# Research in NCTU-AM

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### Brief Introduction

The department currently has five research groups: "Mathematical Modeling and Scientific Computing", "Dynamical Systems and Differential Equations", "Discrete Mathematics and Optimization", "Financial Engineering and Probability" and "Number Theory, Geometry and Analysis". Their research includes pattern formations, neural networks, mathematical biology, and chaotic dynamics in Dynamical Systems; nonlinear partial differential equations, fluid mechanics, quantum mechanics, classical mechanics in Differential Equations; numerical PDEs, computational fluid mechanics, interface problems, porous media, and fractured media problems in Scientific Computations; combinatorial optimization, communication networks, algorithms, graph theory, algebraic combinatorics, finite geometry, and combinatorial designs in Discrete Mathematics; contingent claims pricing, credit risk modeling, randomized algorithms and Markov chains in Probability/ Financial Mathematics. The expertise of other faculty include functional analysis, number theory, geometry, and mathematical physics.

These research areas are also well presented at the National Center of Theoretical Sciences (NCTS) hosted by Chiao Tung University and Tsing Hua University. The department faculty members are actively involved in research activities including regular seminars, short courses, workshops, conferences, and summer programs at the NCTS. The faculty members include one national chair professor, two university chair professors, and four professors who were awarded the National Science Council (NSC) distinguished researchership. Almost every faculty member is supported by NSC research project grants.

Main Research Groups (*click header for more information*)

- Mathematical Modeling and Scientific Computing

- Members:

Wen-Wei Lin, Ming-Chih Lai, Chih-Wen Shih, Juang Jonq, Li-Ming Yeh, Chin-Tien Wu, Shu-Ming Chang, Ming-Cheng Shiue, Yung-Ta Li. - Topics:

Large-Scale Matrix Computations, Numerical PDES, Computational Fluid Mechanics, Interface Problems, Porous Media, Fractured Media Problems, Computational Conformal Geometry. - Recent Research Projects:

- Computational Conformal Geometry and Its Applications (1/4)--Wen-Wei Lin (NSC 102-2628-M-009-002-)
- Nonlinear Matrix Equations and its Applications in Nano Research -- Wen-Wei Lin (NSC 100-2115-M-009 -010 ¡VMY3)
- Interfacial flows with surfactant -- Ming-Chih Lai (NSC 98-2115-M-009 -014 ¡VMY3)
- Frontier research and development in industrial and applied mathematics -- Ming-Chih Lai (101-2911-I-009 -509)
- Research development od mathematics and statatics division -- Ming-Chih Lai (NSC 101-2114-M-009 -001)
- Mathematical Models and Analysis on Biological Rhythms -- Chih-Wen Shih (NSC 101-2115-M-009-002-MY2)
- Numerical methods for elliptic and parabolic equations in heterogeneous media--Li-Ming Yeh (NSC 102-2115-M-009-014)
- Freefrom reconstruction in three dimensional space using high order conformal finite element method -- Chin-Tien Wu (NSC 102-2115-M-009-011)
- Computing Discrete Sufrace Ricci Flows -- Shu-Ming Chang (NSC 102-2115-M-009-012)
- Numerical computation and analysis of the mathematical problems in geophysical fluid mechanics --Ming-Cheng
- Some large-scale matrix computation problems arising in surface parameterization -- Yung-Ta Li (NSC 101-2115-M-009-006)
- Developping mathematical theory for spinor Bose-Einstein condensates -- I-Liang Chern (NSC 102-2115-M-009-013-MY2)

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- Members:
- Financial Engineering and Probability

- Members:

Yuan-Chung Sheu, Michael Fuchs, Guan-Yu Chen, Lo-Bin Chang. - Topics:

Probability Theory and Its Applications, Contingent Claims Pricing, Credit Risk Modeling, Randomized Algorithms, Markov Chains. - Recent Research Projects:

- Pricing perpetual American options in a Levy model-- Yuan-Chung Sheu (NSC 102-2115-M-009-003)
- The mixing of birth and death chains on trees -- Guan-Yu Chen (NSC 102-2115-M-009-004)
- Conditional modeling and inference with applications in computer vision and finance -- Lo-Bin Chang (NSC 102-2115-M-009-001)

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- Members:
- Differential Equations and Dyanmical Systems

- Members:

Song-Sun Lin, Wen-Wei Lin, Chin-Wen Shin, Ming-Chia Li, C. K. Lin, Juang Jouq, Jong-Eao Lee, Fu-Shiang P. Tsen, Shu-Ming Chang, Chi-Jer Yu. - Topics:

Pattern Formations, Neural Networks, Mathematical biology, Chaotic Dynamics, Nonlinear Elliptic Partial Differential Equations, Nonlinear Hyperbolic Partial Differential Equations, Fluid Mechanics, Quantum Mechanics, Classical Mechanics. - Recent Research Projects:

- Mixing Property and Ergodic Theory of Multi-Dimensional Shift -- Song-Sun Lin (NSC 101-2115-M-009-007-MY2)
- Nonlinear Matrix Equations and its Applications in Nano Research -- Wen-Wei Lin (NSC 100-2115-M-009 -010 ¡VMY3)
- Mathematical Models and Analysis on Biological Rhythms -- Chin-Wen Shin (NSC 101-2115-M-009-002-MY2)
- Symbolic embedding and globla implicit function theorem, II -- Ming-Chia Li (NSC 102-2115-M-009-008-MY2)
- Klein-Gordon equation with electromagnetic field -- C. K. Lin (NSC 101-2115-M-009-008-MY2)
- Stochastically Coupled Maps -- Juang Jouq (NSC 101-2115-M-009-011-MY2)

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- Members:
- Discrete Mathematics and Optimization

- Members:

Chiu-Yuan Chen, Chih-Wen Weng, Hung-Lin Fu, Michael Fuchs. - Topics:

Combinatorial Optimization, Communication Networks, Algorithms, Analysis of Algorithms, Graph Theory, Algebraic Combinatorics, Finite Geometry, Combinatorial Designs, Analytic Combinatorics. - Recent Research Projects:

- The application of Graph Theory to wireless sensor networks (data aggregation and localization problems) -- Chiu-Yuan Chen NSC 102-2115-M-009-010)
- Linear Algebra Applied to Graph Theory -- Chih-Wen Weng (NSC 102-2115-M-009-009-MY3)
- Combinatorics and Discrete Models -- Hung-Lin Fu (NSC 100-2115-M-009 -005 ¡VMY3)

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- Members:
- Number Theory, Geometry and Analysis

- Members:

Yi-Fan Yang, Shiah-Sen Wang, Chi-Kaung Pai, Michael Fuchs, Yi-Jung Hsu, Kuo-Zhong Wang, Ming-Hsuan Kang. - Topics:

Matrix Analysis, Operator Theory, Numerical Range, Metric Number Theory. - Recent Research Projects:

- Automorphic forms on Shimura curves and their related problems -- Yi-Fan Yang (NSC 102-2115-M-009-001-MY4)
- Correlations of Shape Parameters in Random Trees -- Michael Fuchs (NSC 102-2115-M-009-002)
- The boundary of the numerical ranges and compressions of matrices -- Kuo-Zhong Wang (NSC 102-2115-M-009-006)
- On the distribution of poles of zeta functions of complexes -- Ming-Hsuan Kang (NSC 102-2115-M-009-005)

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- Members: