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

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

- Fast Solvers for Three-Dimensional Maxwell's Equations on Topological Photonic Crystals(1/4) -- Wen-Wei Lin (NSC 106-2628-M-009-004-)
- Homogenization of composite materials with high contrast conductivity -- Li-Ming Yeh (NSC 106-2115-M-009-010-MY2)
- TUncertainty quantification for shallow water models with random data -- Ming-Cheng Shiue (NSC 106-2115-M-009-011-MY2)
- High accurate conformal parameterization with applications in solving pdes on 3d surface -- Chin-Tien Wu (NSC 106-2115-M-009-002-)
- The study of the presence of vortex structures in Navier-Stokes flows passing an abstacle -- Chi-Hin Chan (NSC 105-2115-M-009-011-MY2)
- Modeling thin nematic films: Weak anchoring model--Te-Sheng Lin (NSC 105-2115-M-009-008-MY2)

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- Members:
- Financial Engineering and Probability

- Members:

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

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

- Genelization of portfolio master rquation and its applications--Yuan-Chung Sheu (NSC 106-2115-M-009-006-)
- The convergence and dissipation of random walks on specific media--Guan-Yu Chen (NSC 104-2115-M-009-013-MY3)

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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 , Hsin-Yuan Huang , 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:

- On maximum meromorphic dynamic Zeta function -- Song-Sun Lin (NSC 106-2115-M-009-004-)
- Chaos for generalized dynamical systems -- Ming-Chia Li (NSC 106-2115-M-009-003-MY2)
- On the elliptic system arising from rank two gauge theory -- Hsin-Yuan Huang (NSC 106-2628-M-009-005-MY4)
- The Effect of Risk Assessments and Immunization Strategies on Multiple-Route Epidemic Spreading Models -- Juang Jouq (NSC 105-2115-M-009-002-MY2)

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- Members:
- Discrete Mathematics and Optimization

- Members:

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

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

- Relaxed Distance-2 Coloring and Its Applications to Data Collection in Tree-Based Wireless Sensor Networks -- Chiu-Yuan Chen (NSC 106-2115-M-009-007-)
- Subgraphs in Edge-colored Graphs (I) -- Hung-Lin Fu (NSC 106-2115-M-009-008-)
- Equitable colorings of circulant graphs -- Wu-Hsiung Lin (NSC 106-2115-M-009-005-)
- On Bipartite Graphs Analogue of Brualdi-Hoffman Conjecture -- Chih-Wen Weng (NSC 105-2115-M-009-014-MY2)

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- Members:
- Number Theory, Geometry and Analysis

- Members:

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:

- Twisted Poincare Series and Finite Automata -- Ming-Hsuan Kang (NSC 106-2115-M-009-012-MY2)
- The numerical radii of matrix commutators -- Kuo-Zhong Wang (NSC 106-2115-M-009-001-)
- The Shape of Random Digital Structures -- Michael Fuchs (NSC 105-2115-M-009-010-MY2)
- Fractional Gradient PDE and Inequalities--Daniel E. Spector (NSC 105-2115-M-009-004-MY2)

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- Members: