I attended several primary and secondary schools in Hong Kong before I received both my undergraduate (1984) and postgraduate (1991) qualification from University College London. My PhD was directed under the late Professor J. Milne Anderson on topics related to Schwarzian derivatives and linear differential equations. Although special functions are almost as old as calculus, many of the centuries old special functions that appear frequently in applications are still far from being understood. My current interests lie in application of Nevanlinna theory, algebraic and geometric methods to better understand these special functions.
Nevanlinna theory and linear special functions
The research involves the study of monodromy groups and Stokes’s phenomena of special functions applying analytic viewpoint in terms of Nevanlinna theory together with algebraic and geometric viewpoints in terms of differential Galois theory and rigid local systems respectively. The research is intended as a continuation of original Riemann’s approach of Fuchsian differential equations (1857) and its confluent cases.
Difference Nevanlinna theory and interpolatory function theory of special functions
The study applies recently discovered Nevanlinna (complex function) theories of difference operators to various aspects of certain special functions that are more naturally related to appropriate difference operators. E.g. the classical (Jacobi’s) theta functions are shown to be deficient in zeros when interpreted appropriately against the classical view where it contains an abundance of zeros.
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K. M. Cheng, Y. M. Chiang and Avery Ching, "D-module approach to Liouville's theorems for differences operators", arXiv: 2109.06487 (New Zealand J. Math., to appear)
Y. M. Chiang, Avery Ching and Chiu-Yin Tsang, "Resolving singularities of Heun connections and P_VI", arXiv:2009.02871 (published online 27th April 2021) Ann Henri Poincar\'e, 22 (2021), 3051-3094
K. H. Cheng and Y. M. Chiang, "Wiman-Valiron theory of a polynomial series based on Askey-Wilson divided difference operator", arXiv:2001.02120 (Constructive Approximation 2021: https://link.springer.com/article/10.1007/s00365-021-09528-3)
Y. M. Chiang and Guofu Yu, "Galoisian approach to complex oscillation theory of Hill equations", (https://arxiv.org/abs/1610.09757) Mathematica Scandinavica, 124, no. 1 (2019), 102-131.
Y. M. Chiang, A. Ching and C. Y. Tsang , "Symmetries of the Darboux equation", Kumamoto Journal of Mathematics, 31, (2018), 15-48. arXiv: 1509.03995
Y. M. Chiang and X. D. Luo, “Difference Nevanlinna theories with vanishing and infinite periods”, Michigan Mathematical Journal 66 no. 3 (2017), 451-480. arXiv:1510:02576
K.-H. Cheng and Y. M. Chiang, “Nevanlinna theory based on Wilson divided difference operator”, Ann. Acad. Sci. Fenn. Math. 42 (2017), 175–209.
Y. M. Chiang and S. J. Feng, “On the growth of logarithmic difference of meromorphic functions and a Wiman-Valiron estimate”, Constructive Approximation 33 no. 3 (2016) 313– 325. arXiv:1309.4211
Y. M. Chiang, “Estimates on the growth of meromorphic solutions of linear differential equations with density conditions”, Proceedings of the Workshop on Complex Analysis and its Applications to Differential and Functional Equa- tions, Gro ̈hn, J. Heittokangas, R. Korhonen & J. Ra ̈ttya ̈ (Eds), Reports and Studies in Forestry and Natural Sci- ences, No. 14, 45–55, University of Eastern Finland, 2014. arXiv:1311.1874
MATH1012/1013 Calculus I
This is an introductory course in one-variable calculus, the first in the Calculus I and II sequence, designed for students that have taken HKDSE Mathematics Extended Module M1/M2. Topics include functions and their limits, continuity, derivatives and rules of differentiation, applications of derivatives, and basic integral calculus.
MATH 2352 Differential Equations
MATH4061 Topics in Modern Analysis
Examples and properties of metric spaces. Contractive mapping theorem, Baire category theorem, Stone-Weierstrass theorem, Arzela-Ascoli theorem. Properties of normed spaces and Hibert spaces. Riesz theorem. Completeness of Lp functions, continuous functions and functions of bounded variations. Best approximation theorem on Hilbert space.
MATH4822e Fourier Analysis and its Applications
Review of basic properties of analytic functions. Phragmen-Lindelof principle, normal family, Riemann mapping theorem. Weierstrass factorization theorem, Schwarz reflection principle, analytic continuation, harmonic function, entire function, Hadamard factorization theorem, Picard theorem.