# 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, Chi-Hin Chan , Daniel E. Spector Te-Sheng Lin . - Topics:

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

- Discrete Green function and hybrid deformable models with applications on MRI images registration--Chin-Tien Wu (NSC 104-2115-M-009-002-)
- Computational Conformal Geometry and Its Applications(3/4)--Wen-Wei Lin (NSC 104-2628-M-009-001-)
- Dynamics And Oscillations In Coupled Systems -- Chih-Wen Shih (NSC 103-2115-M-009-002-MY2)
- Analysis and computation of the related mathematical problems in the atmospheric and oceanic dynamics--Ming-Cheng Shiue (NSC 104-2115-M-009-012-MY2)
- Numerical computation and analysis of the mathematical problems in geophysical fluid mechanics -- Ming-Cheng Shiue (NSC 102-2115-M-009-015-MY2)
- Elliptic equations in random media -- Li-Ming Yeh (NSC 104-2115-M-009-010-MY2)
- The De-Giorgi's Method As Applied In The Study Of Regularity Of Solutions To Hamilton-Jacobi's -- Chan Chi Hin (NSC 103-2115-M-009-010-MY2)

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- Members:
- Financial Engineering and Probability

- Members:

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

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

- The convergence and dissipation of random walks on specific media--Guan-Yu Chen (NSC 104-2115-M-009-013-MY3)
- Relative Arbitrages and Optimal Arbitrages--Yuan-Chung Sheu (NSC 104-2115-M-009-007-)

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- Members:
- Differential Equations and Dyanmical Systems

- Members:

Song-Sun Lin, Wen-Wei Lin, Chin-Wen Shin, Ming-Chia Li, Juang Jouq, Jong-Eao Lee, Shu-Ming Chang, Chi-Jer Yu, Chi-Hin Chan , Daniel E. Spector . - 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:

- Interactive 3D course & cooperative learning forum implement on WEB--Chi-Jer Yu (NSC 104-2511-S-009-005-)
- Chaotic dynamics for nonhyperbolic systems--Ming-Chia Li (NSC 104-2115-M-009-003-MY2)
- Cluster Synchronization And Synchronized Dynamics In Networks Of Neurons With Chemical Synaped -- Juang Jouq (NSC 103-2115-M-009-008-MY2)
- Multiplicative integer dynamical systems--Song-Sun Lin (NSC 104-2115-M-009-004-)

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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:

- Distance-Two Coloring For The Adjacency Graphs Of Wireless Sensor Networks -- Chiu-Yuan Chen (NSC 103-2115-M-009-012)
- The application of graph theory to the localization and routing problems in wireless sensor networks -- Chiu-Yuan Chen (NSC 104-2115-M-009-008-)
- Linear Algebra Applied to Graph Theory -- Chih-Wen Weng (NSC 102-2115-M-009-009-MY3)
- Group Testing And Its Applications -- Hung-Lin Fu (NSC 103-2115-M-009-013)
- Group Testing And Its Applications (II) -- Hung-Lin Fu (NSC 104-2115-M-009-009-)
- Equitable colorings of Cayley graphs -- Wu-Hsiung Lin (NSC 104-2115-M-009-005-)

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- Members:
- Number Theory, Geometry and Analysis

- Members:

Yi-Fan Yang, Shiah-Sen Wang, Michael Fuchs, Yi-Jung Hsu, Kuo-Zhong Wang, Ming-Hsuan Kang, Chi-Hin Chan , Daniel E. Spector . - 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)
- Discrete Probability Theory, Analytic Combinatorics And Phylogenetics -- Michael Fuchs (NSC 103-2115-M-009-007-MY2)
- Numerical ranges of companion matrices -- Kuo-Zhong Wang (NSC 104-2115-M-009-001-)
- Riemann Hypothesis on uniform lattices and non-unifom lattices on local classical groups -- Ming-Hsuan Kang (NSC 104-2115-M-009-006-MY2)

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- Members: