Research
Current and upcoming research projects
My current interests are centred around applying the Heisenberg-Weyl (HW-)algebra to study special functions and complex function theory. Many of these studies make use of tools from holonomic D-modules theory. Students may find a list of potential research projects here:
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To apply HW-algebra to study classical special functions and their generating functions.
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To apply HW-algebra to study various types of orthogonalpolynomials/functions.
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To apply HW-algebra to investigate complex function theory.
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To apply HW-algebra to study calculus of finite differences
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To apply a deformed HW-algebra to investigate fractional derivativities.
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To apply a q-deformed HW-algebra to study q-special functions.
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To apply HW-algebra to study related combinatorial and stochastic problems.
Research biography - a brief history
I started off working on topics related to zero-distribution of solutions of linear differential equations (LDEs) defined either in the unit-disk or on the whole complex plane. The main tools required for the unit-disk case are comparison principle and quasi-conformal mappings, while for the complex plane case is the classical Nevanlinna theory. The latter is called the Complex Oscillation Theory was initiated by Steven Bank and Ilpo Laine in the 1980s. These topics constituted the backbone of my PhD thesis written under the supervision of the late Professor J. Milne Anderson at UCL. Professor J. K. Langley also gave me a lot of advice during my postgraduate study.
Differential Equations
I have since introduced some classical special functions (SF) into the study of the complex oscillation theory of LDEs. This included new characterization or analytic continuation formulae about some of these SF that are new in the literatures despite their long history. These new results complement the complex oscillation theory.
The more that I investigated into these classical SF, the more that I need to learn to use tools from other areas, such as monodromy groups, differential Galois theory and Stoke’s phenomenon to tackle problems related to these classical SF. Despite of their old age and numerous formulae derived in the literatures, I have also found that these SF are not as well understood as what I would have thought. In particular, algebraic tools appear essential in my approaches. Many of the research projects were the results of collaboration with other Mathematicians. My current research involves the use of tools from holonomic D-module theory to study special functions.
Difference Calculus and Difference Equations
My initial interest in difference calculus was inspired both by the works of Ablowitz, Halburd and Herbst on difference integrable systems and the works of Askey and Ismail on q-difference special functions. Some of my earlier works are closely related to those of Halburd and Korhonen. Difference Picard-type theorems were derived with respect to various difference operators from special functions that are important in the current literatures. Coincidently, my current approach to difference SF also involves holonomic D-module theory in parallel with my work on classical SF. The nineteenth century saw the peak of calculus of finite differences. The theory comprised of mostly algorithms that were applied to many scientific problems that needed efficient algorithms for their disciplines. However, many of these works were either forgotten or abandoned because of their lack of mathematical rigor. We shall Hesienberg-Weyl algebra to revisit these old algorithms with a modern viewpoint.
Publications
Click here for complete publication list.
K. M. Cheng, Y. M. Chiang and Avery Ching, "D-modules approach to
special functions and generating functions", C. R. Math. Rep. Acad. Sci.
Canada, 45, no. 1, (2023), 1-12.
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., 53, (2022), 63-79.
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 Poincaré , 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 and S. J. Feng "Nevanlinna theory based on Askey-Wilson divided difference operator", (Proof) Advances in Mathematics, 329, (2018), 217-272. arXiv: 1502.02238
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
Teaching philosophy
Over the decades of teaching experience, I have come across students with a wide variety of mathematics preparations, some come with very good preparation and some are much less prepared. This is especially the case with students in calculus/differential equation courses from other departments as service teaching. On the one hand, students have different mathematics preparation, and on the other hand, the curriculums that serve other departments demand the coverage of a large number of topics where time is usually limited. Thus I try to strike a balance between delivering the topics that must be covered and to maintain an inquiry-based teaching mode so that I could leave some room that allows for their independent thinking. This is particularly important for students from oriental cultural backgrounds in general, and to students from Hong Kong/PRC in particular where rote-learning is a norm. They seldom ask critical questions such as “what” and “why” when a new concept is being introduced. I believe it is only when a student is being inspired (so the student would start to “think”) rather than how much is being taught is what I would consider successful teaching.
Students who are interested to take my courses could find some of the sample lecture notes used in previous undergraduate courses that I taught. Complete sets of notes are available from CANVAS after you have joined the courses.
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.
MATH1014 Calculus II
This is an introductory course in one-variable calculus, the second in the MATH 1013 – MATH 1014 sequence. Topics include applications of definite integral, improper integrals, vectors, curves and parametric equations, modeling with differential equations, solving simple differential equations, infinite sequences and series, power series and Taylor series.
Sample handout - Numerical Integration:
Mid-point and Trapezoidal rules
Sample handout - Sequences and Series
Infinite sequences I
MATH 2352 Differential Equations
Sample handout - Separable PDEs: Heat Equations
Sample handout - Second Order Equations:
Power Series Solutions I
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.