521-570, 2016-09.
29. (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)
30. (with Jeffrey S. Case, Paul Yang) "Extremal metrics for the Q′-curvature in three dimensions." , C. R. Math. Acad. Sci. Paris , 354 (4), 407-410, 2016-04.
31. "On CR Paneitz operators and CR pluriharmonic functions" , Math. Ann. , 362, 903-926, 2015-10. (27 pages, Math. Ann. DOI 10.1007/s00208-014-1151-2)
32. (with Po-Lam Yung) "Solving Kohn Laplacian on asymptotically flat pseudohermitian 3-manifolds" , Advances in Mathematics , 281, 734-822, 2015-08. (Advances in Mathematics, Volume 281, Pages 734–822, Available at arXiv:1303.6557)
33. "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)
34. "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)
35. (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 (1), 1-108, 2014. (Available at arXiv:1112.5464)
36. (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 (3), 359-375, 2013. (Available at arXiv:1303.6547)
37. "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 (42), 207-245, 2012. (Available at arXiv:1108.0498)
38. (with G. Marinescu) "Szegö kernel asymptotics and Morse inequalities on CR manifolds" , Math. Z. , 2012 (271), 509-553, 2012. (Available at arXiv:1005.5471)
39. "Projections in several complex variables" , Mémoires de la Société Mathématique de France , 2010 (123), 1-136, 2010. (Available at arXiv:0810.4083.)
40. "Projecteurs en Plusieurs Variables Complexes" , Phd Thesis at Ecole Polytechnique , 1-254, 2008. (Available at http://tel.archives-ouvertes.fr/tel-00332787)